Advances in the Flow and Rheology of Non-Newtonian Fluids (Rheology Series, Volume 8)

PREFACE

These two parts bring together a number of authoritative, state-of-the-art reviews and contributions, written by well recognized experts in the field of"flow and rheology of non-Newtonian fluids." Knowledge of non-Newtonian behavior is of vital importance to a variety of manufacturing processes including, for example, mixing, shear-thickening, fibre spinning, coating, and molding. This work covers areas such as bio- and food- rheology, electro-rheological fluids, polymers, flow in porous media, and suspensions. Complex and industrial flow situations are dealt with via analytical, as well as, numerical methods. In Chapter 1, a critical account of advances made in the area of flow-induced interactions in circulation is presented. Chapters 2 & 3 deal with shear-thickening in biopolymeric systems, and with the rheology of food emulsions, respectively. The next six chapters are on complex flows, in particular, Chapter 4 discusses worm-like micellar surfactant solutions. Chapter 5 covers time periodic flows. Chapter 6 communicates on secondary flows in tubes of arbitrary shape. Chapter 7 relates effects of non-Newtonian fluids on cavitation. Chapter 8 discusses viscoelastic Taylor-Vortex flow. Chapter 9 deals with non-Newtonian mixing. This is followed by two chapters on computational methods relevant to homogeneous viscoelastic fluids at the macro-level. The next major section is on constitutive equations and viscoelastic fluids. Chapter 12 discusses recent advances in transient network theory. Chapter 13 deals with theories based on fractional derivatives and Chapter 14 involves kinetic theory. Chapters 15 and 16 put forward new concepts approaching the constitutive structure of polymeric melts. The next chapter communicates the theory of flow of smectic liquid crystals. Part A ends with an overview of extensional flows. Volume B starts with a section on electro-rheological fluids. The first two chapters in the section summarize the constitutive theories for electro-rheological fluids from the continuum and molecular points of view. Chapter 21 relates a comprehensive approach to the constitutive structure of electromagnetic fluids, and the following two chapters deal with the properties of electro-rheological fluids. The next section covers some industrial flows related to drag reduction, and paper coating. Polymer processing and the related rheology are discussed in Chapters 26-29. In particular, the rheology of long discontinuous iber thermoplastic composites, thermo-mechanical modelling of polymer processing, injection molding and flow of melts in channels with moving boundaries are covered in Chapters 26-29 respectively. Free surface viscoelastic and liquid crystalline polymer fibers and jets, and numerical sinmlation of melt spinning of polyethylene fibers are the subject of Chapters )0 and 31, respectively.

vi Section 9 contains two chapters dealing with foam flow and non-Newtonian flow in porous media. Section 10 discusses four chapters on various aspects of suspension.Chapter 34 reviews and puts forth new ideas on the fluid dynamics of fine suspensions. Chapter 35 deals with concentrated suspensions. This section ends with a discussion on fiber suspensions and fluidized beds. The last section of Part B contains a discussion on transport-phenomena involving heat and mass transfer in rheologically complex systems and an account of a new onedimensional model for viscoelastic diffusion in polymers. With the publication of this work we hope to update and complement earlier work in a diverse range of topics. These two volumes should be of interest to all those engaged in basic, as well as applied research. The information presented herein is equally valuable for practising engineers who are constantly dealing with complex situations involving non-Newtonian materials. The contents of the two volumes are accessible to those with a background in engineering and/or pure sciences. We would like to take this opportunity to thank the contributors who, despite their busy schedules, kindly agreed to participate.

Dennis A. Siginer New Jersey Institute of Technology Newark, NJ, USA Daniel DeKee Tulane University New Orleans, LA, USA Raj P. Chhabra Indian Institute of Technology Kanpur, UP, India

vii

LIST OF CONTRIBUTORS Advani, Suresh G. University of Delaware Department of Mechanical Engineering Spencer Laboratory Newark, Delaware 19716, USA

Cairncross, R. A. Department of Chemical Engineering Drexel University Philadelphia, Pennsylvania 19104, USA

Agassant, J. F. Centre de Mise en Forme des Mat6riaux Ecole des Mines de Paris URA CNRS 1374 BP 207 06904, Sophia Antipolis, FRANCE

Carreau, Pierre J. Center for Applied Research on Polymers CRASP Department of Chemcical Engineering Ecole Polytechnique, Montreal QC H3C 3A7, CANADA

Bagley, Edward B. 756 S. Columbus Morton, Illinois 61550-2428, USA

Chhabra, R. P. Department of Chemical Engineering Indian Institute of Technology Kanpur, INDIA 208016

Bakhtiyarov, Sayavur I. Space Power Institute 231 Leach Center Auburn University Aubum, Alabama 36849-5320, USA

Co, Albert Department of Chemical Engineering University of Maine Orono, Maine 04469-5737, USA

Bechtel, Stephen E. Department of Aerospace Engineering Applied Mechanics and Aviation The Ohio State University Columbus, Ohio 432 I0, USA

Conrad, Hans Department of Materials Science and Engineering North Carolina State University Raleigh, North Carolina 27695, USA

Blumen, A. Theoretical Polymer Physics Freiburg University Rheinstr. 12, 79104 Freiburg, GERMANY

Couniot, A. Siemens-Nixdorf Information Systems S.A. LoB "Major Projects" Chaussee de Charleroi 116_ B- 1060 Brussels, BELGIUM

Bousfield, Douglas W. Department of Chemical Engineering University of Maine 5737 Jennes Hall Orono, ME 04469-5737 USA

Coupez, T. Centre de Mise en Forme des Materiaux Ecole des Mines de Paris URA CNRS 1374 BP 207, 06904, Sophia Antipolis, FRANCE

Brito-De La Fuente, Edmundo Food Science and Biotechnology Department Chemistry Faculy "E" National Autonomous University of Mexico UNAM, 04510 Mexico, D.F., MEXICO

Creasy, Terry University of Southern California Center for Composite Materials VHE 602 MC0241 Los Angeles, California 90089-0241, USA

Brunn, Peter O. Universitat Erlangen-Nurnberg Lehrstuhl fur Stromungsmechanik Caueerstr. 4 D-91058 Erlangen, GERMANY

De Kee, D. Department of Chemical Engineering Tulane University New Orleans, Louisiana 70118, USA

Buyevich, Yuri A. Center for Risk Studies and Safety University of California Santa Barbara 6740 Cortona Dr. Santa Barbara, Califomia 93117, USA

Demay, Y. Centre de Mise en Forme des Materiaux Ecole des Mines de Paris URA CNRS 1374 BP 207, 06904, Sophia Antipolis, FRANCE

viii

Dintzis, Frederick R. USDA ARS National Center for Agricultural Utilization Resarch Peoria, Illinois 61604, USA Dulikravich, George S. Aerospace Engineering Department 233 Hammond Building The Pennsylvania State University University Park, Pennsylvania 16802, USA Dunwoody, James Department of Applied Mathematics & Theoretical Physics The Queen's University Belfast BT7 INN NORTHERN IRELAND Dupret, F. CESAME Unite de Mecanique Appliquee Universite catholique de Louvain Avenue G. Lemaitre 4-6 B- 1348 Louvain-la-Neuve, BELGIUM Duming, Christopher J. Department of Chemical Engineering and Applied Chemistry Columbia University New York, New York 10027, USA Fong, C. F. Chan Man Department of Chemical Engineering Tulane University New Orleans, Louisiana 70118, USA Forest, M. Gregory Department of Mathematics University of North Carolina Chapel Hill, North Carolina 27599-3250, USA Franco, J.M. Departamento de Ingenieria Quimica Universidad de Huelva Escuela Politecnica Superior La Rabida, 21819 Palos de la Ftra (Huelva), SPAIN Friedrich, Chr. Freiburg Materials' Research Center Freiburg University Stefan-Meier-Str.21 79104 Freiburg, GERMANY Fruman, Daniel H. Groupe Phenomenes d'Interface Ecole Nationale Superieure de Techniques Avancees 91761 Palaiseau Cedex - FRANCE

Gallegos, C. Departamento de Ingenieria Quimica Universidad de Huelva Escuela Politecnica Superior La Rabida, 21819 Palos de la Ftra (Huelva), SPAIN Goldsmith, Harry L. Department of Medicine The Montreal General Hospital 1650 Ave Cedar Montreal, Quebec H3G 1A4, CANADA Hoyt, Jack W. 4694 Lisann Street San Diego, Califomia 92117, USA lsayev, A. I. Institute of Polymer Engineering The University of Akron Akron, Ohio 44325-0301, USA Kanu, Rex C. Department of Industry and Technology Ball State University Muncie, Indiana 47306, USA Khayat, Roger E. Department of Mechanical & Materials Engineering The University of Western Ontario London, Ontario, CANADA N6A 5B9 Kim, Kyoung Woo Fiber Research Center Sunkyong Industries Su Won, 440-745, KOREA Kim, Sang Yong Department of Fiber and Polymer Science College of Engineering Seoul National University San 56-1, Shinlim-Dong, Kwanak-Ku Seoul 151-742, KOREA Kornev, Konstantin G. Institute for Problems in Mechanics Russian Academy of Sciences 101 (1) Prospect Vemadskogo Moscow 117526, RUSSIA Kwon, Youngdon Department of Textile Engineering Sung Kyun Kwan University Su Won, 440-746, KOREA

Lavoie, Paul Andre Center for Applied Research on Polymers CRASP Department of Chemcical Engineering Ecole Polytechnique, Montreal QC H3C 3A7, CANADA Leonov, Arkadii I. Department of Polymer Engineering The University of Akron Akron, OH 44325-0301, USA Leslie, Frank M. Mathematics Department University of Strathclyde Livingstone Tower Richmond Street Glasgow G 1 1XH, SCOTLAND Letelier, Mario Universidad de Santiago de Chile Santiago CHILE Mal, O. CESAME Unite de Mecanique Appliquee Universite catholique de Louvain Avenue G. Lemaitre 4-6 B- 1348 Louvain-la-Neuve, BELGIUM Neimark, Alexander V. TRI/Princeton 601 Prospect Ave. P.O. Box 625 Princeton, New Jersey 08542-0625, USA

Prakash, J. Ravi Department of Chemical Engineering Indian Institute of Technology Madras, INDIA, 600 036 Rajagopal, K. R. Texas A&M University College Station, Texas, 77842-3014, USA

Rozhkov, Aleksey N. Institute for Problems in Mechanics Russian Academy of Sciences 101 (1) Prospect Vernadskogo Moscow 117526, RUSSIA Schiessel, H. Theoretical Polymer Physics Freiburg University Rheinstr. 12, 79104 Freiburg, GERMANY Shaw, Montgomery T. Department of Chemical Engineering and Polymer Program University of Connecticut 97 North Eagleville Road U-136 Storrs, Connecticut 06269-3136, USA Siginer, Dennis A. Department of Mechanical Engineering New Jersey Institute of Technology Newark, New Jersey 07102, USA Steger, R. Rheotest Medingen GmbH RodertalstraBe 1, D-01458 Medingen b. Dresden, GERMANY

Overfelt, R. A. Space Power Institute 231 Leach Center Auburn University Auburn, Alabama 36849-5320, USA

Tang, P. H. Department of Chemical Engineering and Applied Chemistry Columbia University New York, New York 10027, USA

Padovan, J. Department of Mechanical Engineering The University of Akron Akron, Ohio 44325-0301, USA

Tanguy, Philippe A. Department of Chemical Engineering Ecole Polytechnique Montreal P.O. Box 6079 Station Centre-ville Montreal, H3C 3A7 CANADA

Petrie, Christopher J. S. Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne NE1 7RU, UNITED KINGDOM Phan-Thien, Nhan Department of Mechanical and Mechatronic Engineering The University of Sydney NSW 2006, AUSTRALIA

Tanner, R. I. Department of Mechanical and Mechatronic Engineering The University of Sydney NSW 2006, AUSTRALIA Tao, Rongjia Department of Physics Southern Illinois University at Carbondale Carbondale, Illinois 62901, USA

Vanderschuren, L. Shell Research S.A. Avenue Jean Monnet 1, B-1348 Louvain-la Neuve, BELGIUM

Zhang, Y. Institute of Polymer Engineering The University of Akron Akron, Ohio 44325-0301, USA

Vergnes, B. Centre de Mise en Forme des Materiaux Ecole des Mines de Paris URA CNRS 1374 BP 207, 06904, Sophia Antipolis, FRANCE

Zhou, Hong Department of Mathematics University of North Carolina Chapel Hill, North Carolina 27599-3250, USA

Verhoyen, O. CESAME Unite de Mecanique Appliquee Universite catholique de Louvain Avenue G. Lemaitre 4-6 B- 1348 Louvain-la-Neuve, BELGIUM Verleye, V. TECHSPACE AERO Route de Liers 121 B-4041 Milmort, BELGIUM Vincent, M. Centre de Mise en Forme des Materiaux Ecole des Mines de Paris URA CNRS 1374 BP 207, 06904, Sophia Antipolis, FRANCE Vossoughi, Shapour Department of Chemical and Petroleum Engineering University of Kansas 4006 Learned Hall Lawrence, Kansas 66045-2223, USA Wang, Qi Department of Mathematical Sciences Indiana University-Purdue University at Indianapolis Indianapolis, Indiana 46202, USA Wu, C. W. Research Institute of Engineering Mechanics Dalian University of Technology Dalian 116024 People's Republic of China Yoo, Jung Yul Department of Mechanical Engineering College of Engineering Seoul National University Seoul 151-742, KOREA Yziquel, F. Center for Applied Research on Polymers CRASP Department of Chemcical Engineering Ecole Polytechnique, Montreal QC H3C 3A7, CANADA

Zook, C. Institute of Polymer Engineering The University of Akron Akron, Ohio 44325-030 l, USA

FLOW-INDUCED

INTERACTIONS

IN THE CIRCULATION

Harry L. Goldsmith McGill University Medical Clinic, Montreal General Hospital, Montreal, Quebec H3G 1A4, Canada 1. I N T R O D U C T I O N Many years ago, it was suggested to the English physiologist L.E. Bayliss by the eminent Cambridge authority on fluid mechanics, G.I. Taylor, that rather than work on the highly complex rheology of blood, he would be better spending his time studying the flow of marmalade through sewage pipes. That advice did not deter Dr. Bayliss [1,2] and a host of physiologists, rheologists, chemical and mechanical engineers from engaging in the problem, which had, after all, an honourable parentage in the person of Jean Leonard Marie Poiseuille, who published as early as 1839 [3,4], and in this century, the Swedish pathologist, Robin F~thraeus [5,6]. Quite apart from the clinical value of discovering how blood cells travel in the vessels of the circulation, there is, for the rheologist, a fascination about this fluid, since it is able to flow, even at hematocrits (packed red blood cell (RBC) volume concentration) well above 40%, with such ease compared to other suspensions and emulsions [7]. Thus, it is possible to pack a column of mammalian red cells containing less than 2% trapped plasma into a 2 mm diameter glass tube, and then suck it out quite easily into a micropipette whose tip has a diameter of only 0.5 mm. By comparison, oil-in-water emulsions reach the quasi-solid consistency of margarine at much lower concentrations. Clearly, in such a concentrated suspension, interactions between the blood corpuscles and between the corpuscles and the vessel walls play an important role in determining the mechanics of the motion. This chapter is concerned with a description of these interactions. We begin by taking the reader through a brief description of blood, seen as a dispersion of charged colloidal-size particles in a Newtonian suspending medium, and the main flow regimes to which it is subjected. To understand the cell and cell-wall interactions, one must begin with the macroscopic, overall non-Newtonian rheology of blood,

and then to correlate these rheological properties with the flow properties of the individual cells, studied first in isolation, then at increasing concentrations as interactions become increasingly important, i.e. the microrheology of the blood. This part of the chapter will mainly involve the erythrocytes or red cells, since they occupy >99% by volume and -96% by number of the particulate phase of blood, and thus effectively determine both the macro- and microrheology of the suspension. However, the most recent advances in the field have been made at the submicroscopic level, where the molecules and the forces involved in cell-cell and cell-wall adhesion have been investigated, thus entering the realm of what has been called molecular rheology. At this level, the chapter focusses not on the red cells but on the platelets and leukocytes, since it is they that are involved in key physiological and pathophysiological events through their role in blood coagulation and a g g r e g a t i o n - the platelet, and in i n f l a m m a t i o n - adhesion of leukocytes to the vessel wall and their passage into the extravascular space. In fact, adhesion processes between cells and between cells and the vessel wall are particularly critical in the circulation for all three classes of cells. The biophysics and rheology of certain of these adhesive reactions and the means of measuring forces of adhesion are dealt with in Sections 4 and 5. Where possible, the phenomena described are discussed in terms of fluid mechanical and colloid chemical theory. 1.1 B l o o d as a colloidal d i s p e r s i o n Mammalian blood is a dispersion of three classes of negatively charged particles (the cells or corpuscles; Table 1) in an aqueous solution (the plasma) containing dissolved salts and proteins (Table 2). As pointed out above, from the point of view of its rheological properties, blood is essentially a concentrated suspension of red blood cells 1.1.1 Cell membrane The biconcave red cell has a very thin lipid bilayer membrane (-10 nm) containing proteins linked to a submembraneous skeleton of actin/spectrin. The membrane encapsulates a 33% Newtonian salt solution of hemoglobin having a viscosity ,--6x that of the plasma at 37~ The ease of deformation of the red cell is mainly due to the properties of the membrane acting as a twodimensional incompressible material, which deforms at constant surface area and constant volume [8], and when osmotically swollen does not increase its surface area. The deformed cell remembers its undeformed shape and recovers it within a fraction of a second upon removal of the forces of deformation. By means of micropipette aspiration techniques, the viscoelastic properties of the membrane have been extensively characterized. The elastic deformation is described by three fundamental, independent deformations [9]3 elongation or shear of the membrane having a shear modulus of 6-9 • 0mN m -~, an isotropic expansion of the membrane surface without shear or

Table 1 Mean values of physical parameters of human blood cells Cell

Shape

Major Axis gm

Volume 3 gm

Number g1-1

Volume Fraction

Erythrocyte (Red Cell)

Biconcave disc

8.3

88

5.4x106 4.8xl 0 6

0.46 (m) 0.42 (f)

Thrombocyte (Platelet)

Oblate spheroid

3.1

7.5

2.5x105

1.9x10 -3

Leukocytes (White Cells)

Ruffled spheres

7.0x103 (total)

1.2x10 -3 (total)

Neutrophil

7.8

240

3.7x103

8.8x10 -4

Lymphocyte

6.0

115

2.5x103

2.9x10 -4

Table 2 Mean values of concentrations of ions and proteins in plasma PROTEINS

SALTS Ion

Concentration meq./litre

Name

Molecular Weight

Concentration gm/litre

Na +

150

Albumin

69,000

44.1

K+ Ca++ Mg++

4 5 2

Fibrinogen

340,000

3.0

C1HCO3PO 4-

111 28 3

t~l-globulin ~2-globulin

200,000

2.7 5.9

SO4--

1

13-globulin ),-globulin

90,000 160,000

8.8 14.2

bending, having an area expansion modulus of 450 mN m-', and a bending of the membrane with a very low bending modulus of 1.8 + 0.2x10 -19 N m. The resistance of the membrane to the rate of deformation is characterized by its

viscosity; the area viscosity has been found to have a value of 0.6-1.2x10 -3 mN s m -1 [9].

1.1.2 Charge and stability. The negative charge on the red cell membrane is due to N-acetyl and Nglycolneuraminic acid (sialic acids) residues attached via a gycosidic linkage to other molecules. Measurements of the electrophoretic mobility of red cells in buffers of physiological ionic strength = 0.145 yield values of the ~-potential -15 mV [10]. When treated with the enzyme neuraminidase to remove the sialic acid residues, the amount of charge lost, as calculated from the change in electrophoretic mobility, is underestimated 2-3x, when compared with the amount of sialic acid released. The reason for the discrepancy is thought to be due to the presence of a layer of polyelectrolytes on the surface of the lipidprotein bilayer through which the solvent can flow [11]. The sialic acid charges are distributed within this layer, and not anchored to a rigid surface as in solid colloidal particles such as charged latex spheres. The layer of polyelectrolyte, called the 'glycoprotein calyx' is believed to extend out to -- 710 nm beyond the bilayer, and its thickness is a function of the ionic strength. The presence of such a layer would in part account for the fact that red cells suspended in physiological saline (e.g. Ringers, a solution of ions mimicking those in plasma, to which albumin may be added) do not aggregate. If treated as a rigid colloidal charged particle, calculations of the double layer thickness at physiological ionic strength yield a value o f - - 8 nm, i.e., within the glycocalyx boundary and so close to the membrane that one would have expected the cells to be subject to van der Waals attractive forces. In fact, application of the DLVO theory of colloidal stability [12,13] to red cells clearly indicates that the cells should aggregate in physiological saline [11,14]. However, red cells do aggregate in plasma in the presence of fibrinogen, as well as in buffers containing plasma expanders such as dextran [14,15] and polyvinylpyrrolidone. Within the primary aggregate, or rouleau, the cells lie in a regular array as shown in Figure 1, discoid face to discoid face, and deform so that the adjoining faces are locally parallel and have maximum contact. Studies of aggregation in dextran solutions have been interpreted in terms of cross-bridging by polymer molecules long enough to simultaneously adsorb on the surfaces of two neighbouring cells [16]. In the absence of reliable evidence that dextrans are adsorbed on red cells, this interpretation has been questioned and an alternate mechanism, based on depletion flocculation, proposed [17]. In this scenario, surfaces are initially repulsive and stabilized by an electrostatic stress IIo~exp(-z/2~), where z is distance and ~ is the electrostatic decay constant. With addition of non-adsorbing polymer, an osmotic pressure difference builds up between the mid point of the intermembrane gap and the external bulk solution. The osmotic pressure induces an attractive depletion stress in which the decay constant is now the

Figure 1: A network of rouleaux of human red blood cells photographed at rest on a microscope slide. correlation length of the polymer ~, i.e., IIpeXp(-z/~), where ~, characterizes the range of the depletion zone. At low [polymer] there is no adhesion since the electrostatic stress is less than the depletion stress. With increasing [polymer], the osmotic pressure difference increases, and so does the attractive depletion stress, becoming larger than the electrostatic stress, and aggregation occurs. This continues with increasing polymer concentration. However, at the same time, ~ decreases thereby reducing the range and magnitude of the depletion term. Ultimately, the repulsive electrostatic stress once again wins out and the tendency to aggregation is reduced resulting in electrostatic stabilization. These two competing stresses account well for the observed aggregation vs [dextran] curves [15]. 1.2 Blood vessels and flow regimes Table 3 gives values of the vessel and flow parameters in the human circulation. On leaving the left ventricle of the heart, blood enters the ascending aorta having a resting diameter of-.25 mm, and is accelerated to mean linear velocities as high as 0.9 to 1.4 m s-~. Almost immediately, the blood is subjected to branching flows, the vessel diameters decrease and their numbers increase rapidly, and hence the cross-sectional area of the vessel bed. Values given for the dog show that the area increases from 5 c m 2 in the large arteries, to 20 cm 2 in the small arteries to 1,360 c m 2 in the capillaries [18]. Broadly speaking, one can distinguish three flow regimes:

(i)

A rapid, pulsatile flow in the aorta and larger arteries where inertial effects due to acceleration and deceleration of the fluid predominate. The mechanics of the flow have been modelled regarding blood as a homogeneous fluid, neglecting its particulate character [19]. (ii) A much weaker pulsatile or quasi-steady viscous flow in the smallest arteries and arterioles, in which interactions between the blood cells largely determine the mechanics of the flow. (iii) A viscous flow in the capillaries whose diameter is smaller than that of the red cell, and which can be described as a bolus flow of a train of cells surrounded by a thin lubricating film of plasma at the vessel wall, with pockets of plasma (boluses) between cells [21]. The red ceils readily deform into complex shapes [22,23], and their motion has been modeled [24]. The poorly deformable white cells travel more slowly, causing red cells to accumulate behind them [25,26]. Due to the multiplicity of branching, the non-uniformity of vessel diameters and the existence of curved vessel segments, sudden changes in velocity and direction of the bloodstream result in secondary flows having radial components, and not infrequently in flow separation and the formation of recirculation zones. Such flows have been designated by the term "disturbed" [27], to distinguish them from laminar and turbulent flows, the latter observed near normal aortic and pulmonary valves, increasing markedly in disease when the valves become partially stenosed [28]. Branching occurs with such frequency that the disturbance created by the flow at one bifurcation has not had time to dissipate before that due to another branch comes into play. It is well known that these effects are particularly important with regard to the genesis of thrombosis and atherosclerosis, and clearly that they involve interactions between the corpuscles and fluid with the vessel wall. Such interactions, which will be dealt with in Section 6 below, can result in injury to the vessel wall and to the blood cells themselves. 2. M A C R O R H E O L O G Y OF BLOOD The rheological properties of blood resemble that of a pseudoplastic fluid, being a strong function of the rate of deformation. At low shear rates the apparent viscosity decreases with increasing shear rate as the network of rouleaux (Figure 1) breaks up. At moderate and high shear rate, the apparent viscosity decreases as shear-induced deformation of the red cells increases.

2.1 Viscometry in Couette Flow 2.1.1 Effect of shear rate on apparent viscosity The role of red cell deformation and aggregation is shown in Figure 2 in a plot comparing the apparent relative viscosity, T/r at 37~ of 45% red cell

suspensions in plasma and in albumin-saline with that of aldehyde-hardened (still biconcave) red cells in albumin-Ringer solution [29]. All suspending media had the same viscosity, 1.2 mPa s, and measurements were made in a concentric cylinder Couette device. Since there was no fibrinogen or other aggregating protein in the albumin-Ringer suspension, no rouleaux were present and the difference between rh in plasma and albumin-Ringer suspensions in the range 10-z < G < 5 s-~ was due to the progressive break-up of aggregates. The further decrease in rh is then ascribed to cell deformation, the curve eventually flattening out as the blood assumes a quasi Newtonian character at G > 10 3 s -1. By contrast, r/, for the rigidified cells is almost constant with increasing G, there being no aggregation or deformation. 10 3

>., r,o o

o

10 2

~

..=.,

"

'

~

'

~

R B C in

Aggregation

>

plasma

~

Hardened cells in alb.-Ringer

(D >

...=.

o......-o.........~ ......~ ...... ~.

_ ~'~... ........ v ......,~...... o......-':~..

R B C in

alb Ringer

Deformation

/

~ C> '----',.--~ " - ~ 0

"'"

"--"

(~)

rr

.L,,I

10-2

........

I

1 0 "1

........

I

1

........

I

10

......

A,I

10 2

,

, 9 , .... I

10 3

Shear Rate, s ~

Figure 2. Relative apparent viscosity vs shear rate in suspensions of normal RBC in plasma (aggregation and deformation), and in Ringers-albumin (no aggregation) and hardened RBC in Ringersalbumin (no aggregation or deformation). From Chien [29] with permission. 2.1.2 Constituitive equations A comparison of 11 equations which have been used to describe the rheological behaviour of human blood was made by Easthope and Brooks [30]. The equation which best represented their data obtained in a concentric cylinder measuring system was that of Walbum and Schneck [31]:

~'=a 1 exp a2H +

G -1-a4H

(1)

where H = hematocrit, and a~-a4 are adjustable constants, functions of temperature, macromolecular composition of the suspending phase etc. The

equation was developed from a power law relationship between shear stress and shear rate, using regression curve fitting of trial equations built by successive inclusion of variables of decreasing influence. The model does not allow for a yield stress. Because of the complexity and high concentration of blood, it has not been possible to develop constituitive equations based on a mechanistic model of blood flow, without the use of empiricism or approximations. Such an equation is that due to Casson [32] first used by Scott Blair [33]" ,~.1/2._ aol/2 + a l G a ~ 2 (2) Here, the suspension is modelled as containing particles which can aggregate at low shear rates to form rod-like particles, whose length increases with decreasing shear rate. The similarity between these aggregates and rouleaux (Figure 1) makes the model attractive. However, the yield stress, a0 does not become zero except when H = 0. Nevertheless, Equation (2) represents the rheological data on blood over a limited shear rate range with one set of constants. Also, the experimental determination of yield stress poses real problems, since aggregation at very low shear rates leads to migration of rouleaux away from the walls of the Couette cylinders (syneresis), and sedimentation is then also a problem [34]. Another equation based on mechanistic modelling is that due to Quemada [35,36]" ~r

_

(1

_

0.5kH) -2,

where

k

__

k0 + k o o t-~rl/2 --r 1 + G" " F 1/2

and

Gr-

G

Gr

(3)

Here, the suspension properties are represented by the coefficients k0 and k.., the intrinsic viscosities at zero and infinite shear stress of the particles which predominate at those shear rates, and G~, the critical shear rate, which can be considered to be the inverse of the relaxation time for the dominant structural unit causing the suspension to be non-Newtonian. The coefficients are functions of hematocrit, suspending phase composition, etc. The model has been found to be almost as good as the Walbum-Schneck equation [31] with the advantage that the coefficients have a physical meaning. Further discussion of this model and its application may be found in reference [37], and a comprehensive discussion of blood rheology in reference [38]. 2.2 Blood flow in cylindrical tubes The use of macroscopic rheological data obtained in Couette instruments and continuum models to predict flow behavior in l a r g e vessels appears satisfactory as long as there is no appreciable cell aggregation resulting in syneresis and sedimentation. Both effects will occur in horizontally positioned tubes at mean linear velocities < 1 tube diameter/second [39]. In vertically positioned tubes,

10

aggregation leads to syneresis and the two-phase flow of a core of aggregates surrounded by a cell-depleted peripheral layer, with a reduced pressure gradient at a given volume flow rate [40]. With these exceptions, continuum mechanics appears to be satisfactory for vessels having diameters > 500 ktm. As the tube diameter decreases, however, and approaches the dimensions of the red cells, blood no longer acts as a continuum and the following effects arise.

2.2.1 The Fdhraeus effect As the tube diameter decreases below 500 gm, the measured instantaneous hematocrit in the tube, Hr, is found to be smaller than the hematocrit, HR, in the stirred inflow reservoir, or even in the discharge, Ho (providing no screening effects occur at the tube entrance). The ratio Hr/Ho decreases with decreasing diameter, as shown in Figure 3, until the tube diameter falls below - 15 gm, when it increases again. The effect is due to the existence of a slower moving cell-depleted peripheral layer of low hematocrit, surrounding a faster moving central core of higher hematocrit. When mixed, the tube hematocrit is smaller than the reservoir or discharge hematocrit. It can be shown that: Hr HD = ( 1 - H r )

+H r

(4)

where is the average tube blood velocity, and and are the average plasma and blood cell velocities [6].

!

o,

v

,

i

HD 0.6

04 i ~ Critica,,lDiameter 92

I~)0

I000

Diameter, l~m

Figure 3. F~thraeus effect for human red cells, HR = 40-45%, for all literature data for suspensions in tubes, at flow rates ensuring no RBC aggregation (cross-hatched region; also 9 F~hraeus data [6]). Critical diameter (-2.7 g m ) ~ the smallest tube through which a shuman RBC can flow. From [6], with permission.

11

2.2.2 The Fdhraeus-Lindqvist effect The effect refers to the original observation [5] that the hydrodynamic resistance of blood and other red cell suspensions decreases as vessel diameter decreases below 300 gm diameter. The most obvious explanation for a lower resistance or a lower effective viscosity (computed using the Poiseuille-Hagen equation to distinguish it from the apparent viscosity, determined from the measured shear rate and shear stress, as in Couette instruments [37]), is that it is a consequence of the F~ihraeus effect, i.e. a lower tube viscosity due to a lower hematocrit. It may also be due to the rheological effect of a non-uniform distribution of cells across the vessel lumen or to the failure of the continuum model of the suspension, as will be discussed in the next section. 3. M I C R O S C O P I C C O R R E L A T E S OF M A C R O S C O P I C F L O W BEHAVIOUR The general problem of microrheology is the prediction of the macroscopic rheological properties of a material from a detailed description of the elements of which it is composed. In the case of blood, the elements are the individual corpuscles (effectively the red blood cells) each surrounded by the suspending fluid, the plasma. What is most striking about the mechanics of the motion is the fact that, under physiological conditions, the cell finds itself subjected to shear stresses and considerable particle crowding, such that it is continually distorted from the biconcave resting shape. The flow behaviour and interactions of individual red cells has been studied using microrheological techniques, in particular the travelling microtube, a device for tracking the motions of cells and colloidal particles through vertically mounted precision bore glass tubes of 50-200 gm diameter, while photographing or videotaping through a high resolution microscope [41-43]. The microscope axis is fixed, and the tube is mounted within a chamber attached to a vertically mounted sliding platform supporting a syringe infusion-withdrawal pump. Both the platform and the syringe pumps are driven hydraulically by continuously variable speed electronically-controlled DC motor drives. 3.1 Rotation and deformation of red cells At shear stresses z < 0.03 Nm -2, isolated human RBC rotate with periodically varying angular velocity, but for small perturbations due to Brownian diffusion, maintaining their biconcave shape [43]. The rotational orbits are similar to those previously found for rigid discs [44,45], in accord with theory [46] applied to rigid spheroids, as illustrated in Figure 5 for the angular motion of the axis of revolution of an RBC and a 4-cell rouleau:

dt

= 1G[ 1 + B(re)]COS 2q~

(5)

12

where ~ is the azimuthal angle of the axis of revolution with the diametrical, X2-axis (Figure 4; = + 90 ~ in positions 1 and 5, and 0 ~ in position 3 in Figure 5), B(re)= (r 2 - 1)/(r 2 + 1), r~ is the equivalent ellipsoidal axis ratio (axis of revolution/diametrical axis) and T the period of rotation through 2n, given by [47]: T - 27r (re -

G(r----~

+

re 1)

(6)

Integration of Equation (5) yields: tan~=re(

Gt

re + l / r e

)

(7)

As predicted for an oblate spheroid having = 0 . 3 8 , at any given instant, the largest fraction of red cells in the tube (49%) are found with their major axes within + 20 ~ of the direction of flow (r = 90 + 20 ~ [43]). Similar results have been obtained with isolated platelets, except that here, the effects of Brownian rotary motion are appreciable [48], as has been found with other colloidal-size particles, such as doublets of latex spheres [49]. With increasing shear stress, the rotational motion progressively deviates from that predicted by theory, as cells spend more time aligned with the flow. At "t"> 0.1 N m -2, a large fraction are seen lying in the median plane of the tube without apparently rotating, instead aligning themselves at a constant angle to the flow; moreover, they are deformed with an increase in the major, diametrical axis [43]. Such behaviour, resembling that of the deformation of liquid drops in immiscible viscous fluids [7,50], has been studied with red cells suspended in higher viscosity media such as buffered low molecular weight dextran [43,51]. It has been shown that there exists a critical shear stress above which the membrane begins to rotate about the interior of the cell, in what has been called a tank-treading motion, an unfortunate term, since the membrane motion is likely transmitted into the interior of the cell resulting in circulation patterns within the hemoglobin solution. As predicted by a two-dimensional theory applicable to Couette flow [52], an increase in the ratio of external to internal viscosity promotes the stationary orientation of the particle.

3.2 Lateral migration of cells The redistribution of blood cells in narrow tubes, which is at the root of the Fhhraeus and Fhhraeus-Lindqvist effects, is due to a net lateral migration of red cells away from the tube wall. There exists a substantial body of theoretical and experimental knowledge on the effect of the vessel wall on the motions of suspended rigid or deformable model particles (the reader is referred to reviews by Brenner [53] and Leal [54]) and of blood cells, including platelets and leukocytes [55,56].

13

I I l

X

3

r '

X

"-1'

U ~

GX

2

2

Figure 4. Rotation of the axis of revolution of a spheroid (heavy line) defined by the Cartesian (Xi) and polar (0, ~0) coordinates constructed at the particle centre of rotation and origin of a Couette shear field.

360 A

i

_

B,| I

......

I

270 _

i

! 4~~ i

i/1

i1) -o

Ot~" r

~.'~_._.

180

~ I

red cell

i

~r:;0;35 .....

Zol i

90

1~ -

,~

4-cell

"-

i I

rouleau

r'i 1"1

-

i I

0.25

0.50 t T

0.75

1.00 FLOW

Figure 5. Rotation of a single human RBC and 4-cell rouleau in Poiseuille flow at G < 20 s-~. A: Variation of the angle ~ with time t during an orbit having the period T. The line drawn through the solid circles was computed from Equation (7); the dashed line corresponds to uniform angular velocity. B" The same particles, drawn from cinemicrographs at orientations corresponding to positions 1 to 5. From [43], with permission.

14

The vessel boundary exerts its influence on the flowing blood cells not only by retarding their translational and rotational velocities, an effect appreciable within one or two cell diameters from the wall, but also by generating radial components in the cell velocity. There are two established mechanisms by which migration across the streamlines can occur [45,47,53-55,57]: migration due to particle deformation at low Reynolds number, and migration due to inertia of the fluid at moderate and high Reynolds number. Here, we concern ourselves mostly with the former mechanism. Deformable particles (liquid drops, flexible fibers), suspended in Newtonian media undergoing Poiseuille flow in the creeping flow regime, migrate away from the wall towards the axis [45]. In the case of a fluid drop, it has been shown to be an effect of the particle disturbance flow which generates a flow with a radial component in the neighbourhood of the drop. In the absence of fluid inertia, rigid spheres and spheroids do not migrate laterally across the streamlines [45]. Under conditions of negligible fluid inertia, particle Reynolds number < 10 -6, the latter defined in terms of the particle translational slip velocity (particle [u] - fluid [U]) of a sphere, radius b at the axis of a tube, radius R, in Poiseuille flow [58]:

Single human red cells and rouleaux also migrate radially inward, as illustrated in Figure 6, for red cells in Ringers solution and in buffered dextran solutions [55]. As previously found with fluid drops [45], the rate of migration increases with particle deformation (it is much greater in dextran than in Ringer solution; Figure 6) and increases rapidly with increasing ratio of particle to tube diameter. Such migration, although severely inhibited by the particle-crowded conditions in normal blood, nevertheless appears to result in a thin (-- 4 ktm wide) cell-depleted layer at the wall and a significant lowering of the hydrodynamic resistance in many blood vessels. 3.3 Cell interactions at normal hematocrits Long before hematocrits approach those normally present in the circulating blood (N 40-45%), even at 0.1% there are frequent two- three- and some multibody collisions, and these, as will be seen in Section 4, can be studied to obtain information on the forces at play during such shear-induced interactions. As the hematocrit exceeds 10%, there are constant collisions between all cells, and above 30%, particle crowding becomes a factor and actually contributes to the deformation of the red cells from their resting biconcave shape. Such deformation can be observed even at shear stresses < 0.03 N m -2 at which the isolated single cell in plasma would have rotated as a rigid disc (as in Figure 5). Since one cannot observe the motions of individual blood cells in the interior of whole blood flowing through narrow tubes, due to continual reflection and

15

~ O or)

~,

~ ~

RBC in 40~ Dextran

! r

E

-;

Ceins

0.2

I,,,,X X

A 0.4 (.9 V 0.6

0.4

,

1

,

0.6

I

0.8

....

1.0

r/R

Figure 6. Measured inward migration of RBC in plasma Ringers solution compared to that in 25% buffered dextran having 35x the plasma viscosity. is the mean tube shear rate, x is axial distance from the tube entrance. From [55], with permission. refraction of the transmitted light, it was necessary to render the blood transparent. This was achieved by preparing "ghost red cells" by osmotic hemolysis of normal red cells in buffer of 1/10 the ionic strength of plasma. The cells were washed to remove the hemoglobin solution, and then slowly returned to their normal salt and protein environment, whereupon they assume their former biconcave shape [59,60]. Over a range of volume concentrations from 10-80%, the measured apparent relative viscosities of red cell and derived ghost cell suspensions did not differ significantly. Tracer normal blood cells were added to the ghost cell suspensions and the motions of these, clearly visible in the interior of the tube, photographed and analyzed.

3.3.1 Deformation due to shear and particle crowding Figure 7 illustrates the continually changing deformation of a red cell in a 55% ghost cell suspension. The cells no longer undergo angular rotation but spend much of their time aligned and deformed in the direction of flow. The membrane likely rotates about the interior in an irregular fashion. Deformation of rouleaux in ghost cell suspensions is also observed [60]. The ability of red cells to deform and squeeze past each other in flow is the microscopic correlate of the macroscopic flow behaviour of blood in Couette viscometers which exhibits such remarkably low viscosity at moderate and high shear rate, relative to that of concentrated suspensions of rigid particles and even that of concentrated emulsions.

16

0 sec.

0.36sec.

0.72 sec.

1.08 s e c .

O.072sec.

' - - - - 10 ~ - - - - ~ I v"

II - 5 0 p m ~--

I I I E

Figure 7. Tracings from photomicrographs of the deformation of a tracer RBC at 7.2 ms intervals in a 55% ghost cell suspension being tracked in robe flow. From [61], with permission.

3.3.2 Velocity distributions at high concentrations The effects of particle crowding on the velocity distribution had previously been explored in the tube flow of concentrated suspensions of macroscopic size rigid spheres and discs. Here, the suspensions were made transparent to transmitted light by matching refractive indices of particle and suspending phase, and then adding visible tracer quantities of particles of the same size and shape but having a different refractive index [62]. It was shown that for 0.05 < b/R < 0.15, when particle volume concentrations exceeded 20%, the velocity distributions became blunted in the centre of the tube with a core of radius r~ in which particle velocities, u(r) were maximum and constant, = UM, and < U(O), the centreline velocity in Poiseuille flow (as illustrated in the upper panel of Figure 8). This region was designated as partial plug flow although this did not imply that the profile was mathematically flat, only that there was no

17

1.0 ~

.

l

l

l

'

R

" ~ ........... -

Complete

Par

Plug Flow , PlugFlow____~____;t~ 0

0.5

-

L....

1.0 '

"-

:

......~

I 1.0

I

-I

,~

o

I

Spheres ~

0.5

r

]

~

Rigid

......~

.....~

................i ~

i

;

00000fO~176176

~ e u , , , e

.o~"

-

Flow

F'o'~

- Drops

~"

9

-

9I

o

,

s

I t Cells~

0.5- 9

-~

c = 32%

....~'0~

0

Plug Flow -

0.5

~

I .__L_

~]__t_~~ /

.(7""

D

iscs Q

1.0 ........ O-

........ 0.25

I 0.50

I 0.75

1.0

u(r)

U(O) Figure 8. Dimensionless velocity profiles of visible tracer particles in 32% transparent suspensions of model rigid particles and ghost red cells: plot of relative radial distance vs. particle translational velocity + centreline velocity in Poiseuille flow at the same volume flow rate. Upper panel: Rigid spheres (b/R = 0.11) exhibiting complete plug flow compared to emulsion droplets (b/R = 0.09) with only partial plug flow. Lower panel: Rigid discs (b/R = 0.08) compared to ghost cells (b/R = 0.11). From [62], with permission. measurable velocity gradient. Nevertheless, these suspensions were still quasiNewtonian in that the velocity profile was independent of flow rate at low R%, and the pressure drop per unit length, AP, was directly proportional to volume

18 flow rate, Q [62]. In fact, it was shown that, despite the continual erratic (Brownian diffusion-like) radial displacements of the spheres in those regions of the tube in which shear-induced interactions occurred, over relatively short time periods, the displacements were reversible in time and space [62,63]. At a given b / R , the degree of blunting increased with increasing volume fraction, c, of the disperse phase, and at a given c, it increased with increasing b/R. Blunting of the velocity distribution was not due to aggregation of the spheres or discs whose interactions are due solely to hydrodynamic forces, nor due to a redistribution of particles in the tube which could have resulted in the two-phase flow of a more viscous and concentrated central core of suspension surrounded by a less concentrated and less viscous peripheral layer. With the exception of the exclusion layer closer than one particle radius from the wall, sphere centres were shown to be uniformly distributed in the tube [62]. Rather, the effect is due to particle interactions in the crowded suspension, and has been treated by Skalak [64] who has interpreted the interactions of the particles in terms of passing versus non-passing motions of adjacent particles. It should be noted, however, that more recent work has shown that redistribution of rigid particles in variable shear fields can and does occur in the creeping flow regime providing particle interactions proceed for long enough times [65-69]. As shown in the lower panel of Figure 8, at comparable particle volume concentrations, the ghost cell suspensions exhibited blunting of the velocity profile, similar to that Of the discs, but with significantly lower r c [59,70]. As with suspensions of rigid particles, r c increased with increasing b / R and concentration. However, as expected, given the shear and concentrationdependent deformation of the ghost and red cells, there is no quasi Newtonian behaviour: the degree of blunting decreased with increasing Q as the parabolic Poiseuille velocity profile was approached, accompanied by a decrease in effective viscosity [71 ], and the motions were not reversible in time and space. In this respect, the non-Newtonian behaviour of red cells and ghost cell suspensions resembled that of concentrated emulsions of deformable liquid droplets in which the droplets are also deformed not only by the shear stress, but also through particle crowding. As shown by work in transparent oil-in-oil emulsions, the droplets were distorted from prolate ellipsoids into irregular shapes [72]. Moreover, the shear-induced deformation led to some migration of the droplets away from the wall (opposed by an outward dispersive force due to the continual interactions of droplets in the core of the emulsion), and as with the red blood cells, such migration was not reversible in time and space. In addition (see upper panel of Figure 8) the velocity distribution was less blunted than that of a suspension of rigid spheres at the same particle volume fraction and b / R , and was flow rate dependent: as the flow rate increased and the droplets became increasingly deformed, the velocity distribution gradually approached the parabolic Poiseuille velocity profile [72,73].

19

3.3.3 Convective dispersion of cells As pointed out above, the resistance to crowding of the blood corpuscles at normal hematocrits opposes any substantial inward radial migration of red cells, such as observed at low hematocrits. At the microscopic level, this resistance is seen as a series of continuous collisions between red cells which results in a marked dispersion of all blood cells and the surrounding plasma, as illustrated in Figure 9 for a tracer red cell and a leukocyte in a 40% ghost cell suspension. Measurements of the mean square radial fluctuations over small time intervals At of the paths of tracer red cells and 2 ktm diameter latex microspheres were used to compute dispersion coefficients D~ defined as:

Dr= <AP>/2At

(9a)

At shear rates from 2-20 s -~, Dr for RBC and latex spheres ranged from - 1 0 -12 to >10 -11m 2 s -1 [55,59], values 2 to 3 orders of magnitude greater than the

0.9 _ 0.8

/ - 40

"~

~PMN

o.7

"l ~

5O

2

7O

Q Q

"

5

~ ~ "0

0.

I

1,0~

u

u

,

r

.-i. Q I

"~

n

35

"-

0.9

"1: 3

tr

0.8

30

20

,

0

5

lO

15

20

Time,s Figure 9. Radial dispersion of the paths of a polymorphonuclear leukocyte (PMN) and an RBC in 40% ghost cell suspensions in tube flow. Asterisks denote times when particles collided with the wall. The slightly deformed PMN and markedly deformed RBC are shapes seen while tracking the cells. From [74], with permission.

20

Brownian translational diffusion coefficient, Dr, for the isolated red cell suspended in plasma, viscosity 7/- 1.8 mPa s at 23~

Dr = kBTr/K, ri = 4.4

x 10 -14 m E S -1

(9b)

where kB is the Boltzmann constant, TK the absolute temperature and K, = 5.23x10 -5 m, the translational resistance coefficient for motion along or transverse to the axis of revolution (2.4 l-tm) of a rigid spheroid, re = 0.38 [75]. The above values of radial diffusion coefficients in ghost cell suspensions are in fair agreement with blood platelet translational diffusion coefficients obtained from measurements of their dispersion in flowing blood, which increased from < 1 0 -12 m 2 s -1 in plasma to 2.5 x 10 -~ m E s 1 at 50% hematocrit in flowing blood at a mean tube shear rate - 100 s -~ [76]. As a consequence of the enhanced radial dispersion of the other blood cells by the red cell motions, is an increase in the two-body collision rate between platelets in shear flow, which is discussed in Section 4. More important is the effect of the red cells in increasing the frequency of platelet-wall collisions with increasing hematocrit, an effect of considerable significance in hemostasis (the normal plugging of cut or damaged vessel wall, a process initiated by platelets) as well as in thrombosis (abnormal, pathological adhesion and growth of platelet thrombi on the walls of arteries), particularly in regions of disturbed flow, as described in Section 6. It has been shown that adhesion of platelets to various artificial surfaces [77,78] as well as artery subendothelium [79,80] (artery stripped of the endothelial layer thereby exposing collagen fibres to which platelets strongly adhere) in flowing blood is markedly enhanced by the introduction of red cell or ghosts cells into the suspending medium. The extent to which red cell deformation or size may affect the increase in the effective diffusion constant of cells and solute has also been studied. The transport of molecular solute was found to increase slightly with cell size and rigidity [81], and the wall adhesion of platelets observed to increase substantially with red cell size [81]. Augmented transport in the shear flow of concentrated suspensions of model particles and red blood cells has been thoroughly reviewed by Zydney and Colton [82]. They proposed a model of augmented solute transport based on shear-induced particle migrations and the concomitant dispersive fluid motion induced by these particle migrations. Augmented solute transport was defined as (D~e/DsF ) -- 1, where Dee is the effective solute diffusion coefficient measured in the sheared suspension, and D SF i s the solute diffusion coefficient in the absence of flow. If particle rotations are assumed to be unimportant, D ~ - DSF + Dp, Op being the particle diffusion constant. Augmented diffusion is predicted to vary as the Peclet number, Pe - b2G/OsF.

3.3.4 Redistribution of blood cells in tube flow At normal hematocrits and in vessels of diameters > 1 mm, the thin

21

peripheral layer of--4 gm thickness will likely have a negligible effect on the distribution of red cells across the lumen. As mentioned in Section 2.2, however, at low flow rates (corresponding to mean velocities < 1 tube diameter s-~) aggregation of red cells becomes an important factor. Paradoxically, it is at low, and not high flow rates, that inward migration of red cells in plasma or high molecular weight dextran buffer is most pronounced in tube flow. Here, the formation and rapid inward drift of rouleaux of red cells results in the two phase flow of an inner core of a network of rouleaux surrounded by a peripheral cell-depleted layer in which can be seen single red cells, small rouleaux and an apparently large number of platelets and leukocytes [40,56,60,83-85]. Such two-phase flow has also been induced in small vessels in the microcirculation of animals by intravenous injection of gelatin or fibrinogen and of dextran. Both in vivo and in vitro it has been shown that the formation of the red cell core is associated with the 'margination' (displacement to the periphery) of white blood cells [56,84,85]. Cine films of blood flow taken in tubes of 100-340 gm diameter clearly show that the effect is due to the outward displacement of white cells by the inwardly migrating network of packed red cell rouleaux [56,85]. Here, an "inverse" F~hraeus effect occurs: since is less than [Equation (4)], the leukocyte concentration in the tube is greater than that in the infusing reservoir [56]. That such two-phase flow results in a decrease in hydrodynamic resistance, R - AP/Q, had earlier been shown in studies of the oscillatory flow of concentrated suspensions of rigid neutrally buoyant spheres where inertial effects led to a small but significant inward migration of the particles from the tube wall [86]. The effect was later demonstrated in the flow of mammalian blood, where it was also shown that in vertically positioned tubes, enhanced red cell aggregation in the presence of 250 kDa dextran resulted in a lower effective viscosity at < 2 s-~ [39]. By contrast, in horizontally positioned tubes, where there is an asymmetric distribution of red cells and aggregates due to sedimentation of the core, the effective viscosity continues to increase with decreasing . The relation between the radius of the core, r~, and hydrodynamic resistance has been studied in buffered 110 kDa dextran suspensions [40], and as shown in Figure 10 in a tube, R = 172 gm, hydrodynamic resistance at first increased with decreasing until aggregation brought about syneresis and a shrinking diameter red cell core. That the effect is due to aggregation was confirmed by experiments in 10% buffered albumin in which rouleau formation was totally absent, and R continued to increase with decreasing down to < 0.2 s-~ [40]. As expected, in buffered albumin suspensions there was no margination of leukocytes [56]. Redistribution of platelets in flowing blood has also been observed. In the case of platelets in arterioles [87,88], and 1-2.5 gm diameter latex microspheres in robes of-- 200 gm diameter [89,90], particle number concentrations near the vessel wall have been shown to be higher than in the core of the flowing

22

1.0

~

,

,

0.9 re I-I 0.8

0.7

i

i

i

i

9

Citrated Blood

O

Heparinized Blood

9

RBC in 1.5% Dextran

i

i

i

i

3.0 0

co ,2.5

E t~ Q- 2.0

.4

1.5 ----it 1.0

i

9 1.5% Dextran i

i

i

1

~ i

i

i

0.05 0.1 0.2 0.5 1.0 2.0 5.0 10 20 , Tube Diameters s 1

50

Figure 10. Development of a peripheral cell depleted layer in a 34% suspension of RBC in 1.5% buffered dextran 110 in a 172 gm diameter vertically mounted tube. The decrease in the relative width, r JR, of the red cell core of aggregates (upper panel) occurs in parallel with a decrease in hydrodynamic resistance (lower panel) as mean tube velocity decreases below 1 s -~. For RBC in albumin buffer (no aggregation) R continues to increase with decreasing . From [40], with permission. suspension ("near wall excess"). The effect, which requires the presence of red cells, has been modelled by adding a lateral drift term to the convective diffusion equation for platelet transport in flowing blood [91]. The reason for the net outward drift is believed to arise from the inward migration of the more rapidly migrating red cells resulting in a marginal layer of lower red cell concentration, whose width continuously fluctuates and should be viewed in a statistical sense. The outward lateral drift of platelets or microspheres occurs

23 because of a net flux of particles from a region of higher red cell concentration and hence higher red cell collision rate to a region of lower red cell concentration and lower collision rate. Such a drift has been shown to occur in alloys [92]. The fact that the location of the near wall excess of microspheres occurs a few microns away from the wall is due to the fact that the particles are physically repelled, and at sufficiently high Rep, fluid dynamically repelled from the wall through inertial effects. It should be noted that the above hypothesis does not attribute the motion of platelets toward the wall to their exclusion by red cells in the interior of the tube, as is the case for leukocytes.

4. SHEAR-INDUCED TWO-BODY I N T E R A C T I O N S : F R O M CHARGED C O L L O I D A L P A R T I C L E S TO BLOOD CELLS

Microrheological techniques, both in tube (travelling microtube [41-43]) and in Couette flow (Rheoscope [93]) have been used to study two-body interactions between charged colloidal particles and between blood cells. The left panel of Figure 11 illustrates a two-body collision between equal-sized rigid spheres as it appears when the particles are tracked by moving the tube upward with a velocity equal to that of the downward flowing fluid at the mid-point of the axis between the two spheres. The collision shown is one in which the trajectories of the spheres are symmetrical, as previously observed in the case of neutral macroscopic particles in viscous media [47,94]. The particles separate along paths having the same radial coordinate, r, as those of the paths of approach. In the case of colloidal-size charged latex spheres, however, interaction forces due to double layer repulsion and attraction due to van der Waals forces come into play when sphere surfaces approach to within a distance h - 100 nm [95]. This results in asymmetric collision trajectories [96], as illustrated in the right hand panel of Figure 11. These can be analyzed and hydrodynamic theory used [95,97] to show that net interaction forces as small as 10-~3 N are detectable. In the case of latex spheres in aqueous solutions of simple electrolytes, the interaction forces have been interpreted by applying the DLVO theory of colloid stability [12,13,95] and thereby obtain values of the Hamaker constant and the retardation parameter of the van der Waals force [96]. 4.1 Theoretical Considerations The fluid mechanical problem of predicting the trajectories of two interacting, neutrally buoyant rigid spherical particles of equal size, b, suspended in a Newtonian fluid of Viscosity 77, undergoing simple shear flow has been solved [97-99], and extended to the case of unequal-sized spheres [100]. More importantly, it has been extended to the case when interaction forces, Fi, t(h), other than hydrodynamic operate at h < 100 nm [95,101]. The relative velocity of the sphere centres a distance s apart, is then given by [95]:

24

-~ = A ( s * ) G b sin20 sin2q~ + dt

C(s*)Fint(h)

(10)

3 rcb rl

and the angular motion of the doublet axis by: dO 1 dt = -4 B(s*)G sin 20 sin 2q~

d~

(11)

1

(12)

dt = -2 G[ 1 + B(s*) cos 2q~]

-u(r)

-u(r)

t

i i i i i

t X3

Repulsion

i

i

1

,, i

,, Xm 8

X~

X3

Figure 11. Left: Two-body symmetrical collision in Poiseuille flow between rigid spheres forming a transient doublet. The collision is tracked by moving the tube with a velocity-u(r), equal but opposite to that of the fluid at the midpoint of the doublet where Cartesian and polar coordinates have been constructed. Right: Projection in the X2X3-plane of the asymmetric paths, due to double layer repulsion, of the centres of two colliding 4 ktm polystyrene latex spheres in 50% aqueous glycerol with l mM KC1. At the centre is the collision sphere which cannot not be penetrated. From [96], with permission.

25 Here, 0 and r are the respective polar and azimuthal angles relative to X1 as the polar axis, as shown in Figure 11; A(s*) and C(s*) are known dimensionless functions of s* = s/b, which have been documented [95,97,99]. Equation (12 is identical with the Equation (5) describing the angular velocity of a rigid prolate ellipsoid [46] with B(s*)= (r~(s*)- 1)/(r~(s*)+ 1). Asymptotic expressions for r~(s*) have been given for large and small s* [97]. Rearrangement of Equation (10) yields the force equation:

3rtbrl ds = A(s*) 3JrbqGb 2 sin20 sin2q~ + Fint(h) C(s*) dt C(s*)

(13)

in which the term on the left represents the hydrodynamic drag force resisting approach of the particles, and the first term on the right the normal hydrodynamic force, Fn, between spheres acting along the line of their centres, being maximal for 0 = Jr/2, i.e. rotation within the X2X3-plane (Figure 11, left), at r = - ~ / 4 when the force is compressive, and at r = rt/4 where it is tensile. Equations (10)-(12) have been shown to apply in Poiseuille flow providing b/R = 4.0. From [93], with permission.

5.2.4 Locus of bond rupture: use of antigen-linked latex spheres There remained the question of the locus of bond rupture in the SSRC studies. Micropipette aspiration experiments have shown that it is possible to extract antigen under force from a normal red cell [132]. Subsequently, experiments in shear flow showed that blood group antigen extraction can occur even with fixed cells [ 138] which cast doubt on the assumption that breakup of the SSRC crossbridge is due to failure of antigen-antibody bonds. Therefore, studies in which a synthetic blood group B antigen was covalently linked to carboxyl modified polystyrene latex spheres and the spheres crosslinked by monoclonal IgM antibody were undertaken [139]. Here, break-up of doublets of microspheres should occur through the rupture of antigen-antibody bonds, and would be expected to lead to a significantly different force dependence on doublet lifetime. As with the SSRC over a comparable range of normal force, there was a distribution in times to break up. However, significantly higher forces were required to achieve the same degree of breakup for doublets of antigen-linked spheres than for SSRC.

39

5.2.5 Monte Carlo simulation of doublet break-up Using the Bell theory [128], a computer simulation of doublet break-up under shear was developed in which to and x, [Equation (25)] were varied to fit the data collected in the Poiseuille and Couette flow experiments. Since bond formation is thought to be a Poisson process, specification of the average number of bonds, , is all that is required to characterize the distribution of bond numbers used in the computation of the force per bond when fitting the experimental data. Each rotation was divided into 1000 equal time steps, and for each step Pb was computed from Equation (27) and the force per bond calculated from Equation (31) for the current instantaneous values of 0, ~0 and nb. A random number between 0 and 1 was chosen from a uniform distribution for each bond remaining. If the number drawn was less than Pb, the number of bonds was reduced by one, and the force per bond on the remaining bonds recalculated. Thus, bonds are postulated to break sequentially. The results obtained for the best fit of the Couette flow data with 75 pM IgM are shown in the fight hand panel of Figure 17, with x,= 0.40 nm, to = 100 s and - 2.5 bonds. It is evident that the general features of the real experiments are quite well reproduced. Computer simulation of the break-up of doublets of antigen-linked latex spheres indicated that differences between these and populations of SSRC were due to a change in bond character. The antigen sphere-IgM bond was less sensitive to the applied force, had a lower value of x, (0.12 nm) and required a higher value o f f , (1000 pN) for instantaneous rupture compared to that (360 pN) for the SSRC [139]. This supported the notion that antigen-antibody bonds were ruptured in the case of the antigen-linked spheres whereas antigen was able to be extracted from the membrane of the SSRC. 5.3 A Biosurface Force Probe In Section 5.1 it was pointed out that the strength of a single bond is governed by the rate of loading, a factor which could not be controlled in the break-up studies described above. Evans et al. [126], describe an ultrasensitivetunable force transducer that can measure forces over an incredible range from 0.01 pN to > 1000 pN - the strength of a covalent b o n d - and in which the rate of loading can be controlled. As illustrated in Figure 18, the transducer is a microbead probe attached to a pressurized membrane capsule (e.g. RBC or lipid vesicle). The pressure, P, is controlled by micropipette suction and sets the membrane tension Zm:

Rp Zm = P2( 1 _ Rp/Ro )

(33)

where R 0 and Rp are the radii of the membrane capsule and suction pipette, respectively. When a small force, f, is applied to the probe, the capsule is

40 -p

Tm

Azt

Y Figure 18. Schematic diagram of an ultrasensitivetunable force transducer formed by a pressurized membrane capsule. From [140], with permission. elongated by a displacement, Az,, proportional to the force. The stiffness constant, kf for the transducer ( f - kyAz,) is given by:

kf

=

27t-z"m ln(2Rp/R o) + ln(2R 0/rb)

(34)

r b being the radius of circular contact between capsule and microbead. Since stiffness is proportional to tension, the force sensitivity can be tuned in operation between 1 l.tN m -~ and 10 mN m -~ simply by changing P, and Azt is measurable down to 0.01 ~tm using optical techniques. The microbead is chemically conjugated to separate ligands, one for macroscopic attachment to the capsule surface and the other for focal bonding to receptors on a biological surface. The microbead is brought towards the surface beating the receptor, bond formation is signalled when fluctuations in height (which depend on the rigidity of the receptor interface) diminish markedly; similarly bond release shows up when the fluctuations return to the original level. The relative frequency of formation and release then quantitates on/off rates. Finally, the transducer can be retracted to load the force on the attachment until bond rupture occurs.

5.4 Adhesion and detachment of cells from surfaces: leukocytes In the last 15 years, much work has been devoted to elucidate the mechanism whereby leukocytes, in particular polymorphonuclear cells (neutrophils) and lymphocytes, are able to be arrested on the endothelium of post-capillary venules and subsequently to migrate out into the extravascular space. The process appears to occur in three stages: "rolling" of leukocytes along the vessel wall, followed by firm arrest of the cells, and finally transmigration through the vessel wall [ 141 ].

41

5.4.1 Rolling vs firm adhesion of leukocytes A series complex interactions involving fluid mechanical forces and the formation and breakage of receptor-ligand bonds are involved when circulating leukocytes adhere to the venular vessel wall [141]. Blood cells flow through post-capillary venules of diameter from 15-25 lam at mean velocities ranging from 0.3-1 mm s -~. Wall shear stresses are said to be of the order of 0.2 N m 2. Normally, leukocytes flowing in close proximity to the wall do not adhere to the endothelial cells, i.e., no bonds are formed between the leukocyte and an endothelial cell. However, during inflammation, stimulation of the endothelium and/or the leukocyte together with the associated low flow state results in the appearance of more peripherally located white cells [56,84]. Many of these cells appear to be "rolling" along the vessel wall with translational velocities from 10-40 ~m s -~ [142-145], markedly lower than predicted for rigid spheres very close to a plane wall [146]. Rolling is not a smooth process" the translational velocity varies continually and the cell is frequently arrested. Some of the rolling and arrested cells rejoin the mainstream, then flowing at much higher velocities; others become firmly adherent, likely because of activating stimuli received while in contact with the endothelium. These cells can later proceed to enter the interendothelial cell junctions and migrate out [147]. Each of the 3 steps of the cell-wall interaction process is mediated by different sets of receptor-ligand pairs (adhesion molecules)with different kinetics of bond formation and rupture. To elucidate the separate roles of the various adhesion molecules and the effect of fluid mechanical stress on adhesion, recourse was had to in vitro experiments in which leukocytes were allowed to roll on, and adhere to surfaces bearing adhesion molecules in a parallel plate flow chambers [148]. Except near the side walls of the chamber, there is a Poiseuille velocity distribution between upper and lower surfaces, and the wall shear rate Gw = 3Q/2wh 2, where w is the width and h the height of the chamber.

5.4.2 Leukocyte rolling: selectin mediated adhesion Since leukocytes, unimpeded by interactions with endothelial cells, likely flow past the vessel wall at velocities of the order of 10 venule diameters s ~, a rapid rate of bond formation is necessary to arrest such a cell. As regards rolling, which involves the continual making and breaking of bonds, the kinetics of bond formation and dissociation are clearly more important than the so called bond affinity, i.e., the equilibrium association constant K A = K'o,/n'oy, where too, (moles s ~) and 1r (s -~) are the on and off rate constants. The ability of the bonds to withstand high strain before rupture (high "tensile strength"; low value of xt~) will also be important in initial adhesion and rolling, an issue addressed in two mathematical models of the rolling process [131,149]. The initial slowing down of leukocytes in the bloodstream involves a class of cell transmembrane adhesion molecules known as selectins (L-selectin on the

42

leukocyte and E - and P-selectin on the endothelial cells). The ligand binding sites on selectins are calcium-dependent lectin-like domains, carbohydrate structures which recognize fucose containing oligosaccharide moieties known as sialyl-Lewis x and sialyl-Lewis ~ on the leukocyte or endothelial cell. In the case of neutrophils, L-selectin is concentrated on the tips of the microvillus-like projection of the ruffled cell membrane, favouring the formation of the selectin-carbohydrate bond with the endothelial cell. Attempts have been made to determine the parameters of Bell's model from experiments with rolling cells as well as receptor-linked latex spheres. Alon et al. [ 150], using neutrophils rolling along lipid bilayers containing P-selectin on the lower surface of a parallel plate flow chamber, measured the force dependence of ~;oZ for the P-selectin-carbohydrate ligand bond using the distribution of times during which cells were arrested. They found xt~ = 0.05 nm, an extremely small value, suggesting the linkage to be close to an "ideal" bond, the lifetime of which varies little with applied force [131,151 ]. The work also showed that both too, and tCog~ were fast compared to other known macromolecular interactions [150]. This may be compared with xa = 0.40 nm, obtained from computer simulation of the observed force dependence of rupture of a protein-protein bond between doublets of latex spheres [152].

5.4.3 Leukocyte adhesion mediated by integrins After a variable period of rolling, if activation of the leukocyte has occurred, bonds between the activated leukocyte 13/-integrins and their counter-receptors on the endothelial cell belonging to the immunoglobulin superfamily, can form after transient cell arrest and eventually induce firm adhesion. Integrins are heterodimeric cell-surface proteins consisting of one of several o~-subunits and one of several [3-subunits bound non-covalently. The ~2-integrins, Mac-l, LFA-1 and VLA-4, are the known leukocyte integrins. The immunoglobulin superfamily of receptors is defined by the presence of the immunoglobulin domain, which is composed of 70 amino acids arranged in a well characterized structure [141]. The immunoglobulin adhesion molecules implicated in leukocyte-endothelial cell interactions are cellular adhesion molecules ICAM-1 and VCAM-1 and mucosal addressing cell adhesion molecule, MAdCAM-1. The effect of wall shear stress on leukocyte-endothelial cell adhesion has been extensively studied. Selectin-mediated rolling followed by integrin-mediated cell adherence is highest in the range of venular wall shear stress, ~:w, between 0.1 and 0.4 N m -z, but decreases rapidly at higher Tw. By contrast, integrinmediated adhesion, in the absence of rolling, leads to cell adhesion only at Vw < 0.1 N m 2 [145,148], emphasizing the necessity of the selectins for successful cell arrest to occur. The physiological importance of the selectins is underlined by the recently described leukocyte adherence deficiency 2 (LAD-2) syndrome, characterized by impaired immune function as well as developmental anomalies traced to a defective fucose metabolism. The other leukocyte adherence

43

deficiency syndrome, LAD-l, characterized by life-threatening bacterial and fungal infections, is due to a structural defect in the leukocyte integrins, making the leukocyte unable migrate through the interendothelial cleft. 5.5 Force dependence of detachment of cells from surfaces Measuring the detachment of cells from a substrate to which they specifically bind is the most commonly used technique for evaluating the strength of adhesion. In the presence of an applied force f per bond, such as generated in shear flow over the substrate surface, the lifetime tb of a cell-surface bond is given by Equation (25) [128], and the value of f = fa for instantaneous bond rupture can be usefully identified with the "strength" of a single bond. A measure of the adhesion strength can also be inferred from the force dependence of the time course of the detachment of a population of cells. Measurements of particle adhesion and detachment have been made using stagnation point flow chambers - as in the "impinging jet" [153] and "radial flow detachment assay" [154] methods, in which a narrow confined stream of suspension exits from an orifice and impinges on a plane surface where it is observed under a microscope. One can then study the role of hydrodynamic, colloidal and other forces on deposition and detachment of particles on surfaces under well-defined fluid flow and mass transfer conditions. A stagnation point flow chamber was first used with native whole blood to study platelet and leukocyte deposition in order to test the thrombogenicity of surfaces (155,156).

5.5.1 Impinging jet technique Dabros and van de Ven [157] first obtained solutions for the flow field and mass transfer equations in the region of the stagnation point flow. Studies have since been carried out on the deposition and detachment of latex spheres in aqueous electrolytes [157], live and fixed E. coli bacteria on glass surfaces [158], and SSRC on glass covered with monoclonal antibody [138,159]. The coordinates of the stagnation point flow field are shown below in Figure 19. A jet of suspension of radius R impinges on a surface (glass or plastic coverslip coated with receptor) a distance h away, and is contained between the two surfaces flowing out through both sides. The flow field is described by: U~=~z

; Uz=-Yz 2

[35]

where U~ and U~ are the radial and normal components of the velocity field, r and z being the radial and axial distance from the stagnation point, S, and ~, defines the strength of the flow field, for which expressions have been given as a function of Re (153). The wall shear rate on the cover slip, Gw, is given by:

Gw = ~Ur/~z = 'yr;

[36]

44

f I~ \

\ \

\

\

\

\

\

\

\

\ \

\

\

\ \

\

lI

I

I S ~ \ \ \ \ " ~ \ \ \ \ \ i

h

\ \ \ \ ~

I

t . , 4 I 2R - - ~ Figure 19. Stagnation point flow chamber: A jet of suspension of radius R impinges on a surface a distance h away and is contained between the two surfaces, flowing out through both sides. Thus, Gw is zero at the stagnation point and increases linearly with radial distance. The region where y is c o n s t a n t - a pure stagnation point flow, extends out to r = 0.1h.. For greater r, corrections to the flow field can be used. The particles are observed in epi-normal or interference contrast illumination and events recorded on videotape. Suspensions of cells or derivatized microspheres beating ligand flow onto the surface, and the number of particles per unit surface area expressed as: n, = n= [1 -exp(-t/cr)]

[37]

where n is the number at t - oo and cr is given by the relation: l/or =

1/cr,~ +

l/crbt

[38]

cr~ being the characteristic escape time of the cells and crb~that of slowing down of the deposition by the presence of deposited particles [160]. Measurements of n~ with time in various regions of the surface yield values of the initial adhesion rate and the above parameters. The escape time (average lifetime of a cell on the surface) is a measure of the cell-surface bond strength, and can be compared for the different receptor-ligand systems. Detachment experiments are carried out with the surface covered with spheres, and then exposed to a stream of suspending medium of known viscosity at various flow rates. The tangential hydrodynamic force, F~, exerted on a particle of radius b adhering to a surface is given by [146]: F~ = 1.7x6mS;wb 2 = 1.7x6n:yrb 2

[39]

can be related to the surface bond strength through the force dependence of the time course of detachment, which is dependent on the surface density of ligand.

45

An example is shown in Figure 20 for the detachment of populations of SSRC from surface-bound antibody using the impinging jet system, under forces from 20 to 100 pN. The maximum number of cross-bridges per cell was estimated to be 7, and the average rupture shear force per bond to be 17 pN [138]. 100~" = r.~ =

80

= ""

60

E

40

o

r

~

50

"~

40

E

30

~

"~ 1~

20 10

0

20

40

60

80 "

100

120

pN

0 0

10

20

30 40 Time, minutes

50

60

Figure 20. Shear-induced detachment of SSRC on a glass surface having covalently-bound monoclonal IgM. Plot of the normalized surface density against time. The average tangential hydrodynamic force was 75 pN. Inset: plot of the surface density vs the average hydrodynamic force. From [138], with permission. 5.5.2 Radial flow detachment assay

Similar stagnation point flow chambers have been used to study receptormediated cell attachment and detachment in a radial-flow detachment assay [161,162]. The assay has been used to study the relationship between the detachment force per bond and the bond affinity, K a [163]. In this device, detachment was observed at large distance from the stagnation point flow, where Gw decreases with increasing r. Antigen-coated latex spheres were allowed to adhere to antibody-coated surfaces under static conditions, and then exposed to steady shear flow. Spheres detached from the surface up to a critical radial distance from the stagnation point, corresponding to a "force required to break the bonds" beyond which the force was too low to detach the particles. As expected, given the stochastic nature of bond rupture, the contour of the ring separating adherent from non-adherent spheres, was not a sharp one. The shear stress to detach 10 ~m latex spheres covalently coated with rabbit anti-IgG on a surface covalently coated with rabbit IgG corresponded to

46 a force of 3 nN [161], 30x greater than that found in the SSRC-IgM antibody impinging jet experiments [138]. However, the surface density of rabbit IgG antibody was 1-3 orders of magnitude greater and the number of bonds, rib, was thought to be very high.

6. B L O O D C E L L I N T E R A C T I O N S IN R E G I O N S OF D I S T U R B E D FLOW Fluid mechanical factors play an important role in the localization of sites of atherosclerosis, the focal deposition of platelets resulting in thrombosis, and the formation of aneurysms in the human circulation. The localization is confined mainly to regions of geometrical irregularity where vessels branch, curve and change diameter and where blood is subjected to sudden changes in velocity and/or direction. In such regions, flow is disturbed and separation of streamlines from the wall with formation of eddies are likely to occur. In this Section the flow patterns and fluid mechanical stresses at these sites, their effect on cell and cell-wall interactions are described, and their possible involvement in the genesis of the above mentioned vascular diseases are considered. 6.1 Model System- Blood cells in an annular vortex The flow behaviour and interactions of red cells and platelets were studied in the annular vortex downstream of a sudden tubular expansion of a 150 ktm into a 500 l.tm diameter glass tube, serving as a model of a region of flow separation and an arterial stenosis [ 164 - 166]. 6.1.1 R e d blood cells

As predicted by fluid mechanical theory [167], when dilute suspensions of cells were subjected to steady flow through the model stenosis, a captive annular ring vortex was formed downstream of the expansion. Figure 20 shows paths and orientations of the red cells in the median plane of the tube at inflow tube Reynolds number, Re0, based on upstream mean fluid velocity, , and diameter. During a single orbit, the measured particle paths and velocities, as well as the locations of the vortex center and reattachment point, were in good agreement with those predicted by the theory applicable to the fluid [164]. Over longer periods, however, single cells and small aggregates < 20 ktm in diameter migrated outward across the closed streamlines, describing a series of spiral orbits of continually increasing diameter until they rejoined the mainstream. In contrast, aggregates of cells > 30 ktm in diameter remained trapped within the vortex, assuming equilibrium orbits or staying at the center. In pulsatile flow (a sinusoidal oscillatory flow superimposed in parallel with the steady flow), the observed phenomena were qualitatively similar to those described in steady flow. The vortex varied periodically in size and intensity;

47

Re =

37.8

~

< U > = 0 . 2 3 m s "1

I i

Q

Figure 21: Single orbits and orientations of glutaraldehyde-hardened human RBC in the median plane of the annular vortex formed downstream of the expansion of a 151 gm into a 504 gm diameter tube at Re0 = 37.8. Arrows indicate location of the reattachment point; particle velocities were 2.4 (O), 3.6 (A), 3.0 (B), 6.9 (C), 94 (D), 0.84 (E), 0.95 (F) and 272 mm s-1 at (G). From [164], with permission. the axial location of the vortex center and reattachment point oscillated in phase with the upstream fluid velocity between maximum and minimum positions about a mean which corresponded to that in steady flow. At hematocrits from 15 to 45%, migration of cells still occurred and resulted in a lowering of the vortex hematocrit. In part, particle migration in the vortex was likely due to the dilution effect of cell-poor plasma taken into the vortex from the fluid layer adjacent to the vessel wall upstream of the expansion. The mechanism for the trapping of large aggregates in the vortex was qualitatively explained by using existing fluid mechanical theory [57] for lateral particle migration near a tube wall due to inertia of the fluid, and by the operation of a mechanical wall effect [ 168]. 6.1.2 Platelet aggregation in the vortex The flow behaviour and interactions of human platelets in the annular vortex were studied in platelet-rich plasma (PRP) and in washed platelet suspensions. The vortex provided favourable conditions for the spontaneous aggregation of normal human platelets through repeated shear-induced collisions of cells while circulating in its orbits [165]. In a given suspension, the formation and growth of platelet aggregates was observed in a narrow range of Re0. In PRP

48

anticoagulated with heparin, containing significant numbers of platelets with pseudopods (Figure 12) and microaggregates of 2-6 cells, the rate and extent of aggregation were greatest, with aggregates > 100 ~m in length forming in ,

o,.~.

10 0

x o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

or) O O

x

i

0

x

o

x

o o o

x

x

~ x x ,,,

:"Ooc b x ~,,,, "o

>

o o

1 0 -1 10 0

. . . . . . . .

o

o c>! o 0.~. om~

, 10 ~

. . . . . . . .

i

10 2

(~

. . . . . . . .

10 3

Shear Rate (s -1) Figure 6. Viscosity/shear rate behavior in Couette flow at 60 ~ C of a 10% (wt/vol) sample of waxy maize starch autoclaved with stirring at 140 ~ C for 15 min at pH - 7.0. Because the initial low stresses in these aqueous systems are below the operating range of the sensor, the first three data points on each of the curves AB are unreliable and probably not valid. O = freshly cooked sample; X = sample stored in Dewar flask for 3 hr prior to flow measurement. (From Dintzis et al. [34]). a first approximation followed the initial downcurve BC, from approximately 750 to about 9 s ~ of Figure 6 though the viscosity level did drop slowly on

77 successive recyclings. The result was shown not to be an artifact of the instrument (see Figure 2 in Dintzis et al. [34]). These results have been extended to waxy rice and waxy barley starches and it has been established that it is the amylopectin component that causes these effects (Dintzis et al. [35]). With careful sample preparation one can detect a very attenuated form of the shear thickening and structure formation effects in regular maize starch. This very sizeable shear thickening effect, in which the viscosity increased by almost half a decade, occurred only with freshly prepared solutions. Once the higher viscosity level had been generated, the starch solution showed, on subsequent shear rate cycling, shear thinning behavior only. Structure formation during the shear thickening portion of the initial curve was suspected and phase microscopy pictures confirmed this suspicion as shown in Figure 7. The fresh, unsheared samples of waxy maize, dull waxy maize and normal maize starches showed little evidence of any structure. Shearing of the regular or normal starch solutions generated little, if any, structure. However, shear-generated "structure", as indicated by refractive index inhomogeneities in the micrographs, is quite evident in the dull waxy maize sample and predominates in the case of the waxy maize starch. The later sample appeared quite clear, water-white, but after shearing through the shear thickening region the solution developed an obvious opalescence. Results in these studies are very sensitive to sample preparation procedures. This can be illustrated with reference to time effects observed in some solutions. These effects can be seen (Figure 7 of Dintzis et al. [34]) where shear thickening was observed occurring over surprisingly long times at constant shear rate for a gently dispersed 3% waxy maize sample in 90%DMSO-H20. However, vigorous stirring changed this time behavior significantly, eliminating the shear thickening effect. Normal maize solutions generally appear to be simple shear thinning materials. A more detailed study of 3 g waxy maize starch in 100 g of 90% DMSO showed three different types of behavior. When the dispersion/solution in a flask was stirred with a stirring bar for 1 hour at 100 rpm, one initially observed an opaque dispersion of low viscosity that soon became transparent as the viscosity increased during stirring. Finally, the fluid became viscous, elastic, and water white, but with "fish eyes" (visible regions of refractive index inhomogeneity) readily apparent. This fluid, when examined using a Rotovisco R rheometer with shear rates held at 10, 50, 100 and then again 10 s1 for 20 minutes at each rate, showed no shear-thickening behavior. This starch stock fluid was subjected to further stirring and then tested again in the Rotovisco R. After sufficient stirring of the stock fluid, shear-thickening behavior was observed in the rheometer. When the stock fluid was again subjected to additional stirring, the shear-thickening effect and the "fish eyes"

78 disappeared and the solution became shear thinning.

.

.i~ ....

9,: ~,, is minus one, which in terms of shear stress x means that there exists a critical shear stress ~ ~ ( = 11p ~,~), at which rl ,,jumps" from its plateau value 1"1p to some lower value (see figure 2).

10-1 . . . . . . . . .

. .

o

Surfactant 1000 ppm T=35 C DA45

13.,

r x

4h 2 0 min 4 min 1 min

},

~ r-.,I

1 0 "2

r

r,/3

0 0 o

~

>.

0

-

~ xXXXX

IO-

- ....

I 0-3

.

.

...-.,,=

10 2

Shear

,

x

r

...... . , , . , . ,

X X ....... , - , - . . . . . ,

10-I

Stress

T in

Figure 2. The results of figure 1 as a function of shear stress

'w-

r

w: i : w T C

I

lO

I

Pa

122

It is of interest to note that while the quantities ~ , rl p and ~ are time dependent (as far as the sweep time is concerned) ~ r is not. Thus, the rheological characteristics of the solution seem to be i) time dependend behavior (which has been shown to be rheopectic [10], ii) a sudden increase of rl at some (time dependend) apparent critical shear rate ~ and iii) a sudden decrease of 11at some ( time independent ) shear stress ~ . In order to avoid a sweep time dependence all viscometric results to follow have been obtained with a sweep time of more than four hours. For completeness one should note that the sudden increase in flat some apparent critical shear rate ~ is found for many surfactant solutions in the concentration range where rod-like micelles dominate [4,5]. For that reason it is not astonishing to see that it has received considerable attention in the literature. The picture that at "~ a new phase is suddenly ,,created", the so called s_hearinduced-s_tate (SIS) is intriguing since it allows one to consider the q , ~

Surfactant 1000 ppm T=35 C

co

t3-,

a DB45 * ME4-5 A ME46

10-1 o000~

9r-',,I

t~

rl o

10

o o~o

-2 "

O

~

A

Q

9 r-,,,ll

& & A A A A

A

D 0 o

0 r r~

10

-3

kAk - dk'

**~0~

I~II

AAA A A

9r,-,I

b,. 10

-4 11 - -

It

"l-- 9 : I

I-i

n1011 ! ....

Shear

Iv '" Iv-~-I --IV -II ~11 i ~

2

- -W~ '' W-- 9 ~

W W W31~01

Rate ~ in I/s

Figure 3. Influence of the relative gap width ~con the apparent flow curve A: K=1.037, *" K=1.078; ~ K=1.176

123

jump on the basis of a phase transition (like a sol-gel phase transition). Readers interested in this subject are referred to the excellent papers of H. Hoffmann and, repectively, H. Rehage [5,11,12]. The viscometric results of our solution show a dependence of 1"1upon the"type and the size of the measuring system used. For example, by using three systems of different relative size (indicated by the fact that K:, the ratio of the outer to inner cylinder, 1 1, differs) we find that )~ increases while "iqp simultaneously decreases with decreasing 1< (figure 3). If for 1 1"1s) at all then this jump is bound to be rather small. Interesting is another, and totally unexpected, fact namely that there is also a dependence upon the actual gap width. Figure 4 shows that three systems with the same K - 1.078 furnish (rather unsystematic) results. Two Mooney Evert systems (termed ME) imply an increase of 11with increasing gap width, the double gap device (DA) contradicting that somewhat, since the gap widths (1.375mm and 1.63 ram) are between the others. We conclude from these results that the apparent flow curves depend upon the details of the measuring system used. 10-1

_

o * e

Surfactant 1000 oppm T=35

ooo

.

12., 17~

DA45 ME45 ME30

oooooo O

%'*******~o ~ Oo 0 ~ ~ oo o

10 -2-

B I,,,,-f

0

0 B

~r,,,I

~

O to r~

10

10

-3

-40-1 1

......

'i'

' '

Shear Figure 4.

'

' ' ~' 10'"

. . . . .

R a t e ~ in

"i~)

2

'

' "'

'"10

3

1//s

Influence of actual gap width AR- R - R, on the apparent flow curve o. AR=l.085 mm; , AR=l.63 mm; o- AR=1.375 and 1.63 mm (K = 1.078 in all three cases)

124

2.2. Theoretical considerations

In a concentric cylinder viscometer (Couette type) the outer cylinder (radius Ro) rotates with angular velocity f2 relative to the inner one, (radius R~, height h). The torque M, needed to hold the inner cylinder at rest, is measured. If viscometric flow prevails the shear surfaces are concentric cylinders. Thus, the shear rate ) will be a function of r, r being the distance from the axis of rotation, do (1) dr

Here o = o(r) is the angular velocity of a cylindrical surface of radius r. A torque balance reveals that the shear stress x varies like r-2, so that

with "r., t h e shear stress at the inner cylinder. Expressed in terms of measurable quantities it is given by M (3) x ~ = 27r.hRiZ

For the measuring systems employed by us the shear stress thus varies across the gap width by 7.5, 16.2 and by 38.3 % (~: =1.037, 1.078 and 1.176). Assuming the fluid to adhere at a solid surface the other measurable quantity, namely f~, is given by Ro

do

1

Rj

dr

2

(4)

"ci Ti/

r" 2

T,

It is important to note that this implies f~ - f~('t,, "t o ) = f~('r,,, K 2 ) . Principally it is possible to invert this relation (and thus to get the shear viscosity rl ) if experiments are performed with different systems (different K's) using the same shear stress at the outer cylinder, i.e. fixed x o = "r., / K : . For in that case we get from equation (4) the shear rate at the inner cylinder, ~, Of~ (5) c31ogK "tO

125

Thus, in a plot of f2 versus log~: the slope will be ~,. It is clear that such a procedure will be extremely time consuming (to obtain from experimental data the slope rather accurately will require many measurements) and we know of no attempt that this procedure has ever been tried. An alternate way is to employ a mean value theorem to equation (4) with the result 2K2 K2

~

(6)

xi

-1

where u is the appropriate mean shear stress, x o _ , T < l l > - T and 7' T. Hence, in these time dependent curvilineal flows the physical components of the rate of strain tensor

(22)

1 [Vv + (Vv) T] and stress tensor T have the property D " D -- T " T ( 1 3 > - a - f l

(23)

2.1 T i m e P e r i o d i c Flow

By a suitable choice of Q in (18) it is readily established that the shear stress functional in (19) and (21) is odd in the history k(.; t), i.e. :r, o

t)) -

t))

(24)

142 Also, if the non-zero components of the velocity are time periodic, i.e.

271 i -- V (X 1 t) a2

v i ( x 1, t ~- - - 1

for

i -- (2, 3),

(25)

then q ( x 1, t "F- -271" ) -- q ( x 1 , t ) a2

(26)

and k ( s ; t + 2a- ) ~d

-7

-- --'7

~o ~ q ( x 1, t -~- -27r - CO

/o

r)dr (27)

q ( x 1 , t -- r ) d r - k ( s ; t)

Hence, time periodic curvilineal flows have the property that the shear stress components defined in (21) are periodic:

271 T

(x 1, t -}- - - ) -- T

(x 1, t) a)

for

i -- (2, 3).

(28)

In passing it is noted that SlsC~__0 (k(8; t)) and 8 2 ~ 0 (k(s; t)) are even functionals of their arguments, so that all the stress components are periodic. If the velocity is anti-periodic, v i ( x 1 t + -7r) ~d

-v i(xl

(29)

t)

and therefore periodic, then the history k(8; t -~- - - ) -- - - 7 ~d -- "7

for all 0 _ s < ~ .

/o s q(x 1, t +

/o

~d

r)dr

(30)

q ( x 1, t -- r ) d r - - k ( s ;

t)

It follows from (21) that if (29) applies then

71" T (X 1, t -~ - - ) -- - - T < l k > (X 1, t) a2

fo r

k - (2,3),

(31)

i.e. the shear stresses are anti-periodic. However the normal stress differences are not(cf. Coleman & Noll[8]).

143 It is evident that the converse of the statements that (25) implies (28) and (29) implies (31) is true if and only if

(32)

ceT -~-/~T - ~~ 0 (k(8; t))

is invertible. While this is to be expected in the neighbourhood of the zero history k(s; t) for all 0 ___ s < ~ , it should not be taken for granted elsewhere (el. Hunter & Slemrod [9] and Truesdell & Noll, w Thus the generation of time periodic flows of arbitrarily large amplitude by application of time periodic forces must be treated with caution. The observations of Giacomin, Hatzikiriakos et al. in a series of papers ineluding references [7,10,11 ] and also those of Durand et al. [12 ] and their introduction of 'wall slip' are also pertinent to this point. Because of the above observations, it is considered prudent to consider firstly those flows as above which give rise to strain histories such that Ik(s; t)l is moderate for all 0 _ s < co. To this end it is assumed that k(s; t) C E where

)U--

{

/0

k(s;t)"

e-P~k(s;t)ds < o o , ~ ( p ) > 0

}

,

(33)

and k(t; p) --

e-P~k(s; t)ds < e ~ , ~(p) > 0

j~0~176

(34)

is an element of a Fr(~chet space of functions analytic in a right half-plane of the complex plane. Further, it is assumed that E is a subset of the domain of the functionals in (19). All k(s; t) such that [k(t;P)l<M0

(35)

belong to/C, and in particular all periodic flows with transform 27r

k(t;- p) - 1 -Pfe- :~____e~fo -z- e-P~q(t - s)ds

(36)

satisfy this requirement for all p such that ~(p) > 0 through proper choice of v q ( t -

1

s) -- 2 (t rD 2) ~. In addition 27r

-

q(t + - - - s) - q ( t - s) 4, k(t + cO

27r CO

;p) - k(t; p)

(37)

144 Among the materials covered are the Coleman-Noll[8] fading memory fluids requiring that for some real/3 > 0

1 > fO c~ e -;3~ { k2(s;t) + -~ lk4(s;t)} ~ ds > fo c~ e-Z~[k(s;t)lds,

(38)

the K-BKZ class for which

7-(k(~; t)) -

~- (k(~; t), ~)d~

j~0(X)

(39)

exists on (33), and various special models such as those named for Maxwell, which form sub-species of the above (cf. Crochet, Davies & Walters [13]). 2.2 Equations of M o t i o n It is assumed that each of the stress functionals appearing in (17) and (19) is Fr~chet differentiable over its domain in some normed vector space of functions on [0, ~ ) , which intersects with (33). For example, corresponding to T~0(. ) there exists a linear functional dT~=o(.) defined by

%~o (k(~; t) + ,~xk(~; t)) - LTo (k(~; t)) e----~0 6 (40)

d T ~ o (k(s; t)). Ak(s; t) - lim

for all Ak(s; t). Formally, the relation (40) may be replaced by dT (/c(t; p)). Ak(t; p) - lim 7 (k(t; p) + eAk(t; p)) - r (k(t; p))

e----~0

(41)

6

where

T (k(t; p)) -- ~'s~ (/~-1 [k(t; p)]) , } dT (~(t; p)) -- d~/-s~ (/~-1 [k(t; p)])./~-1

(42)

are compositions of operators. Also, if

Vi (xl , t; p) _ --[c~ e -psv i(x I t -- s)ds Jo

(43)

145

it follows that

Vi =

O~ i

Ot

_~_pOi

and

~1~

(44)

0~3 = -flpk(t;p). g•_33_ v lOxl

0 ~2

Ox 1

i -- (2 3) '

'

= -apk(t; p)

(45)

In general the extra stress tensor (46)

S-T+pI has the physical components

S -

I g( ii) i g(JJ) Sj - v/g(ii)g(jj)S ij

(47)

and in terms of them the equations of momentum balance are

flv/g(jj)

=~

OvJ

Ovj

v/g(j)) 0 i

+ ~ir

v/g(~) Ov/g(~) v~v~ +

- - ~ -t- v s Ox------~ -- g(JJ)

~

1

Ov/g(JJ)ox s vJ Vs ]

v/g(jj)

S~/g ) -t- 10v/g(jj) ~<jj>

Oxi

v/g(ii)g(jj)

1

{ Ov/g(JJ)

V/'g(ii)g(jj)

OxJ

2 ~SKij> OX i

g(jj) --

OxJ

Ov/g(~) OxJ

SKii>

}

--

1 op v/g(jj) OxJ

(48) Whether or not the preceding purely kinematic considerations are empty is determined by whether or not the equations (48) for specific choice of coordinates admit appropriate solutions for the velocity components (3) with rate of shear strain histories (11). It is evidently true that the form of the constitutive equation (17) must have a bearing on this, but the question of existence is most often avoided by the convenience of equating the inertial terms making up the left hand side of (48) to zero on the grounds that they are of no consequence. Solutions to the resulting divergence equation, involving the stress tensor only, may then be sought by the inverse, semi-inverse methods well known to elasticians. However, the errors that can arise from this simplification are all too commonly overlooked(cf. Schrag[2]).

146 2.2.1. Plane Flows. For these flows the chosen coordinate system is rectangular Cartesian with the cordinates of a fluid particle (x 1 , x 2 , x 3) identified with (x, y, z), while (~-1, ~-0,

V3 -- 0 ==~

k(S; t) -- -- ~0 ~ 0V2(X,

t -- r) d r

(49)

5x

so that (48) reduce to

0S ~x

0S Ox

OP = o Ox @

Ov 2

N

P-gi-

Op

o

(~0)

Oz

It follows from (50) that if a pressure p

-

(51)

a(t)y + b(t)+ S

is applied the solution of equations (50) reduces to the solution of aS _ a(t) -- p

Ox

~v 2

(52)

Ot

subject to ak(~; t) Ova(x, t - ~) (53) Os Ox If it is assumed that the inertial term in the right hand side of (52) is neglible and then set to zero, solutions to (52) and (53) may be sought in the separable form v~ - ~ {~y(x)},

(54)

~- r

which implies that

S ----~/-s~

( ~ If( x) --~(1

-

e-'~)

Then there are solutions if and only if

])

f(x)-

V'(x),.

(55)

147

(i) a(t) = 0 = f ' ( x ) = 0 while dT~=o exists, ~ - a (t) # 0. (ii) T~--0 is linear, f'(x) is constant and a(t + -5-) The first of these conditions relate to oscillatory plane shear flow and are appropriate to gap loading(cf. Ferry[I]). However, in order that (52) yield a unique value of S corresponding to each f ( x ) it is necessary and sufficient, since the frequency w must be absolutely small for neglible inertia, that lim

w--~0

~

,, (1 - e

r O.

b(M

(56)

If this condition does not hold then the possible effects of hysteresis must be considered(cf. Hunter &: Slemrod[9]). The second set of conditions relate to plane pulsatile flow, but the condition that T ~ 0 is linear is very restrictive. Among the fluid types satisfying it are the Navier- Stokes fluids and certain 'second order fluids' (cf. Truesdell & Noll[6]). Of course if the strain amplitudes are infinitesimal a linearized theory, which is compatible with both sets of conditions, is applicable. Existence of periodic solutions to (52), for a special model proposed by Slemrod[14], is assured by a theorem of Rabinowitz [15] (cf. Dunwoody[16]), if the nonlinearity in the equation is sufficiently weak. 2.2.2. Cylindrical Flows. For these flows the curvilinear coordinates (x 1, x 2, x 3) are identified with the cylindrical coordinates(r, 0, z), while in (11) - ~(~)

,

~-

~(~)

,

~-

Z(~)

,

(57)

so that the physical components of the extra stress tensor S are from (17) s - ~(~).

(58)

Hence, the equations (48) become for these flows

1 0 rot

(rS)-

S r

1 0 (~'S)-~- S

r Or

r 10 rot

(rS)

Op = Or

-pro 2

1 0p = - p r o

r O0 Op Oz = - p z

oo

oo

(59)

148 where t~ - o0 etc. It follows from these that the pressure required to support such flows is of the form

P--~-t-f

~--~r - ~ - p ( r b )

dr + a(t)O + b(t)z.

(60)

The velocity components (t), i) are obtained as the solutions, if such exist, to the equations

OS 2 pro.9 + a(t) Or + r S -r ' 0S 1 Or + -Sr - p2 + b(t) ,

(61)

where S = c t ( r ) T ~ o (k(s; t)) ,

S

0S Or

-- 2 S < 3 3 >

}

Op Or

r

--pr sin 2 0 ~2.

(86) It is clear from (85) and (86)2 that if

0p or

= o,

(87)

which is the condition necessary for the pressure to be single valued in r there is incompatibility between (83) and (86)2 unless the inertial term is zero, which is the case for steady flow. In addition,the relations (19) and (83) imply that 0 (S- S)o-7

~0 ( s < ~ >

- s)

so that there is compatibility between (83) and

- 0

(86)1,3 if and

(88)

only if

0 (~----~{Sis%0 (]9(8" t)) -~- $2c~ 0 (]9(8" t))} - 2 p r 2 sin 2 0r O~ ' ~= ' 00'

(89)

which cannot be satisfied for all r and 0 since O0 r O.

(90)

This incompatibility has not always been recognised(cf. MacDonald, Marsh & Ashare [19]). On equating all the inertial terms to zero in the right hand sides of the equations (86) they and (83) are compatible if and only if

c(t)

~ =

- 2 sln 0

0 {SiCs=O ~ (k(8", t))-~- $2 s=O c~ (k(8" t))} - O. .,

00

(91)

153

These conditions are satisfied for all fluids, but only approximately, when

2

c___0_~ 7~,

0r = - a (t) , (90

(92)

T~=o (fo a ( t - u)du) - c(t), where e is small. In particular, the time periodic form

- (aO + b)e ~ t

(93)

is possible under these stringent conditions, which again corresponds to a form of of 'gap loading'. The stresses 7', T < l l > and T at the boundaries required to maintain the flow are then readily obtained from (86) 1,3 once r is prescribed. As noted by Ericksen[20], there are two distinct approximations involved in the above analysis, one dynamical in nature and the other geometrical.

3. S M A L L A M P L I T U D E

FLOWS

In these flows a time periodic viscometric flow with small amplitude of displacement is assumed superimposed either on a rest state of the fluid, or a steady viscometric flow of it of the same type. The contribution that the neglect of inertia might make to error of measurement is the basic consideration.

3.1 Gap Loading and Inertia Since inertia is not being ignored, the discussion must be restricted to either plane or cylindrical flows, and to avoid the unnecessary complications arising from the geometry of the flow the former of these two types is chosen. A comprehensive treament of 'simple shear' has been given by Schrag [2], but a revised version influenced by the desire to extend it to the nonlinear theory of viscoelastic materials has been given by Dunwoody [21] and is preferred. It is assumed that in a time time periodic plane shear flow m~x Ik(~;t)l ) - t " -- - p r o (121) r

and

P - po + . / p r O 2 d r

(122)

is single valued in 0. The strain history is from (63)

t - S*)ds, _ ek(s; t). k ( s ; t ) - - / r O 0 (Os*

(123)

The flows envisaged are contained by two cylinders of radii a and b, a < b, so that the gap width is L - ( b - a). (124) Therefore, there are two fundamental lengths which must figure in the considerations of dimensional analysis, viz. the the inner radius a and the gap width L. It is evident from (41), (97) and (123) that, as in the case of plane flows, a separable solution to (121) may be sought which is time periodic. Since equation (121) is explicit in r it is appropriate to choose a as the more fundamental of these two lengths, and so corrresponding to (104) the non- dimensional set of variables

a~ -- r,

ea~av -- tO,

a~-lt-

t,

elU*(a~)la -- S

(125)

is introduced for these flows. In terms of them the equation (121) is replaced, on setting v - ~ [V(~)e ~t] , (126)

160

by the ordinary differential equation (127)

,~v"(, ~) + ,~v(,~)+ ( Z ~ - 1 ) - 0 where

~ _ _~R:I,*(~)I ,*(~)

and

R;

=

pwa 2

(128)

(~)1

I,*

"

The boundary conditions which are applied may be V(1)-0,

(129)

V(I+L)-I+L,

which correspond to the standard Couette flow. Alternatively, they may be V(1 + L ) - (1 + L)~ [~e'~] (130) d (~_lv(?~)) -- fl2:~V(1) -- 0 d§ § _

_

,

where the non-dimensional parameter S defined by the relation aS=

K-w2I 27rPw2a3L

(131)

depends on K, the restoring constant of a torsion wire attached to an inner cylinder of radius a, length L and moment of inertia I. Thus the conditions (130) are representative of a popular device in which the oscillations of an inner cylinder, created by the forced oscillations of an outer cylinder through the medium of a fluid contained between the cylinders, are resisted by the torsion wire (cf. Oldroyd et. al. [3], Markovitz [4], Walters [24]). The phase constant c in (130) allows for a difference in the phase angle of the oscillations of the two cylinders, and in arriving at (130) the identity b- ~0 (132) has been employed to eliminate 0 in favour of O. The solution corresponding to the boundary conditions (129) is (1+L) V ( ~ ) -- -- 2-W-(L-) { J l ( ~ r ) Y l ( f l ) -

tL/-

Jl(~)

[ t1+ Lt]

Yl(flr)Jl(fl))

v~(~)

[ t1+ Lt]

,

(133)

161 That corresponding to the the boundary conditions (130) is more complex,but the amplitude of the velocity at the inner cylinder and the phase angle of the outer cylinder relative to it are obtained from

V(1)- ~(1 + L ) e *c [Y2(/3)J1(/3)- J2(/3)Y1( z ) ] / D , D - Sl [/3(1+].)] Y1 [/3(1+ L)] v~(~) J~(9) + ~ J1 [/3(1+ L)] Y1 [/3(1+ L)] r Jl(/~) YI(~)

(134)

and the requirement that V(1) is real. The conditions that W and D are non-zero guarantees that there are no 'normal modes' in either case. Because the arguments of the Bessel functions in (133) and (134) are complex, asymptotic expansions have been employed in their evaluation(cf. Oldroyd [25]). In practice this causes no serious error as it is most likely that the fluids tested in cylindrical viscometers will be 'short memory' fluids of low viscosity. Interest will therefore be concentrated on reasonably high frequencies. As noted by Joseph[26], high frequencies in the argument of rj* (co) correspond to small times in the arguments of the relaxation modulus G(s) in (101), from which the asymptotic formula

G(0) ~ , G'(0) (~)~ ~ o(~co2)

~*(~)~E~

(1 35)

is readily derived, assuming G(s) is sufficiently smooth at s - 0. For small enough values of co

(136) and then the Bessel functions may be replaced by their power series expansions to obtain a power series in/3 to replace (133) say (cf. Walters [27]). On the other hand, for intermediate values of w Waiters [27], in his critique of previous work by Markovitz [4], has shown how to obtain a Taylor series in powers^of ~L for V(1) by expanding its expression in (134) about its value at L - 0. The condition for strong convergence of this series is ^

19IZ, 1 r

>> 1,

R* = ( 1 - r 2) F 1 + w~r" 1-r 2 sinnO,

(32)

w ( 2 > - ( 1 - r 2) F2 +r"F~ sin nO,

where, ~3v~ v2

V 2 w (i) = 0

"

'

i = 1,2, .. " '

An appropriate solution for

W (1)

V2 -

d2 1d + -~ + 3r 2 r Or

1 d2 r 2 ~5~2

(33) "

in equation (33) can be found by defining,

W(~)= (.4 (') eit+ A(')e-;' )sin nO+ Dr" sin nO,

(34)

where A ~1) is a complex function of r, and D is an arbitrary constant. Upon substitution of equation (34) into equation (33) it follows that, d2A(')2dr + rl dA(1) (dr

i ~ n_~) A = C (I) I, (i~r).

(36)

189

The constant C~) is complex, requirements. Defining,

to be determined

from physical

continuity

C (l) = C(l) + iC) ') , I, (~/if2 r) = In1 (r) + In2 (r), we obtain, A (1) - - a(~) + ia~ ~) , a(~) = C(~) I~, ( r ) - C ~ ~) 1.2 ( r ) , a~ ') = C~ l) Inl (r) +

C(1) 1,2

(37)

(r).

The first order velocity is now given by, w (') = 2 (a( 'J cos cot - a z(~) sin cot) sin n 0 + D r n sin n 0.

(38)

Setting D = 1 and equating equations (38) and (32) an equation for F/is found,

0-r2)Fl =2 [(a~1) -rnpl)COS t-(a~ 1) - r n p 2 ) s i n al

P ~ - l_r 2 ,

t] sin

nO,

(39)

a2

P2-

The functions p~ and

l_r 2. P2

are continuous functions of r. Since for r = 1 the

LHS of equation (39) is zero, this condition leads to, a~') (1) - p~(1),

a 2~) ( 1 ) = P 2 (1)

(40)

which determines the constants in equation (37). In this way F, becomes a continuous function of r throughout the domain. Equation (32)3 defines the second order velocity, where, w ~2) = (1-r 2) F 2 + rn(F~ cos t-F~2 sin t) sin 2nO,

F~(r) = 2(a~) - r " p , ) F~2(r) = 2(a~ ~)- rnp2) l_r z ' l_r 2 9

(41)

190

Equation (41) may be rendered linear in trigonometric functions by using the relation, sin 2 nO

= 1- cos2n0

Physical continuity at r = 1 demands that w ~2) should be of the form,

w '2' = 2(b~ c o s t - b 2 sin t ) + 2(a( 2' c o s t - a ~ 2' sin t ) c o s 2 n 0 . The unknown functions equation (33),

B(r)

e i' + - B ( r )

e -it ,

(42)

b, (r), b2 (r) can be obtained from a 0 - free solution of

B(r) = C o I o (x]~r), C O = Co, + i Co2, I o = Iol +ilo2,

(43)

and similarly, A ~2' : C ~2' 12,

(x/-f-t-~r) = a~2) + ia~2' , C ~2) : C(2) + iC~2' ,

I2n : 12n I .qt_ //2n2 "

The four c o n s t a n t s Col , C02 , C~ 2) and C~2) can be determined by equating equations (41) and (42), and setting r = 1. In this way, the expression for F 2 is derived as, 2 F2 - 1 - r 2

2 1-r

bl

r "F~I 4

cos t

-

b2

[(a(2)+ r n F ~ ) c o s t - ( a ~ 2 ) + 2

4

Higher order terms and several values of oscillating flow. It is in this case for e, = intervals.

4

sin t +

rnF~z)sint 1 cos 2n 0. 4

(44)

can be found in a similar manner. Computations for n = 6 .O were made by Letelier et al [8], for the case of a purely noted that the pipe contour approaches a regular hexagon, 0.1481. Wall shear stress was also computed at finite time

191

This method can be applied to flows other than oscillatory, as long as it is not necessary to impose an initial condition. Such a condition cannot be satisfied by the present analysis. 3. VISCOELASTIC FLOW IN CIRCULAR PIPES

Steady, fully developed, viscoelastic pipe flows are rectilinear when the pipe is a circular cylinder [11, 12]. This is true even when the flow is unsteady, regardless the constitutive behavior. Some special results of practical interest arise from non-linear effects. One of these results is flow enhancement in periodic flows. Several authors have explored this important aspect both theoretically and experimentally [1, 2, 14 - 18]. Integral models seem to simulate experimental data better than other models. An example of integral models is Green and Rivlin's [19] simple fluid model of the multiple integral type. For incompressible fluids, the total stress T, can be expressed in terms of the strain history G, on a particle X, T + p l = Fs:o[G(X,s)] ,

s = t-r,

(45)

where p, t and r are the pressure, time, and past time, respectively. Following Green and Rivlin, the functional F is expressed as a Fr6chet series, F [ G ( X ; ~)] : e.S 25 kHz).

229

14

l O - 3 x Pac ' P a

12 10

c

= 0 ppm

c = 3 ppm m

-

(

zone

zone

one V

-

}~

o~ 0.5

10-SxRe t

0.6

0.7 I

15

0.8 I

0.9

1.0

1.1

1.2

1.3

1.4

I

I

I

I

I

I

I

I

10 9

8

7

6

5

4

3

2

(7

Figure 11 : Acoustic pressure as a function of the cavitation number and of the Reynolds number for water and a 3 ppm PEO homogeneous polymer solution flowing around a circular cylinder. From Reitzer et al. [36].

Reitzer et al. [36] have also investigated the flow around a cylinder in an open loop cavitation tunnel. A 1000 ppm PEO solution was ejected upstream of the test section with a flow rate such that the concentration of the homogeneous solution in the test section reached an homogeneous concentration of 3 ppm after mixing. Their main results are summarized in Figure 11, where the acoustic pressure is plotted as a function of the cavitation and Reynolds numbers for water and for the homogeneous polymer solution. For water, there is a region, zone I, where the flow is non cavitating, up to a Reynolds number, defined with the free stream velocity and the diameter of the cylinder, of 85 000, for which cavitation is initiated and isolated bubbles can be seen. Beyond this Reynolds number, cavitation develops and the acoustic pressure increases up to a maximum (zone II), after which it decreases over a limited range of Reynolds (zone III). After a relative minimum, the noise increases very sharply, reaches another maximum much larger than the previous one (zone IV) and then decreases as sharply as it had increased (zone V). By comparison, the polymer solution completely inhibits zone II, which is replaced by the continuation of zone I, and zone III, where the noise was decreasing in the case of water, is replaced by an increasing region that fits perfectly zone IV. The polymer has no effect whatsoever on zones IV and V where cavitation occurs respectively in the form of alternating bursts and an elongated vapour cavity. From these and other data, the authors conclude, as in other previous works, that the cavitation retardation associated with the

230

presence of the polymer molecules is more likely due to the modification of the flow field around the cylinder than an effect of the macromolecules on the dynamics of individual bubbles.

0.65 -

Inception parameter

0.60

9

0.55 -O

00

oemo o o o

O

PEO-FRA

9 A

[]

9

v~mvvV

9 Am~ mm

0.50

9

t o~ .. 9 9

9

[]

~pr Ipr V

A

m m [] A Ak A A y y y v

0.45

A

V

0 ppm 100 ppm 250 ppm 500 ppm

A

V"

9

IPr

0.40 1 0 - S x Re 0.35 0.8 0.65 -

I 1.0

I 1.2

I 1.4

I 1.6

I 1.8

I 2.0

I 2.2

Inception parameter

0.60 -

9

O0 9

i~ 9 9

9

-o e ~ 9 emo oo 9

mum

0.55

PAM-273

nnn

0 []

[][] mmmam m 9 0.50

! 2.4

At

A

mum

T

~?V V 0.45

_

k

A

0 ppm 100 ppm 300 ppm 500 ppm

V

0.40

10-5• Re

0.35 0.8

I

I

I

I

I

I

I

I

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Figure 12 9 Inception cavitation numbers as a function of the Reynolds numbers, computed using the local azimuthal velocity and the diameter of the protrusion, for pure water and three concentrations of PEO and PAM. From Ting [40].

The experiments with cylinders can be compared to those conducted by Ting [40] in a specially designed unit allowing him to perform experiments using very small amounts of test fluids. It consisted of a circular plate, on the surface of which were two protrusions having a diameter and height of 2.9 mm, rotating inside a casing. The protrusions were observed through a transparent wall in order to determine, visually, the initiation and the extent of the

231

cavitation. The inception cavitation numbers as a function of the Reynolds numbers, computed using the local azimuthal velocity and the diameter of the protrusion, for pure water and three concentrations of PEO and PAM, are plotted in Figure 12. For the largest concentration, PAM seems to be more effective over a larger range of Reynolds numbers as a result, probably, of its larger capacity to sustain high shear rates with moderate degradation. Ting also signalled that the appearance of the polymer cavities was more transparent than in water and showed a regular and smooth wavy pattern, as beautifully demonstrated by Brennen's [38] visualizations. 4.3 Vortex cavitation It is well known that viscoelastic fluids cause very strong effects in rotating flows. In particular, while in a Newtonian (or even inviscid) fluid the rotation of a rod makes that the surface of the liquid is depressed near the rod, in a polymeric fluid the liquid will climb the rod. This is the famous Weissenberg effect. The non-Newtonian liquid develops a normal stress contribution which acts in such a way that the depression induced by the rotation of the liquid is overcome, and even exceeded. In fact, the so called "vortex inhibition phenomena" has been used as a method to estimate quantitatively and qualitatively the viscoelastic properties of liquids (Gordon and Balakrishnan [41]). It seems therefore natural to assume that the contribution of normal stresses may have a significant effect on the cavitation which may occur in confined (vortex chambers) or unconfined (tip vortices) situations. For an axisymmetric line vortex, the radial component of the momentum equation, assuming that the radial component of the velocity is zero, is given by, =

Or

R

OTrr

Trr -- TOO

Or

r

(17)

where V o is the tangential velocity, and Tii the extra stresses (i = r, 0). The pressure on the axis of the vortex is obtained by integrating equation (17), from infinity to r = 0. It is easy to see that the contribution of the rotation, whatever the velocity distribution, is to decrease the pressure on the vortex axis. In a Newtonian fluid the extra stress terms cancel. In non-Newtonian viscoelastic fluids, the experience shows, as stated above, that these terms contribute to diminish the centrifugal effect. Inge and Bark [42] conducted experiments with a specially designed elliptical wing having a maximum chord of 0.16 m and a half span of 0.238 m. They reported results obtained by ejecting concentrated polymer solution into the water stream through a multiperforated injector situated one meter upstream of the foil and for homogeneous polymer solutions with concentrations

232

between 0.01 and 12 ppm of PEO Union Carbide WSR-301. In Figure 13, the values of a modified cavitation number, taking due account of the different values of the incidence angle and of the free stream velocity, are plotted as a function of the polymer concentration in the homogeneous solutions for three conditions of cavitation development. The results show very clearly that the incipient cavitation occurrence is significantly delayed for concentrations larger than 1 ppm. The same is apparent for the other conditions of cavitation although the changes are proportionally smaller. Interestingly enough, the permanent cavity and the attached cavity situations cannot be distinguished for the largest concentration. The experiments with the polymer ejection gave qualitatively similar results.

o"n O o

O Incipient cavitation [] Appearance o f a p e r m a n e n t cavity ZX Cavity reaches the w i n g

~

0

o

[]

El

....

c, ppm 0

I 0.01

I 0.1

I 1

]

,,,

10

Figure 13 9 Values of a modified cavitation number, taking into account the effect of the different values of the incidence angle and of the free stream velocity, as a function of the polymer concentration for three conditions of cavitation development. From Inge and Bark [42]. Fruman [43] investigated the behavior of the tip vortex of a NACA 16020 cross section elliptical foil in water and in water with polymer solutions ejected from a 0.5 mm diameter tube at a distance of 20 mm upstream of the tip. Prior to the polymer tests, experiments conducted by ejecting a 50% water + glycerin mixture, with a viscosity of 10-2 Pa.s, did not show any significant contribution of the augmentation of the viscosity of an otherwise Newtonian fluid. Ejecting a 500 ppm PEO solution when a quite large vapour tube occupies the vortex core considerably modifies the appearance of the cavitation. At an ejection velocity of about half of the free stream the

233

continuous very long cavity is reduced to a very short cavity of about half of a maximum chord length and only scattered isolated bubbles are carried downstream. By doubling the ejection velocity these entrained bubbles are eliminated and only the shortened cavity remains. In terms of the incipient cavitation number, Figure 14 shows inception results obtained with an ejection of a 250 ppm PEO WSR 301 solution at an ejection velocity slightly larger than the free stream.

2.5

cYi

no ejection polymer

2.0

.....

1.5

1.0 o

0.5 5

I

I

I

7.5

10

12.5

Figure 14 9 Inception cavitation numbers as a function of the wing incidence angle during the ejection of a 250 ppm PEO WSR 301 solution from a 0.5 mm diameter tube situated 20 mm upstream of the wing tip. From Frurnan [43].

Thus, it appears that the polymer solutions have a strong effect on the tip vortex cavitation occurrence and development. However, it is not clear what is the mechanism responsible for such changes because, in these experiments, neither the tangential velocities of the vortex nor the lift of the wing have been measured. The latter is a very important factor since the tip vortex intensity is directly related to the mid-span circulation of the foil and, as shown by Wu [44] and Kowalski [45], lifting surfaces immersed in flowing homogeneous polymer solutions display a reduction of lift. In order to elucidate these aspects, experiments were conducted by ejecting the polymer solution from the tip of the wing through a small diameter orifice (Fruman et al. [46]), measuring both the tangential velocity distributions and the incipient cavitation numbers. The results show a significant reduction of the incipient cavitation number and, more interesting, a decrease of the

234

maximum tangential velocity and of the slope of the velocities in the core region when the polymer solution was ejected, Figure 15. Measurements of the lift did not show any change during polymer ejection (Aflalo [47], Fruman and Aflalo [48]) in agreement with the fact that the tangential velocities in the potential region were unchanged.

V, m/s qi, m 3/s

2

0 0 . 7 x 10 -6 2.2 x 10 -6

1

0

-1

-

U o o = 5 m/s

~=10 ~ ~2

-12 1.5

I

I

I

I

I

I

I

I

I

I

-10

-8

-6

-4

-2

2

4

6

8

10

V, m/s

1.0 0.5 !

--0.5

-1.0

U -

-1.5

-12

=5m/s 0~=5 ~

I

I

I

I

I

-10

-8

--6

-4

-2

0

I

I

I

I

I

2

4

6

8

10

Figure 15 : Vertical (tangential) velocities as a function of distance to the vortex axis for different ejection rates of a 1000 ppm PEO WSR 301 solution. Measurements conducted at 0.20 m (five maximum chords) downstream the tip. From Aflalo [47] and Fruman and Aflalo [48].

235

(P ~ - P o)P / (P ~ - P o)w

1.0

u

[] []

0.8

5 5.5 5 5.5

[] []

[]

0

0

0.6

a;

, m/s

o

5 5 10 10

[]

0.4

0

0

0.2 107x q-/ 0 0

I 0.5

I 1.0

I 1.5

I

2.0

Figure 16 9 Ratio of the pressure difference between infinity and the vortex axis for the ejection of the polymer solution and the pure water as a function of the reduced injection rate. 1000 ppm PEO solutions. From Aflalo [47]. Using these velocity distributions it was possible to integrate between a position far from the vortex, where the pressure is p=, to a position on the vortex axis, where the pressure is P0, the Vff/r term of equation (17) in the case of pure water flow, index w, and polymer ejection, index p. The ratio of the resulting pressure difference contributions are plotted in Figure 16 as a function of the non dimensional ejection flow rate, m

qi =

qi c

U~ b Cma x

The results show that the polymer ejection causes a reduction of the pressure difference as compared to the pure water situation. This reduction is largely sufficient to justify the delay on the cavitation occurrence even if the contribution of the normal stresses in equation (17) is ignored. An investigation on the effect of homogeneous polymer solution on tip vortex cavitation was conducted by Fruman and Aflalo [48]. The homogeneous solution, of about 10 ppm, was obtained by injecting, upstream of the water tunnel recirculating pump, a concentrated polymer (1500 ppm).

236

, Cavitationnumber at time t umber at t = 0

1.00 (

0.95

0.90

0.85

-

Time after beginning mixing, s 0.80 0

i

i

i

1

I

200

400

600

800

1000

At time t=O

Attime t= lOOs

At fin~ t = 9OO s

Figure 17 : Top : evolution of the cavitation number as a function of the time elapsed after the beginning of the mixing of a 1500 ppm PEO solution into the recirculating water of the cavitation tunnel. Bottom : photographs showing the elliptical foil (left) and the tip vortex cavity initiated at the tip and extending downstream for pure water (t =0) and with increasing homogeneous polymer concentrations (t =100 and 900 s). From Fruman and Aflalo [48].

237

The photographs of Figure 17 show that, in spite of the fact that the modification of the flow rate and the reference pressure caused a significant reduction of the cavitation number during the process of mixing, for an initially well developed cavity the build-up of the polymer concentration in the cavitation tunnel recirculating water was responsible for a significant decrease of the cavity length. It is interesting to note also that the polymer seems to be more effective at distances from the tip of several maximum chords and do not causes the cavity to detach from the foil.

3.0

8

2.5

iO

--

0

water polymer

@0

w~

2.0-

(9 []

0

Om

o

0

[]

1.5-

u

0 0

o

0~

o~ o

[]

1.0-

O~

~ 0.5 0.1

c

Z

I

I

I

I

I

t

I

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 18 9 Desinent cavitation numbers as a function of the lift coefficient for the NACA 16020 elliptical foil in an homogeneous polymer solution. Open and closed symbols refer to two free stream velocities, 8 and 8.4 m/s. From Fruman and Aflalo [48]. The measurements of the lift of the foil showed that it was markedly reduced in the homogeneous polymer solution. The tangential velocities were also considerably reduced in both the core and the potential region. The latter is in direct agreement with the reduction of the lift and hence of the mid-span circulation. Figure 18 shows the desinent cavitation numbers as a function of the measured lift coefficients in water and homogeneous polymer solution. The effect of the polymer solution is to degrade the performance of the foil since, for the same lift coefficient, the critical cavitation number is larger. The effect of the polymer solution on the lift is largely more important than that on the inhibition of cavitation inception. Since the boundary layer of the NACA 16020 section used in these tests is very sensitive to laminar-to-turbulent layer transition (Pichon et al. [49]) it may well be that the extremely large decrease

238

of the lift coefficient is due, in part, to the already mentioned destabilizing effect of the homogeneous polymer solutions. The effect of doubling the size of the foils and increasing the free stream velocity, increasing thus the Reynolds number, on the effectiveness of the polymer ejection at the tip of the foil was investigated recently by Fruman et al. [50] with results comparable to those already reported, provided the ejection flow rate was scaled with the Reynolds number. A very detailed survey of the tangential and axial velocity distributions at different positions along the vortex path was also conducted in the very near region (less than a maximum chord) downstream of the tip. The original results, showing a decrease of the maximum tangential velocity and of the slope of the solid body rotation core, were confirmed. Again, the increase of the pressure on the vortex axis due to the velocity modification gives a decrease of the desinent cavitation number, Figure 19.

2.0

w Cp

llum

I

/

"~-- crd withoutejection

J

~

1.5

L

0,~ .. ~

_

m

I0 r

e

no ejection

cre with polymer ejection

1.0

III

I

0.5 0

I 0.2

I

I 0.4

I

I 0.6

I

I 0.8

X /C max I I

1.0

Figure 19 : Pressure coefficient on the vortex axis, computed using expression (17) ignoring the stress terms, as a function of the distance to the tip. The values of the desinent cavitation numbers for the conditions of pure water and polymer ejection flows are also indicated. The minimum of the pressure coefficients are very close to the desinent cavitation numbers. From Fruman et al. [50]. The ejection of polymer solutions at the tip of the foils to delay cavitation has been applied with success by Chahine et al. [51 ] to a five blade screw propeller of about 29 cm diameter. Because of the relatively low aspect ratio of the blades, the position of the tip vortex detachment was not as easy to establish as

239

in the case of the elliptical foils. Therefore, a preliminary investigation was conducted in order to determine the positions of ejection ports allowing the ejected fluid to be carried out into the tip vortex. Once these positions were chosen, experiments were conducted to determine the effect of the polymer concentration and solution flow rate. As an example, Figure 20 shows the cavitation number for inception as a function of the polymer concentration for the pure water conditions, for water/glycerin injection and for polymer injection. The most important result is that, depending on the position of the ejection ports, the effect of polymer injection can be unnoticeable or achieve a decrease of the incipient cavitation number of up to 35%.

17 - C a v i t a t i o n n u m b e r

water + glycerin injection blade #5, no injection blade #1 }injection blade #3

16 15 14 13 12

~ ~ o l y m e r 10

concentration, ppm

I

I

I

I

I

I

I

1000

2000

3000

4000

5000

6000

7000

Figure 20 9 Inception cavitation number as a function of the polymer concentration for the pure water conditions, for water/glycerin injection and for polymer injection. From Chahine et al. [51 ]. Inge [52] investigated the contribution of the stresses due to an Oldroyd Bfluid to the pressure within a weak vortex. The numerical computations showed that the pressure on the vortex axis was increased with only a small influence on the velocity field. This finding is not confirmed by experiments, which show that, whatever the conditions - polymer ejection or homogeneous solutions -, the azimuthal velocity profiles are modified enough to give an increase of the pressure on the vortex axis. Hoyt [53] investigated the effect of homogeneous polymer solutions of PEO on the onset of cavitation in a vortex device where the liquid was injected tangentially from a single port and evacuated through an axial pipe. The results are summarized in Table 2. Incipient cavitation numbers are defined as

240

the difference between the discharge pressure and the vapour pressure divided by the pressure difference across the device, and the air content is non dimensionalized by the value at saturation. There is a large inhibition effect for the PEO solutions, even at very low concentrations. However, a non drag reducing polymer, Carbopol, does not display an inhibition effect. Table 2 : Incipient cavitation number and air content for tests in a vortex chamber. From Hoyt [53]. Test liquid Dearated water Tap water PEO, 8.2 ppm PEO, 12 ppm PEO, 16 ppm PEO, 17.5 ppm Carbopol, 20 ppm

Incipient Cavitation Number 0.253 1.480 1.330 0.547 0.513 0.100 0.107 1.300

Air content/Air content at saturation 0.284 0.701 0.755 0.709 0.709 0.785

Bismuth [54] conducted tests in a long vortex chamber where the fluid was introduced tangentially at one end through eight rectangular slits and evacuated axially at the other end. The reference pressure was measured in a large reservoir situated downstream of the vortex chamber and used to determine the incipient cavitation number using, instead of the pressure drop across the chamber as in Hoyt [53], the kinematic head at the exit section. For a 10 ppm concentration of PEO, the cavitation onset was advanced, as compared to the pure water results, Figure 21, in complete disagreement with the earlier results by Hoyt [53]. It has to be pointed out that, because of the different definition of the cavitation number, direct comparison of these results is difficult. However, if it is accepted that the drag reducing properties of the polymer solution should increase, at equal pressure drop, the flow rate, the results by Hoyt [53] should correspond to much larger inhibition effects. On the other hand, velocity measurements made by Bismuth [54] did show that the tangential component of the velocity increased, as compared to pure water, when moving from the wall towards the center of the vortex. This increase was large enough to justify, by integration of the simplified radial momentum equation (equation (17) without the normal stress terms), the enhanced cavitation characteristic of this flow. Moreover, since the flow rate determines the ejection tangential velocity, and hence the circulation, Figure 21 can be compared to Figure 18, where the lift coefficient is also proportional to the tip

241

vortex circulation. In both cases there is augmentation of the incipient cavitation numbers as compared to pure water.

20

[]

[]

19

18

17

-

16

-

EO]

f

I

I

IQ' 10-3m3/~

15 1.1

1.0

1.2

1.3

1.4

I

I

I

I

11

12

13

14

Ejection tangential velocity, m/s

Figure 21 9 Inception cavitation number in the vortex tube as a function of the flow rate for water and a 10 ppm homogeneous polymer (PEO) solution. From Bismuth [54].

Bismuth also performed experiments by ejecting, through an orifice of 1 mm diameter situated at the end opposite to the evacuation, semi dilute solutions of PEO. In Figure 22, the ratio between the incipient cavitation number with polymer ejection and the one for pure water are plotted as a function of the ratio of flow rates, ejection/total, for different values of the reference pressure in the downstream reservoir. As shown, the polymer ejection significantly inhibits cavitation onset. This figure is very much analogous to Figure 16 and leads us to think that polymer ejection in the center of a confined vortex or of a tip vortex has the same qualitative effect. In well developed cavitation conditions, the vapour tube on the chamber axis is made to disappear over increased distances when the rates of ejection of the polymer solution increase.

242

~Rp CrRw PR

1.00

0.86 0.98 1.17

0.95 _

[]

0.90

1.23

0

1.35

o 0.85 []

0.80

qi /Q x103 0.75 0

I

I

I

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Figure 22 9 Ratio between the incipient cavitation number with polymer ejection and the one for pure water as a function of the ratio of flow rates, ejection/total, for different values of the reference pressure in the downstream reservoir. From Bismuth [54].

4.4 Cavitation in confined spaces The flow occurring between a moving and a fixed wall separated by micron size gaps is of the lubrication type and is characterized by very large pressure gradients in the flow direction and, within the usual assumptions, nearly constant pressures in the direction normal to the flow. Moreover, because the minimum pressure can be made equal or less than the vapor pressure, cavitation can locally occur. It can be theorized that the elongational flow prevailing in the vicinity of the minimum gap will promote the occurrence of a viscoelastic contribution, if polymer solutions are employed, and modify the conditions for cavitation onset and development. Ouibrahim et al. [55] have conducted experiments in a micron size space, with a minimum gap, e, comprised between 5 and 20 ~tm, confined by a rotating drum of radius, R, equal to 7 cm, and a fixed flat wall with a Newtonian fluid (water) and a nonNewtonian viscoelastic fluid (600 ppm aqueous solution of PEO WSR 301). In this peculiar situation, the product of (e/R)Re, where Re is the Reynolds number computed with the drum tangential velocity, mR, and e, are much smaller than unity as lubrication theory requires. The cavitation number, is defined as,

Prey -- Pv or- I -~- P ( ( o R ) 2

(18)

243

where Prey is a reference pressure. In the case of water at inception conditions, intermittent cavitation spots, scattered along a very thin band parallel to the cylinder axis, appear downstream of the minimum gap. If the cavitation number is decreased, the cavitation spots increase in number, become less scattered, adopt the shape of an arrowhead with the tip facing the upstream of the rotating cylinder and show bubbles escaping from the trailing edge. When the cavitation number is further reduced, a string of nearly regular cavities is formed. They have characteristic dimensions which are of the order of a few millimetres, separated by thin lateral liquid films. m

O' i

(:rip= 2514 Re -1.88 lO

•iw =

f

O o

7287 Re

-1.84

polymer water Re

5

I llll 5 6 789

10

I 2

I 3

456789

2

10

I 2

I 3

Figure 23 9 Incipient cavitation numbers as a function of the Reynolds number for water and polymer solution and e - 10 ~tm. From Ouibrahim et al. [55]. In the case of the polymer solution and of incipient conditions, intermittent tiny cavitation spots in the shape of rod-like cells scattered along a very thin band parallel to the cylinder axis, appear downstream of the position of the minimum gap. When the reference pressure is decreased below the one for cavitation inception, a nearly continuous vapour cavity, characterized by a practically straight leading edge and a more or less undulating trailing edge, develops parallel to the axis of the rotating cylinder. This continuous cavity results from the lateral coalescence of the arrowhead cavities, present only for a very small range of the reference pressure. The difference in morphology is not a consequence of the modification of the shear viscosity, since tests conducted with a water/glycerine solution, having a viscosity of 11 mPa.s, much larger than the one for the polymer solution, 2.05 mPa.s, shows

244

arrowhead cavities analogous to the ones observed in the case of water. When the cavitation number decreases, the trailing edge waviness increases while oscillations occur in the flow and axial directions. For extreme conditions the movement of the cavity trailing edge becomes chaotic. In Figure 23, the inception cavitation numbers as a function of the Reynolds number (computed using the solvent viscosity and the minimum gap) have been plotted in logarithmic coordinates for e = 10 ~tm. As shown, the initiation of the cavitation is delayed in the case of the polymer solution but the slope remains the same as that in the case of water. It can be hypothesized that the inhibition effect is associated with the elastic contribution of the normal stresses, which has the net effect of reducing the incipient cavitation number, that is to say, to increase the pressure over the Newtonian contribution. The added pressure, APe, can be estimated thus from the difference of incipient cavitation numbers at an equal Reynolds number, by, 1

APe -

2 t%

fie,

9 8 7

tYiw l.

-

(19)

APe, Pa

Pa.s ~

,-

-, --0.8236

3

-

6 5 2

4 3

2

Ou/Ox, s -1 10 10 3

I

i

I

I

i

i I tt

2

3

4

5

6789

:

105 2

10 4

Figure 24 9 Estimated elastic contribution to the pressure and elongational viscosity as a function of the estimated elongational strain rate. From Ouibrahim et al. [55].

245

In Figure 24, the values of APe have been plotted, together with an estimate of the elongational viscosity defined as, ~e = ~

APe

c)u "/ Ox

(20)

where c)u / Ox (=70 co) is the maximum elongational strain rate, as a function of the latter. The elastic pressure contribution increases with the strain rate increasing while the elongational viscosity decreases. The latter is up to four orders of magnitude larger than the shear viscosity (a Trouton ratio as high as 104). These Trouton ratios and elongational viscosities are comparable to the ones obtained for a variety of polymer solutions displaying an elastic behavior. It seems that the effect of the polymer solution in the present situation is due to the development of normal stresses in the elongational flow in the space confined between the moving and the fixed wall. 5. C A V I T A T I O N

EROSION

The inhibition or enhancement of material erosion caused by cavitation in non-Newtonian liquids has been a subject of much conflict and debate. This is not surprising since, in the case of Newtonian fluids, the mechanism of cavitation erosion is not well understood either in stagnant or flowing liquids. Indeed, erosion seems to result from the combined action of the pressure wave, generated during the near spherical phase of collapse of the bubbles, and the reentering jet generated during the terminal, highly non spherical, phase of collapse. Moreover, in flowing liquids, the intensity of erosion (rate of material removal) increases very rapidly when the free stream velocity increases. Finally, the interaction between the mechanical and crystalline properties of the eroded material and the hydrodynamics of the bubble can be quite strong. Tests in stagnant fluids have been conducted by inducing ultrasonic cavitation on metallic samples and determining the weight loss as a function of time. Ashworth and Procter [56], with copper test specimens placed at 1.3 mm below the tip of an ultrasonic probe (horn) obtained, as compared to the results in pure water, a decrease of the incubation time and an increase of the erosion rate with a 1000 ppm concentration of the PAM agent in distilled water, Figure 25. For a concentration of 100 ppm, no significant change in behavior occurs. It has to be mention that, because polymers in solution are partially degraded by extensive exposure to ultrasonic cavitation, these investigators took the precaution to have a fresh solution continuously fed into the test reservoir in order to achieve an equilibrium solution. In a glyceroldistilled water mixture, having a viscosity comparable to that of the "equilibrated" 1000 ppm PAM solution, the incubation time increased by a

246

factor of three as compared to pure water and the erosion rate decreases slightly.

W e i g h t loss, mg

J

[]

50 Fluid

40

J

water glycerol-distilled water mixture

"o

)J

100 p p m P A M 1000 p p m P A M

20 10 E r o s i o n t i m e , rain I I

0

10

20

30

40

50

60

Figure 25 9 Weight loss as a function of time for copper specimens subject to ultrasonic cavitation. From Ashworth and Procter [56].

Weight loss, mg

f = 19.5 k H z A = 38 ~tm t I = 295 K

50 ..O

J

40

Fluid

30 0

.~/

20 -

[] A 0

10

0

I

i

I

I

I

I

10

20

30

40

50

60

water 100 p p m P E O 500 p p m P E O 1000 p p m P E O

Figure 26 9 Weight loss as a function of time for aluminium specimens subject to ultrasonic cavitation. From Shima et al. [57].

247

Ten years after these early results, Shima et al. [57] conducted very similar experiments, but used PEO instead of PAM and had the aluminium test specimen attached to the vibrating rod. Their results are very much different of those of Ashworth and Procter [56]. First, no incubation time seems to be necessary either in water or polymer solution; second, a 100 ppm solution shows a behavior similar to that of water with a slight increase of the weight loss; third, for 500 and 1000 ppm solutions the weight loss increases for a short time after initiation and decreases significantly thereafter, Figure 26. When the 1000 ppm polymer solution is moderately degraded the weight loss increased in the early stages and decreased afterwards while for a fully degraded solution the results are very close to those for pure water. Again, a water/glycerol mixture decreased the material removal. Tsujino [58] pursued the research by Shima et al. [57] and investigated the effect of the peak-to-peak amplitude. He showed that when the amplitude is decreased, from 38 to 25 ~trn, water and a 1000 ppm PEO solution have exactly the same behavior during a very short duration (-10 min) after initiation and then differ, the aggressiveness being much decreased in the polymer solution. 50

Weight loss, mg

f = 19.5 kHz t I = 295 K

40 30

A= 38 l

-

20

a

m

~

~

A, vibratory amplitude

~'0

-

A Shimaet al. [57] 9 Tsujino[58]

-

/

_

/

/

/

~

' \ A = 25 gm

O Ashworth and

Procter [56]

10 I Test ti~ne, min I 0

0

10

20

30

40

s0

60

Figure 27 9 Weight loss as a function of time in water. Adapted from Ashworth and Procter [56], Shima et al. [57] and Tsujino [58]. As shown, the results obtained by Ashworth and Procter [56] on one side, and Shima et al. [57] and Tsujino [58] on the other, are markedly different in spite of the fact that the two polymers used in the tests are very effective drag reducing agents, display analogous viscoelastic effects, and have, for equal concentrations, comparable shear viscosities in solution. The major differences

248

have to do with the materials and with the position of the test specimens in the ultrasonic pressure field. However, in the case of pure water, the weight loss versus time remains nearly linear regardless of the specimen material, the position of the sample and the amplitude of the ultrasound, Figure 27. In the polymer solutions, a linear behavior has been shown to exist by Ashworth and Procter, while a power law behavior, with a power of less than unity, has been demonstrated by Shima et al. and Tsujino. In a quite recent work, Hoyt and Morgan [59] have reported results analogous to those of the Japanese researchers when a 200 ppm PAM was added to a 6% mixture of tomato puree in water. How then to interpret the changes in the case of the polymer solutions ? No explanation has been proposed as yet, although the visualisation conducted by Shima et al. and Tsujino show conclusively that the development of the cavitation bubbles in the more concentrated PEO solutions is very different from the one observed in pure water. In the latter case, a cloud of bubbles develops in an orderly fashion on the surface of the horn with a thickness increasing from the rim to the center and, away from this surface and around the axis, a column of small bubbles extending quite far away (one to two times the diameter of the horn). When the erosion progresses, the thickness of the cloud seems to increase on the surface and, in particular, around the center of the sample. In 500 and 1000 ppm polymer solutions, isolated bubbles, clouds or filaments, occupying a cylinder of at least the diameter of the horn, occur away the sample surface and on the side wall of it, in what can be said to be a disorderly fashion (as compared to water). When time, and hence erosion, progresses, bubble clouds become denser and thicker on the surface and on a column extending away from it. Therefore, it seems that, in the polymer solutions, low pressure regions, where cavitation can occur, are developed because of macro scale (of the size of the sample diameter) changes in regions which are not even supposed to be affected by the vibration, such as the side wall of the sample and the column mentioned above. For the test conditions of Shima et al. [57] and Tsu~ino [58], it seems reasonable to hypothesize that the very dense and thick cavitation, developing on the test specimen surface when erosion progresses, has a cushioning effect that reduces the damage due to bubble (cloud) collapse. For the Ashworth and Procter test geometry, this cushioning effect is reduced because of the distance between the sample and the tip of the vibratory horn and the modification of the macro scale (which is in this case of the order of magnitude of the space between the sample and the tip). Ting [60] offered a completely different explanation of the Ashworth and Procter results. He claimed that the enhanced erosion resulted from the "solidlike" behavior of the reentering jet and the additional impact stress developed in the extremely rapid deformation process undergone by the fluid. In view of the results by Shima et al. [57], Tsu~ino [58] and Hoyt and Morgan [59] this reentering jet erosion enhancement effect does not seem to be of primary

249

importance. In fact, because all of the single bubble behavior investigations indicate that the effect of viscoelasticity is to keep the bubbles spherical much longer than in the Newtonian solvent, it seems reasonable to expect, if no macro scale changes of the flow occur, a reduction of cavitation erosion. Only one paper (Shapoval and Shal'nev [61]) refers directly to erosion in a flowing PAM solution. In their tests, cavitation was generated in the wake of a cylinder attached to the wall of a rectangular section. They determined the cavitation numbers corresponding to the onset of cavitation and to the situation corresponding to a cavity having a given length. In the polymer solution onset is quite significantly delayed as compared to water. For a cavity having a length of the cavitation zone equal to the cylinder diameter, the cavitation number in the polymer solution was also reduced but proportionally less than for the onset condition. The removal of material from a sheet of rolled lead placed on the wall of the test section behind the cylinder was reduced in the case of the polymer solution when the length of the cavity was twice the diameter of the obstacle. Assuming that the pattern of the eroded regions, limited to the shear layer, reflects the structure of the flow behind the cylinder, the authors note that cavitation in the polymer solution appears to be analogous to that in water at larger values of the Reynolds number. Again, in these Russian experiments it is impossible to conclude the reasons for the apparent decrease of the erosion. This is more so considering that the region where the erosion imprint was analyzed corresponds to the shear layer in the wake and to conditions which seem to be far from those prevailing at the closure of an attached cavity where a stagnation point occurs. This latter situation was discussed by Ting [60] for travelling bubbles. 6. C O N C L U S I O N A considerable number of investigations have been conducted towards the understanding of non-Newtonian fluid effects on cavitation occurrence and its consequences. The major conclusions can be summarized as follows 9 9 for single bubbles, non-dimensional time evolution of spherical bubbles in a stagnant and unbounded domain is unaffected by the non-Newtonian properties of the fluids, - in a domain bounded by a solid or free surface and in a shear layer, the non-spherical collapse of bubbles is affected by the non-Newtonian properties which delay the departure from sphericity, 9 in moving liquids such as 9 in submerged jets, results show that cavitation can be either delayed or unaffected by the non-Newtonian fluids and that major changes seem to be associated with modifications of the structure of the free-shear layer, -

-

250

-

in flow around blunt bodies, major changes appear to be correlated with the early occurrence of laminar to turbulent transition in drag reducing polymer solutions, without any major contribution associated with bubble dynamics, in tip-vortex flow situations with homogeneous polymer solutions, the non-Newtonian effects seem to enhance, for equal foil lift coefficient, the cavitation occurrence, while, in pure water and polymer solution ejection from the wing tip, there is a systematic cavitation inhibition, - in confined vortex flow in homogeneous solutions, the results show either cavitation enhancement or inhibition, without an explanation for the causes of such conflict, - in lubricating type flows, cavitation is delayed as a result of the development of a normal stress contribution to the pressure. Erosion due to cavitation can also be either augmented or diminished depending on the way the tests are performed. In summary, besides for a few controlled flow situations, such as the ones utilized for investigating single bubble behavior, there is not a definite answer concerning the favorable or unfavorable effect of non-Newtonian fluid properties on cavitation occurrence and the consequences of its development. The major obstacle to a complete understanding of this problem lies on the fact that polymer solutions have, at least for the range of flow parameters encountered in laboratory test situations, strong effects on the boundary layers developing along streamlined bodies, on the structure of large eddies in shear layers of separated flows, on the lift of hydrofoils, etc. These large scale effects are further complicated by the behavior of the individual cavitating bubbles, essentially determined by the local pressure and the presence of solid boundaries or fluid interfaces at distances smaller than the bubble diameter; thus relatively small scales. It is our opinion that some of the many questions that remain unanswered may receive an appropriate response if test can be conducted at much larger Reynolds numbers than those achieved up to know and with very dilute polymer concentrations. However, the degradation of the polymer molecules may have then a negative impact on the results of the cavitation tests. -

ACKNOWLEDGEMENT The authors wishes to thank the support he has received over many years by the Ministry of Defence, France, to conduct research on polymer solutions effects in hydrodynamics. He is deeply indebted to Professors Jack W. Hoyt and William R. Schowalter for taking the time to read a drat version of this chapter and provide the author with many correction and wise comments.

251

NOMENCLATURE r

D e

G k L m,n P po pc Pd Pg PgO p(R) Pv p~(t)

Prez qi

Q R

Ro RA Re

Rc Rch R(t) (r, O, (p) t U

U

vj vo l~ts Y 7? r

concentration diameter of headforms minimum gap in lubrication tests non-dimensional injection rate of polymer fluid polytropic index distance between the center of a bubble and a wall power law parameters pressure pressure at infinity at t = 0 critical pressure pressure in a reservoir non-condensible gas pressure within the bubble non-condensible gas pressure at t = 0 pressure at the interface of a bubble on the liquid side vapour pressure pressure far from the bubble reference pressure non-dimensional flow rate volume flow rate of ejected solution radius of the rotating drum in lubrication tests radius of a nucleus distance between the position of the bubble interface, opposite to a wall, and the center Reynolds number critical radius initial radius for a collapsing bubble bubble radius at time t spherical coordinate system time radial velocity component free stream velocity bulk velocity of a jet tangential velocity boundary layer displacement thickness surface tension and shear rate apparent viscosity viscosity

252

P O"d O"i "t'R

Trr, TOO, T ~ CO

APe

elongational viscosity liquid density desinent cavitation number incipient cavitation number Rayleigh time normal components of the deviatoric stress tensor angular velocity of the drum in lubrication tests added pressure in lubrication tests

REFERENCES ~

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6. ~

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R.T. Knapp, J.W. Daily and F.G. Hammit, Cavitation, McGraw-Hill, New York, 1970. D.H. Trevena, Cavitation and tension in liquids, 1987, Adam Hilger, Bristol and Philadelphia. W. Tillner et al., The avoidance of cavitation damage, Mechanical Engineering Publications Limited, London, 1993. C.E. Brennen, Cavitation and bubble dynamics, Oxford University Press, Oxford, 1995. A.J. Acosta and B.R. Parkin, J. of Ship Research, 19, 1975, pp. 193-205. A. Shima and T. Tsujino, Chemical Engineering Science, 31, 1976, pp. 863-869. H.S. Fogler and J.D. Goddard, Phys. Fluids, 13 (5), 1970, pp. 1135-1141. H.S. Fogler and J.D. Goddard, J. Appl. Phys., 42 (1), 1971, pp. 259- 263. A.T. Ellis and R.Y. Ting, NASA-SP 304, Pt. 1, 1974, pp. 403-420 (with discussions). W.J. Yang and M.L. Lawson, Journal of Applied Physics, 45 (2), 1974. G.L. Chahine, 1974, Thesis, Univ. Paris VI, 1984, Rapport de recherche ENSTA 042. R.Y. Ting, Phys. Fluids, 20 (9), 1977, 1427-1431. P.S. Kezios and W.R. Schowalter, Phys. Fluids, 29 (10), 1986, pp. 31723181. G. Riskin, J. Fluid. Mech., 218, 1990, pp. 239-263. G.L. Chahine and D.H. Fruman, Phys. Fluids, 22 (7), 1979, pp. 14061407. E.A. Brujan, C.-D. Ohl, W. Lauterborn and A. Philipp, Acustica, 82, 1996, pp. 423-430. G.L. Chahine and A.K. Morine, ASME Cavitation and Polyphase Flow Forum, 1980, pp. 7-8. A.K. Morine, 1982, Thesis, Univ. Paris VI, Rapport de recherche ENSTA 179, 1983. G.L. Chahine, Applied Scientific Research, 38, 1982, pp. 187-197.

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20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

P.T. Kezios, PhD dissertation, Princeton University, 1984. P. Ligneul, Phys. Fluids, 30 (7), 1987, pp. 2280-2282. J.W. Hoyt, ONR Drag Reduction Workshop, Boston, 1970. J.W. Hoyt, 16th American Towing Tank Conference, Sao Paulo, Brazil, 1971. J.W. Hoyt, ASME Cavitation and Multiphase Flow Forum, pp. 44-47, 1973. J.W. Hoyt, ASME Journal of Fluids Engineering, 98, 1976, pp. 106-112. C.B. Baker, J.W. Holl and R.E.A. Arndt, ASME Cavitation and Polyphase Flow Forum, 1976, pp. 6-8. J.W. Hoyt and J.J. Taylor, ASME Journal of Fluids Engineering, 103, 1981, pp. 14-18. R.E.A. Arndt, ASME Cavitation and Polyphase Flow Forum, 1980, pp. 9-10. R.Y. Ting, AIChE Journal, 20 (4), 1974, pp. 827-828. J.W. Hoyt, 1l th International Towing Tank Conference, Tokyo, 1966. A.T. Ellis, J.G. Waugh and R.Y. Ting, Journal of Basic Engineering, 92, 1970, pp. 459-466. C.B. Baker, R.E.A. Arndt and J.W. Holl, 1973, App. Res. Lab. Tech. Memo. 73-257, The Penn. State Univ. J.H.J. van der Meulen, ASME Cavitation and Polyphase Flow Forum, 1973, pp. 48-50. J.H.J. van der Meulen, 1976, Doctoral Thesis, Netherlands Ship Model Basin. E.M. Gates and A.J. Acosta, 12th Symposium on Naval Hydrodynamics, National Academy of Sciences, Washington, 1978. H. Reitzer, C. Gebel and O. Scrivener, J. of Non-Newtonian Fluid Mech., 18, 1985, pp. 71-79. J.H.J. van der Meulen, ASME Cavitation and Polyphase Flow Forum, 1976, pp. 4-5. C.E. Brennen, J. Fluid Mech., 44, 1970, pp. 51-63. V.A. Bazin, Y.E.N. Barabanova and A.F. Pokhil'ko, Fluid Mechanics Soviet Research, 5, 1976, pp. 79-82. R.Y. Ting, Phys. Fluids, 21 (6), 1978, pp. 898-901. R.J. Gordon and C. Balakrishnan, Nature Physical Science, 231,1971, pp. 177-178. C. Inge and G. Bark, TRITA-MECH-83-12, 1983, The Royal Institute of Technology, Sweden. D.H. Fruman, ASME Cavitation and Polyphase Flow Forum, 1984, pp. 73-76. J. Wu, Lift reduction in additive solution, Journal of Hydronautics, 3 (4), 1969, pp. 188-200. T. Kowalski, Journal of Hydronautics, 5 (1), 1971, pp. 11-14.

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46. D.H. Fruman, D. Bismuth, S.S. Aflalo, in The influence of polymer additives on velocity and temperature fields, B. Gampert ed., SpringerVerlag, Berlin, 1985, pp. 399-4. 47. S.S. Aflalo, Thesis, Univ. Paris VI, 1987. 48. D.H. Fruman and S.S. Aflalo, ASME Journal of Fluids Engineering, 111, 1989, pp. 211-216. 49. T. Pichon, A. Pauchet, A. Astolfi, D.H. Fruman, J-Y. Billard, Journal of Ship Research, 41 (1), 1997, pp. 1-9. 50. D.H. Fruman, T. Pichon, P. Cerrutti, J. Mar. Sci. Technol., 1, 1995, pp. 13-23. 51. G.L. Chahine, G.F. Frederick and R.D. Bateman, ASME Journal of Fluids Engineering, 115, 1993, pp. 497-503. 52. C. Inge, TRITA-MECH-83-05, 1983, The Royal Institute of Technology, Sweden. 53. J.W. Hoyt, ASME Cavitation and Polyphase Flow Forum, 1978, pp. 1718. 54. D. Bismuth, Thesis, Univ. Paris VI, 1987, also, Rapport de Recherche ENSTA 215. 55 A. Ouibrahim, D.H. Fruman and R. Gaudemer, Phys. of Fluids, 8 (7), 1996, pp. 1964-1971. 56. V. Ashworth and R.P.M. Procter, Nature, 258, 1975, pp. 64-66. 57. A. Shima, T. Tsujino, H. Nanjo and N. Miura, ASME Journal of Fluids Engineering, 107, 1985, pp. 134-138. 58. T. Tsujino, Ultrasonics, 25, 1987, pp. 67-72. 59. J.W. Hoyt and J. Morgan, ASME Cavitation and Multiphase Flow Forum, 1989. 60. R.Y. Ting, Nature, 262, 1976, pp. 572-573. 61. I.F. Shapoval and K.K. Shal'nev, Sov. Phys. Dokl. 22 (11), 1977, pp. 635-637.

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LOW-DIMENSIONAL DESCRIPTION OF VISCOELASTIC TAYLORVORTEX FLOW

Roger E. Khayat Department of Mechanical & Materials Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9

1. INTRODUCTION The influence of inertia and elasticity on the onset and stability of Taylor vortex flow (TVF) is examined for an Oldroyd-B fluid. The rigid-flee (stick in the azimuthal direction and slip along the axial direction) and rigid-rigid boundary conditions are used. Both formulations lead to essentially the same qualitative stability picture, but the former conditions allow a much simpler formulation, more easily amenable to algebraic manipulation and analysis. The resulting equations reduce to the Lorenz equations for a Newtonian fluid and small number of modes. A truncated Fourier/Chandrasekhar representation of the flow field, and the Galerkin projection, lead to a closed nonlinear dynamical system with sixteen degrees of freedom. In contrast to the six-dimensional system derived previously [Khayat, Phys. Fluids A 7 (1995) 2191 ], the present model is capable of capturing the dynamical behavior observed experimentally for a (Boger) fluid with negligible inertia. The improvement stems mainly from the proper choice and inclusion of higher-order modes that account for normal stress effects. For flow with dominant inertia, the stability picture is similar to that for a Newtonian fluid: steady TVF sets in as the Reynolds number reaches a critical value, Reo which decreases with fluid elasticity or normal stress effects, and is strongly influenced by fluid retardation. As elasticity exceeds a critical level, a subcritical bifurcation emerges at Re o similar to the one predicted by the Landau-Ginzburg's equation and the previous six-dimensional model. It is found that slip tends to be generally destabilizing. The coherence of the model is established through comparison with existing linear stability analyses, and experimental measurements and flow visualization.

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Recent linear stability analyses and experiment indicate a dramatic departure in the stability and bifurcation of the Taylor-Couette flow of viscoelastic fluids in comparison to Newtonian fluids. The Taylor-Couette flow of Newtonian and viscoelastic fluids involve a rich sequence of fundamental flow regimes that are observed during experiment, coveting the range from laminar to turbulent motion. For a Newtonian fluid, it is observed that at a sufficiently small Reynolds number, Re, there is a unique stationary (Couette) flow, which is globally stable, except perhaps near the ends of the cylinders. When Re is raised in the range near a critical value, R e o the stationary flow loses its stability, and develops a regular cellular vortex structure in which closed ring vortices alternating in sign are wrapped round the axis of rotation. In this range of Reynolds number, the flow remains essentially time independent and axisymmetric. This is the well known Taylor vortex flow (TVF) after G. I. Taylor, who was the first to examine this flow regime both theoretically and experimentally [ 1]. At higher Re value, the stationary cellular structure loses its stability in turn to another structure with a different number of vortices [2,3]. The upper limit of the TVF regime coincides with the emergence of time-dependent flow and the breaking of axisymmetry. Simultaneously, there appears a uniformly rotating pattern, or wavy vortex flow (WVF), of tangential azimuthal traveling waves superimposed on the cellular structure of TVF [4]. The TVF-WVF transition was carefully investigated by Fenstermacher et al. [5] and Gorman & Swinney [6] through flow visualization and spectral techniques. They obtained an accurate picture of the spatio-temporal flow structure which was later theoretically examined by Rand [7]. As the WVF sets in, the amplitude of the azimuthal waves is shown in the experiments to grow continuously with Re. The velocity power spectrum indicates the appearance of a single discrete frequency component with its harmonics. As Re is further increased, a second fundamental appears, with a gradual increase in the smallscale structure. At higher Re value, there is a broadening of the base in the power spectrum, indicating the onset of aperiodicity or chaos. For highly elastic (Boger) fluids, normal stress effects appear to prohibit the onset of steady TVF. The experimental measurements of Muller et al. [8] clearly demonstrate the existence of a purely elastic overstable mode when the Deborah number exceeds a critical value for a fluid with vanishingly small inertia. Periodicity appears to be lost as the level of elasticity is further increased. The influence of inertia was later examined by Baumert & Muller [9] through flow visualization. More extensive experiments are, however, needed that could elucidate on the possible route(s) to instability and transition to turbulence. One of the main objectives of the present paper is to actually predict the conditions

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under which transition may occur in the Taylor-Couette flow of viscoelastic fluids. Fluid turbulence is among the most challenging areas in the physical sciences, and has been the subject of intensive research over the past hundred years or so. Similarly to any flow in the transition or turbulent regime, the Taylor-Couette flow involves a continuous range of excited spatio-temporal scales. In order to assess the effect of the motion of the arbitrarily many smaller length scales, one would have to resolve in detail the motion of the small scales. This issue remains an unresolved one since, despite the great advances in storage and speed of modem computers, it will not be possible to resolve all of the continuous range of lengths scales in the transition regime. It is by now well established that loworder dynamical systems constitute an alternative to conventional numerical methods as one strives to understand the nonlinear behavior of flow [ 10]. For the Rayleigh-Benard thermal convection problem, the simplicity of the Lorenz equations [11], and the rich sequence of flow phenomena exhibited by their solution, have been the major contributing factors to their widespread use as a model for examining the onset of chaotic motion [12]. Despite the severe level of truncation in the formulation of these equations, some of the basic qualitative elements of the onset of thermal convection and the destabilization of the cellular structure have been recovered through the model. More general problems in fluid dynamics have also been treated using low-dimensional descriptions and the theory of nonlinear dynamics [13]. These methods are based on the expansion of the flow field in terms of a complete set of orthogonal functions, Fourier series or other standard basis functions, and the Galerkin projection technique whereby the initial set of partial differential equations is decomposed into an infinite set of ordinary differential equations governing the time-dependent expansion coefficients. The system is then truncated, leading to the finite-dimensional dynamical system. These methods have mainly been applied to Newtonian fluids, and the author appears to be the first to extend their application to nonNewtonian fluids [ 14-19]. While the problem of Taylor-Couette flow has been extensively investigated for Newtonian fluids, relatively little attention has been devoted to the case of viscoelastic fluids. In addition to the relatively recent interest in polymeric flows, the onset of chaos and turbulence is far less widespread in viscoelastic than in Newtonian fluids. To our knowledge, there has been no experimental evidence of the existence of (deterministic) chaos for highly elastic polymeric solutions similar to the one found for Newtonian fluids [5,6]. Recently, however, there has been a growing interest in the Taylor-Couette flow of viscoelastic fluids [20,21 ] for a recent review). Some of the early experimental work on the stability of rotating viscoelastic fluids is that of Giesekus [22,23]. Using polyacrylamide

258

solutions [23], he showed that the critical value of the Reynolds number at the onset of Taylor vortices may be smaller or higher than the Newtonian value depending on polymer concentration. At high polymer concentration, when elastic effects become dominant, the critical Reynolds number was found to decrease with the flow Deborah number, De. However, at high polymer concentration, solutions may exhibit significant shear thinning. Thus, ideally, both viscoelastic and shear thinning effects must be accounted for through a suitable constitutive model. The later experiments of Haas & Biahler [24] tend to show similar results. The experiments of Muller et al. [8] clearly demonstrate, for the Taylor-Couette flow of a Boger fluid, the existence of a purely elastic time-periodic instability at a critical rotation rate. The experiments were conducted using Laser Doppler velocimetry (LDV) measurements of the axial velocity component of a polyisobutylene-based fluid between two concentric cylinders, with the outer cylinder being at rest and the inner cylinder rotating. The results show an oscillatory flow at a vanishingly small Reynolds number (Re -~ 7x 10-3). The flow appears to undergo a transition from the purely azimuthal Couette flow to time periodic flow as the Deborah number exceeds a critical value, which is in good agreement with the value based on earlier linear stability analysis of an Oldroyd-B fluid [25]. The LDV measurements show that the oscillatory behavior is not localized but appears to be spread throughout the flow. As the Deborah number increases from the critical value, the amplitude of oscillation increases monotonically with De. The corresponding power density spectra show peaks, which are instrumentally sharp at the fundamental frequency, the growth of harmonics, and eventually subharmonics, reflecting, perhaps, the presence of period doubling or emergence of a second fundamental frequency (quasiperiodicity). The influence of inertia (shear rate) on the evolution of TVF was later examined by Baumert & Muller [9] who performed flow visualization on Boger fluids using reflective mica platelet seeding. Parallel theoretical work, for axisymmetric and non-axisymmetric flows, has also been conducted on the linear stability of viscoelastic fluids [25-27] and nonlinear calculations [28-30], with new phenomena being constantly attributed to fluid elasticity [31,32]. Unlike Newtonian fluids, which obey Newton's law of viscosity, viscoelastic fluids are not governed by a similar universal constitutive law. The dependence of the predicted flow behavior on the particular choice of a constitutive model adds another difficulty in our attempt to interpret an already complex flow situation, particularly in the transition and turbulent regimes. Larson [33] carried out a linear stability analysis in the narrow-gap limit,.using the Doi-Edwards and K-BKZ constitutive equations. He found that the dependence of the critical Reynolds number on De is generally non-monotonic, but the flow is increasingly destabilized by fluid elasticity in the higher De range

259

[33]. Numerical calculations were carried out by Northey et al. [30] for the Taylor-Couette flow of an upper-convected Maxwell (UCM) fluid neglecting inertia. The non-axisymmetric flow solution was obtained by Avgousti & Beris [28] and Avgousti et al. [29]. Elastic overstability was also proved through various linear stability analyses [25,26], showing that the base (Couette) flow can lose its stability via a Hopf bifurcation as the Deborah or Weissenberg number exceeds a critical value. Most of existing analyses are, however, limited to the linear regime. In our quest to probe the nonlinear regime, we have undertaken the current series of studies on the influence of fluid elasticity on Rayleigh-Benard thermal convection and Taylor-Couette flow. Khayat [14-17] developed a fourdimensional dynamical system for the thermal convection of strongly elastic fluids of the Oldroyd-B type. Such a system constitutes a generalization of the Lorenz equations [11] to include viscoelastic fluids. The critical Rayleigh number at the onset of the convective cellular structure was found to be the same as for Newtonian fluids. This is a direct consequence of the fact that the nonlinear terms (at least to the degree of truncation adopted) are the same as those in the Lorenz equations; no convective or upper-convective nonlinear terms survive in the constitutive equations. The conductive state thus loses its stability to the two steady convective branches C1 and C2, say, through a supercritical bifurcation similarly to Newtonian fluids. The two convective branches lose their stability in turn through a Hopf bifurcation as the Rayleigh number exceeds a value that, this time, depends strongly on fluid elasticity and retardation. It was also observed that fluid elasticity tends to precipitate the onset of chaotic motion, while fluid retardation tends to delay it. Above a critical value for the Deborah number, the flow behavior departs significantly from that of a Newtonian fluid. All three fixed branches remain unstable in the supercritical range for any value of the Rayleigh number. Thus, for D e > D e c, no steady convection can set in, and the cellular structure is always periodic in time (overstable), with the corresponding Fourier spectrum showing a sequence of period doubling as D e is increased. The existence and stability of the Hopf bifurcation corresponding to the onset of overstability were established in a later paper [15] using center manifold theory. These findings have prompted us to adopt a similar approach to the Taylor Couette flow of viscoelastic fluids in the narrow gap limit [18,19]. Our earlier work [ 18] focused on the influence of elasticity and retardation on the sfability and amplitude of the Taylor vortices with inertia dominating the flow. A truncated Fourier representation with Galerkin projection of the flow field similar to that of Kuhlmann and associates for a Newtonian fluid was proposed. Kuhlmann [33], and later Kuhlmann et al. [35] examined the

260

stationary and time-periodic TVF, in the narrow-gap limit and arbitrary gap width, respectively, with the inner cylinder rotating at constant and harmonically modulated angular velocity. The solution to the full Navier-Stokes equations was obtained by implementing a finite-difference scheme, and an approximate approach based on the Galerkin projection. Comparison of flows based on the two methods led to good agreement. In the approximate Galerkin projection [ 18], the velocity and stress components assume a truncated Fourier representation in space, with the expansion coefficients being functions of time alone. Since elastic effects were assumed to be weak, higher-order normal stress terms were neglected, thus leading to a six-dimensional nonlinear system, and reducing to Kuhlmann's three-dimensional system for a Newtonian fluid [18]. In both models, the velocity is assumed to adhere to the cylinder surfaces in the azimuthal direction, while it slips in the axial direction (rigid-free boundary conditions). The corresponding rigid-rigid boundary model leads to more realistic values of the critical Taylor number and the corresponding wave number, but to a less realistic value for the torque. Kuhlmann's three-dimensional system turned out to be equivalent to the Lorenz system with the Prandtl number equal to one. In this case, Kuhlmann's model cannot predict the destabilization of the Taylor vortices, and therefore cannot account for the onset of chaotic behavior. The situation is exactly similar for the Lorenz system when the Prandtl number is less than or equal to one [36,37]. In the previous work [18], a similar level of truncation in the Fourier representation for the velocity and stress was adopted for an Oldroyd-B fluid. Similar levels of truncation have been widely used for the Navier-Stokes and energy equations [13,38]. Examination of the influence of additional modes on flow and temperature fields [39-42] indicates that many of the gross features predicted by low level models are essentially recovered by the higher-order models. In the viscoelastic model, there are two Fourier modes in the azimuthal velocity and shear stress components that are expected to lead to nonlinear terms in the stress equations. This is in contrast to thermal convection [14-17] where the nonlinearities originate from the convective terms in the energy equation. While the stability and bifurcation picture at the onset of the cellular structure in the Rayleigh-Benard convection of viscoelastic fluids is similar to that of Newtonian fluids, the presence of nonlinearities in the stress equations of the Taylor-Couette flow of viscoelastic fluids has a drastic influence on the stability of the Taylor vortices and the transition to chaos. While the two models suffer from a similar level of truncation, the addition of viscoelastic effect, even to a very weak extent, appears to alter dramatically the stability and bifurcation picture. Most importantly, unlike the Newtonian model, elasticity tends to destabilize the TVF leading to chaotic behavior and the emergence of a Lorenz

261

type attractor. As the Deborah number increases, the onset of TVF occurs at a Reynolds number that decreases with De. Beyond a critical De value, the exchange of stability takes place via a subcritical instead of a supercritical bifurcation. Since the previous formulation [18] is not valid in the presence of dominant elastic effects, most results from existing linear analyses, finite amplitude numerical calculations, and experiments on the purely elastic overstability could not be recovered. The aim of this chapter is to investigate the onset and stability of finite amplitude TVF with elastic effects dominating over inertia. It is shown that the purely elastic overstability can only be predicted if the higher-order normal stress terms, neglected in the previous formulation [18], are properly accounted for. Particularly, the addition of the azimuthal normal stress component z00 leads to additional coupling with higher-order eigenmodes that can no longer be neglected. The resulting nonlinear dynamical system involves sixteen instead of six degrees of freedom. Although more cumbersome, and therefore less amenable to algebraic manipulations, the present expanded model is more accurate in its predictions, and leads to good agreement with existing formulations and experiments. Both the rigid-free and rigid-rigid boundary conditions are used and the resulting flows are compared. It is found that the two formulations lead essentially to similar qualitative flows. The former conditions allow a much simpler formulation. It also allows the examination of the influence of the higher-order normal stress terms when compared with the previous formulation of Khayat [18]. Additional calculations are also carried out in the absence of inertia, based on a dynamical system that accounts for yet higher order normal stresses, in an attempt to reach a quantitative agreement with the measurements of Muller et al. [8]. The chapter is organized as follows. In w the derivation of the nonlinear dynamical system is discussed. The stationary solutions and their stability are examined in w The influence of inertia for weakly and strongly elastic fluids is examined through numerical calculations, which are presented in w Discussion and concluding remarks are covered in w 2. DERIVATION OF THE NONLINEAR DYNAMICAL SYSTEM The derivation of the nonlinear dynamical system for a viscoelastic fluid is summarized in this section. The governing equations are solved by assuming an infinite discrete Fourier/Chandrasekhar representation in space for the flow field, which, upon truncation and application of the Galerkin projection, leads to the sixteen-degree-of-freedom nonlinear dynamical system (16NDS) that governs the expansion coefficients.

262

2.1. General equations and boundary conditions Consider an incompressible viscoelastic fluid of density 9, relaxation time X1 and viscosity q. In this study, only fluids that can be reasonably represented by a single relaxation time and constant viscosity are considered. The fluid is assumed confined between two infinite and concentric cylinders of inner and outer radii R i and R o , respectively, and the flow is axisymmetric. The inner cylinder is taken to be rotating at an angular velocity fl, while the outer cylinder is at rest. The conservation of mass and linear momentum equations are, respectively:

V . u = O,

Re

+ u . Vu

)

= - V p -/-Rv V 2 u - V. z"

(1)

where V is the gradient operator, t is the time, u is the velocity vector and x is the elastic part of the deviatoric stress tensor obeying the following constitutive equation [43]"

De

+

u. V r- (17u) t . z" -

1

r. Vu = - r -

(2)

Vu- ( V u ) t .

The various dimensionless groups in the problem, namely the Reynolds number, the Deborah number, D e , the solvent-to-polymer viscosity ratio, R v , and the gap-to-radius ratio, ~ (not explicitly appearing in the equations above), are given by:

Re,

Re -

dRi.O w

'

De =

2 R .O d

"

Rv

=

22 A1 - / Z 2

-

, tip

c=

d

,

(3)

Ri

where qs and rip are the solvent and polymer viscosities, respectively, and 22 (0 O, but the resulting stability picture is essentially the same as that based on the Lorenz equations. In the case of the Lorenz model (10), one easily deduces that at r = r c = 1, t w o additional fixed branches emerge, corresponding to the onset of (axisymmetric) Taylor vortices in opposite directions" Ws = r -

U s = G = + [ b ( r - 1)] 1 / 2 ,

1.

(11)

For the 16NDS, the critical value of r = r c (or T a c for the Taylor number) at the emergence of the two fixed branches is not as transparent as for equations (8) or (10). The value of r c tends to generally decrease as the level of elasticity increases. The value of r c corresponding to system (8) gives a reasonable estimate of this tendency [ 18]"

rc = ~

"

a

or

k2+.21,i 1 i ,

Ta c =

k )2

1 + a rE

(12)

where the term between square brackets constitutes the Newtonian contribution from the Lorenz system. In this case, T a c has a minimmn at !

k = km = ~[aE

.

.

.

.

n. 2

lr 2

(aE

+ 1) + aE

1] 1/2 1

Clearly,

in t h e c a s e

of a Newtonian fluid,

= 1 and k m = h a l 2 . Similar conclusions are reached when the formulae are cast in terms of the Reynolds number, R e , and Deborah number, D e . We shall use whatever notations are convenient as the present results will be compared rc

271

with those based on existing linear stability analyses. Comparison will be made with existing formulations that use exact solutions for the eigenvalue problem, and with the previous model that involves a lower number of eigenmodes [ 18]. Joo & Shaqfeh [27] carried out the numerical solution of the linear stability problem, using the RR boundary conditions, and examined the influence of inertia and elasticity for a UCM fluid. It is thus helpful to assess the accuracy of the present approximate method against their results. Fig. 1 displays the neutral

300

I

I

I

I

I

I

'I

De - 0.0 2.5 5.0 7.5 10.0

250

I

..... ..... .......... .....

/

200

~D

150 9f,,,,\ "('..', ,,

100

I

..-"

I"

UNSTABLE ~ "r

..'" ...... . .... ""'"""

_.."'I""

,,

, i-'"

"~ \ " ", ~ 9 "\'..... ".... 9 "\ "... " . ".-..

..--" _.~-'"

...-'"

...- ~

,,.

..-""

.i"

9...........

. ....... .'" . .......... .-'~'~"

..

. ............ . ~ . ~ . ~ " ........... ~ . . ~ . ~ ' ~

. . . - o"

50 "

'

I

I

I

I

I

1

2

3

4

5

,

,

,

7

8

I

6

Fig. 1. Influence of fluid elasticity and marginal stability curves for a UCM fluid (Rv = 0) based on the RR boundary conditions and e = 0.01. stability curves in the (Re, k)-plane for the onset of steady TVF for 0 < De 10, the neutral curves show again a very localized minimum, with an increasingly narrower range of wave numbers for the Couette flow to lose its stability to steady TVF. For a given k value, Rec decreases generally with De roughly like 1~De. This is, perhaps, the most important result in fig. 1" fluid elasticity tends to precipitate the onset of axisymmetric Taylor vortices at any value of the wave number in the axial direction. The results of Joo & Shaqfeh (reported in [27]) display the (Re, k) neutral stability curves for the same range of the Deborah number. Their results should be regarded as exact. They show good agreement with the curves based on the current approximate method. The approximate curves tend to give slightly lower values for Re c and ktn . Neutral stability curves in the (Re, k) plane were also obtained using the RF conditions. This enabled the assessment of the influence of the boundary conditions on the stability picture. The RF marginal curves lead to essentially no qualitative disagreement with the curves in fig. 1. Both figures show a localized minimum at k -- km that increases with De, a tendency of the curves to flatten around the minimum for the more elastic fluids, a general decrease in the critical Reynolds number as fluid elasticity increases, and the narrowing range of practical k values for the onset of steady TVF. Consider now the important question whether there is any (at least qualitative) change in the stability picture due to the inclusion of higher-order modes in the 16NDS as opposed to keeping only six modes as in system (8) [18]. A direct quantitative comparison was carried out by examining the influence of fluid inertia, elasticity and retardation through Re, E and Rv, respectively. Comparison shows that the stability picture is qualitatively the same in both cases. The inclusion of higher-order modes appears to make little difference for Newtonian and weakly elastic fluids. In fact, the curves corresponding to small values of E (E = 0 and 0.01) were essentially identical. Deviation between the two sets of curves became more evident for the larger E values. This is expected, as discussed above, since higher-order modes, previously neglected in [ 18], become important as the level of fluid elasticity increases. In this case, the neutral stability curves appear to be flatter as a result of the inclusion of higher-order modes. Similar conclusions are reached as to the influence of fluid retardation on the critical Taylor number Tac for an Oldroyd-B fluid with E = 0.5. Fluid retardation

273

tends to delay the onset of Taylor vortices. Note that in the limit Rv ~ 0% one recovers the Newtonian marginal stability curve (E = 0). The influence of Rv on the loss of stability of the base flow is to be expected since retardation tends to delay the onset of instability. Comparison shows a growing deviation and flattening of the curves for the lower Rv values considered. Recall that as retardation decreases, elasticity becomes relatively more dominant, and the influence of the higher-order (normal stress) modes is expected to be stronger. It is interesting to note that the curves in fig. 1 are reminiscent of those corresponding to the onset of overstability obtained from the linear analysis of Larson et al. [25] (see also fig. III-11 by Larson [20]). It is clear in both cases that fluid retardation tends to prohibit the onset o f steady and oscillatory Taylor vortices. Although the mechanism of onset of instability is different in the two situations (given the absence of inertia in the analysis of Larson et al. [25]), the role of fluid elasticity (at least for an Oldroyd-B fluid) appears to be very similar to that of inertia. This is particularly obvious from fig. III-11 in [20], which clearly shows the destabilizing influence of both inertia and elasticity. Additional calculations and comparisons with existing results were also carried out to closely examine the influence of fluid elasticity and retardation. The results will only be commented upon here but not shown for lack of space. The behavior of the minimum value of the critical Taylor number Tacm(E,Rv)-Tac(k=km,E,Rv) as function of the Elasticity number and viscosity ratio reveals a very similar trend as established previously (fig. 3 in [18]) on the basis of model (8). It is found that Ta m drops sharply with E at small elasticity in the case of a UCM fluid, in a manner similar to that predicted by earlier linear analyses (see fig. 111-6 in [20] and fig. 8 in [57]) for the case of corotating cylinders). The decrease in Ta m is considerably attenuated at the larger E values. As to the influence of fluid retardation, there is a flattening in the (Ta m , k) curves as Rv increases. This behavior agrees with that obtained from the linear analysis of Thomas & Walters [58] in the case of a Maxwell fluid, and those of Ginn & Denn [59] for a second-order fluid corresponding to a second normal stress of zero value. In this case, the formulation based on the secondorder fluid is expected to lead to the same stability picture as that based on a Maxwell fluid with small E value. Similar agreement is reached, as to the influence of fluid elasticity on the value of the wave number km at which the critical value of the Taylor number has a minimum, upon comparison with earlier formulations [ 18].

274

3.2. Stability of the Taylor vortex flow and bifurcation The previous section (3.1) deals with the loss of stability of the (purely circulatory) Couette flow to steady or oscillatory TVF depending on whether inertia is significant or negligible. In this section, attention is focused on the stability of the TVF itself. For a Newtonian fluid, the emergence of the two nontrivial branches at r = 1 is accompanied by an exchange of stability between the origin and the two fixed branches via a supercritical bifurcation. An exactly similar situation is encountered in the Rayleigh-Benard convection of a Newtonian fluid [37], and even in the convection of a viscoelastic fluid of the Oldroyd-B type [ 18]. For the Taylor-Couette flow of a viscoelastic fluid, the two solution branches correspond to the nontrivial steady solution of the 16NDS. A closed form solution such as (11) is difficult to obtain in this case because of the nonlinear coupling in the flow field and the large number of equations involved. The solution is thus obtained numerically using the damped Newton-Raphson method. On the other hand, the steady-state solution of system (8) is possible to obtain in closed form, and is written here for reference as [ 18]: (U 2 - 5b)(U 2 - 8 ) + ( a R v + a fa+ 5 ) ( b a R v - a f a + 5 ) U 2 r

~

a ( b R v - (a- 1)U 2 + b 5(a + 5 ) Ws = Xs

U Z + (a + 5 ) r - tY baRv - a fa + tY '

= -aUs,

Ys = ( a R v - r - W s ) U s ,

Vs = Us, Z s = aRv Ws -

(13)

b

It is thus clear that the solution of the first equation in (13) leads, similarly to the Lorenz system, to two nontrivial solution branches corresponding the onset of Taylor vortex flow. The steady-state solution of the 16NDS also leads to the emergence of two nontrivial branches at r = rc(E, Rv, k), which will be referred to as C 1 and C2, that are symmetric with respect to the r, and correspond to the onset of Taylor vortices in the two opposite directions. Unlike the case of a Newtonian fluid, the pitchfork bifurcation at r = rc is not always supercritical. In fact, our previous study based on system (8) shows that the bifurcation is supercritical only for weakly elastic fluids. When the Elasticity number E exceeds a critical value, E = Esub, the bifurcation becomes subcritical. This bifurcation picture is confirmed when higher-order modes are included as we shall see next.

275

For weakly elastic flows (E < E s u b ) , the 16NDS leads to an exchange of stability similar to the case of a Newtonian fluid, which takes place at r = rc(E, Rv, k), except that the critical value r c now depends strongly on fluid elasticity and retardation. As was established above, rc is different from one, and becomes increasingly smaller as E increases. At r = r c, a supercritical bifurcation emerges. The base flow, which is stable for r < r c, loses its stability to the two steady branches C 1 and C2 as r exceeds r c. Thus, for r > r c, the solution evolves to either one of the two branches depending on the initial conditions. As r increases and reaches a critical value, the two branches lose their stability as will be seen below. This stability and bifurcation pictures remain essentially unchanged until E exceeds E s u b , when the supercritical bifurcation gives way to a subcritical bifurcation at r = r c. The influence of the Elasticity number on the bifurcating branches at r = r c is depicted in fig. 2 for a UCM fluid ( R v = 0) and k = 3. These curves correspond the RF conditions, and should be compared to the bifurcation diagrams based on system (8) as shown in fig. 5 of [ 18], and included here in the inset of fig. 2 for reference. Because of symmetry with respect to the r-axis, only one branch (C 1) is shown. The curves in the figure are obtained numerically, contrary to those based on the analytical expressions (11) and (13). There are several important aspects to be observed from fig. 2, and the inset therein, in comparison to the Taylor-Couette flow of a Newtonian fluid. It is first observed that there is a strong dependence of r c on fluid elasticity. For small E, there is a supercritical bifurcation at r = rc, with r c becoming increasingly smaller as E increases, and decreasing all the way to zero. At the critical value E = E s u b ( R v , k), which in this case ( R v = 0, k = 3) is equal to 0.018, the bifurcation changes from super- to subcritical. This corresponds to a change in concavity at (r = r c, U] = 0). In the supercritical regime, when r < r c, a small disturbance of the base flow decays exponentially according to linear theory. The nonlinear terms in the 16NDS remain small in this case. As r exceeds r c, linear theory predicts an exponential growth of a small disturbance of the base flow. This growth is, however, halted by the stabilizing nonlinear effects. Although linear theory may well describe the onset of secondary flow, it fails to give the magnitude of the steady disturbance. In the subcritical regime (E > E s u b ) , when r < r g (shown here for the curve E = 0.05), the base flow is globally asymptotically stable [44]. When r g < r < r c, t w o bifurcation solutions exist for the disturbance from the base flow. In this case, any disturbance below a certain threshold decays to the origin. Above the threshold value, the base flow is destabilized. In this range of r values, the flow is commonly defined as metastable [60]. Whether the bifurcation at rc is subcritical or supercritical,

276

i

1.8

I

I

I

I I

I

I

I

I .-

"

!

I

;

E = 0.000 0.006 ..... 0.018 ...... 0 . 0 4 0 ........... 0.050 ..... 0.100 .....

...............:.;.::.-.....................

1.6 ..., ........

1.2

:..

_

9 9

'.

....

...-.

//

1.6 0.8

.... --- .......................

1.4 0.4

1.2

o.o.

.~ . . - o. I

0 0

i

0.2

I

i

0.4

/.

..-'"

j . ../

s/.

il

1

0.8

0.6

1.2

,'"

1.4

E sub = 0 . 0 1 8

.-"

,,"

,,,./"

,'

/1"

.................................................... - ......................................................?.':" ............................................ s

0.8

~

',

I'".,,

I

Esub. For the subcritical curves, the stable regions are indicated by breaks in the curves and are delimited by two arrows. Inset shows bifurcation diagrams based on lower-order theory eqs. (8) with Esub = 0.014. the TVF loses its stability at some critical r value r h > r c (shown here for the curve E = 0.05) Consider the stability of the steady branches C 1 and C2 as r is increased from r c. It is helpful, however, to first recall the situation for a Newtonian fluid. The discussion is focused on the Lorenz system (10) since the inclusion of higherorder mode does not alter the qualitative picture. Linearization of the Lorenz equations around the steady-state solution leads to a characteristic equation of the third degree. The conditions for the loss of stability of the toroidal TVF is of particular interest here. For a Newtonian fluid, the two nontrivial fixed points C 1

277

and C2 are always sinks for any value of r. At r = 1, a pitchfork bifurcation occurs, while the origin remains a saddle point with a one-dimensional unstable manifold. Note that this is exactly the same situation for the Lorenz equations with the Prandtl number equal to one. In this case, no Hopf bifurcation occurs since the characteristic equation does not possess a pair of purely imaginary roots [61 ]. The roots have a real part that remains negative for any value of r. Thus, for r > r c = 1, the three-dimensional Newtonian model cannot predict the destabilization of the TVF (through a Hopf bifurcation, for instance), nor can it entertain a chaotic solution. This situation remains unchanged when higher-order modes are retained in the solution. The presence of fluid elasticity in the constitutive equations appears to be a sufficient condition for the existence of a Hopf bifurcation. Indeed, at r = r h ( E , R v , k) > r e, linear stability analysis around C 1 and C2 indicates that these two branches lose their stability via a Hopf bifurcation emerging at r - r h. In fig. 2, sub

the range of r values for which the branch C 1 is stable (for E > E ) is indicated by breaks in the lines between two arrows. It is observed from the figure, that for highly elastic fluids (E > 1), the range of stability of the C1 and C2 branches decreases to zero, so that the (steady) TVF is unstable for any postcritical r value. While the existence of the Hopf bifurcation may not be difficult to establish, the investigation of its stability can be algebraically quite involved [62]. The difficulty stems from the hyperbolicity of the fixed point at r = rh, and center manifold theory must be applied in order to determine the stability of the fixed point [63]. The situation is similar for r near r o The stability picture will be thus established through the numerical solutions of the 16NDS. 4. FINITE A M P L I T U D E TVF AND C O M P A R I S O N W I T H E X P E R I M E N T Linear stability analysis, such as the one presented in the previous section, determines the flow field as it is slightly perturbed from the base flow or from the (steady) TVF. However, it fails to give the flow structure for a large disturbance. The influence of the nonlinear terms must thus be examined through the numerical solution of the 16NDS. The influence of inertia and elasticity on weakly and strongly elastic flows is examined in some detail. A weakly (strongly) elastic flow is defined as one whose Elasticity number is small (large) enough for it to undergo a supercritical (subcritical) bifurcation at the onset of Taylor vortices. Particularly, the emergence of super- and subcritical Taylor vortices, as well as the onset of chaos, are investigated by determining the flow spatio-temporal structure in phase space, through power spectra and/or time

278

signatures, whichever representation is most insightful. The present nonlinear formulation is assessed against experiments for a weakly elastic flow and a flow with negligible inertia. 4.1. Influence of inertia on weakly elastic flow

The influence of inertia on a moderately weakly elastic Oldroyd-B fluid is examined for a fixed value of E by varying the value of D e (or, equivalently, Re). This is equivalent to fixing the level of fluid elasticity and increasing the shear rate. Baumert & Muller [9] performed experiments on the flow visualization of the Taylor-Couette flow of dilute solutions of high molecular weight polyisobutylene in oligometrie polybutene. Rheologieal measurements of these solutions confirmed that they are of the Boger fluid type (highly elastic with constant viscosity). They monitored the temporal evolution of TVF over a range of shear rates using reflective mica platelet seeding. The transition to steady TVF was observed to occur with or without oscillatory flow depending on whether elasticity was dominant or not. In this section, the calculations are carried out for a weakly elastic flow of moderately small inertia (Re .~ 100), and the transient flow behavior is compared with the observations of Baumert & Muller [9]. The aim of the present comparison is to reproduce some of the experimental observations as the base flow loses its stability to TVF. Given the uncertainty in the experimental values of the relaxation time and the wave number, and the assumptions adopted in the derivation of the low-dimensional dynamical system, only a qualitative agreement can be expected. The results are also compared with the numerical calculations of Avgousti et al. [28] whenever possible. The calculations performed in this section are based on the rigid-flee boundary conditions. For a fixed level of fluid elasticity (E fixed), experiment suggests that transition from the purely azimuthal flow to steady TVF occurs once the shear rate reaches a critical value, i.e., D e = D e c (or Re = R e ) . The transition from simple sheafing to the large toroidal axisymmetrie vortices, which span the gap, is monotonic with time as D e first exceeds D e c. At higher D e value, an oscillatory secondary flow is observed prior to the formation of steady vortices. Axially migrating mica alignment bands begin to appear, but the band formation ceases with steady vortices setting in after some time. At this point, steady TVF is observed to persist for a long time. Consider the ease of a moderately weakly elastic fluid with E = 0.16. This corresponds roughly to the low-viscosity Boger fluid used in the experiment of Baumert & Muller [9], with p = 0.856 g/c m3, rls = 2.6 poise and I"1 = 3.0 poise.

279

The value of the relaxation time ~1 was found to be equal to Ls = 0.021 s or ~t-0.11 s (with the corresponding Elasticity number equal to E = 0.164 or E t = 0.94), depending on whether steady shear or transient experiments were used [9]. The value of the viscosity ratio is set equal to R v = 6.5. Figure 3 shows the time evolution of the (dimensional) axial velocity component U z at z = k/2, where it is maximum, and x = 0.5, midway through the gap. The evolution of the velocity is displayed for various values of the shear rate: 13 < D e < 14 (81.25 < R e < 87.5), above the critical shear rate, D e c ~ 13 (Re c ~ 81.25). The curves in the figure are similar to the results reported in fig. 1 of Avgousti et al. [28], and the trend is in agreement with the observations of Baumert & Muller [9] for low and medium

0.01

0.009 -

....................................................................................................

9 ..../

0.008 -

"''".... ................................................................................................................................................................................................

0.007-

0.006-

0.005 :

/,'-

: i

0.004-

....

i /

:

tI

0.003 -

De = 13.09 13.18 13.27 13.49 13.70

0.002 i

9 :;

i

0.001 J

i

ii

/

, ,'

,'"

,"

,i /

/

,'

o I 0

5;0

10b0

!

15;0

2000

25'00

..... ..... .......... .....

30'00

3500

Time (s)

Fig. 3. Monotonic time evolution of the axial velocity for different shear rates (13 < D e < 14) and small initial perturbation of the base flow (E = 0.16, R v = 6.5, k = 3.5 and e = 0.093).

280

viscosity fluids. The initial condition taken for all the curves in fig. 3 is U1 = 0.01 with the remaining variables equal to zero. In this case, the transition from the base flow to TVF occurs via a supercritical bifurcation similar to the branches E < 0.018 in fig. 2 for a UCM fluid. In the precritical range (De < Dec), the origin in phase space (base flow) is asymptotically stable to any perturbation, and the numerical integration of the 16NDS, in the vicinity of the origin, follows simply an oscillatory decay to the origin. As De (Re) exceeds Dec (Rec), an exchange of stability occurs between the origin and the steady-state branch C1 (or C2), coinciding with the emergence of steady TVF, with increasing amplitude as De (Re) increases.

100

Theory

+ + +

+ -i-

....,

[-.,

Experiment

10

100

De

Fig. 4. Time after shear initiation of appearance of steady vortices as function of the shear rate (De). The values in the figure are based on the curves in fig. 3. The inset shows the experimental measurements of Baumart & Muller [9] for a medium viscosity fluid.

281

Typically, the flow evolves monotonically from the base flow to steady TVF over a period of time that decreases with increasing shear rate as depicted from fig. 3. This appears to be in agreement with the observations of Baumert & Muller [9] who monitored the dependence of onset time on the shear rate for a fluid with medium viscosity. Figure 4 displays the time after shear initiation of appearance of steady vortices as a function of the Deborah number (shear rate). An inset showing the values from experiment is also included for comparison, and reveals good qualitative agreement with theoretical predictions. The values in the inset are based on measurements performed for a medium viscosity fluid, with E = 44. As will be seen next, at higher shear rates, however, theory predicts that the time TVF takes to set in begins to increase with De, a trend in disagreement with what experiment suggests.

0.o45[ '~ o

0.0as

.

li

.

.

.

.

.

.

.

,

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i. i

De = 17.47

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17 81 17194

5~o

lo'oo

~;oo

2000

Time (s)

Fig. 5. Oscillatory time evolution of the axial velocity for different shear rates (17 < D e < 18) and small initial perturbation of the base flow. The parameters used are the same as in fig. 3.

282

0.028

-

(a)

0.026:

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18000

Time (s)

Fig. 6. Time evolution of the axial velocity for 16 < De < 21. The flow for each shear rate level is obtained using the flow of the previous (lower) level as initial conditions. The parameters used are the same as in fig. 3. The figure is subdivided into three segments of 18,000 s period each. Steady TVF is always reached during the first period (a) for the range 16 < De < 18.7. Oscillatory instability develops towards the end of the second period (b) for the range 18.6 < De < 19.27. Transition to periodic behavior some time during the third period (c) for the range 19.26 < De < 21.

283

The evolution of the flow loses its monotonicity as shearing exceeds a certain level. This is depicted in fig. 3 for the curves corresponding to De > 13.2, which show a local depression in the flow amplitude before steady TVF sets in. In fact, as De is further increased, an oscillatory behavior emerges before the onset of steady vortices as shown in fig. 5 for 17 < De < 18. The secondary flow gains strength initially at a much faster rate than for the lower range of shear rates (fig. 3). The amplitude of oscillation decays with time until the onset of steady of TVF, but the frequency appears to remain constant. As De increases, the overall amplitude of oscillation increases, accompanied by a phase shift and delay in the onset of steady TVF. Eventually, oscillatory behavior is sustained for longer time for higher De values; in the limit, no steady TVF is found. Before examining the onset of sustained oscillations, it is desirable to address the issue of the influence of the initial conditions on the ensuing flow behavior. Baumert & Muller [9] report that "flow behavior was not found to be particularly sensitive to the type of velocity ramp imposed" initially. They indicate that the flow behavior commencing at the start of the final velocity was indistinguishable from the case when a plateau was imposed over a certain period. This observation is presently confirmed through additional calculations carried out by taking as initial conditions the steady TVF reached before increasing the shear rate, instead of starting again from near the origin as in figs. 3 and 5. The resulting sequence of initial flow transients, after the imposition of the higher (new) shear rate, and subsequent final (steady or oscillatory) TVF are depicted in fig. 6 for the range 16 < De < 21. The figure displays the evolution of flow over a period of 54,000 s, subdivided into three equal periods of 18,000 s. The first segment is shown in fig. 6a for the range of shear rates corresponding to 16 < De < 19. Comparison between the curves in fig. 5 and those in fig. 6a indicates that the flow behavior is essentially unaffected by initial conditions. It is interesting to note the similarity in the emergence and decay of oscillatory behavior between one steady level and the other. The figure clearly shows that the time it takes steady TVF to set in increases with shear rate. This is a result of the oscillatory transition that takes a longer time for the higher shear rate. This is indeed even more evident from fig. 6b, which shows, for the range 18 < De < 20, that steady TVF does not even set in over the time period a given shear rate is applied. The oscillations tend to decay at a slower rate as De increases, and eventually start to grow with time as indicated in the last portion of fig. 6b, corresponding to De = 19.26. This growth, which is linear upon onset, does not remain unstable over time as linear stability analysis suggests. At some time, when the signal amplitude is large enough for nonlinear effects to become significant and halt the (exponential) growth.

284

The bifurcation diagrams in fig. 2 suggest that steady TVF is reached as long as r is below rh(E, Rv, k). This corresponds to the onset of a Hopf bifurcation at De - Deh when steady TVF loses its stability to periodic behavior. For the present flow in fig. 6, Deh is found to be approximately equal to 19. The existence of a Hopf bifurcation at De = Deh is easily confirmed through linear stability analysis around the nontrivial steady-state solution(s) when two eigenvalues are purely imaginary, and no other eigenvalue possesses a zero real part. The stability of the Hopf bifurcation, however, is difficult to establish for a complex system like the 16NDS. Such an analysis is algebraically involved, but a similar analysis, based on center manifold theory, was carried out previously for the thermal convection of viscoelastic fluids [15]. Numerical integration, on the other hand, shows that the Hopf bifurcation indeed exists, and is stable as will be seen below. This situation is in sharp contrast to the case of the Lorenz system (10) whereby no loss of stability of the steady TVF is predicted. In figs. 6a and b, and for the range 16 < De < 19.15, the computation has been limited to flows with final state, after transients have died out, that corresponds to a standing wave. This also corresponds to most of the range of shear rates examined in the experiment of Baumert & Muller [9] that led to steady TVF for the low and medium viscosity fluids. However, they reported the observation of coexistence of migrating bands and distorted vortices at the highest shear rate considered. It is thus desirable to numerically examine the transition from steady to oscillatory TVF. The sequence of flows displayed in fig. 6c is obtained for De above Deh subject to the initial condition corresponding to the steady TVF at De slightly below Deh. Avgousti et al. [28] reported having numerical difficulties for a UCM fluid when they took the slightly perturbed standing wave as their initial condition. Figure 6a shows the transition from steady to oscillatory TVF, that is, from the flow at De = 19.26 to that at De = 19.38. The figure also shows the transition to another oscillatory state at De = 20.56. The first transition is preceded by a destabilization of the steady TVF to a traveling wave of amplitude (exponentially) growing with time, which is reminiscent of the behavior reported in figs. 6 and 9 of Avgousti et al. [28]. The exponential growth (instability) ceases immediately upon increase of the shear rate, giving way to a regular and stable oscillatory vortex structure. This structure is sustained for as long as the shear rate is maintained at De = 19.38. Upon further increase of De, another regular oscillatory structure sets in with a larger amplitude. The modulation of the oscillation indicates the emergence of period doubling. Additional theoretical and experimental investigations are obviously needed if the flow at still high shear rate is to be explored. It is possible that the flow may not remain

285

axisymmetric, a fact that does not seem to be suggested by the experiments of Baumert & Muller [9]. 4.2. Weakly inertial flow and influence of boundary conditions The nonlinear behavior is further examined by focusing the calculations on the influence of fluid elasticity in the presence of very weak inertial effect. One of the most interesting phenomena encountered in the Taylor-Couette flow of a viscoelastic fluid is the emergence of overstability that is attributed to fluid elasticity, which is otherwise absent in the case of a Newtonian fluid. The existence of such a purely elastic overstable mode was proved through linear stability analysis [25] and numerical calculations [30] carried out in the absence of inertia. Recently, Muller et al. [8] conducted Laser Doppler velocimetry (LDV) measurements of the axial velocity component of a Boger fluid between two concentric cylinders (~ = 0.0625), with the outer cylinder being at rest, and the inner cylinder rotating at constant angular velocity. The measurements show an oscillatory flow at a vanishingly small Reynolds number. The flow appears to undergo a transition from the purely azimuthal Couette flow to time periodic flow as the Deborah number De exceeds a critical value, Dec, which is in good agreement with the value predicted by earlier linear stability analysis of an Oldroyd-B fluid [26]. The LDV measurements show that the oscillatory behavior is not localized but appears to be spread throughout the flow. As the Deborah number increases from the critical value, the amplitude of oscillation increases like (De - Dec) 1/2. The corresponding power density spectra show peaks, which are instrumentally sharp at the fundamental frequency, the growth of harmonics, and eventually subharmonics, reflecting, perhaps, the presence of a period doubling or quasiperiodic motion. A similar sequence of flow behaviors is also obtained from the finite-element calculations (with inertia neglected) of Northey et al. [30] for a UCM fluid. These authors, however, report having numerical difficulties in obtaining the solution at the higher Deborah numbers (possibly coinciding with the onset of period doubling). Their calculations are thus limited to an extremely small range of postcritical Deborah numbers. As discussed earlier, the system (8) used in our previous study [18] could not possibly reproduce any of the reported experimental results for vanishingly low-Reynolds number. This fact is easily verified if the inertia terms are dropped from system (8). In this case, the acceleration terms on the left-hand side, together with the curvature term (V) on the fight-hand side of the first equation, are neglected. It is then easy to deduce that one recovers the trivial solution or the purely azimuthal flow. Thus, the level of truncation adopted in the previous model makes the formulation inadequate

286

for the investigation of the purely elastic overstability or even a flow at very small Reynolds number. In general, however, the presence of inertia, no matter how small it may be, prohibits the base flow from losing its stability to the overstable mode. Instead, the base flow loses its stability first to steady (and not oscillatory) TVF since there is always a finite range of r values over which the branches C 1 and C2 are stable (see fig. 2). The influence of fluid elasticity is now examined for vanishingly small inertia in the postcritical range of the Deborah number, in an attempt to recover the flow sequence observed in the experiment of Muller et al. [8]. One cannot expect full quantitative agreement between theory and experiment given the level of approximation in the 16NDS. Moreover, the wave number k needs to be imposed in the present calculation. This quantity is not known from experiment; the wave number is difficult to establish under unsteady conditions of flow [9]. The comparison between theory and experiment is covered next for the RF and RR formulations. Consider the flow with negligible inertia, and set r = 10-6. Also let Rv = 3.75 and e = 0.0625 corresponding, respectively, to the viscosity ratio of the fluid and the gap-to-radius ratio used in the experiment [8]. Although the wave number is not specified in the experiment, it is set k = 4 based on wave numbers reported in other experiments on Taylor-Couette flow of viscoelastic fluids [9]. Thus, only the Deborah number will be varied. The flow is examined as De is increased from zero, that is from the Newtonian level. Referring to fig. 2, it is seen that for De = E = 0 the base flow is unconditionally stable to any perturbation since r 0.1 in fig. 2) and for vanishingly small Reynolds number, the base flow loses its stability to oscillatory and not to steady TVF. This prediction confirms the experimental observation of Muller et al. [8]. Unlike the previous model [18], the present system led to favorable comparison with experiment (figs. 7 and 8). The present theory predicts the sequence of periodic behaviors observed as the Deborah number is increased: (1) loss of stability of the base flow to an oscillatory flow at a critical Deborah number (Dec = 29.3 and 29.7 for the RF and RR formulations, respectively, as predicted by the model vs. 32 from experiment), (2) growth of amplitude of the velocity signature

298

like (De - Dec) 1/2, in agreement with asymptotic analysis, with the emergence of higher harmonics in the Fourier spectrum, and (3) emergence of subharmonics as De is further increased, reflecting (most likely) the bifurcation into period doubling and, eventually, chaos. Finally, an augmented twenty-dimensional dynamical system, 20NDS, is also used to describe highly elastic TVF. The model is derived without inertia effect, and higher normal stress terms are added. This approach constitutes a first systematic and accurate theoretical prediction for the purely elastic overstability observed by Muller et al. [8]. It is shown that the finite amplitude TVF can be effectively described if higher-order normal stress terms are properly accounted for. Particularly, the addition of the azimuthal normal stress component leads to additional coupling with higher-order eigenmodes that are of O(eDe). The resulting nonlinear dynamical system involves only twenty degrees of freedom. The sequence of flows predicted by the present model is comparable to that reported by Muller et al. [8]. The model predicts the sequence of periodic behaviors observed as the Deborah number is increased: (1) loss of stability of the base flow to an oscillatory flow at a critical Deborah number (Dec = 32 as predicted by the model vs. 32.3 from experiment), (2) growth of amplitude of the velocity signature like (De - Dec) 1/2, in agreement with asymptotic analysis, and (3) the emergence of higher harmonics in the Fourier spectrum as De is further increased [ 19].

REFERENCES ~

2. 3. 4. 5. .

7. 8. .

10.

G. I. Taylor. Phil. Trans. Roy. Soc. A 223 (1923) 289. T. B. Benjamin. Proc. Roy. Soc. Lond. A 239 (1978) 1. D. E. Shaeffer. Math. Proc. Camb. Phil. Soc. 87 (1980) 307. D. Coles. J. Fluid Mech. 21 (1965) 385. P. R. Fenstermacher, H. L. Swinney & J. P. Gollub. J. Fluid Mech. 94 (1979) 103. M. Gorman & H. L. Swinney. J. Fluid Mech. 117 (1982) 123. D. Rand. Arch. Ration. Mech. Anal. 79 (1982) 1. S. J. Muller, E. S. J. Shaqfeh & R. G. Larson. J. Non-Newt. Fluid Mech. 46 (1993) 315. B. Baumert & S. J. Muller. Rheol. Acta 34 (1995) 147 G. R.Sell, C. Foias and R. Temam Turbulence in Fluid Flows: A Dynamical Systems Approach (Springer-Verlag, 1993).

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11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

38. 39. 40.

E. N. Lorenz. J. Atmos. Sci. 20 (1963) 130. C. Sparrow. The Lorenz Equations, (Springer-Verlag, New York 1983). H. N. Shirer & R. Wells. Mathematical Structure of the Singularities at the Transitions Between Steady States in Hydrodynamic Systems (Springer-Verlag, Heidelberg 1980). R. E. Khayat. J. Non-Newt. Fluid Mech. 53 (1994) 227. R. E. Khayat. J. Non-Newt. Fluid Mech. 58 (1995) 331. R. E. Khayat. Phys. Rev. E 51 (1995) 380. R. E. Khayat. J. Non-Newt. Fluid Mech. 60 (1996) R. E. Khayat. Phys. Fluids A 7 (1995) 2191. R. E. Khayat. Phys. Rev. Letts. 78 (1997) 4918. R. G. Larson. Rheol. Acta 31 (1992) 213. E. G. S. Shaqfeh. Ann. Rev. Fluid Mech. 28 (1996) 129. H. Giesekus. Rheol. Acta 5 (1966) 39. H. Giesekus. Prog. Heat Mass Transfer 5 (1972) 187. R. Haas & K. Btihler. Rheol. Acta 28 (1989) 402. R. G. Larson, E. S. G. Shaqfeh & S. J. Muller. J. Fluid Mech. 218 (1990) 573. E. S. G. Shaqfeh, S. J. Muller & R. G. Larson. J. Fluid Mech. 235 (1992) 285. Y. L. Joo & E. S. G. Shaqfeh. Phys. Fluids A 4 (1995) 2415 M. Avgousti, B. Liu & A. N. Beris. Int. J. Num. Meth. Fluids 17 (1993) 49. M. Avgousti & A. N. Beris. J. Non-Newt. Fluid Mech. 50 (1993) 225. P. J. Northey, R. C. Armstrong & R. A. Brown. J. Non-Newt. Fluid Mech. 42 (1992) 117. A. Groisman & V. Steinberg. Phys. Rev. Letts. 77 (1996) 1480. M. K. Yi 7& C. Kim. J. Non-Newt. Fluid Mech. 72 (1997) 113. R. G. Larson. Rheol. Acta 28 (1989) 504. H. Kuhlmann. Phys. Rev. A 32 (1985) 1703. H. Kuhlmann, D. Roth & M. LOcke. Phys. Rev. A 39 (1988) 745 E. Ott. Chaos in Dynamical Systems (Cambridge University Press, Cambridge 1993) J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York 1983). G. Veronis. J. Fluid Mech. 24 (1966) 545. J. H. Curry. Commun. Math. Phys. 60 (1978) 193. H. Yahata. Prog. Theor. Phys. 59 (1978) 1755.

300 41. 42. 43. 44. 45. 46. 47. 48. 4.9. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 63. 64. 65. of 66. 67.

H. Yahata. Prog. Theor. Phys. 61 (1979) 791. H. Yahata. Prog. Theor. Phys. 59 (1978) 1755. R. B. Bird, R. C. Armstrong & O. Hassager. Dynamics of Polymeric Liquids, vol. 1, 2nd Ed. (John Wiley & Sons, New York 1987). P. G. Drazin & W. H. Reid. Hydrodynamic Stability (Cambridge University Press, Cambridge 1981). H. Harder. J. non-Newtonian Fluid Mech. 36 (1991) 67. J. B. McLaughlin & P. C. Martin. Phys. Rev. A 12 (1975) 186. A. Davey. J. Fluid Mech. 14 (1962) 336. J. T. Smart. J. Fluid Mech. 4 (1958) 1. P. Chossat & G. Iooss. The Couette-Taylor Problem. Springer-Verlag (1991). M. Nagata. J. Fluid Mech. 169 (1986) 229. P. Tabeling. J. Phys. Lett. 44 (1983) 665. D. V. Boger. J. Non-Newt. Fluid Mech. 3 (1977/78) 97. R. D. Richtmyer. NCAR Tech. Note TN-176+STR (1981). P. S. Marcus. J. Fluid Mech. 146 (1984) 65. S. Chandrasekhar. Hydrodynamic and Magnetohydrodynamic Stability, Dover (1961). R. E. Khayat. J. Fluid Mech. (in press). D. W. Beard, M. H. Davies & K. Walters. J. Fluid Mech. 24 (1966) 321. R. H. Thomas & K. Walters. J. Fluid Mech. 18 (1964) 33. R. F. Ginn & M. M. Denn. AIChE J. 15 (1969) 450. C. Normand, & Y. Pomeau. Rev. Mod. Phys. 49 (1977) 581. S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York 1990). J. E. Marsden & M. McCracken. The HopfBifurcation and Its Applications (Springer-Verlag, New York 1976). J. Carr. Applications of Center Manifold Theory (Springer Verlag, New York 1981). K. Walters. Rheometry: Industrial Applications. Research Studies Press (1980). R. I. Tanner. Engineering Rheology. Oxford University Press (983). R. B. Bird, R. C. Armstrong, C. F. Curtis & O. Hassager. Dynamics Polymeric Liquids, vol. 2, 2nd Ed. John Wiley & Sons (1987).. M. Doi & S. F. Edwards. The Theory of Polymer Dynamics. Oxford University Press (988). B. C. Eu & R. E. Khayat. Rheol. Acta 30 (1991) 204.

301

NON-NEWTONIAN MIXING WITH HELICAL IMPELLERS AND PLANETARY MIXERS

RIBBON

Philippe A. Tanguy ~ and Edmundo Brito-De La Fuente 2

1 Department of Chemical Engineering, Ecole Polytechnique Montreal P.O. Box 6079, Station Centre-ville, Montreal H3C 3A7, CANADA 2 Food Science and Biotechnology Department, Chemistry Faculty "E", National Autonomous University of Mexico, UNAM, 04510 Mexico, D.F., MEXICO

1. INTRODUCTION Mixing is a very common processing operation, which accounts for about 15 % of all unit operations in the chemical and food industries. The range of possible mixing duties is extremely wide and may involve a single phase (in which case agitation is a more appropriate wording) or several phases. Moreover, the phases may be of a different nature, solid, liquid and gas. All combinations may be found depending on the process. Table 1 summarizes some important practical mixing applications. Table 1. Typical mixing applications Kneading of Pastes Dispersion of Agglomerates Flocculation Gas-Liquid Dispersion Liquid-Liquid Dispersion Reaction Enhancement Slurrying Storage Blending

Crystallization Emulsification Flotation Heat Transfer Liquid Blending Polymerization Solids Suspension Uniformity Maintenance

302

A mechanical mixer is a relatively simple device. It comprises a vessel, an impeller mounted on a shaft and a drive assembly. In mixing process development, a number of issues must be addressed for a given mixing application, namely 9 the selection of mixing equipment (impeller, drive) and operating conditions (speed, temperature, mixing duration in batch mode) 9 the design of the vessel (shape, aspect ratio, type of bottom, position of feed and side streams) and the internals (baffles, draught tube, heat exchangers) 9 the evaluation of performance (power consumption, mixing time, circulation patterns, circulation rate, distribution of shear rate) 9 the control of composition non-uniformity (segregation). Turbulence is the main physical mechanism responsible for mixing with low viscosity fluids. The impeller power is split into the power dissipated by the circulating fluid and the power dissipated by the sheafing action of the impeller in the vicinity of the blade tip. The respective amount of these two power components depends on the impeller type. When the viscosity is high, turbulence can hardly be achieved and when it is, the power drawn by the impeller is extremely large, yielding high operating costs and a significant temperature increase in the fluid bulk. Viscous mixing is a laminar process based on the "stretching-foldingbreaking" principles described in [1 ]. Mixing is obtained by the mechanical decrease of the striation thickness by shear and/or extensional forces up to a scale where molecular diffusion may fully play its role. This is a rather "gentle" flow process, which usually requires time. As with other fluid mechanics problems, the Reynolds number is the relevant number to characterize the significance of the viscosity effects. The Reynolds number is defined as: Re

-

pND2

(1)

77 where p and 1] are the fluid density and dynamic viscosity respectively, N is the rotational speed of the impeller and D its diameter. Below Re = 10, the viscous effects predominate, yielding a quasi-creeping flow. The mixing regime is laminar. At the other end of the Reynolds number scale, the mixing regime is turbulent. Depending on the impeller, the turbulence

303

threshold is comprised between Re = 104 and Re = 3.104. The intermediate mixing region above the laminar regime and below the turbulent regime is called the transition regime. In principle, mixers can be operated in any of this regime, provided adequate power is available. In practice, for viscous fluid, as turbulence cannot be achieved in the vessel, the mixing regime is at best in the transition region and often in the laminar region. Experience shows that the mixing regime is always laminar when the fluid viscosity is above 50 Pa.s. High viscosity fluids usually exhibit non-Newtonian properties. NonNewtonian flows are very common in the process industry. Typical examples include the manufacturing of rubber, plastics, petroleum, detergents, cosmetics, pharmaceuticals, cement, food, paper pulp, paints to name a few. A wide variety of flow situations are encountered in practice (flow in transfer lines, in processing equipment). Our discussion in this chapter will focus on non-Newtonian mixing and we refer the reader to the literature [2] for the treatment of the general principles governing nonNewtonian fluid mechanics. The most common non-Newtonian behavior is shear-thinning, i.e. the viscosity decreases with an increasing shear rate. This behavior is very often encountered in polymer manufacturing or in fermentation. Yield stress is often present especially with gels and highly concentrated suspensions. The yield stress phenomenon is associated with the presence of structures in the fluid that requires a minimum amount of deformation energy to break up. The mixing of yield stress fluids (or B ingham fluids as they are often called) is really difficult due to the particularly complex fluid mechanics in the vessel. A well-mixed cavern forms in the fluid bulk around the impeller [3] due to the shearing action of the blades. Far from the impeller, the rate of deformation is not sufficiently high to break the structure and the fluid is at rest. The same phenomenon may be encountered with shear-thinning fluids although in a less severe manner [4]. Another particular behavior is thixotropy, which is manifested by a time-dependent shear-thinning. From a physical standpoint, it is believed that thixotropy is related to the presence of organized microstructures whose conformation changes under an applied stress. Examples of such fluids are chocolate and paper coating colors. Thixotropy can be described in engineering terms as substances with several yield stresses [5][6]. Likewise B ingham fluids, thixotropic fluids may be difficult to mix. Shear-thickening is the opposite behavior of shear-thinning, i.e. the viscosity increases with the shear rate. This behavior is observed with

304

colloidal suspensions at very high concentration like high solids mineral slurries and with starch. Fluids exhibiting this property must be handled very carefully. Indeed, as the power consumption is proportional to the viscosity (as it will be seen later in this chapter), if shear-thickening properties develop during the process, the impeller may stall in the vessel or, worse, the motor may pull out. Viscoelasticity may add another level of complexity in the mixing of non-Newtonian viscous materials. Like shear-thinning, viscoelasticity is encountered in polymerization and fermentation. Apart from the wellknown shaft climbing phenomenon by the fluid (known as the Weissenberg effect in rheology) viscoelasticity increases significantly the power consumption in the vessel and strongly alters the circulation patterns [7].

2. S U M M A R Y OF MIXER DESIGN PRINCIPLES The mixer performance, expressed in terms of energetic, dispersing sheafing or blending efficiency, is directly related to the impeller performance. The selection of the appropriate device requires a very careful appraisal of the mixing duty, and a systematic stepwise approach based on: 9 Identification of the operations to be carried out 9 Detailed physical characteristics of the phases (Table 2) 9 General features of the impeller (shear rate, circulation rate, pumping pattern) 9 Selection of impeller candidates 9 Final selection Table 2. Phase characteristics

Liquid

Solid

Gas

specific gravity viscosity curve yield stress temperature range wt. %

Specific gravity Particle size distribution Settling velocity Wettability- solubility wt. %

Flowrate Pressure Solubility

There are two possible ways of classifying an impeller, according to the discharge flow it produces, or according to its size with respect to the

305

vessel. In the first category, there are radial discharge impellers (flat turbine), axial discharge impellers (propellers) and tangential flow impellers (anchors). If the diameter of an impeller is significantly smaller than the diameter of the vessel, this impeller is an open impeller. On the contrary, if the impeller exerts a scraping action at the vessel wall, it is called a closeclearance impeller. We show in Figure 1 typical examples of open and close-clearance impellers.

Figure 1: Flat blade turbine (open impeller) and helical ribbon screw (closeclearance impeller) In viscous and non-Newtonian mixing, close-clearance impellers are by far the best choice due to their superior top-to-bottom pumping capacity and wall scraping action as compared with open impellers. If the medium is a paste, multiple impellers comprising high speed tools for intense shearing in the bulk (emulsification head, dispersing disk) and low speed wall scraping blades may be advantageously used. We show in Figure 2 an example of kneading equipment. Sometimes, the various impellers are mounted on a rotating carousel (planetary mixers). With this technology the kinematics of the impellers is such that all the vessel volume is swept at regular intervals. There is therefore no dead zone. There is not a unique choice of mixing technology for a given application and several impellers may likely satisfy the mixing requirements. The final selection will be based on both process operating conditions and economics. Design parameters to consider include operating

306

mode (batch or continuous), volume to be processed, mixing time, and pressure in the vessel. The design of the vessel and the internals is also critical from mechanical and economic reasons: shape, dimensions, open or closed tanks, positions of feeds and outlets, baffles, heating coils are all important technical decisions that will directly impact the overall mixing system efficiency.

Figure 2: a multiple tool kneading head In industry, the use of empirical correlations has been the standard for the design of mixing systems. Nowadays, stringent process efficiency requirements make the use of pilot experiments in association with computer modelling an essential step of the development phase. From the formulation developed in the laboratory by the chemists, the process is first developed at pilot scale (typically 200 to 500 1) and then some scale-up guidelines are used to design the industrial unit. Computer modeling can be used at both steps, for the pilot scale and for the eventual production line. The selection of the mixing scale-up criteria to be used for the design of industrial facilities is an open issue. The fundamental question to address is how a process changes when the scale is changed, the objective being obviously to maintain process similarity irrespective of the scale. There are two broad classes of similarity: flow and transport phenomena. Flow similarity may involve the geometry (shape similarity), the kinematics (speed similarity) and the dynamics (force similarity). As for

307

the transport phenomena, one may have thermal similarity (same temperature gradients), chemical similarity (same concentration gradients) and rheological similarity. In the latter case, the same distribution of deformation rates (shear and extension) is sought between the pilot process and the scaled-up installation. In practice, when the process scale is changed, the new design will be based on keeping some parameters equal. The most common scale-up parameters are: 9 mixing times (in laminar regime, the mixing time is inversely proportional to the impeller speed) 9 power per unit volume 9 impeller tip velocity 9 effective shear rate or process viscosity. An approximate similarity can be obtained if the same geometry and the same kinematics is used, while using proportionality rules. For instance, if we want to keep the impeller tip speed ND constant, if D is increased, N should be diminished proportionally. An improved similarity would consider identical energy dissipation scales. The main difficulty is the effect of walls and bottom surface whose influence decreases drastically as the vessel size is increased. This becomes a serious constraint when heat transfer is involved during the mixing.

3. NON-NEWTONIAN MIXING F L O W S From an historical perspective, non-Newtonian mixing flows in agitated vessels have been first dealt with simple models based on simple physics principles and correlations, and then with computer fluid dynamics (CFD) approach. These two approaches will be described here, the scope being restricted to close clearance impeller mixers and planetary mixers. 3.1 Flow models in a mixing vessel Several flow models have been proposed in the literature to predict power consumption with close clearance impellers. They fall in two main groups, namely flow between two coaxial cylinders and drag-based analysis As it was mentioned in Section 1, the mixing of highly viscous and rheologically complex fluids is a slow process both at the macro and micro scale. As the flow regime is laminar, the theoretical notions given in the following paragraphs are valid mainly for this regime.

308

3.1.1 Coaxialflow model The basic idea is to depict the impeller as a solid cylinder rotating inside another cylinder (the vessel). This geometry is known as the Couette geometry. Let us consider the case of a shear-thinning fluid obeying the power law model:

O~rl)-- 2(n-1)12Klrl n-'

(2)

where K is the consistency index, n the shear-thinning index and T the shear rate. It is easy to show that in the Couette geometry [8], equation (3) expresses the variation of the shear rate with the radius, namely:

[rd(Vo/r)l_[ 2~]n 1(1)2 /n

dr

(3)

r1-27n --- r7-21n

Equation (4) gives the shear rate at the inner rotating cylinder (or impeller):

I 1-(rl[r2)a/n2)1[ 2or2] nO-(r, l r 2

(4)

r22-rl 2

In these equations, f~ is the rotating speed (in rad/s), V0 the angular component of the fluid velocity, and r~ and r2 are the inner and outer radius respectively. From a mixing perspective, two cases of practical interest are included in equation (4). The first one occurs when the impeller shear rates are only contained in a small volume very close to the blade. This situation is common in laminar mixing, where the flow generated by the agitator does not reach the tank wall as if the impeller was rotating in an infinite medium. For this case, the usual procedure is to assume that rl/r2 essentially approaches zero. Then, equation (4) becomes:

7, -

4n:N 1 n

(5)

309

According to equation (5), the shear rate is linearly related to the impeller rotational speed, N, and the rheology. The other interesting case arises when r~/r2 approaches 1. This situation can be found when using very small clearances in the mixing geometry. Then, equation (4) becomes:

2 ~2 r 2 ~/

~

2

r 2

I

(6)

2

--

r1

The flow patterns around mixing impellers are fairly complex and the controlling factors of the drag on an impeller are difficult to appraise. In practice, simplified flow situations are used to estimate the viscous energy dissipation, and then deduce the drag or power requirements. In nonNewtonian mixing technology, the flow model around an impeller blade must obviously include the shearing rates or shearing stresses. This is a direct consequence of the fact that the viscosity of these fluids to an imposed stress is not constant but depends on the magnitude of the deformation rate. The knowledge of shear rates in stirred mixing tanks is central to power input calculations and equipment scale-up. Empirical and theoretical approaches have been developed in the literature to estimate shear rates. In the following paragraphs, these approaches are discussed and compared.

3.1.2 Metzner and Otto Approach In late 50's, Metzner and Otto [9] introduced the following rule to define an apparent viscosity in the mixer. If a Newtonian fluid and a nonNewtonian fluid are agitated in the laminar regime under the same operating conditions and in the same equipment (so that the power input measured is the same), then, because all variables are identical, the average viscosity is the same for both fluids. This quite important result for mixing practice was expressed by equation (7):

y'- KsN

(7)

where Ks is a constant of proportionality which has to be determined experimentally for each impeller geometry of interest. In other words, the average fluid shear rate is related only to the impeller speed.

310

It is worth noting here that equation (7) is only valid for laminar mixing, although it has been used for the transition regime. The value of Ks is typically estimated by performing duplicate power input measurements, one with a Newtonian fluid and one with a non-Newtonian fluid, in the same mixing system. Thus, if the rheological behavior of the fluid is expressed by a power-law model, then, from the experimental Newtonian power input curve, at fixed N, an apparent or effective viscosity defined as:

~a -- K(Yav )n-1

(8)

is calculated by:

770-77

(9)

where 1"1is the Newtonian viscosity, PnN and PN are the non-Newtonian and Newtonian power respectively. From the apparent viscosity calculated in equation (9), the corresponding value of )'av is estimated by equation (8). Upon repeating this procedure for pairs of (N, ~/av)values, the value of Ks is determined. It is worth nothing here that as n tends towards the limiting Newtonian case (n=l), equation (5) gives 7 = 41-IN = 12.6N or Ks = 12.6. This is very close to the universal value of Ks = 13, originally proposed in the literature [9] for several fiat-blade turbines. The Metzner-Otto correlation has been extensively used to characterize the average shear rates of non-Newtonian inelastic fluids [ 10], and it is routinely recommended as a standard procedure in mixing and unit operations textbooks. This algorithm is considered a useful approach for the estimate of hydrodynamic similarities on scale-up although it might not represent the rheological properties existing in the mixing vessel. It is a current practice to consider Ks as a constant for given impeller geometry. However, some authors have reported that this constant might depend on the power law index [11][12][13], in particular for highly shear thinning fluids. A compilation of experimental Ks values for helical ribbon (HR) and helical ribbon screw (HRS) impellers is shown in Table 3. A detailed analysis on the correlations proposed for Ks clearly shows that the effect of wall clearance might be the most important one. The value of Ks increases as the wall clearance decreases and this becomes more pronounced as the ratio r~/r2 approaches the limiting value of 1.0.

311

Table 3. Ks data available in the literature (HR and HRS impellers)

Type ,

D/d

n

Ks

i

Comments

HR

1.02- 1.12

0 . 4 - 1.0

66.06 a

Weak Ks(n) Ks(geometry)

[14]

HR

1.059

0 . 2 - 1.0

25

Ks = constant

[15]

HR

1.10- 1.11

0 . 3 5 - 1.0

27

Ks - constant

[16]

HRS

1.03

0 . 3 5 - 1.0

24.58

Weak Ks(n) Ks(geometry)

[17]

HR

1.053

0 . 5 - 1.0

36.73

Ks = constant

[18]

HR

1.056

0 . 2 7 - 1.0

30

Ks = constant

[19]

HR

1.042- 1.19

0 . 5 - 1.0

27.60 a

Ks(geometry)

[20]

HRS

1.056 - 1.118

0 . 2 6 - 1.0

30.6

Ks = constant

[21]

HR

1.11 - 1.37

0.17 -0.65

79.85 b

Strong Ks(n) Ks(geometry)

[22]

HR

1.05 - 1.163

0.35 -0.75

24.68 a

Ks(geometry)

[23]

HR

1.05- 1.33

26.80 a

Ks(geometry)

[24]

HR

1.11

17-40

Weak Ks (n)

[13]

0 . 1 8 - 1.0

Ks (geometry) HR & HRS

1.135

0 . 0 9 - 1.0

7 - 36

Ks (n) for n < 0.5 Ks (geometry)

[12]

This value is calculated for D/d = 1.10. Value estimated for n = 0.6; D/d = 1.135 and 1/d - 3.24, where 1 = length of impeller blade. Regarding the functional dependence of Ks on the fluid flow properties, for most authors, Ks may be considered as independent of the rheological properties. This is particularly true for those authors who have strictly followed the Metzner and Otto assumptions. However, for authors who developed correlations based on the Couette flow analogy principle, the resulting Ks was found to be a function of the shear-thinning level.

312

3.1.3 Extended Couette Flow Approach Bourne and Butler [25] investigated the mixing of viscous Newtonian and shear thinning fluids with helical ribbons, using flow pattern visualization and liquid velocity measurements. They proposed the existence of three main flow zones: 9 the core flow surrounding the impeller shaft 9 the region between the blades of the impeller 9 a highly sheared zone located between the tank walls and the impeller blades.

The authors used the Couette flow analogy to describe the highly sheared zone substituting the impeller by a solid cylinder. Setting as a characteristic fluid velocity the tip speed velocity ND, a particular form of the Reynolds number for power law fluids was proposed:

pN2-nD2 R e p~ =

K

(10)

Thus, the familiar expression for the dimensionless power input is given by:

Np Re pl

P

KNn+ ] D3 =Kp(n)

(11)

In the above equations, no Ks values must be known a priori to handle experimental power input data. Equations (10) and (11) have been the basis for the work of Chavan and Ulbrecht [ 17], and Rieger and Novak [ 18]. Unlike in [25], Chavan and Ulbrecht considered only the geometry of the helical ribbon impeller to define an equivalent cylinder diameter to be used in the Couette flow analogy. More recently, the use of an equivalent diameter based on the ribbon geometry and the fluid rheological properties has been suggested as improvements over previous studies [26][13]. Figure 3 is a typical representation of power input experimental data obtained with a HR impeller for a wide range of shear thinning fluids, using equations (10) and (11). Details regarding the experimental setup and geometry of the HR impeller are given elsewhere [27]. As suggested by the results of this figure, the power consumption decreases as the fluid becomes more shear-thinning (decreasing values of n) at the same Reynolds number. It is also clear that as n tends towards the limiting Newtonian value (n=l), the power data tends to the Newtonian power correlation.

313

10 8

--

i.

l

I

I

I

I

I

I I

'

'

'

'

' ' ' 1

'

'

'

'

'

' ' ' 1

'

'

'

'

'

' ' ' 1

_

_

10 5

Q

-

-

"'..

_

.%

_ _ _

_

_

10 4

_--

o

_

Z

~

"O

e o

_ _ _ _

_

10 3

__ _ _

_

[]

n = 1.0

*

n = 0.77

/x

n = 0.70

0

i

10 2

n

9

_ _ I

1 0 .4

=

0.5

(Newtonian)

"'-.O.'.-.+ :-: ~-~" -, . ' O . ~I k . ' &

3

"

I

I

I

I

'll

1

I

I

I

0 .3

'

' ' I l l

I

I

I

,

, , , , I

1 0 .2

1 0 ~

_ ..

"',II~"Lx "O

n = 0.28 I

.

"'',

I

i

"J, " .

6An

i

i

i

n

i

l

-

1 0 ~

Repl

Figure 3. HR impeller power curves Figure 4 is the plot of the Kp(n) function corresponding to the data shown in Figure 3. The Kp(n) function can be represented by a non-linear model [11][ 12], b and c being fitting parameters, namely: n-1

Kp (n) - Kp(n=l) bn-1 c

n

(12)

It should be noted that the representation of the non-Newtonian power input as shown in Figure 3 is not very convenient since at least two dimensionless parameters, n and Re, influence the power number. Extensive experimentation would thus be necessary to include the effect of other variables. Another drawback is that this type of representation is not general and unique. A unifying principle can, however, be established by comparing the Metzner and Otto approach with equation (11). Indeed, it can be shown [ 11 ] that:

_

Np

Repl - -

P KN,,+~ 9 3 "- K p (n) - Kp(n=l) Ks n-1

(13)

314

or

II-

1

1

Kp(n)

Ks-

mN--2-+]d3

180

160

-

140

-

Q 9 [] 120

-

100

-

80

-

A

n-1

(14)

gp

G e l l a n 1.5 % Xanthan 3% X a n t h a n 2 % - C M C 1% Xanthan 1.5%-CMC 1.5% Xanthan 1.75%-CMC 1.25% Xanthan 1.65%-CMC 1.35% Xanthan I%-CMC 2% CMC 2.5% CMC 1.5% Newtonian fluids Kp

-

J

'

// /

/ /

/ / / /

-

a= /

60

40

20

-

0 0.0

,

I 0.2

,

I 0.4

,

I 0.6

=

I 0.8

, 1.0

Figure 4. Variation of the power constant with the power law index

As suggested by equation (14), Ks plays the role of a shift factor which should coalesce all the non-Newtonian power input results into the Newtonian curve, yielding a unique power input (master curve). We show in Figure 5 the result master curve corresponding to the data of Figure 3. As it can be observed, the whole set of non-Newtonian power input data are superimposed on the Newtonian curve when using the experimental Kp(n) function as shown in Figure 4 and equation (14).

315

10 6

dl 10 5

- 1

Np Re = 162.55

-!

[exp. Newtonian correlation]-

U3 "=t 10 4 -

Z Q.

II

10 3

-

Z

10 2

-

10 ~ 10-3

9 9 9 9 9 []

Xanthan 3 % "~1~ -i Gellan 1.5% ' ,I~~, Xanthan 2%-CMC 1% Xanthan 1.5%-CMC 1.5% Orn Xanthan 1.75%-CMC 1.25% CMC 2.5% I

1 0 -2

Re = p

10 1

10 ~

I

I

I

I

I I

10 ~

N 2"n d 2/In Ks n'l

Figure 5. Power master curve for HR impeller It is worth noting that the functional form of equation (14) makes the Ks function very sensitive to small variations in power consumption measurements, particularly in the highly shear thinning region (0.5 < n). This is illustrated in Figure 6 for a power measurement uncertainty of 10%. It can be seen that the interpretation of the results on the actual value of Ks is delicate. With the measurement uncertainty, all conclusions are possible: Ks may be taken either as a constant (independent of the fluid rheological properties), or as a decreasing or even an increasing function of the flow index. The most significant variations of Ks are found in the highly shear thinning region, 0.1 < n < 0.5. Undoubtedly, these results shed some light on the apparent opposite findings shown in Table 3. In the opinion of the present authors, the differences found in the literature could be easily explained if the impact of experimental errors on power measurements had been fully accounted for.

316

40

''''1''''~'''1''''1''''1''''1''''1''''1''''1'''

\

35

30

-.

c=l.1

~Oo

c=l.O 25

20

/

c = 0.913 (exp.

data)

15

10

5 i 0.0

,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 6. Ks sensitivity to power consumption uncertainty

3.1.4 Drag force-based analysis This approach considers that the torque exerted on the rotating impeller is due to the drag force exerted by the fluid flow around the impeller blade. The forces acting on the blade are namely [28]: 9 the dynamic pressure due to the normal velocity at the blade (known as normal drag or form drag), 9 the tangential stresses caused by the fluid friction (skin drag) along the surface of the blade. If the impeller tip speed is taken as the characteristic velocity, the mixing power can then readily obtained with the following equation:

P - z ( 2 ~ N ) - 2~rbNFdo

(15)

where rb is the radius of the impeller blade, and Fd0 the drag force in the 9 direction. The drag force can be written in terms of the drag coefficient, Cd, as"

317

( l pVo2~

(16)

F d - C d -~

where A is a characteristic blade area and V0 is the angular component of fluid velocity. In the literature, only a small number of correlations based on the drag flow analogy has been proposed. The existing ones differ mainly in the expression used for Cd. It is worth noting that most of these correlations are limited to Newtonian fluids. For non-Newtonian fluids, one correlation is presented in Table 3 [22]. For other works on this topic, the reader is referred to [29]. In the following section, we now turn to the computer modelling approach of mixing flows. 3.2 Numerical simulations We consider the flow of a viscous fluid generated by an impeller rotating in the center of a cylindrical vessel. Using symmetry of revolution arguments, the velocity and the pressure fields can be readily obtained by resolving the standard equations of motion in the non-Galilean frame of reference of the impeller; in other words by considering a stationary impeller and a rotating vessel. In this Lagrangian viewpoint, the equations of change read as: - Vp + divT - p(v.Vv

f

divv = 0

+ co o (co o r ) + 2o9 o v )

(17)

-, (Irl)r

where v is the velocity vector, p the pressure, r the radial coordinate and ~0the angular velocity. The symbol o denotes the standard vector product. This equation is written without considering the unsteady term. This approach is valid provided the flow in the Eulerian viewpoint is periodic. For completeness, the viscosity function in equation (17) must be made explicit with a rheological model. Typical models used in mixing simulations include the Newtonian model (constant viscosity), shearthinning (power law and Carreau models) and yield stress (HerschelBulkey) fluids.

318 Power law model

O~'l)-- 2(n-1)/2

girl'-'

(2)

where K is the consistency index and the n the shear-thinning index. These parameters can be readily obtained by performing a logarithmic regression on the flow curve. Carreau model

r/~?'l) r/= + (770 r/oo)(1+ 2(1t,I ~s _

_

2

(18)

where 7/0 is the zero-shear viscosity, r/_ the high-shear viscosity, t, a fluid characteristics time and s the slope of the viscosity curve. This slope is related to the shear-thinning index of the power-law model by s = (1- n)/2. Herschel-Bulkley model

TO

r/~'l)- 21/217,--~+

2 (n-l)/2

Klrl" '

(19)

where ~:0 is the yield stress, K the consistency index and n the shearthinning index. When the viscosity of the fluid is constant, the familiar B ingham model is obtained:

TO

+ r ] rTOy])-~21,2it I

(20)

For mathematical well-posedness, the equations of change must be completed with appropriate boundary conditions. In the Lagrangian frame of reference chosen, the boundary conditions read as [30]: 9 at the vessel wall and bottom: v = Xwallo r 9 on the impeller: v=0 9 at the free surface: Vz = 0 The last boundary condition implies that the free surface in the vessel is fiat. Experience shows that this assumption is valid in viscous mixing.

319

The above equations are solved using the Galerkine finite element method. Let us recall briefly the principle of this method. We first rewrite the equations of motion using tensor notations, namely:

f div FI = f div V = 0

(21)

II = - P 5 + 271)' With help of the variational calculus, these equations may be written as a saddle-point problem, namely:

Inf

Sup ~

~t (r(v)) 2 d ~ -

~a P div v d f ~ - ~ f v d ~

(22)

The equilibrium equations of this saddle-point problem are:

a(v, lg)-b(lg, p) = ( f , ~ ) , V ~

f

~ [H~ (f~)] 3

b(v, O) - O, V O e L 2 ( a ) a (v, N) - ~a/1 grad v grad N d ~

(23)

b(v, ~) - fa ~ div v df~

where H and L are appropriate Sobolev spaces and (u, v) - ~~ uvdf~

(24)

The principle of the finite element method is the transformation of the above variational problem into a set of algebraic equations that can be readily solved with the tools of linear algebra. For this purpose, suitable approximations are used for the velocity, the pressure and the test functions. In order to guarantee stability and convergence of the numerical solution, these approximations must be carefully chosen. In three-dimensional mixing simulations, the super-linear tetrahedral element PI+-Po and the tetrahedral quadratic element P2+-P1 are state-of-the-

320

art elements [31]. Tetrahedral elements are a much better choice than quadrilateral elements for several reasons: 9 they are more flexible for unstructured grids 9 their accuracy suffers less from distortion 9 they are less costly to use that their equal-order quadrilateral counterpart. After discretization, and using the penalty-Uzawa algorithm for the treatment of incompressibility, the matrix problem to solve is of the form: A

rv

i+1

--

BT p i = F

A r = A + rB rB

(25)

pi+l _ p i _ r n v i + l

where A and B are the elliptic (viscous) matrix and the divergence matrix respectively and r is the penalty parameter. The reader is referred to [32] or an advanced textbook on finite element methods in fluid flow problems for a detailed exposure of the above principles. For very viscous fluids and pastes, a single impeller may not be sufficient to mix the media and stagnant regions may occupy most of the vessel volume. In such a case, multiple impeller kneaders and planetary mixers must be used. The fluid mechanics of mixing inside a planetary mixer is much more complex than with a single impeller. From a simulation standpoint, the complexity arises from the fact that there is no symmetry of revolution usable to simplify the flow description. The most convenient viewpoint for flow description is the classical Eulerian viewpoint of the laboratory and the problem must be dealt as fully transient. In order to tackle kneading flows, a new approach has been developed called the virtual finite element method [33]. This method is nothing but a subset of the broad category of domain imbedding finite element method. In short, the idea is to represent in the volumic finite element mesh, the impeller as kinematics constraints. For this purpose, the impeller is discretized with control points on which the kinematics is known. Then these additional constraints are implemented into the finite element discretization. Mathematical optimization methods (penalty method and Lagrange multipliers) are used to enforce these constrainnts. The final matrix problem is modified accordingly and an extra step is required to update the Lagrange multiplier iteratively.

321 The flowfield is obtained after resolution of the set of governing equations. This field adequately post-processed may yield very useful information for design purposes (power consumption, shear and extensional force profiles, mixing segregation) and process understanding (dispersion and distribution mechanism, chaotic features). We propose to illustrate some of these capabilities in the next section.

4. R E S U L T S A N D D I S C U S S I O N 4.1 Validation and Couette flow analogy The validation of the above numerical method is carried out for the mixer shown in Figure 7. The impeller is a single flight helical ribbon closeclearance impeller.

D s = 0.013 rn W=0.03

m

C = 0.0125 m

P = 0 . 1 8 5 rn

D = 0.210 m

H = 0.24 m

~I

Figure 7: Single flight helical ribbon impeller mixer We first show in Figure 8 the variation of Np vs Reg for Newtonian and shear-thinning power law fluids (CMC solution, 1.5 wt. %) at room temperature up to a Reg value of about 2. It can be seen that the agreement between the numerical predictions and the experimental data is excellent, which confirms the findings of an earlier study [30].

322

1111111

I :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: . . . . . . . . . .................. . . . . . . . . . . . . .9. . .

' ........... ..:~ . .. . ... . ...

' ....... .: . . . . . . . .

9............... .............. .~ ~ ........... ................... ..... ~~-.:---i--!"-i/ .:....... ,~~.... ........ :...,......

~. . . . . . : . . . . . ~ . . . . i . . . ~ . . x..:" ................... : ~ . . . . 1 .~. . *: . ..' . :. i .!. ' :~ . . . . . -. . . . . . . . . . . . . . ,. .~. ., .:

................. ~........ ..~.-.~

..... i--..;...,..,..--~.

: ........... :i . . . . . . . . . . .

.................. .:. .......

c. . . . . . . ,~: . . . . . .

.~ . . . . . . . . . . . ~ . . . ~ . . . . ~ . . . : . . :" . . . . 2~. . . d . , . .:~ . . . ~ .: . J . - : "

~........ --...*...+.--t---~--.,--

I ................i........~-...".....~-.4-.~--~-!--i-..........4 ......~ ........i.....~'----i'"'-'"'!"§ ..............................:.-....... . !.....i---i----i---!-i-; ...................~...........i.......~....~-----~---!---J-i-.................. ~.......... -~ ...... ~-.... "...~-'..i-*-i

: Z"

~ ......... i ....... .~ ..... ~-4 ...... .~-~--

!

9

i i ! i:~!

9

100

. .................. .......................

; ........... , ~i ....... .......

:..

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: *. . . . . . . .

: " ......

-L....~-~..,..'.~...-'-"..,.~....:-.1..~,-~ . .

tf

0.10

. . x. . . ...... .

§i ........... .......

~~........ . . . . . . . ,~ i ,..... - . . . . : . i-.--:------i--.*---,~ ........... ...-

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:-~..~_

: ! : "'"'~"'T":

................. ~........... ~. ..... - . ~ ' ~ - - . ~ - ~

10

..............

i i ii.l~'

9 i ~": ...................

' ~ ...........

: ~ .......

~. ~ :. : ..... :'""~

~ 3.-.:., : ." :

................ ~ ........... ! ..... i ..... i--.-~ .......

i 1.00

......

num. n--1.00

[ ]

num. n=0.651 exp. n-l.00 ,exp. n~O.esJ

i---i-

i! iii i 1 10.00

Re. F i g u r e 8: P o w e r c u r v e in the l a m i n a r m i x i n g r e g i m e W e p r e s e n t in F i g u r e 9 the i s o - s u r f a c e ( s u r f a c e o f c o n s t a n t v a l u e s ) o f the p r o c e s s v i s c o s i t y for the s a m e C M C s o l u t i o n a n d a r o t a t i o n a l s p e e d o f 10 R P M . In this case, a v a l u e o f 3.37 s -~ w a s o b t a i n e d for the e f f e c t i v e rateo f - d e f o r m a t i o n y i e l d i n g a p r o c e s s v i s c o s i t y v a l u e o f 4 . 8 6 Pa.s.

F i g u r e 9: L o c a t i o n o f the p r o c e s s v i s c o s i t y

323

It can be seen that the process viscosity is located along the impeller blade and not in the gap between the ribbon and the wall, as it could be expected with the Couette flow analogy. This is to our knowledge the first time such a result is presented. The numerical prediction of the variation of Kp(n) vs the shearthinning index is compared with the experimental results in Figure 10. 150 -.-e- numedcal 1 -..a-- experimental 100 r

:Z 50

.... 0

~ ..... 0.2

i .... 0.4

:, . . . . . . . 0.6

0.8

1

Figure 10: Variation of the power constant with the power law index It can be seen that Kp(n) does not vary linearly with n. Indeed, although the slope is constant above n=0.6 as already noted in [ 17], the slope tends to decrease for higher shear-thinning fluids. The same trend is observed numerically and experimentally, although to a lesser extent in the latter case. In Figure 10, there is a discrepancy between the numerical predictions and the experimental determination of Kp for the lower values of n. The difference could be imputable to the larger numerical errors inherent to the evaluation of stiff velocity gradients in the simulation. We present in Figure 1 l a a plot of the normalized angular velocity Vz/(Vr 2 q- Vz2)1/2. It corresponds to an isometric view of the cross section plane intersecting the central shaft. Figure 1 lb shows a top view of the same field at mid-height. The spectrum goes from dark grey (small angular velocity) to light grey (high angular velocity). These two figures were obtained for the CMC solution (1.5 wt.%, room temperature) at N= 10 RPM. In Figure 11 a, the downward pumping zone appears clearly in the vicinity of the shaft, where the rotational speed is negligible.

324

//

,jJ Figure 11" Pumping zone of the HR impeller - a) side view; b) top view In Figure 11, two pumping regions can be observed" a (downward) pumping zone in the center and an (upward) pumping zone near the helical ribbon impeller. The downward pumping zone is bounded by an offcentered circular envelope which can be regarded qualitatively as the Couette equivalent cylinder.

4. 2 Mixing flows in Dual helical ribbon impellers We now consider the flow in a typical polymerization reactor provided with a dual helical ribbon impeller (Figure 12). Usually, in industry, these impellers are preferred over single helical ribbons due to their improved performance. The simulation has been carried out using the Galerkine finite element method described before and the Lagrangian frame of reference approach. The surface finite element mesh is shown on Figure 6 as well. We show in Figure 13 a typical circulation profile. The axial velocity has been plotted at two horizontal cross sections. The scale is from-0.12 m/s (pumping down velocity) to 0.13 m/s (pumping up velocity). This picture gives a fairly good representation of the circulation in the vessel. The helical ribbon blades push the fluid at the top in the vessel periphery while scraping the wall. The downward circulation region is located along the shaft, the flow being mainly horizontal at the top (pointing inward) and at the bottom (pointing outward).

325

Figure 12: Double HR impeller

Figure 13" Vertical velocity profiles

Figures 14a and 14b represent a typical dispersion pattern obtained with such an impeller. These results are obtained by tracking with time the position of particles initially clustered and injected in the bulk in the vicinity of the linking rod between the impeller and the shaft. The trajectories are computed using an adaptive 4th order Runge-Kutta method with time-variable steps. In Figure 14a, the "stretching-folding-breaking" mechanism can be clearly identified. The particles initially agglomerated gradually separate one from the next (like a widening ribbon) while turning around the shaft during their travel towards the vessel bottom. In the bottom region, they break apart and disperse in the bulk, circulating outward. Figure 14b shows a similar behavior after a longer tracking time. The dispersing-homogenizing capability of the impeller appears very clearly.,

Figure 14: Dispersion pattern- a) initial stage; b) homogenizing stage

326

Figure 15 shows the pumping pattern in the vessel from the top. This figure is another way of looking at the circulation pattern. The pumping down region (on the left) covers most of the region confined within the helical ribbons while the pumping up phenomenon is restricted to the ribbon region only. The "Couette" cylinder appears remarkably.

Figure 15: Downward (left) and upward (fight) circulation region

4.3 Mixing flows in planetary kneaders Planetary kneaders can be seen as a special type of closeclearance mixers. They are mixing systems of choice for pastes, thickenerbased fluids and for cross-linked polymers. The modeling of the kneading flow is a new application of CFD which is receiving an increasing attention. In this work, we consider the flow in a twin-blade planetary kneader from APV. The system comprises a slow plain blade (twisted paddle) and a fast hollow blade mounted off-centered on a rotating carousel (Figure 16).

Figure 16: Planetary kneader computer model

327

The carousel to plain blade speed ratio and plain blade to hollow blade speed ratio are both 0.5 by construction. The results shown hereafter were obtained with a carousel speed of 15 RPM, a plain blade speed of 30 RPM and an hollow blade speed of 60 RPM. Contrary to the previous close-clearance impellers, the kinematics of the blades is such that no convenient simplifying frame of reference can be used. The virtual finite element method is then required to tackle the flow simulation. We show in Figures 17a and 17b the typical flowfield in the kneading vessel for an industrial cross-linked polymer. In Figure 17a, a detailed flow pattern in the region of the blade is shown. Although the blade has a twisted shape similar to a screw, the circulation appears rather horizontal, an indication of a poor top-to-bottom pumping. Figure 17b shows the general circulation pattern at mid-height in the kneader. Intense shearing appears between the blades and in the clearance between the blade tips and the wall. Far from the blade, the flow is slow and almost stagnant near the wall. .

"~

.'~

-.'~

.

.

.-,

.

.

c

.

~.

.

.

,--

.

.

.

.

.

.

.

.

.

.

.

.

~

. . . .

Figure 17" Velocity profile in the kneading vessel We show in figure 18 the dispersion pattern in the vessel. This result was obtained in the same way as with the double helical ribbon impeller. The clustered particles are initially injected at mid-height in the bulk close to the region where the blades mutually interact in an intense shearing region. It can be observed that the axial dispersion is limited to the central core of the kneader, confirming the deficiency of the top-to-bottom circulation observed in Figure 17. Radial dispersion appears, however, very

328

good as shown by the very chaotic trajectory patterns that covers most of the cross-section surface of the vessel. This proves if necessary the relevance of using more than one impeller to break the flow symmetry in viscous mixing, and therefore enhance the process efficiency.

Figure 18: Dispersion pattern; left: side view; fight: top view.

5. CONCLUSION In this chapter, we have shown how the design of mixers and kneading equipment for non-Newtonian viscous fluids can be carried out using published correlations and/or numerical techniques. It is now possible to predict mixing performance during the process development phase, before building up equipment. It is worth mentioning that this field is still knowing significant developments, and it is expected that more complex problems involving multiphase systems and more complex rheology will be tackled in the near future. ACKNOWLEDGMENTS The technical contribution of Dr. F. Bertrand and Mr F. Thibault from Ecole Polytechnique is gratefully acknowledged. Thanks are also directed to DGAPA (UNAM) and NSERC for the financial support that made possible several contributions in this chapter.

329

REFERENCES ~

2. .

4. .

6. .

~

.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Ottino J., 1989, The Kinematics of Mixing, Cambridge Univ. Press. Skelland A.H.P., 1967, Non-Newtonian Flow and Heat Transfer, Wiley. Eklund D.E. and J.E. Teirfolk, 1981, TAPPI J., 64, 63. Solomon J., T.P. Elson, A.W. Nienow and G.W. Pace, 1981, Chem. Eng. Comm., 11, 43. Barnes H.A., 1997, J. Non-Newt. Fluid Mech., 70, 1. Rauline D., P.A. Tanguy and P.J. Carreau, Mixing of Thixotropic Fluids, in preparation. Hocker H., G. Langer and U. Werner, 1981, German Chem. Eng., 4, 133. Brodkey R.S., 1967, The Phenomena of Fluid Motions, The Ohio State University Press, Columbus. Metzner A.B. and R.E. Otto, 1957, AIChE J, 3, 3. Doraiswamy D., R.K. Grenville and A.W. Etchells, Ind. Eng. Chem. Res., 1994, 33, 2253. Brito-de la Fuente E., J.C. Leuliet, L. Choplin and P.A. Tanguy, 1992, AIChE Symp. Ser., 286, 28. Brito-de la Fuente E., L. Choplin and P.A. Tanguy, 1997, Trans. IChem. E., 75, 45. Carreau P.J., R.P. Chhabra and J. Cheng, 1993, AIChE J., 39, 1421. Bourne J.R. and H. Buffer, 1969, Trans. IChem. E., 47, T263. Coyle C.K., H.E. Hirschland, B.J. Michel and J.Y. Oldshue, 1970, AIChE J., 16, 903. Hall K.R. and J.C. Godfrey, 1970, Trans. IChem. E., 48, T201. Chavan V.V. and J. Ulbrecht, 1973, Ind. Eng. Chem. Process Des. Develop., 12, 472. Rieger F. and V. Novak, 1973, Trans. IChem. E., 51,105. Nagata, S., 1975, Mixing" Principles and Applications, Kodansha and Wiley. Sawinski J., G. Havas and A. Deak, 1976, Chem. Eng. Sci., 31,507. Chowdhury R. and K.K. Tiwari, 1979, Ind. Eng. Chem. Process Des. Dev., 18, 227. Yap, C.Y., W.I. Patterson and P.J. Carreau, 1979, AIChE J., 25, 516. Kuriyama M., K. Arai and S. Saito, 1983, J. Chem. Eng. Japan, 16, 489. Shamlou P.A. and M.F. Edwards, 1985, Chem. Eng. Sci., 40, 1773.

330

25. 26.

27.

28. 29. 30. 31. 32. 33.

Bourne J.R. and H. Butler, 1969, Trans. IChem. E., 47, T11. Brito-de la Fuente E., L. Choplin, A. Tecante, and P.A. Tanguy, 1994, in Progress and Trends in Rheology IV, C. Gallegos, J. Munoz and M. Berjano (Eds.), Darmstad-Steinkopff, Germany, 272. Brito-De La Fuente E., J.A. Nava, L. Medina, G. Ascanio and P.A. Tanguy, 1996, Proc. XII International Congress on Rheology, A. AitKadi, J.M. Dealy, D.F. James and M.C. Williams (eds.), Canadian Rheology Group, 672. Patterson W.I., P.J. Carreau and C.Y. Yap, 1979, AIChE J., 25, 508. Tatterson G.B., 1991, Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill. Tanguy P.A., R. Lacroix, F. Bertrand, L. Choplin and E. B rito-De La Fuente, 1992, AIChE J., 38, 939. Bertrand F., M. Gadbois and P.A. Tanguy, 1992, Int. J. Num. Meth. Eng., 33, 1251. Tanguy P., L. Choplin and M. Fortin, 1984, Int. J. Num. Meth. Fluids, 4, 441. Bertrand F, P. A. Tanguy and F. Thibault, 1997, Int. J. Num. Meth. Fluids, 25, 719.

331

VISCOELASTIC FINITE VOLUME METHOD N . P h a n - T h i e n a n d R.I. T a n n e r

Department of Mechanical and Mechatronic Engineering The University of Sydney NSW 2006, Australia 1. I N T R O D U C T I O N

Analytic solutions to non-trivial viscoelastic flow problems are rare due to the complexities of the constitutive equations and the nonlinearities of the conservation equations. To make any progress, we have to abandon the search for the analytic solution and seek an approximate solution via a numerical method, which is either a finite difference (FDM), a finite volume (FVM), a finite element (FEM) or a boundary element method (BEM); see for example, Crochet, Davies and Walters [1], Reddy and Gartling [2], Phan-Thien and Kim [3] and Patankar [4]. Earlier numerical schemes failed to converge at a relatively low level of flow elasticity when elastic effects become comparable with viscous effects [5]. The level of the flow elasticity is characterised by either the Weissenberg number Wi (product of a characteristic fluid relaxation time and a typical shear rate), or the Deborah number De (the ratio of a characteristic fluid relaxation time to a characteristic time scale for the flow, which is usually taken to be the reciprocal of the wall shear rate in a fully developed flow region). Recent progress in FEM, such as the streamline-upwind (SU) [6], the explicitly elliptic momentum equation (EEME) [7], the elastic-viscous split stress (EVSS) formulation [8], and the adaptive viscoelastic stress splitting (AVSS) scheme [9], have been reviewed elsewhere in this book. Therefore we will be concerned with the Finite Volume Method here, which has been very popular in high-Reynolds number Newtonian flows [4], but only made its presence felt in computational viscoelastic fluid mechanics recently. Our review begins with the formulation, the implementation, and some two and three-dimensional problems that we have had recent success with, including the channel flow past a cylinder, and the three-dimensional entry flow.

332

2. F O R M U L A T I O N

2.1 Governing Equations We are concerned with a general time dependent and isothermal flow of an incompressible viscoelastic fluid where the governing equations take the form: C o n t i n u i t y e q u a t i o n (conservation of mass)

V-u--O,

(1)

M o m e n t u m e q u a t i o n s (conservation of linear momentum in absence of body forces) p ( ~ - + V- (uu)) - V . a ,

(2)

where t is the time, u is the velocity vector with components {u, v, w}, and a the Cauchy stress tensor, given by a = - p l + S,

(3)

with p being the hydrostatic pressure, 1 the unit tensor, and S the "extra" stress tensor (not necessarily traceless), which is related to kinematic quantities by an appropriate constitutive equation. C o n s t i t u t i v e equations In most studies, either a differential (Maxwell-type) or an integral constitutive equation is used to model the fluid rheology. Integral constitutive equations are less popular because of the need of particle tracking. A review of constitutive equations, covering both microstructural and continuum view points is given in Huilgol and Phan-Thien [10]. Here, we will be mainly concerned with the P T T model [10, 11], where the extra stress tensor is written as S - 2~ND + 7",

(4)

where ~N is the Newtonian-contribution viscosity, D - (Vu + Vu T)/2 is the strain rate tensor, with T denoting the transpose operation, and 7- is the polymer_contribution stress, which evolves according to the following constitutive equation g r + A v = 2r/m0D'

(5)

333

where A is the fluid relaxation time, r/m0 is the "polymer-contributed viscosity", (.v) represents the following convected derivative: v Or v - ~ + V - ( u ~ ' ) - (L - ~D)7" - v (L - ~D) T ,

(6)

and Ae g = 1 +--tr

(r),

(7)

r/m0

where L - V u T is the velocity gradient tensor, and ~, e the material parameters for the P T T model [11]. By introducing the retardation ratio defined as/3 - ~,n0/~0 with r/0 = 7/N + 7/m0 being the total viscosity, the Oldroyd fluid B [12] (when ~ and e are set to zero) is recovered with the retardation time being A2 - (1 - / 3 ) A; in addition, the Upper Convected Maxwell (UCM) model is recovered with /3=1. The Reynolds number R~ defined as R~ - p U h / ~ o , and the Weissenberg number W~, defined as We = A U / h , are used to characterise the flow and the fluid elasticity, where h and U are typical length and velocity scale in the flow, respectively. To compare numerical to experimental results, a zero shear rate relaxation time, defined by N1 ('~)

(8)

2r

is used for the experimental fluid. The Deborah number D e - A ~ based on the maximum wall shear rate "~w in the full developed downstream channel is sometimes used instead of W~. As an example, for the planar contraction flow (constant viscosity elastic fluids [13]), this is thrice We when the upstream aspect ratio of the duct is sufficiently large.

2.2 Split stresses Our numerical experimentation leads us to adopt the EVSS formulation of Rajagopalan et el. [8]. That is, the extra stress (Eqn. (4)) is re-written as:

Sij - 271odij +

Y]ij.

(9)

Substituting (9)into (3) and (5)yields

OUi

0 (

OUi~

OP OY]ik

334

(11)

gEij + A ~ i j - 2~7/0 [(1 - g)dij - A ~ij

Thus the dependent variables are now ui, p and Eij. With this coupling between the kinematics and the stresses, we find that the kinematics calculations are less sensitive to the gradual loading of elasticity via the pseudo-body force term OEik/OXk, thereby improving the stability of the calculations. 2.3 B o u n d a r y C o n d i t i o n s To the set of governing equations one must add a set of relevant boundary and initial conditions.

Slip or no-slip on a solid surface The no-slip boundary condition at a solid surface is usually adopted in most studies, where the fluid velocity assumes the velocity of the solid surface. This assumption works well for viscous fluids, but there is a large amount of experimental data suggesting that it may not be relevant for polymeric liquids in some circumstances. There are extrusion experiments with polymer melts [14-18], which suggest that wall slip may be responsible for melt fracture. In these experiments, the occurrence of the extrudate irregularities occurs above a critical wall shear stress, which is accompanied by a fluctuation in the pressure drop. A phenomenological approach to the slip boundary condition has been proposed by Pearson and Petrie [19] where the slip velocity is taken as an empirical function of the wall shear stress. A polymer network model has been proposed recently to account for the dynamic slip velocity [20]. Here, we just simply adopted the no-slip boundary condition at a solid surface, and note that real progress in this area will be made by a careful consideration of the microstructure near a solid surface. With this (Dirichlet) boundary condition, the velocity field is prescribed as u=u0,

OnSu,

where u0 is known on the boundary S~.

Traction boundary conditions Sometimes the traction vector is given on a part of the boundary, say St. This type of (Neumann) boundary conditions take the form t=er-n=t0,

on St,

where t is the traction vector. This yields the normal traction, tn

"

-

-

n - O ' - n -- --p + S- n = n - to,

o n St,

(12)

335

and the tangential tractions

t-t-nn=t0-t0-nn,

on St.

Note that we use Su to denote parts of the boundary where velocity is prescribed, and St, parts of the boundary where the traction is prescribed. These surfaces need not be singly connected, but may consist of several patches. Indeed both velocity and traction boundary conditions can be prescribed at a given location, but in different directions, as in the case of the extrusion problem. Robbins boundary conditions arise in some slipstick problems, where the slippage velocity is determined by the tangential traction (shear stress) at the wall.

Free surface boundary conditions For problems with a free surface, e.g., bubbles, extrusion, etc., the location of the free surface is not known and must be found as part of the solution procedure. Here, a kinematic constraint can be used as the condition to locate (implicitly) the free surface. In a steady flow, the kinematic constraint for a free surface is

u-n=0.

(13)

For unsteady flow, if r t) = 0 is the location of the free surface, then always remains zero on the free surface and its material derivative must also be zero there. The free-surface kinematic constraint thus takes the form ar 0--7 + u - V r - 0. (14) Since n - + V r is a normal unit vector on the free surface (the sign can be chosen so that n is the outward normal unit vector), this constraint also takes the form

1 0r IVr a t

+ u..

= 0.

(15)

In addition to this, the traction on the free surface is known from the physics of the problem. If there is no surface tension, for example, then the traction vector is zero on the free surface. For the case where the surface tension is not negligible, we recall that the surface tension 9/is postulated to be the force per unit length acting along the edges of the free surface. The equilibrium condition on an arbitrary surface element A S reads

fAS [er]. n dS + fAC "~q dl -- 0,

336

where [er] 9n - (er + - er-) 9n is the jump in the normal traction, with er + being the stress on the positive side of n, and or- the stress on the negative side of n, and q is the unit vector normal to the boundary curve AC, but tangential to the interface AS. This is the mathematical statement of zero force on AS. From Stokes' theorem, the surface integral over A C can be done by parts,

/~c7 q dl - /~sV7 d S - /As7

(V-n)n

dS,

leading to (p- - p + ) n + (S + - S - ) . n + V 7 - 7 ( V . n ) n - 0.

(16)

Note that the mean curvature of the surface is given by [21] V n .

.

.

1 ~~

1 .

.

R1

(17)

R2

Symmetry boundary conditions Very often the computational domain can be reduced with the help of symmetry, these sylmnetry boundary conditions are quite easy to state: a plane of symmetry has no normal velocity component, and no tangential stress (tangential tractions or vorticity are zero), i.e.,

u-n=O,

t-t.nn=O,

(18)

where t is the surface traction.

Inflow outflow boundary conditions At the inflow to the solution domain, we know something about the flow, usually the velocity field. For viscoelastic fluids, the stresses are also required there (in essence, these represent the information carried with the fluid from its previous deformation history). Therefore, the boundary conditions at the inlet are usually Dirichlet boundary conditions. Note that all components of the stress cannot be arbitrarily prescribed. First, the stress components should be consistent with the specified kinematics (but need not be), otherwise steep stress boundary layers will be set up. Secondly, if traction boundary conditions are also given there, then the traction component tangential to the boundary would involve the stress components alone without the pressure terms, and therefore these stress components cannot be prescribed arbitrarily. Finally, Renardy [22] has pointed out that specifying all the stress components at the inflow may lead to an over-determined mathematical problem. It may be prudent in practice to retain the time derivative terms in the (differential) constitutive equations,

337

even though we may be dealing with steady-state solutions, and integrate the equations in time through an initial stress state until a steady state solution is reached, including the stress components at the inflow. This ensures that a physical solution is obtained. At the outlet, the flow is usually well developed and arranged so that a uni-directional flow results. Outflow boundary conditions therefore usually take the form of no transverse velocity and no axial traction.

Boundary Conditions for the Pressure Boundary conditions on the traction vector arise naturally from a weighted residual method. However, in the finite volume method the equation for the pressure is usually solved separately from the velocity field, by using the continuity constraint on the linear momentum equations, and therefore boundary conditions for the pressure may be required. The correct boundary conditions for the pressure can be derived either from the traction boundary condition, or from the momentum equations. Thus, a traction boundary condition (to) will turn into a Dirichlet boundary condition for the pressure:

p=S:nn-t0.n,

on St.

(19)

Otherwise, a Neumann boundary condition for the pressure will result from the momentum equation, if the kinematics are fully prescribed, = '1

-p

+ u. Vu

+ v.

s + pb

,

on

(20)

Note that a traction boundary condition at one point on the boundary will implicitly set the pressure. If there is no traction boundary condition, then the pressure can only be determined up to an arbitrary constant. It is important to keep in mind that a traction boundary condition is not the same as the pressure boundary condition; the former is a physical quantity that we can actually impose on the fluid, the latter is a derived quantity, arising only because we are interested in solving the Poisson's equation for p in isolation. If the set of equations, continuity, momentum, are solved jointly, then p would inherit the correct boundary conditions from the boundary conditions for u and t, and there is no need to impose anything on p at all.

Initial Conditions For time-dependent flows, a set of initial conditions is required. The initial velocity prescribed must be divergence free. For viscoelastic fluids, a set

338

of initial values for the stress components is also needed. Note that the incompressibility constraint also forbids an impulsive start and stop, normal to the boundary, since

fs

u.n

dS-O.

3. F I N I T E V O L U M E M E T H O D (FVM) The finite volume method has been very popular in computational fluid dynamics dealing with Euler and Navier-Stokes flow problems. There are two main approaches in the finite volume method.

3.1 Chorin-Type Methods The first approach uses an artificial compressibility condition to satisfy the continuity equation. A pseudo-transient formulation is then adopted in the momentum equations and the steady state solution is considered as the asymptotic solution of a time-dependent problem with time-independent boundary conditions (if need be), and these are computed by a timemarching scheme. The method is due to Chorin [23] and is very commonly used in computational fluid dynamics dealing with high Reynolds number flows. Integration in time can be carried out either implicitly or explicitly. In implicit schemes, it is necessary to solve a system of equations at each time step, making the scheme very expensive for large three-dimensional problems. However, such schemes can be made unconditionally stable and relatively large time steps can be used. On the other hand, in explicit schemes, it may be necessary to solve only a simple mass matrix, or it may not be necessary to solve any system of equations at all. The main disadvantage of explicit schemes is that the time step is restricted by the Courant-Friedrichs-Levy stability condition. However, there are several ways to accelerate convergence such as residual averaging, local time-stepping and multigrid schemes. Phelan et al. [24] use a similar scheme, but with a modification to suit hyperbolic types of constitutive equations, for solving cavity-driven flow problems for the UCM model. The same method has also been used to solve the slip-stick problem of Maxwell-type constitutive equations [25].

3.2 S I M P L E R - T y p e Methods The second approach in finite volume method is much more popular and forms the basis of some commercial packages dealing with Navier-Stokes equations. The resulting algorithms are known as SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), SIMPLER (SIMPLE Revised) [4],

339

SIMPLEC (SIMPLE Consistent) [26], or PISO (Pressure-Implicit with Splitting of Operators) [27], to name a few. In these schemes, the momentum equations are solved sequentially, and the pressure is corrected from the continuity, equation and the linearised momentum equations. Applications of the SIMPLER algorithm to viscoelastic flows include the 4-to-1 entry flow and the die entry problems [28-30], the non-isothermal flow past a cylinder [31], the flow past an eccentric cylinder problem [32]. The PISO method has been applied to the non-circular pipe flow of the Criminale-Ericksen-Filbey model to study the effects of the weak secondary flow on the pressure drop [33]. A variant of the SIMPLER method, termed SIMPLEST (SIMPLE with Splitting Technique) has been used recently to study secondary flows in non-circular pipe flows, and the 4-to-1 threedimensional flows [30, 34]. Formulation In the FVM, the flow domain is subdivided into a set of non-overlapping control volumes (CV). The grid nodes (storage locations for the dependent variables) are located at the centre of each of the control volumes. The discretised equations of the dependent variables are obtained by integrating the mass, momentum conservations over each control volume as follows: 1. First, all of the governing equations are cast into the form of a general transport equation: AO(I) -b-/- +

0 (Auk,I)) -

a

k

+

where (I) is the dependent variable which can be a component of a vector or a tensor and even a constant. The coefficients A, F have different meanings for each different dependent variable, and Sr is called the source term, which lumps all the terms that cannot be accommodated in the convective and diffusion terms, and is specific to the particular variable (I). 2. Next, the general transport equation is integrated over the control volume surrounding the node, namely node P, in the flow domain and the time interval St, using the divergence theorem whenever possible,

340 Here, A V is the volume of the control volume and ~ v the corresponding face area with n k being the unit outward normal to the face. 3. Finally, a proper discretisation scheme (the control volume-based interpolation functions) is adopted for the temporal and spatial approximations of the dependent variable. Generally, the first-order backward Euler implicit formula is used for temporal differences because of its simplicity and its unconditional stability for numerical calculations. As to the volume integral of the source term S~, the value at the central node of the control volume, namely S~, is assumed to prevail over the whole control volume, and it can be linearised in term of the nodal value Op as usually assumed in FVM. In this way, the final algebraic approximation equation which relates Op to its neighbouring nodal values can be written as o ap~p -- ~ anb~nb -4- Sc --[-ape~p, nb

(23)

where o_ A

ap

~-~AV,

o

-

ap -- E anb + ap -- Sp. nb

(24)

Here, the subscript p refers to the central node P, and the summation is to be taken over all of the neighbor nodes nb of the node P. An overbar means the applied values are evaluated using the known fields from the previous time (or iteration) level, and the coefficients anb are the functions of the dependent variables, and their structures depend on both the form of the control volume chosen and the approximation scheme used. It is these coefficients that determine the spatial accuracy of the final solution. In our calculations, the power law scheme proposed by Patankar [4] is employed for the formulation because it covers central difference and upwind difference, and gives an excellent approximation to the exact exponential solution for the one-dimensional convective-diffusive equation. In addition, FVM with power law scheme, instead of central or upwind difference, has better conservational properties and thus produces physically realistic solutions even for coarse meshes. In the power law scheme, the coefficients are chosen as follows: anb -- D n b f ([Pnbl) + [sign(nb)Fnb, 0~ ,

(25)

where Dnb -- (FA/ Xi)nb is the local diffusion conductance; Fnb -- (Au~A)nb the "mass" flux passing thought the corresponding face A normal to i

341 direction of the control volume, sign(rib) is +1 for upstream face and - 1 for downstream face, and f (I Pnb I) is the function of the local Peclet number defined by Pub -- Fnb/Dnb, which is given by

f ([Pub[) -- [0, (1 -- 0.11Pnbl)5],

(26)

where the symbol [a, b~ means the greater of a and b.

Constitutive equation discretisation The viscoelastic model is also in the form (21), without the diffusion term (F - 0). To ensure numerical stability a first-order upwind difference is used for spatial discretisation. Thus, the discretised constitutive equation will take the similar form to (23) (thereafter we use the symbol "1- instead of E for the elastic stress tensor Eij 2flTIodij to avoid confusing with the summation symbol E)" --

7"ij

--

ij ij ij ap Tp -- E anbTnb -~- Sc ~, nb

(27)

where the superscripts ij refer to tensor components while reserving subscripts for the grid node and the overbar for the values from the previous time (or iteration) level. The constant part of the source term, in which the stresses and deformation are approximately piecewise-constant in each control volume, takes the form (no sum in i and j)"

S:~ -

ap~'p~ + A V {2/3~0 (1 - g)dp j + )~ (1 - ~)[(dpi + d jj) ,0] CpJ + )~ [~k (1 - 5kj)+ 2/3~0d~k] [(OuJ I

(28)

\Oxk/p \Oxk]p

0 while the coefficients ai.j and and SpJ - - A V [g

ap0

take a form like (24) with F - 0, A -

)~ (1 - ~ ) [ - (dpi + dJpJ), 0]] - E sign(nb)Fnb,

A,

(29)

nb

where Fnb -- ()~uiA)nb can b e thought of being the "mass" flux passing through the corresponding face A normal to i direction of the control volume. The term Esign(nb)Fnb in 5'~J is actually the continuity constraint nb

which should approach zero when the solution converges.

342 We note that there is an extra convective term in the source term in the constitutive equation; this extra source term should be consistent with that of the convective term of the stresses. Thus, upwind difference should be applied to its discretisation. This leads to 0

-

With upwind difference, the resulting coefficients anb will be anb -- [sig~(nb)Fn~, 0].

(31)

Therefore, the sufficient condition for convergence for the adopted numerical algorithm-TDMA (typically alpj > E anb) is satisfied even for the

nb

steady-state calculation with a ~ being zero. Some work [35] used a pseudotransient constitutive equation for the steady-state calculation to ensure a resulting diagonally dominant matrix, and so obtained convergent solutions for the inertialess flow of UCM fluid (g - 1) through a 4 9 1 abrupt axi-symmetric contraction with a much higher value of the Weissenberg number (up to W i - 6.25). The pseudo-transient method for the steady-state calculation is actually equivalent to a local under relaxation: the positive inertia coefficient ap0 damps out possible oscillation during the stress iteration. In our previous work [32, 34, 36], an artificial diffusion term -g~-2~k\ ox~] with c~ being an artificial diffusion coefficient is introduced on both sides of the constitutive equation, and in discretisation, the current values are taken for Tij on the left hand side; while the known values from previous iteration level for that on the right hand side. As a result, depending on the spatial discretisation scheme used, the coefficients anb in (27) will take the form

anb- Dnbf

~

-4- [-sign(nb)Fnb, 0],

(32)

where Dub -- ((~A/6xi)nb denotes the local diffusion conductance, and function f ( I D ~ b l ) h a s the different form for different schemes. For example, when the central-difference scheme is adopted, we have \1

I/

1 05,

,

343

Thus, with the scheme adopted, the convective term of the equation can be thought of being always discretised using upwind difference, and the diffusion term is discretised using central difference, but the resulting local diffusion conductances are different for different schemes used, that is ' + [sign(nb)Fnb, O~ - anb d + anb ~ , anb -- Dub

(34)

with D'nb -- D ~ b - 0.5 IFnbl for central difference and D'nb -- Dub for upwind d to denote the contribution of the artificial difference. Here, we use anb diffusion term discretisation, and a~b the contribution of the convective term discretisation, that is:

d _ D n' b ~ anb c _ [sign(nb)Fn~, O~ anb

(35)

To ensure all of the contributions to be positive, we need !

(36)

Dnb > O.

Obviously, this is always the case when upwind difference is used. However, for the case when central difference is employed, we require that

Dub ~ 0.5 IFnbl.

(37)

Thus, we have

(

Dub--

1)

1-- ~

Dub,

(38)

with Dub -- IFnbl and ~; is an arbitrary constant satisfying ~; _> 0.5. This can be thought of being the criterion for the proper selection of the artificial diffusion coefficient c~. Similarly, the source term will contain an extra artificial diffusion term _ f z x u _ ~ko \(c~~176 dV, and central difference is applied for its discretisation. The discretised form for the extra source term will have the form:

~dj -- _ y~ DnbTnb _ij + "fp3 ~ Dub. nb nb

(39)

Thus, the final discretised constitutive equation can be cast into the form c + adnb) Tub ij + s~J 2t- ~d -"j, (ap T ad T apO_ ~ipj) ~ "" _ E (anb (40) nb where C apC :~-~ anb and a d :~--~ anb,d nb nb

(41)

344

which can be re-arranged to yield ....

(ap + a ~ - ~ipj)

+ [(a;

+ aO -

I v " _c

ij __ O~J

n~ ttnbTnb-1- ~c

+

)

-

+A,

(42)

where /~

__

~ d _.i j ~.d _--dj q_ ~adJ Y~ t.nb.nb ~ t~p'i p (a~ + a ~ - ~ J ) + a d

(43)

nb

will tend to zero when the solution converges. Therefore, by introducing an artificial diffusion term, the spatial accuracy of the solution is less than second-order, but an effective local underrelaxation factor

ap + ap0 _ ~ j OiPJ -- (a~ + a ~ - ~pJ) + ~ D'nb

(44)

is introduced for the stress calculations so that possible oscillation during iteration is alleviated. Structured and Unstructured M e s h U

W

w_ , ' ..O" / _ G e - O- - ,-s- =0- - - I ""'P [

@

E

D

Figure 1" The control volume for node P. In the S I M P L E R method and most of its invariants, a staggered mesh is invariably used, where the pressure nodes (and the nodes for the extra stress components), are positioned along x, y, and z directions first, then the control volume faces, at the centroids of which the velocity components are calculated, are placed midway between neighbouring grid points. Thus the x-direction velocity u is calculated at the faces that are normal to the x-direction. The same rule is applied to v and w. An example of a control volume for node P, with neighbouring nodes U, D, N, S, E and W is illustrated in Fig. 1.

345

Figure 2" An unstructured finite volume mesh for a flow past a cylinder in a channel. In the figure, the lower case letter refers to the interface between two neighbouring cells, e.g., e refers to the interface between the control volumes for P and for E. One ends up having four different types of control volumes, three for the velocity components, and one for the pressure. The staggered grid is designed to eliminate the checker-board pattern in the pressure field [4]. With structured mesh, a linear interpolation scheme is often used, resulting in an O(h) accuracy, where h is a typical linear dimension of the control volume. Although one ends up with four different types of control volumes (three for the three velocity components and one for the pressure), structured mesh has been very popular because of the ease in implementation; in addition, iterative solvers like the tri-diagonal Thomas algorithm (TDMA) can be readily applied along the coordinate lines defining the mesh. Structured mesh is not very suitable for complex geometry, however. To provide the flexibility in fitting complex computational domains, unstructured non-staggered triangle mesh have been developed by a number of authors, for example, Prakash and Patankar [37], Masson et al. [38], and Davidson [39]. An example for such an unstructured mesh for the two-dimensional flow past a cylinder in a channel is given in Fig. 2. To retain the basic structure of the iterative line-by-line tri-diagonal Thomas solver (TDMA), lines of nodes from the entry to the exit boundaries are introduced (Huang et al. [32]). If this is not possible, then the line is terminated in the domain. The nodal information (neighbouring ID's, etc.) is stored in a sweeping line array. Similar to a re-ordering of the nodes in the finite element method to minimise the bandwidth of the system matrix, this array stores all the information needed for the line-by-line tridiagonalmatrix algorithm, which only requires computer storage and computer time

346

of O(N), where N is the number of unknowns. The method has been used successfully on viscoelastic flows between two eccentric cylinders [32], and past a cylinder in a channel [36].

Solver In the S I M P L E R method and most of its variants, the kinematics are determined by solving the momentum equations, assuming that the pressure field in known. The pressure correction is then applied, by enforcing mass conservation [4]. For the viscoelastic flow computations, the source terms containing the extra stress OEik/OXk in the momentum equations are treated as pseudo-body forces with the known dynamics field obtained from the previous time (or iteration) level by solving the discretised constitutive equations. In each cycle of the algorithm, no system matrix needs to be solved, and the coupled discretised equations for the dependent variables are solved sequentially from an initial guess for all field variables (typically quiescent field for Newtonian computations). For the momentum equations, two T D M A sweeps are performed, and for pressure correction and stress equations, four T D M A sweeps. In the iterative procedure, the calculations of velocities and stresses are under-relaxed by a global factor of 0.85 ~ 0.5 depending on the elastic level, but no relaxation is need for the pressure calculation. The convergence criterion for terminating the calculation is that the integral residual of the discretised equations over all control volumes for any dependent variable is less than the input tolerance, of the order 10 -6 10 -s, and the relative changes in the values of flow field (typically the velocities) near the solid wall from one iteration to the next are of the order 10 -5 ~ 10 -7. For viscoelastic computations, the corresponding Newtonian result (A - 0) is used as the initial guessed field, and Wi is increased gradually by increasing A. The under-relaxation mechanism due to adding artificial diffusion is very effective in the stabilisation of the calculations at high elasticity. To speed up the convergence rate and save some C P U time, a three-dimensional block T D M A solver can be used, see [30]. The two-dimensional implementation was done with both structured and unstructured mesh. The three-dimensional version was implemented with structured mesh only, as three-dimensional automatic unstructured mesh generation is still an active area of research. Both versions have been well tested with simple flow problems where analytical solutions exist, including Couette and Poiseuille flows, and proved to be very robust. In the 2D Poiseuille flow, there seems to be no upper limiting Weissenberg

347

number, and an agreement to four significant figures with analytical results was demonstrated at We = 40 in [36]. The 3D version has been tested for the Poiseuille flow of the Oldroyd-B fluid in a straight pipe with square cross section. Numerical results show that no upper limit in the Weissenberg number is encountered, and the convergence is improved with mesh refinement. It is verified that adding the artificial diffusion has a similar function to increasing the under relaxation factor [34]. We now consider the flow past a cylinder in a channel, and the threedimensional entry flow. 4. F L O W P A S T A C Y L I N D E R

The flow past a cylinder in a channel has been well investigated, both experimentally and numerically, and therefore is a good candidate to test the finite volume method. Here, the flow past a cylinder, of radius R, placed on the centreline, or offset from the centreline by a distance e in a channel of width 2L, is considered. No-slip boundary conditions are applied at the walls. Along the centreline symmetry conditions are applied. The length of the downstream section is 9L, and that of the upstream section is 6L. All distances are normalised by cylinder radius R. Having fixed the model parameters and the problem geometry, the only parameter left to vary is the average velocity U of the fluid at the channel entry. The Deborah number (De) and Reynolds number (Re) are defined as

De = AU/R,

Re = pRU/Vo.

(45)

Dhahir and Walters [40] reported some experiments and finite element calculations (using Polyflow TM) on the effects of viscoelasticity and of the wall on the flow of non-Newtonian liquids past a cylinder confined by two plane walls, with particular attention to the drag force on the cylinder. Inertia effects were neglected in the calculations, and they found the streamlines nearly independent of the mean flow rate. In addition, they also found the streamlines to be essentially independent of the Weissenberg number, at low values of the latter. Flows past an asymmetrically confined cylinder have also been investigated and the effect of the eccentricity on the drag force has been obtained both numerically and experimentally. Baaijens et al. [41] investigated the same flow of a shear thinning solution PIB/C14 both numerically and experimentally. A number of constitutive equations were used in their FEM simulations, including the PTT, the Giesekus and the UCM models. The field variables were compared to experimental values, along the centre line

348

and over cross sections of the channel. In general, there was a good agreement between measured and computed field variables. Measured and computed birefringence was also presented in [42] and a qualitative agreement was found. The wall effects were considered by placing the cylinder off the centreline, and it was found that the elongational thickening of the viscoelastic fluid causes a significantly larger flow rate through the broader gap compared with inelastic fluids. Barakos and Mitsoulis [43] investigated viscoelastic flow past a cylinder symmetrically confined by two parallel plates using a K-BKZ integral constitutive equation, using FEM with a path tracking scheme. A good agreement between their results and those of [41] was found. Huang and Feng [44] also reported some numerical results, using PolyflowTM, for the flow of the Oldroyd-B fluid past a confined and non-confined cylinder, retaining inertial terms. In the finite volume simulation, the computational domain is divided into non-overlapping polygonal control volumes, as shown in Fig. 2, and the SIMPLER algorithm is adopted for the UCM model. First, the accuracy of the Newtonian calculations is assessed by comparing the drag coefficient defined by K=

4~0UR' where F~ is the axial force acting on the cylinder, with mesh refinement. The extensive mesh refinement studies are performed with different meshes by reducing both the polar mesh size hp and the maximum mesh size hm according to Table 1. By comparing the results within and across mesh groups, the refinement of the polar mesh size is found to play a more important role in producing a more accurate solution. In the table, the errors are calculated relative to the result of the finest mesh L4. Polar Maximum Mesh Nodes Drag Force Error Mesh Size Mesh Size M2 0.069813 0.4 1067 10.0634 0.03740 M4 0.0349065 0.2222 10.3178 3434 0.01306 M6 0.02617 0.1538 6504 10.3060 0.01419 M7 0.02416 0.1818 7711 10.2927 0.01546 N1 0.021 0.2566 2392 10.3260 0.01228 N2 0.0175 0.2566 2806 10.3313 0.01177 N3 0.00873 0.2566 4296 10.3513 9.86e-3 N4 0.00714 0.2566 10.4172 3.55e-3 4996 L1 0.021 0.127 6991 10.4306 2.27e-3 L3 0.00873 0.127 9695 10.4543 8.19e-7 L4 0.00714 0.127 0 10977 10.4543

349

Table 1" Drag force and relative error computed for a Newtonian fluid.

3000 2500

..... FEM . 0 6 s O uCVM 0.0s

!

2000 O,I

E o 1500

V

LI_•

1000

500

%

30 40 50 60 70 80 FLOWRATE (cma/s)

Figure 3: A comparison between the FVM (symbols) and the FEM results of Dhahir and Walters (solid and dotted lines). The best fit polynomial to the calculation of K is K(hp)

-

10.5245 - 6.4120hp,

showing an approximately linear convergence with decreasing mesh size. An extrapolation of this formula gives K - 10.5245 at zero mesh size, which is comparable to the FEM result of 10.5313 [45], and asymptotic result of 11.0199 [46]. For viscoelastic flows, mesh refinement is the only check of the consistency and accuracy of the numerical results. A monotonic decrease in drag force with the decreasing polar mesh size is found, with the maximum mesh size fixed at hm - 0.1818. For a given mesh, a decrease in K with increasing D e is found. The results indicate a good mesh convergence with 0 < D e _< 1. At higher values of D e , the convergence is slower. More iteration is needed for higher Deborah numbers. For Newtonian flow 350 iterations are needed for a convergent result, the number of iterations increases to 670 at D e - 0.914 and 800 at D e - 4.0225. Figure 3 shows a comparison of the FVM results (symbols) and the F E M results of Dhahir and Walters [40] (lines, taken from Fig. 19 of their paper).

350

6

....

,..-d:".'-.,

....

6

....

,..~:':..',

....

6, . . . .

' 4-

-

~:''''',

....

F

4[-

2

,"

'-

4-

-

2 +

'

o

:~.~ a

-2

. ~

Iz~.z-a o

o,.-~~. - 2 .~---

. ,

9

q

~a

-

b

4

,I,

. . . . . . . . .

-6_ -1 o

I""

12

-6

'"

"

~

"

_

1 2

-,JI

"

I.

I

. . . . . . . . . . .

6-2 lo

12

Figure 4: A comparison between the computed (FVM: solid lines, FEM: triangles and squares) and experiment (circle) results by H. Baaijens [42] along several cross sections for P T T fluid with A - 0.0431 s, p - 0.8 g/cm 3 and e - 0.39. Left: axial velocity, middle: normal stress difference and right: shear stress. The shear rate at entry is so chosen that D e - 2.31. We have chosen the same geometry, with L = 5/3R, and 2113 control volumes were used over the full flow domain. Using the UCM model, the force per unit length on the cylinder is computed at varying flow rates and two values of A (0.0 and 0.06 s). Inertia effects are neglected by assuming creeping flow. Figure 3 displays the variation of normalised force components F* - F = / % (cm2/s) with the flow rate. The Deborah number at the highest flow rate of 80 cm2/s is 0.64. It is seen that F= increases with the flow rate in both Newtonian (A - 0) and UCM (A - 0.06 s) cases. Increasing elasticity reduces the force on the cylinder. The direction of the force is along the flow direction, as it must. The drag force in both the Newtonian and UCM cases are in excellent agreement with the F E M results of Dhahir and Walters [40]. The flow of a 5% P I B / C 1 4 solution past a symmetrically confined cylinder between two plates of a distance 2L apart is simulated. Following [41], the P T T model is chosen. The density of the liquid is taken into account and fixed at 0.8 g / c m 3, the relaxation time is fixed at 0.04313 s and P T T

351

model parameter is fixed at e = 0.39 and ~ = 0 (these values were given in [41]). The computation is done on half of the flow domain and with mesh M4 (with a total nodes of 3434, closed to the total nodes of 3981 in [41]). Figure 4 shows a comparison between the measured and computed (finite volume) results at cross sections x = - 5 R , x = - 2 R , x = - 1 . 5 R , x = 1.5R and x = 2R at the highest measured Deborah number of 2.31, where the velocity has been normalised with the mean velocity U, and stresses with 7o - 3~oU/R. To provide a visual comparison with the measured data, the computed results in the half domain are either symmetrically or asymmetrically mapped to the other half of the flow domain. Due to experimental errors, the measured results are not symmetric about the centreline. Figure 4 shows a good agreement for the velocities at all cross sections. The measured and computed first normal stresses agree well at all cross sections except at x = - 2 R , especially near the walls, although a reasonable agreement is still found inside the flow domain. The computed and measured shear stresses also agree reasonably well in all cross sections except at x = - 5 R . At x = - 5 R and near the entry, the flow is still fully developed, the shear stress is linear with respect to y / R , and tends to zero at centre line. The computed and measured shear stress agree better in the left domain than in the right domain. The finite volume solutions in the previous subsection are also checked against F E M results [41], plotted in Fig. 4 as triangles for a l-mode and squares for a 4-mode P T T model (the F E M data were taken from Fig. 4.14 and Fig. 4.16 from [41]). In Fig. 4, F E M results are plotted at the cross sections Ix/RI - 1.5 for the first normal difference (middle figure) and shear stress (right figure) where significant differences are found between measured and computed results. The shear stress profiles are displayed on the right of Fig. 4. An overall agreement with F E M results is noted. In summary, the unstructured finite volume results are mesh convergent, comparable to F E M results, and agree well with available experimental data for the flow past a cylinder in a channel. 5. T H R E E - D I M E N S I O N A L

ENTRY

FLOW

In the study of the entry flow problem, two types of the cross-sections, circular and rectangular, are of particular interest. The corresponding flows are referred to as axi-symmetric contraction and planar contraction flows, respectively. A large number of experimental works and two-dimensional numerical simulations on the flow behavior of various types of viscoelastic fluids in both types of geometry have been performed. Most earlier work has been reviewed in the literature (see, for example [47-49]). Here, we

352

only briefly list some of the main observed phenomena. Z

Figure 5: Three-dimensional planar contraction geometry and graded meshes (one quarter of the domain).

| '0[

l)JJtll

3

(a) w =0.82

4

.,

1

2

3

(b) W = 1.29

4

/// JJli

3

(c) w.-- 2.75

Figure 6: Streamlines for the Boger fluids fitted using the UCM model with different elasticity under the same flow condition (Re - 2.1), (a) W~ - 0.82; (b) We - 1.92; (a) We - 2.75. The aspect ratio is one. In the axi-symmetric contraction flow for Newtonian fluids, it has been clearly shown experimentally and computationally [50-51] that the presence (or absence) of the upstream corner vortex is determined by fluid inertia only.

353

0.05

IIII II t i t

w.

--

3.40E-3

1/

0.00

/ / //

-~~;'[2

- - ~ -0.05

-

2.7

-

-

.----'m

2)I

-

2.9

1.0

(a) Wmi.

0.05

_

ItfJJ,{tf

-8.73E-3

li

0.00

-0.05 - ' - - ~ - -

"~-"~~'~

~"'-

,

i/

/

/ / /! / I

~".,'1

-2_~ 7 - - ~ - ~ / ! / /

--~.____._ ~

2.6

/I

lilltfltt/t

. - I ~ - - ~

i

i-.~ ~

.

ii

9

2.7

.,i

1.0

(b) 0.05

Wm.~n--- 2 . 1 4 E - 2 0.00

........

,

--..----'--'7"--

2.6

2.7

218

' -

2.9

3.0

(c) Figure 7: A close-up view of the meshes and flow fields in terms of streamlines near the re-entrant corner of Fig. 6. However, for viscoelastic flow, experimental work has demonstrated many unusual flow phenomena which distinguish it from the corresponding Newtonian counterpart, such as the vortex enhancement [52], the appearance of the lip vortex [53], and the flow transition from the steady-state to three-dimensional, time-periodic and aperiodic flow states as the flow rate (or the Weissenberg or the Deborah number) increases to some level [54]. The dependence of these flow phenomena on the fluid properties is highly non-linear, and it seems that fluid elasticity based on steady and dynamic

354

shear properties alone is not adequate to explain the different behavior for different viscoelastic fluids. It was argued that the extensional behavior of the fluid plays a more important role on the vortex activities for viscoelastic fluids: strong vortex enhancement is expected to occur for fluids with strain-rate thickening extensional viscosity; on the other hand, strainrate thinning extensional viscosity will reduce the strength of the vortex activity [55-56]. In contrast to the axi-symmetric contraction flow, where a strong vortex enhancement is expected for some Boger fluids, the vortex activity is weak in planar contraction flows [57-59]; only under some extreme flow conditions [53] does a lip vortex appear. While for some shear thinning viscoelastic fluids, such as polyacrylamide aqueous (PAA) solutions, vortex growth has been observed in both cases. More remarkably, at very high flow rates, a non-trivial 3D unstable and alternate flow pattern is observed

40

-

.

, -=

35 . - - - o ~ +

,

-

-

,

.

.

.

.

,

-

140 . . . .

-

r

= 0.4

We=0.83

We = 0.4

120 ~

we = o.83 ;

*~gzz~tb ~

We = 1.39 We = 1.88

We = 1.88 X

2s

. we

We = 1.39

30

"~

-

.

We=2.9

~

100

(~=0)

v

We=

2.9(~

-

We=2.9

(~=0)

r~

We=4.15 =O.1

~

20

,~,

80

Z--

60

-----o-----

We=

4.15

~

We=2.9(Exp.)

W e = 2 . 9 ( P, = 0.1

15 4O 20 o .............. 0

-' 2

......

-9

-.----~----,-+ ~---;

.

-6

-3

:

.

i

0

.

. . . . .

3

6 Z

1

2

.

.

-9

. . . .-6 ..

.

.

'

-3

0

"

. .._';;;;;;,_.~l:J

3

6 Z

(a) (b) Figure 8" Computed distributions of (a) axial velocity w; (b) the first normal stress difference N1 along the centreline. Some of the selected results of the three-dimensional simulation of the entry flow [30] are given here (see Fig. 5 for the flow geometry), to illustrate the performance of the finite volume method. First, the 3D flow patterns of the Boger fluid fitted using the UCM model are simulated under the experimental flow conditions used by Walters and Webster [57] and Walters and Rawlinson [58]. The streamlines calculated for the fluids with different elasticity are given in Fig. 6, at a Reynolds number of 2.1 and an aspect ratio of W / H - 1. Here, the Reynolds number R~ defined_ as R~ - pUh/rlo, and the Weissenberg number is defined as We - A U / h , where h and U are

355

the half height and the mean flow rate in the downstream channel, respectively. As we can seen from the figure, there is virtually no corner vortex, which is in good agreement with experimental observation. However, upon close-up inspection of the streamlines near the re-entrant corner, as shown in Fig. 7, there exists a tiny lip vortex which spreads over several meshes and increases in both size and strength with increasing fluid elasticity. AIthough the lip vortex feature for Boger fluids in planar abrupt contraction flows was observed only at very high elasticity (We at order of 100) in the flow geometry with higher contraction ratio [53], the trend of our simulation results is consistent with the experimental observation. There may be two plausible reasons for the discrepancies. First, the lip vortex may be very hard to discern in the flow visualisation due to its being small in size. Secondly, the UCM model may not have all the relevant physics in complex flows of Boger fluids. Recently, Quinzani et al. [61] measured the detailed flow fields of a wellcharacterised shear-thinning polymer solution (polyisobutylene dissolved in tetradecane (PIB/C14)) flowing in a 3.97:1 planar abrupt contraction with the Laser-Doppler velocimetry (LDV) and flow-induced birefringence (FIB). Their experimental results have been partly simulated by some 2D numerical simulations with the fluid modelled by an integral K-BKZ equation with a spectrum of four relaxation times [62], and by multi-mode PTT, Giesekus and UCM models [63]. All of the simulations showed a good general qualitative agreement with experimental data. But the agreement is still semi-quantitative, especially for the extensional stress near the entry section (up to 30 - 40% discrepancy depending on the level of elasticity). The PIB/C14 shear-thinning solution used in the experiment work has been well characterised by Quinzani [64] with several non-linear multi-mode models, and they found that the best quantitative fit to the transient extensional viscosity could be made using the P T T model with p - 0.8 g/cm 3, /~ 1, ~0 - 1.424 Pa.s, A - 0.06 s, ~ - 0 (or 0.1), ~ - 0.25 (or 0.05). A finite volume simulation has been performed with these parameters using a graded mesh. A comparison between the computed values for the centreline velocity (w), and the first normal stress difference (centreline birefringence) is shown in Fig. 8. Generally, the numerical results agree reasonably with experimental data; however, the peak values predicted for N1 are 20-40% higher than the observed values (depending on (). The discrepancies are believed to be mainly due to the inadequacy of the constitutive model. -

356

6. C O N C L U S I O N S

The results of the finite volume calculations presented here are rather encouraging for computations of complex flow with arbitrary geometry. The method is very robust, allowing converged solutions to be obtained at a high Weissenberg number. The results also compare favourably with those obtained with traditional finite element methods. In addition, the use of unstructured mesh allows us to handle more complex computational domain, much the same way as the finite element method. Coupled with "both-side" diffusion, which is basically an under-relaxation mechanism, the finite volume method appears to us a valid way of computing stable solutions at high Weissenberg number. ACKNOWLEDGEMENTS

This research is supported by then Australian Research Council (ARC). The calculations were performed using the facility of Sydney Distributed Computing Laboratory (SyDCom). REFERENCES

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357

10. R.R. Huilgol and N. Phan-Thien, Fluid Mechanics of Viscoelasticity: General Principles, Constitutive Modelling, Analytical and Numerical Techniques, Elsevier, Amsterdam, 1997. 11. N. Phan-Thien and R.I. Tanner, J. Non-Newtonian Fluid Mech., 2 (1977) 353. 12. R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids: Vol i: Fluid Mechanics, John Wiley and Sons, 2nd Edn, New York, 1987. 13. A.R. Davies, S.J. Lee and M.F. Webster, J. Non-Newtonian Fluid Mech., 16 (1984) 117. 14. E.B. Bagley, I.M. Cabot and D.C. West, J. Appl. Phys., 29 (1958) 109-110. 15. A.M. Kraynik and W.R. Schowalter, J. Rheol., 25 (1981) 95-114. 16. A.V. Ramamurthy, J. Rheol., 30 (1986) 337-357. 17. D.S. Kalida and M.M. Denn, J. Rheol., 31 (1987) 815-834. 18. F.J. Lim and W.R. Schowalter, J. Rheol., 33 (1989) 1359-1382. 19. J.R.A. Pearson and C.J.S. Petrie, Proc. 4th Int. Cong. Rheol., Part 3, (1965)265-282. 20. S.G. Hatzikiriakos and N. Kalogerakis, Rheol. Acta, 33 (1994) 38-47. 21. O.D. Kellogg, Foundations of Potential Theory, Dover, New York, 1953. 22. M. Renardy, ZAMM, 65 (1985) 449-451; J. Non-Newt. Fluid Mech., 36 (1990)419-425. 23. A.J. Chorin, J. Comput. Phys., 2 (1967) 12-26. 24. F.R. Phelan, Jr., M.F. Malone and H.H. Winter, J. Non-Newt. Fluid Mech., 32 (1989) 197-224. 25. H. Jin, N. Phan-Thien and R.I. Tanner, Comp. Mech., 13 (1994) 443457. 26. J.P. van Doormaal and G.D. Raithby, Num. Heat Transfer, 7 (1984) 147-163. 27. R.I. Issa, J. Comput. Phys., 62 (1985) 40-65. 28. J.Y. Yoo and Y. Na, J. Non-Newt. Fluid Mech., 39 (1991) 89-106. 29. Y. Na and J.Y. Yoo, Comp. Mech., 8 (1991) 43-55. 30. S.-C. Xue, N.Phan-Thien and R.I.Tanner, J. Non-Newt. Fluid Mech., in press (1997). 31. H.H. Hu and D.D. Joseph, J. Non-Newt. Fluid Mech., 37 (1990) 347377. 32. X.-F. Huang, N. Phan-Thien and R.I. Tanner, J. non-Newt. Fluid Mech., 64 (1996) 71-92.

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33. B. Gervang and P.S. Larsen, J. Non-Newt. Fluid Mech., 39 (1991) 217-237. 34. S.-C. Xue, N.Phan-Thien and R.I.Tanner, J. Non-Newt. Mech., 59 (1995) 191-213. 35. G.P. Sasmal, J. Non-Newtonian Fluid Mech., 56 (1995) 15. 36. X.-F. Huang, N. Phan-Thien and R.I. Tanner, J. Non-Newt. Fluid Mech., in press (1997). 37. C. Prakash and S.V. P a t a ~ , Numer. Heat Trans., 8 (1985) 259-280. 38. C. Masson, H.J. Saabas and B.R.Baliga, Inter. J. Numer. Meth. Fluids, 18 (1994) 1-26. 39. L. Davidson, Inter. J. Numer. Meth. Fluids, 22 (1996) 265-281. 40. S.A. Dhahir and K. Walters, J. Rheol., 33 (1989) 781-804. 41. H.P.W. Baaijens, G.W.M. Peters, F.P.T. Baaijens and H.E.H. Meijer, J. Rheol., 39 (1995) 1243-1277. 42. H.P.W. Baaijens, Evaluation of Constitutive Equations for Polymer Melts and Solutions in Complex Flows, PhD Thesis, Eindhoven University of Technology, Netherlands (1994). 43. G. Barakos and E. Mitsoulis, J. Rheol., 39 (1995) 1279-1292. 44. P.Y. Huang and J. Feng, J. Non-Newt. Fluid Mech., 60 (1995) 179198. 45. A.W. Liu, D.E. Bornside, R.C. Armstrong and R.A. Brown, J. NonNewt. Fluid Mech., in press (1997). 46. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics. Kluwer, Boston (1964). 47. S.A. White, A.D. Gotsis and D.G. Baird, J. Non-Newtonian Fluid Mech., 24 (1987) 121. 48. D.V. Boger, Ann. Rev. Fluid Mech., 19 (1987) 157. 49. B. Tremblay, J. Non-Newtonian Fluid Mech., 43 (1992) 1. 50. M. Viriyayuthakorn and B. Caswell, J. Non-Newtonian Fluid Mech., 6 (1980) 245. 51. M.E. Kim, R.A. Brown and R.C. Armstrong, J. Non-Newtonian Fluid Mech., 13 (1983) 341. 52. P.J. Cable and D.V. Boger, AIChE J., 24 (1978) 869. 53. R.E. Evans and K. Walters, J. Non-Newtonian Fluid Mech., 20 (1986) 11. 54. G.H. McKinley, W.P. Raiford, R.A. Brown and R.C. Armstrong, J. Fluid Mech., 223 (1991) 411. 55. F.N. Cogswell, Polym. Eng. Sci., 12 (1972) 64.

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56. J.L. White and A. Konodo, J. Non-Newtonian Fluid Mech., 3 (1977) 41. 57. K. Walters and M.F. Webster, Philos. Trans. R. Soc. London, A308 (1982) 199. 58. K. Walters and D.M. Rawlinson, Rheol. Acta, 21 (1982) 547. 59. D.M. Binding and K. Walters, J. Non-Newtonian Fluid Mech., 30 (1988) 233. 60. K. Chiba, T. Sakatani and K. Nakamura, J. Non-Newtonian Fluid Mech., 36 (1990) 193. 61. L.M. Quinzani, R.C. Armstrong and R.A. Brown, J. Non-Newtonian Fluid Mech., 223 (1994) 1. 62. E. Mitsoulis, J. Rheol., 37 (1993) 1029. 63. F.P.T. Baaijens, J. Non-Newtonian Fluid Mech., 48 (1993) 147. 64. L.M. Quinzani, R.C. Armstrong and R.A. Brown, J. Rheol., 39 (1995) 1201.

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S E G R E G A T E D F O R M U L A T I O N S IN C O M P U T A T I O N A L ASPECTS OF C O M P L E X V I S C O E L A S T I C F L O W S

Jung Yul Yoo Department of Mechanical Engineering College of Engineering Seoul National University Seoul 151-742, Korea

1. INTRODUCTION The past quarter century has seen a significant advancement in numerical simulation of complex viscoelastic flows. In the early years, the breakdown of numerical algorithms even for modest values of elasticity parameter (such as Weissenberg number) which was observed regardless of the choice of constitutive equation, descretization method and iteration method had been a longstanding problem in the study of viscoelastic flows. Joseph and co-workers [13] suggested that the problem might be associated with, among others, hyperbolicity and change of type in the governing equations, by showing that coupling the constitutive equations of hyperbolic type with the momentum equations of elliptic type for incompressible flow led to a system of mixed character. In this regard, Song and Yoo [4] argued that a type dependent upwinding scheme like the one used in transonic flow calculations might be useful, and performed a FDM simulation of the flow of an upper convected Maxwell fluid through a planar 4:1 contraction using a type dependent upwinding scheme for the vorticity equation. This resulted in an enhancement of the Weissenberg number limit to a higher value than ever, but did not quite resolve the high Weissenberg number problem. More comprehensive discussions on hyperbolicity and change of type in steady viscoelastic flows can be found in the book of Joseph [5].

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In the last decade, several coupled finite element methods have been developed for the solution of viscoelastic flow problems. They include the sub-element method of Marchal and Crochet [6], the explicitly elliptic momentum equation (EEME) method of King et al. [7], and the elastic-viscous split-stress (EVSS) method of Rajagopalan et al. [8]. These techniques enabled the researchers to develop f'mite element methods that were stable enough to obtain converged solutions at Weissenberg numbers as high as possible. It seems that the success of all these strategies resulted from applications of streamline upwinding technique to the constitutive equations of hyperbolic type. However, it is noted that these finite element methods were based on fully coupled algorithms, in which the velocity, pressure and stress were solved simultaneously. In addition, they were restricted to very simple geometries requiting modest number of elements even with supercomputers. For example, in one of the calculations reported in [6], the degrees of freedom for stresses were 14116 with a mesh containing 210 elements. In the meantime, another group of researchers have studied viscoelastic flows in terms of decoupled strategies, in which the velocity, pressure and stress are solved separately. In decoupled algorithms, the momentum equation is solved separately from the constitutive equation, which is significantly less expensive than their coupled counterparts. Decoupled technique is also a natural choice for integral viscoelastic models since it is not a simple task to devise coupled algorithms for them [9]. Furthermore, a properly constructed decoupled scheme should in general have a larger radius of convergence than a coupled scheme which often relies on Newton type iterations [10]. The numerical accuracy and computer efficiency of a decoupled algorithm depends on three aspects: (i) solution of the constitutive equation subject to a given velocity field; (ii) solution of the Navier-Stokes equation subject to a given viscoelastic stress field; (iii) the way how the two equations are decoupled. There has been a traditional difficulty associated with solving the NavierStokes equation for an incompressible flow, that is, the incompressibility constraint, or the coupling of the pressure and velocity. For an incompressible flow, the continuity equation in itself does not have an explicit reference to the pressure. Therefore, a pressure equation, which enables the segregation of the solution procedure for the pressure from that for the velocity, must be devised by some further manipulation of the governing equations. A very well-known way of devising such an equation is the SIMPLE (semi-implicit method for

363

pressure linked equations) algorithm [11] used in the finite volume methods. Another approach calls for a 'pressure correction', in which an initially guessed pressure is corrected successively at each iteration by adding the pressure correction obtained from the associated equation. This pressure correction is also common to artificial compressibility [12] and augmented Lagrangian [13, 14] techniques. In the framework of the FEM, this difficulty can be easily overcome by using integrated formulations in which the velocity and the pressure are obtained simultaneously. However, the global matrices become inevitably larger than those of the segregated formulations. Therefore, in segregated formulations, one can save more memory and CPU time since the pressure is obtained separately from the momentum equation. Some time ago, Boger [15] pointed out that, experimentally, flows in twodimensional geometries tend to become time-dependent and three-dimensional when viscoelasticity is important [16]. As was also discussed in the work of Evans and Waiters [17], concerning the lip-vortex mechanism of vortex enhancement in planar contraction flows, it is clear that any numerical simulation aimed at predicting the vortex growth in such flows should also be extended to three space dimensions and time dependent behaviour. These features of the flow have to be taken into account in the numerical simulation of such flows [18]. Recent researches pay more and more attention to computationally efficient algorithms, since the flows of recent interest are ultimately three-dimensional and time dependent [16]. Decoupled and segregated methods, which deemed insufficiently accurate when they first appeared, have been developed so that they now can produce credible results. Furthermore they remain very attractive because of significantly reduced core memory and CPU time requirement, which are crucial in transient and three-dimensional problems. The objectives of the present article are to examine the segregated formulations currently adopted in numerical simulations of viscoelastic flows and to discuss the possibilities of emerging algorithms. To avoid possible confusions in the usage of terminologies in this article, the term 'decoupled' is to be used for the decoupling of the extra stress into the viscoelastic component and the optional Newtonian component as they appear in the momentum equations (Keunings [9]), and the term 'segregatea~ is now used for the segregation of the pressure from the momentum equations. In section 2, governing equations for viscoelastic flows in conjunction with basic decoupled techniques for the extrastress are presented. In section 3, several strategies handling the hyperbolic

364

nature of the constitutive equations are discussed. The segregated formulations of the Navier-Stokes equation are treated in section 4. The possibilities of emerging segregated algorithms are considered briefly in the conclusion.

2. G O V E R N I N G EQUATIONS FOR V I S C O E L A S T I C F L O W S For an incompressible viscoelastic flow, the governing equations are the continuity, momentum and constitutive equations. Since the main concern of the present article is to take a general view of segregated algorithms in numerical simulation of viscoelastic flows, we do not intend to elaborate on the viscoelastic modelling. In this regard, we will simply adopt the 'upper-convected Maxwell model', which is one of the most widely studied differential models for viscoelastic flows. Then we are to consider the following set of equations:

V.u=O,

(1)

~gu

(2) v

r+A,~=2r/d,

(3)

where u is the velocity, p is the pressure, d is the rate of strain, lr is the extra stress, p is the density,/7 and A, are the viscosity and relaxation time, respectively, and v denotes the upper-convected derivative def'med by v

9

lr = - ~

+ u . V l r - V u r . l r - lr. V u

.

(4)

When equations (3) and (4) are taken in conjunction with the momentum equation (2) and the continuity equation (1), it is apparent that the extra-stress components have to be calculated together with the velocity components and the pressure. The concept of the decoupled method is based on splitting the extra-stress into the Newtonian part and the viscoelastic part as

r=2rld+

(5)

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By substituting equation (5) into equations (2) and (3), the momentum and constitutive equations are now written as 81/

(6)

P-~

v

v

I'E + A,l-e = - 2 ~ r/d .

(7)

The above alternative formulation was first introduced by Perera and Waiters [19-20], Mendelson et al. [21] and Crochet and Keunings [22]. This was further modified by Rajagopalan et al. [8] to construct the elastic-viscous split stress (EVSS) formulation, which is categorized as a coupled method. In this coupled formulation of the EVSS, the convected derivative of d in the Galerkin discrete form of equation (7) involves second-order spatial derivatives of the interpolating function ~, which can be eliminated by performing v an integration by parts on the term O d, with the resulting boundary integral containing the flux of d through the boundary. In a decoupled formulation, however, this complication may be avoided by first calculating r/d at node points from the known velocity field and then consistently interpolating r/d (as part of the extra stress) in the same manner as 1:~ [10]. Now, the momentum equation turns out to be the Navier-Stokes equation with a pseudo-body-force V. ~:~,so that the solution always reduces to that of a Newtonian flow when A, vanishes. The velocity and the pressure can be obtained by solving equation (6) with the continuity equation in the given stress field, and the stress can be obtained by solving equation (7) in the given velocity field, iteratively. The main disadvantage of decoupled techniques lies in the iterative procedure because Picard schemes usually used in decoupled methods are often slow in convergence, which is not even guaranteed no matter how closely the initial estimates are chosen to the solution. However the decoupled technique generally requires less core memory than a coupled method, which turns out to be more efficient for larger computations. Another attractive feature of decoupled methods is the breakup of the problem into the solution of an elliptic Newtonian-like flow with pseudo-bodyforce, and the integration of constitutive equations using fixed flow kinematics.

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Thus classical methods can be used to discretize the Navier-Stokes equation, while special techniques for the extra-stress computation may be developed to take into account the special mathematical nature of the constitutive equation involved [9].

3. NUMERICAL TREATMENT OF CONSTITUTIVE EQUATIONS IN DECOUPLED METHODS

3.1 Background The main difficulty in dealing with the constitutive equation (7) is that its type is hyperbolic. Even though this advection equation is linear in the stress with its structure being much simpler than the Navier-Stokes equation, it is still a challenge to computation because the advection equation has no built-in mechanism to smooth discontinuities in directions normal to the flow [23]. This problem is similar to that encountered in high Reynolds number flows of Newtonian fluids where high velocity along streamlines generate wild oscillations in the numerical solution. Before the advancement of the upwinding technique, the only way to eliminate these oscillations seemed to be to refine the mesh, such that convection was made not to be dominant on an element level. However, Keunings [24] showed in a study on the high Weissenberg number problem that the use of a very f'me mesh rather reduced significantly the range of the Weissenberg number for which the viscoelastic solutions could be obtained. It is because the local Peclet number, which represents the ratio of the convection to the diffi~ion, remains to be infinite from equation (7), no matter how we ref'me the mesh. Furthermore, it is reminded that one must calculate the stress closer to the comer singularity with a f'mer mesh in the case of the problem involving corner singularity, such as the 4:1 contraction flow problem.

3.2 SUPG (Streamline Upwind/Petrov-Galerkin) Method There have been several methods to overcome this difficulty, regardless of the coupled or decoupled method. In a finite element framework, the most popular approach for this problem is the Streamline Upwind Petrov-Galerkin (SUPG) method [25]. Some earlier upwind formulations for multi-dimensional problems often exhibited numerically false diffusion in the direction perpen-

367

dicular to the flow direction. The basic idea of the streamline upwind method is to add diffusion which acts only in the flow direction. Although the SUPG method has been highly successful in solving the Navier-Stokes equations for Newtonian flows, it was never introduced into solving the constitutive equations of the viscoelastic flows, until the first attempt was made by Marchal and Crochet [6] who adapted the SUPG method for their coupled method. Following Brooks and Hughes [25] and Marchal and Crochet [6], we define modified weighting function q~ as .

9 =T+ku~-v~P,

_

k =

luAl+

,eh

(8)

2

where q~ is the Galerkin weighting function, u~ and Uy are the components of the velocity vector u h at the center of an element, h~ and hy are the characteristic lengths of the element. The normalization of the velocity field is introduced so that the weighting function remains to be O(1) in regions even where the velocity field vanishes. Note that in the coupled formulation of Marchal and Crochet [6], the form of k in equation (8) is not used because it is not differentiable at u~ = U y - 0 , and one cannot calculate the Jacobian matrix with Newton type iterations. Instead, they adopted the following form:

%=

U hx)

(9)

2

Applying the modified weighting function to all terms in the constitutive equation, we end up with the consistent streamline upwind method:

q~+]T~-~-V~P ~ + , ~

2~,

=

.

An alternative method for the solution of the constitutive equation is the inconsistent streamline upwind method, in which the modified weighting function is applied solely to the advection term u. V~g of the constitutive equation:

368

~n 9 ~:e+~:E+2,q,r/d +

.V~'E d O = 0 .

(11)

Comparing (11) with the conventional Galerkin weighting form, it can be considered as if an extra term is artificially added to the Galerkin formulation. This method is equivalent to the artificial diffusion method commonly used for hyperbolic equations in the FDM or FVM. Note that the artificial diffusion method is second-order accurate in space, while conventional upwinding method is first-order accurate.

3.3 DG (Discontinuous Galerkin) Method It is well known that a cure for the numerical oscillations by introducing a certain amount of artificial diffusion into the usual upwinding technique only results in a loss of accuracy in the numerical solution. Discontinuous Galerkin (DG) method of Fortin and Fortin [26] was developed for a decoupled finite element approach to avoid smooth and stable but physically unrealistic solutions. This technique first introduced by Lesaint and Raviart [27] for the neutron transport equation was successfully applied to the numerical simulation of viscoelastic flows at high Weissenberg numbers. However, at first it was considered to be impossible to solve the steady flow problem by using the Picard iteration scheme. Therefore a time stepping scheme had to be employed, possibly because steady-state Picard's scheme does not provide information on the qualitative behaviour of the numerical solutions as Keunings [9] pointed out. Later, Fortin and Fortin [28] showed that the steady flow problem could be solved by applying the Generalized Minimal Residual (GMRES) algorithm of Saad and Schultz [29]. The value of the Weissenberg number thus obtainable was limited again and the extra-stress field showed an oscillatory behaviour. It was subsequently shown by Basombrio et al. [30] that the use of a linear interpolation for the extra stress tensor, rather than a quadratic as in Fortin and Fortin [26], significantly reduce the oscillations. However, it is quite unfortunate that for the DG method, the stress variable cannot be eliminated on the element level as, for instance, in the work of Baaijens [31] and that oscillation free solutions are only obtained at limited values of the Weissenberg nmnber. Baaijens [31] and Baaijens [32] combined the DG method with the Operator Splitting (OS) method [23], which was originally designed to decouple the

369

advection treatment and the rest of the procedure for a time-dependent problem. A shared feature of the OS and DG method is the use of discontinuous interpolations of the extra stress tensor. This allows the elimination of the extra stress variables on the element level, yielding an efficient algorithm for multi-mode fluids. More recently, Fortin et al. [33] and Gurnette and Fortin [34] have combined the DG method with the EVSS method, by which they could obtain good convergence and oscillation-free solutions for higher Weissenberg numbers. 3.4 Method of Characteristics Another method for handling the hyperbolic type equation in the decoupled finite element approach is the method of characteristics, which was first introduced in the numerical simulation of viscoelastic flows by Fortin and Esselaoui [35]. Hadj and Tanguy [36] also employed the weak formulation of the method of characteristics to compute the full convected derivative of the stress tensor and obtained solutions at a higher Weissenberg number with an Oldroyd-B fluid than that attained by Fortin and Esselaoui with a Maxwell fluid. In a recent work of Basombrio [37], the hyperbolic equations for the advective transport of stresses were integrated directly at each node using nonconservative strong formulation of the method of characteristics [38]. This method was also combined with the EVSS method by Kabanemi et al. [39] in which the hyperbolic nature of the constitutive equation was coped with using a weak formulation. 3.5 Other Methods in FDM and FVM In the finite difference or finite volume framework, the applicable methods for the hyperbolic type equations are the upwinding techniques and the artificial diffusion techniques. Hu and Joseph [40] used a conventional, first order upwind scheme for the constitutive equations. Choi et al. [41 ] and Yoo and Na [42] used the deferred correction method, which is an upwind corrected scheme involving an artificial diffusion term to attain second-order accuracy and unconditional stability, while Xue et al. [43], and Huang et al. [44] used an artificial diffusion method only. Sato and Richardson [45] used a finite volume approach for the constitutive equation, while a finite element approach was used for the momentum equation. They introduced the FCT (flux-corrected transport) concept into solving the constitutive equations, since numerical

370

schemes for advection should be TVD (total variation diminishing) in order to obtain stable results. Recently, Mompean and Deville [18] applied, so-called, QUICK (Quadratic Upstream Interpolation Scheme for the Convective Kinematics) to the constitutive equation, which was proposed by Leonard [46]. For a uniform grid system this scheme is of third-order accuracy, and for a nonuniform grid system it is of second-order accuracy [46]. The staggered grid system has been widely used in the FDM or FVM approaches, where the pressure and the stresses are located in the center of the control volume and the velocities are located on the faces of the control volume as shown in Figure 1. The staggered grid is first designed to eliminate the checkerboard pattern in the pressure field for the Newtonian flows of the FDM or FVM, which is quite similar to the mixed f'mite element methods satisfying the Babuska-Brezzi compatibility condition [47, 48] on the spaces for the velocity and pressure. This formulation of the viscoelastic flow calculation offers a major advantage in avoiding the stress calculations, for example, at the re-entrant corner of the 4:1 contraction flow problem, while the method correctly preserves the flow physics in its neighboring control volumes.

I

u

I L .

p,l: .

.

.

I _1

Figure 1 Staggered grid in the computational domain In recent works of Saramito [49, 50], the original finite volume element (Figure 2) was proposed for the mixed FEM which was somewhat similar to the staggered grid in the FVM. It is noticeable that the staggered formulation is relatively difficult to discretize the governing equations and it nearly cannot be used for an unstructured grid mostly because one cannot define a control volume consistently for arbitrarily shaped meshes. Therefore the non-staggered

371

grid system in which all dependent variables are located at the grid points becomes more and more popular in the Newtonian flow calculations. In the unstructured FVM approach of Huang et al. [44], they used a non-staggered formulation, which corresponds to equal-order element in the FEM. 19

0

E

)

9

()

l

0

E

[-] ~2

Ou 0

/

"r "t'22,p

Figure 2 Modified finite element

4. SEGREGATED FORMULATIONS IN THE NAVIER-STOKES EQUATION 4.1 Pressure Equation One of the complications encountered in solving the Navier-Stokes equation is that it includes the pressure gradient term, whereas there is no independent equation for the pressure. Furthermore, the continuity equation does not have a dominant variable in it. Mass conservation is a kinematic constraint on the velocity field rather than a dynamic one. In the FEM approach, this difficulty is simply avoided by using an integrated formulation in which the momentum and continuity equations are solved simultaneously. However, this formulation requires a large memory and computing time. An alternative way to overcome this difficulty is to construct a pressure equation in order that the pressure obtained from it intrinsically satisfies the continuity equation. In this segregated velocity-pressure formulation of the Navier-Stokes equation, velocities and the corresponding pressure field are computed alternately in an iterative sequence, in contrast to the integrated formulation. The most attractive aspect of the decoupled approach in the numerical analysis of viscoelastic flows is that it works very well with the segregated strategy for the Navier-Stokes equation which requires a much smaller memory and computing time.

372

This segregated formulation has been one of the most commonly used approaches in the finite difference procedure [51], since it does not solve the momentum and continuity equations simultaneously as in the integrated formulation of the FEM. For the purpose of solving them simultaneously, Song and Yoo [4], Choi et al. [41] and Sasmal [52] applied the stream function-vorticity formulation to viscoelastic flow problems. However this approach cannot be extended to three dimensional flows, and it needs assumptions on stream function and vorticity in treating the boundary conditions. Derivation of a pressure equation is based on combining the momentum and continuity equations. A pressure equation of Poisson type can be obtained by taking the divergence of the momentum equation (2): = v.

+v.

.

(]2)

Sato and Richardson [45] used this formulation for their viscoelastic flow calculation, in which MAC (Marker And Cell) method of Harlow and Welch [53] was incorporated into a finite element solution. A similar procedure to this formulation is also used in the three-dimensional FVM calculation of Mompean and Deville [ 18].

4.2 SIMPLE (Semi-Implicit Method for Pressure Linked Equations)-like Algorithm The well-known approach in the FVM for the pressure equation is the SIMPLE algorithm of Carretto et al. [54], which can be simply described as follows: (i) Predictor stage In this stage a guessed pressure field is used, usually denoted by p*, and by solving the momentum equation the velocity field u* is obtained. (ii) Corrector stage To obtain the velocity and pressure fields which satisfy both the continuity and momentum equations, the corrections u' and p' are obtained in this stage, and then u = + u p = p + p' This stage is repeated by treating the corrected pressure p and velocity u as a newly guessed pressure p* and velocity u* until a converged solution is obtained.

373

However, the SIMPLE algorithm does not converge rapidly because the velocity correction is obtained improperly. Its performance depends greatly on the size of the time step, o r - for steady flows - on the value of the under-relaxation parameter used in the momentum equations. It was found by trial and error that convergence characteristics can be improved by controlling the under-relaxation. Thus some modifications to this conventional SIMPLE algorithm were proposed. They are SIMPLER (SIMPLE Revised) of Patankar [11], PISO (Pressure-Implicit with Splitting Operators) of Issa [55] and SIMPLEC (SIMPLE Consistent) [56] which do not need under-relaxation of the pressure correction [57]. Recently, the FVM approaches based on these algorithms were adopted for viscoelastic flow computations, because of their attractive features of space and time savings, as well as their numerical stabilities. For steady flow problems, Hu and Joseph [40], Yoo and Na [42] and Huang et al. [44] used SIMPLER algorithm, and its modification is used in the work and Xue et al. [43], while Gervang and Larsen [58] used PISO algorithm and Luo [59] used classical SIMPLE algorithm. Note that in these algorithms the pressure and velocity fields obtained at the end of each time step (or iteration step for steady flow problems) do not satisfy one and the same momentum equation so that for time-dependent computations, sub-iterations are necessary at each time step, or very small time steps must be chosen, which accordingly degrades the efficiency of these algorithms.

4.3 Augmented Lagrangian Method In the frame work of the FEM, there is an alternative approach for the segregation of the pressure, that is augmented Lagrangian method [13, 14] which was derived from the variational formulation of the Stokes flow problem and turned out to be a powerful iterative method for solving it. Using the variational formulation of the Stokes equation, one can obtain the symmetric positive-definite functional equation, which is equivalent to the minimization problem. For the solution to this minimization problem, there have been many variants from the following very simple algorithm known as Uzawa algorithm [13, 14]-

P St - V ' ~ = - V p ' ,

(13)

374

p"+~ =p" - e V . u "

,

(14)

where m denotes the present iteration step and e is a step length for the iterative procedure. Hadj and Tanguy [36] and Kabanemi et al. [39] used this classical Uzawa algorithm for the momentum and continuity equations. Fortin and Fortin [26] and Gu6nette and Fortin [34] solved the Stokes flow problem by this algorithm combined with a condensation method [60] that eliminates the velocity degrees of freedom associated with the center of each element and the pressure gradients. This is achieved by using the continuity equation at the element level, thanks to the discontinuous approximations of the pressure. This method greatly reduces the number of unknowns, resulting in a very efficient Stokes solver. However, the convergence properties of this algorithm are not so good because the determination of e is not easy so that e is usually fixed with the iterative procedure. The way around this is quite simple but requires a few developments [13, 14, 23]. Fortin and Fortin [28] showed that an introduction of the preconditioned residual for the dependent variables enhance the convergence, but the computations still had to be performed in double precision. In fact, the algorithm (13)-(14) can be interpreted as a gradient algorithm applied to the minimization of the functional. With this interpretation in mind it is natural to seek the more effective iterative methods for the minimization of quadratic functionals, such as the steepest decent method, the minimum residual method or the conjugate gradient method. In these modified methods, the conjugate gradient method is especially attractive for solving quadratic problems because theoretically it converges in a finite number of iterations and moreover, it leads to quadratic convergence in the general case. These algorithms require the additional presence in memory, but this increased memory requirement will be justified if the automatic determination of the step length e leads to a very clear improvement in the speed of convergence compared with original algorithm (13)-(14) [ 13, 14, 23 ].

4 . 4 0 S (Operating Splitting) Scheme Clearly, the augmented Lagrangian method is not applicable in itself to the Navier-Stokes equation containing the non-linear advection term for the velocity. As mentioned in section 3, the Operator Splitting (OS) scheme has been used in order to decouple an advection treatment and the rest of the procedure

375

for a time-dependent problem. This method is also called the alternating direction implicit technique [13, 14, 23, 61], which separates the main two difficulties in the computation of the Navier-Stokes equation: the non-linearities in the momentum equation and the incompressibility constraint. In the conventional numerical simulation of viscoelastic flows, the non-linear term is usually neglected. In an attempt to answer some of the unresolved questions in their earlier work, Evans and Walters [17] investigated aqueous solutions of polyacrylamide and concluded that a lip-vortex mechanism may be responsible for vortex enhancement for all planar contraction ratios. These experimental studies consist of observations and measurements of the phenomena resulting from the change of the flow rate of the same viscoelastic fluid. Therefore, in order to simulate such flows numerically, the inertia must be retained in the momentum equations. Glowinski and Pironneau [23] proposed the 0-scheme based on the abovementioned OS method to treat the nonlinearity and the incompressibility of the Newtonian flows. It consists of three step time marching technique such that (i) First step gl n+O _ II n

P

Ok

_aV..g,+o +Vpn+O =fn+O ..[_~V. ,[,n (/gn .V)/gn ,

V.u "+~= 0 ,

(15) (16)

(ii) Second step un+l-O _ 11n+O P

(1 - 2 0 ) k

- f l Y . Zn+l-O "b (U n+l-O" V ) U n+l-O : fn+l-O + O~V..[n+O _ V p " + ~ ,

(17) (iii) Third step

P

U n+l

ign+l-O Ok _ o ~ V . ,[n+l q_Vpn+l = f n + l + ~ V . , [ n + l - O

_

V-/g n+l = 0 ,

(lln+l-O.v)lln+l-O ,

(18) (19)

376

where a = (1- 20)/(1-0)and/3 = 0/(1-0), and 0 can be selected by a numerical experiment. One can see that the first and third steps are the Stokes-like problems in which the nonlinear terms are eliminated by the second step. Thus the Uzawa type algorithm can be used in the first and third steps and the nonlinear term is treated in the second step by using the method mentioned in section 3, such as the upwinding technique. This scheme is of order two in time, and allows one to also compute steady solutions efficiently. Glowinski and Pironneau [23] introduced a preconditioning stage to the Uzawa type conjugated gradient method for an acceleration of the convergence in the Newtonian flow computations. In the case of the Stokes flow computation, the second step can be skipped since it needs not the advection treatment. However, when the 0scheme is combined with the constitutive equations of viscoelastic flows, the second step can not be skipped because the advection term in the constitutive equation is always involved. Recently, Saramito [49, 50] applied this 0-scheme to the calculation of the viscoelastic flows, but he neglected the inertia in the momentum equations, while Luo [62] extended it to the case of including the inertia in the momentum equations. Until now, no one used the 0-scheme for time-dependent flow problems. As Szady et al. [63] mentioned in their recent work of coupled approach, the 0-scheme can be efficiently utilized for the three-dimensional transient flow where it can save much CPU time and guarantee numerical stability.

4.5 Fractional Step Method The alternating direction method is closely related to the fractional step method [14]. Actually the alternating direction or fractional step method has been extensively used for quite a long time for solving time-dependent partial differential equations. Concentrating particularly on the Navier-Stokes equation for incompressible viscous fluids, the first works were performed by Chorin [64] and Temam [65]. However, the fractional step method for the Navier-Stokes equation had not been used extensively until successful result of its application to the three-dimensional unsteady turbulent flow calculation was reported by Kim and Moin [66], which was later used for the Direct Numerical Simulation (DNS) [67, 68]. A typical formulation of the fractional step method is as follows"

377 (i) First step

P At

+

3(un'V)un-(un-l"V)un-1)=-2

(ii) Second step V 2~bn+l = P V.u At

(21)

(iii) Third step

P

Un+l --fl _v~n+l At = '

(22)

where r is a pseudo-pressure and V~b#+~- Vp #+~+O(At2). One can see that the third step is derived from taking the divergence of equation (21) and requiting u n+~ to satisfy the continuity equation. There are many variants of the fractional step method such as a fully implicit fractional step method [69], due to a vast choice of approaches to time and space discretization, but they are all based on the principles described above. The fact that the pressure is segregated from the velocity in the fractional step method was utilized in the finite element analysis of the incompressible Navier-Stokes equation by several researchers [70-74]. Because the fractional step method does not include any approximation procedure, such as adopted in the SIMPLE algorithm based finite element methods [75-78], this approach is more accurate than the SIMPLE algorithm based approach for the same grid. Furthermore, in the fractional step method, the coefficients of the pressure equation are fixed and the inverse matrix of the pressure equation does not need to be calculated at every time step like the SIMPLE algorithm based approach [79]. Carew et al. [80] and Baloch et al. [81] used this fractional step method combined with the Taylor-Galerkin method for the advection treatment. The details of that scheme is described in the work of Hawken et al. [82]. Recently, Baloch et al. [83] were able to carry out a numerical simulation of a three-dimensional viscoelastic 40:3:3 expansion flow, using the Phan-Thien and Tanner

378

(PTT) constitutive fluid model, where they showed the streamlines in the center plane of the 3D expansion and made comments on the differences in the vortex development from the 2D simulation. Recently, Sureshkumar et al. [84] applied the fractional step method of Kim and Moin [66] to the DNS of the turbulent channel flow of a polymer solution by spectral method. The major difference between the 0-scheme with preconditioned conjugate gradient (PCG) method (15)-(19) and the fractional step method (20)-(22) is how to advance one time step. The two methods are both of order two in time and the Poisson equations for the pressure are involved respectively. However, sub-iterations for satisfying the convergence criteria of the momentum and continuity equations, that is, equations (15)-(16) and equations (18)-(19), are needed in the PCG method of the 0-scheme, while the Poisson equation is solved just one time at the second step (21) in the fractional step method. It is well-known that the Poisson equation for the pressure is the most time consuming one in the segregated formulation of the Navier-Stokes equations. In view of this, comparing with the fractional step method, the 0-scheme may be inefficient, particularly for three-dimensional problems, despite the fact that the Poisson equation is solved in the discrete pressure space whose dimension is much smaller than the dimension of the discrete velocity space in mixed FEM formulation. Note that the 0-scheme with Uzawa type PCG method cannot avoid the Babuska-Brezzi condition [13], and thus it must be discretized via the mixed formulation where the discrete pressure on a grid is defined twice coarser than the one used to discretize the velocity. On the other hand, the fractional step method can avoid the compatibility condition as shown in the recent work of Choi et al. [79] which combined the SUPG with an equal-order finite element formulation. This work was extended to the numerical analysis of a Bingham plastic [85]. However, the boundary condition treatments for the intermediate velocity ti and the pressure in the fractional step method is somewhat unclear and some further assumption is needed, while such an assumption is not needed in the 0-scheme.

5. CONCLUDING REMARKS Recently, the streamline upwind (SU) scheme for the control volume finite element method (SUCV) [86] has been proposed. When we consider that the

379

FEM in conjunction with the SU technique gives satisfactory results in analyzing viscoelastic flows, adopting SUCV into the existing algorithm based on the FVM seems to be promising. Furthermore, in the last decade the segregated finite element formulations [70-79, 82] have been steadily developed, which combine the merits of both the FEM and segregated FVM using SIMPLE-like or split methods. Therefore, future research efforts implementing these segregated FEM in the numerical simulation of viscoelastic flows are also highly recommended. It can be said that the earlier difficulties in the high Weissenberg number problem have been now overcome by several coupled formulations such as a sub-division method [6], EEME [7] and EVSS [8] methods, which can obtain the converged solutions at Weissenberg numbers as high as possible. In the meantime, another group of researchers using a decoupled and segregated formulation showed that they can also handle the high Weissenberg number problems. In particular, some of those segregated methods utilized special techniques used in coupled methods for hyperbolic type constitutive equations. The new trend of recent researches of a viscoelastic flow computation is the analysis on unsteady and three-dimensional flow including the inertia [16] , since those flows are related to the most interesting phenomena of nonNewtonian fluids such as the lip-vortex growth or the turbulent drag reduction by polymer additives. Therefore fully decoupled, segregated formulations of viscoelastic flows in the FDM, FVM and FEM seem to be more and more attractive nowadays due to their capability of handling those problems as well as their cost-effective properties.

ACKNOWLEDGEMENT I would like to express my deepest thanks to Mr. Taegee Min for his invaluable efforts in preparing this manuscript.

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385

CONSTITUTIVE THEORY

EQUATIONS FROM

TRANSIENT NETWORK

C.F. Chan Man Fong and D. De Kee

Department of Chemical Engineering, Tulane University, New Orleans, LA 70118, USA

1. I N T R O D U C T I O N An understanding of the flow properties of non-Newtonian fluids is important in many areas of application. To solve flow problems, it is required to introduce a constitutive equation that relates the stress tensor to various kinematic tensors. The determination of a constitutive equation which describes adequately the rheological properties of a fluid under all flow conditions is one of the central problems in rheology. So far constitutive equations have been developed based on either continuum mechanics or on the molecular structure of the fluids. In the continuum approach, no explicit consideration is given to the molecular structure of the material. A relationship between suitable dynamic and kinematic variables is postulated. The conditions that this relationship has to satisfy were stated by Oldroyd [ 1]. A suitable dynamic variable is the extra stress tensor v and a suitable kinematic variable is the relative right Cauchy-Green tensor C t. The tensor C t relates the square of the relative distance between two arbitrary material points at two different times, one at the present time t and the other at a past time t'. The assumption that __zdepends on the history of _C t_ from time t' - - ~ to the present time t can be expressed mathematically as

1;-

t ~ t ! "-" ~

(Ct) oo

(1)

386

where _~ is an isotropic tensor-valued functional. Equation (1) is the constitutive equation of a simple fluid. The constitutive equation for a Newtonian fluid is 3C (2)

=

t' =t

where 770 is the constant viscosity. Various simplified forms of equation (1) have been proposed and are discussed in [2-5]. In the molecular approach, the molecular structure of the fluid is taken into consideration. In modeling a polymeric material, we first represent the polymer molecules by mechanical models. We then introduce a probability distribution and an evolution equation for the distribution. Finally we calculate the average of all quantities so that a relationship between the macroscopic quantities can be obtained. Three models, namely the bead-rod-spring model, the transient network model, and the reptation model, have been popular among rheologists. We shall discuss the development of the transient network model after a brief description of the other two models. 2. B E A D - S P R I N G M O D E L This model was proposed to describe dilute polymer solutions. The polymer molecule is idealized as a dumbbell consisting of two beads, each of mass m, joined by a spring. The beads are labeled 1 and 2; their vector positions are r l a n d r 2 and R = ( r 2 - r l ) is the vector joining bead 1 to bead 2. The polymer solution is modeled as dumbbells suspended in a Newtonian fluid of constant viscosity r/s. We introduce a d i s t r i b u t i o n f u n c t i o n F ( r 1 , r 2, trl, .7 2, t ) . such . that .F ( r 1,. r 2, 71 . , 7 2, .t ) dr .l d r 2 d r 1 d r 2 is the number of dumbbells in the position range velocities in the range r'i

t o t~i +

F__i t o F_.i + d r i

and the beads have

dT_i (i = 1, 2). It is customary to write F as a

product of a configuration distribution

!/'t(rl,r2

,

t)

and

a velocity

387 distribution ~ (F__'I, /~2) which is often assumed to be Maxwellian. If the solution is homogeneous, 7~ can be written as (L1, r2, t) = n ~ (_R, t)

(3)

where n is the number density of dumbbells. The conservation of the number of dumbbells leads to an equation of continuity of the form (Bird et al. [6]) 3t

- - ~_R -It ~ 1 1 V

(4)

where ~ 11denotes an average in the velocity space. The forces acting on bead i are:

(a)

the hydrodynamic f o r c e Ff h) which is assumed to be proportional to the difference between the average bead velocity and the velocity of the solution at that point. Assuming that the velocity of the solution is not affected by the other beads, _F}h~ can be written as

_F{h) = -~ {I]~i ]]-Vi}

(5)

where ~" is a constant and v i is the velocity of the solution at F_.i . (b)

the Brownian force _F~B~which is assumed to be of the form F{ B)-- - k T ( ~ 2 n W / ~ r i )

(6)

where k is Boltzmann's constant and T is the temperature.

(c)

the intramolecular force F~ ~ and, in the present model, it is the force due to the spring. The forces on the two beads are equal and opposite and a connector force F (c~ can be defined as

388

(7)

F (c)- _F~*)= -F~ *)

In the absence of external forces and neglecting inertia terms, the equations of motion for the beads are --~ {[I i l ]] --V1} -- kT (~)2nW/arl) + _F(c)- 0

(8a)

--~ {[I i 2 ]] -- V2 } -- k T ( O 2 n U d / O L 2 )

(8b)

- _F (c) -

0

On subtracting we obtain {[I ll~]] + (v 1 - v 2 ) } + 2kT (~)2n~/~)_R) + 2F(C) = 0

(9)

By considering the flow to be homogeneous, v 1 - V2 can be written as V_.I--V 2 -- - _ g ' R

(10)

where L is the velocity gradient. Combining equations (9, 10) yields ~1I ~11 = ~ _L- _R- 2kT

(a,env/a_13)-

2F (c)

(11)

Substituting equation (11) into equation (4) yields m

9

~t

(L-R_) ~ - ( 2 k T / ~ ) ( ~ / ~ _ R )

- 2F(C)~/~I }

(12)

The total stress/7 for the solution can be written as =l-=I = 1-=I__s + FI_p_

(13)

where =/-/sand Hp are respectively the contributions from the solvent and the polymer. The solvent is a Newtonian fluid and _/-/s is given by Us - Ps---- ns ~

(14)

389

Ps is the pressure associated with the solvent, 6 is the unit tensor, and

where

is the rate-of-strain tensor. The contributors to __/-/pare the connector, ~___p, (c)/7 and the beads,

--(b) l=lp . It is

shown in Bird et al. [6] that __/_/(pc)and __/_/(pb)are given by (c) lip= = - n < R F =l-=Ip(b) =

(c) )

(15a)

2 nkT 8

(15b)

where < > denotes the average over the configuration space. Combining equations (13, 14, 15a, b) yields __H -

Ps ~ - rls "~ - n < R F (c)) + 2 n k T ~

(16)

It has been shown (Bird et al., [6]) that at equilibrium n ( R F~C)>0 = n k T ~

(17)

From equations (16, 17) we deduce that __H- (Ps + n k T ) ~

+~=

(18a)

~_ = -rls T - n + n k T 8

(18b)

Equation (18b) is the Kramers expression for the stress tensor. To calculate

where n is the number of segments.

(42)

400

To evaluate
we need

to solve for 7~ from equation (40) and this

implies that we need to postulate c and ,e. In the Lodge model, the following assumptions are made c = W0/~

(2, is a constant)

(43a)

2 = W/~

(43b)

_F(c) = H_R (H is a constant)

(43c)

where ~0 is the value of 7~ at equilibrium. The choice of the same constant ~ in equations (43a, b) ensures that at equilibrium there is no net loss or gain of segments. Substituting equations (41, 43a, b) into equation (40) yields OWOt =

~R~ (L= 9RW)_ + ~ ( W 0 - W)

(44)

Equation (44) seems to be quite different from equation (12), however if equation (44) is multiplied by R R and integrated over the R_-space, we obtain (Bird et al. [6]; Chan Man Fong and De Kee [15]) V

= I((RR>

-)

< R R > o = J _8

(45a) (45b)

where J is a constant. Combining equations (42, 43c, 45a, b) yields V

H+LH

= -nil J8

(46)

Noting that

FI = P ~5 + "c

(47a)

z_ = 0

(47b)

at equilibrium

401 V 8 - - ~

(47c)

we deduce from equation (46) that V "c + )~'c - - r l "~

(48)

where 77 (= ~, n H J) is a constant. Equation (48) is the constitutive equation of an upper convected Maxwell fluid (Carreau et al. [4]) and an integral form of equation (48) is

- -

m (t - t') C t (t') dt'

m (t - t') = (yl/~ 2) exp [-(t - t')/~,]

(49a)

(49b)

We note that in this case, as in the bead-spring model, we have been able to derive a constitutive equation without having to solve for 7~. The time constant ~, for the network model is empirical whereas the time constant ~,1 in the beadspring model is related to the structural parameters of the model (equation 22). The two models yield identical constitutive equations if the solvent viscosity is negligible. For clarity we have so far considered the case of a single time constant ~,. We can generalize from one time constant to a multitude of constants ~p. We can associate /~p with the different ages of the junctions (Lodge [16]) and to each ~p corresponds a __rp. Each __rp satisfies equation (48) and =r (= ~ __Vp) is p given by equation (49a) with m ( t - t') defined by m (t - t') - ~ (T~p/)~ 2p) exp [ - ( t - t' ) / ~ p ] P

(49c)

The Lodge model can describe many of the p h e n o m e n a associated with linear viscoelasticity (Carreau et al. [4]) but it does not predict shear thinning as observed in viscometric flows of polymeric fluids. One way of overcoming this shortcoming is to allow c and ,g to be functions of a macroscopic variable, such as an invariant of the shear rate, and this is considered next.

402 4.2. Shear Rate Dependent Models Several authors [15, 17, 22] have assumed that c and ,e are functions of the second invariant of the shear ~ defined by = # (1/2) trl~ 2

(50)

We now consider the multimodes model proposed by De Kee and Carreau [21 ]. Corresponding to the p-mode, the functions cp and ,gp are assumed to be Cp(t) = Lp [~t(t)] Wpo,

~ p(t)

- tlJp/Xp [~(t)]

(51a, b)

where ~up0and ~p are the distribution functions associated with the p-mode at equilibrium and at time t. Substituting equations (5 l a, b) into the balance equation [equation (20)] for the p-mode, multiplying the resulting expression by R R and integrating over the R_-space, we obtain

v < R R) p = Lp =8- < R R)p/~p

(52a)

Lp

(52b)

"

-

(4rt/3)

R4LpdR

Integrating equation (52a) yields < R R)p =

f~oo

{ Lp exp [

--

ftt'

dt"/1;p(t")] C t 1(t') } dt'

(53)

It is assumed that L p and "t'pcan be written as

Lp = [Tlpfp('~)]/~p,

"l;p = ~,pgp('y)

(54a,

b)

where r/p and Z p are constants and have dimension viscosity and time respectively.

403

Following the usual procedure, it is deduced that the extra stress tensor T can be written as

= -ffoo

m (t, t', 3;) Ct 1(t') dt'

(55a) '

YI pfp [~ (t')] exp

m ( t , t', 3~) - Z

2 )~p

ftt

dt"

(55b)

~,pgp [~(t")]

The functions fp and gp cannot be deduced from the model and have to be prescribed empirically. The following assumptions are made (De Kee [23]) fp = exp [-~tp (3 - 2c)] , go = exp [-~tp (c - 1)], fp = gp = 1, where c and

p - 1, 2, ..., k

(56a) (56b)

p - 1, 2 .... , k

p = k + l , k+2, ...

(56c)

tp are constants.

Equations (56a to c) imply that there are various types of junctions, the rates of loss and creation of the first k types are shear-rate dependent and the remaining types are shear-rate independent. Equations (55a, b) have been found to be adequate to describe the rheological properties of polymeric systems and the predictions of equations (55a, b) compared to experimental data will be examined later. Other forms of fp and gp, usually as rational functions of ~, have been proposed and are discussed in Carreau et al. [4].

4.3. Stress Dependent Models Kaye [24] assumed the rates of loss and formation of junctions are functions of the invariants of the stress tensor. He proposed that cp and ,gp are functions of Q1 and Q2 and they are defined by Q1 -

12lx-212~

'

Q2 = 213lx - 9 I

lx

I 2x + 2 7 I

3x

(57a, b)

404

Ilz - tr ~ ,

I2z = (1/2) [(trx=)2 - try=2] ,

I3x - det ~

(57c, d, e)

Note that I 1T, 12v, and 13 r are the invariants of __vand Q1 is positive for all '~"~

Equations (5 l a, b, 54a, b) are now replaced by Cp(t) = Lp [Ql(t), Q2(t)] W0 ,

2p(t) = Wp/1:p [Ql(t), Q2(t)]

(58a, b)

Lp - [lip gp (Q1, Q2)] [ )~p,

"l:p - ~,p gp (Q1, Q2)

(59a, b)

The constitutive equation is given by equation (55a) with the memory function defined by

m (t,t',Q1, Q2) - ~ P

I

TlPgP[QI(t')' Qz(t')] exp ~2 P

Iftt

dt" ~pgp [QI (t,,), Q2(t,,)]

tl

(6O) Kaye [24] considered one mode only (p = 1) and assumed g l to be a linear function of Q1 and independent of Q2. In a viscometric flow, this model predicts a viscosity which decreases with increasing shear rate and the ratio of the first normal stress difference to the square of the shear stress is constant. Phan-Thien and Tanner [25], Phan-Thien [26], and Tanner [27] have assumed that the rates of loss and creation of junctions are functions of p and (IRl2)p0 respectively. Since the segments are assumed to be Hookean (=v is proportional to < R R>),< IRl2)p is proportional to tr v__p. Their model is a stress dependent model. In addition they have assumed that the deformation is non-affine and their model is discussed in section 4.6. The Marrucci model [28] was proposed as a modification of the upper convected Maxwell model but it has been shown that it is derivable from the network theory (Jongschaap [29]) and is of the stress dependent type. Acierno et al. [28] have also introduced a structural parameter related to the segment density in their model and we shall discuss these models in section 7.

405

4.4. Strain Dependent Models A strain dependent model was proposed by Wagner [30]. He assumed the rate of creation Cp to be constant and the rate of loss ,s to depend on 11 -1

(= tr C_t ) and I 2 (= tr C_t ) . Further he proposed the loss of junctions to be due to the Brownian motion (a constant rate of loss) and to the deformation. Instead of equations (59a, b) we now have Lp

=

l]p/~,p

~p = ~,p + g (I1, I2)

,

(61a, b)

where g (associated with the loss of junctions due to the deformation) is independent of p (the type of junction). The constitutive equation is given by equation (55a) and m can be written as m (t,t , ,I1,I2)

2 = E [ (lqp/~p)

exp

P

= h (II,I2) Z

2 ('qp/~p)

IItt

e_(t_t, )/~,p

g

at" -[Ii(t")i I2(t")]

t

(62a)

(62b)

P

The damping function h is defined by

h (I 1' 12) -" expIftt'

dt" g (I 1' 12)

(63)

and is to be determined empirically. Based on relaxation data, Wagner [30] proposed the following form for h h = exp [-[3 4t~ I 1

+ (1--t~)

12 - 3 ]

(64)

where a and fl are positive constants. Wagner and Stephenson [31] added a further restriction on the function h. Equation (64) is valid only when the deformation is a non-decreasing function in time. If the deformation is a decreasing function of time, h is taken to be the minimum value of h over the relevant period. Physically this means that junctions which are lost over a previous increasing period of deformation are lost forever.

406

Comparing equations (55a, 62a) with equations (35 a to c), we observe the Wagner model is a special case of the Doi-Edwards model or the K-BKZ model. 4.5. Y a m a m o t o

Model

In all the transient network models considered so far, the functions c and A? depend on the macroscopic variables and it was possible to deduce a constitutive equation in a closed form. Yamamoto [14] proposed that the rates of creation and loss of segments depend on the end-to-end vector R. It is assumed that c and ,8 can be written as c -

fl (_R) q J 0 ,

~

-

f2(-R)~I j

(65a, b)

Since at equilibrium c = 2 , we deduce from equations (40, 65a, b) that 3q' ~)t

-

~)R

(R W) + f ( R ) ( W o - W) -

(66)

where f = fl - f2If we proceed as in the previous cases, that is by multiplying equation (66) by R R and integrating over the configuration space, we are led to evaluate r/s. This is in accord with our interpretation that q= represents the viscosity of the solvent and the stray and dangling segments.

420

From equation (91a) it is seen that q is a decreasing function of ~ and to predict shear thickening we can use equation (81). Chan Man Fong and De Kee [ 15] have assumed that k~ f l - kl f2 - kc d~/m

(94)

where d and m are constants and ~ is the dimensionless form of ~. Combining equations (81, 87, 94) yields 1"1 = 1"10 (1 + 2~kcd~/m) ] (1 + d~/m) ~

(95a)

= 2r10 [~ + ~ (1 + 2 ~, k~ d ~/m)] ] (1 + d ~ m) -- 1 /

k c

(95b)

(1 + d ~/m)

(95c)

where 7"/0and ~ are constants. Figure 11 compares the values of q and ~l given by equations (95a, b) with experimental values of a partially hydrolyzed polyacrylamide solution containing 2 g/2 of NaC1 reported by Ait-Kadi [50].

0.5 .-,.. tO

i

--2

,,

g.

,,

\A\

/

- 0.5

- 0.2

g. --3-

-0.1 I

0.1 I0 ~

,

, I,

,,,I

I

,

, I ,,,,I

I01

I

,

,

0.05

I0 z

Figure 11. Plot of viscosity (A) and primary normal stress coefficient (A) versus shear rate (~) for a partially hydrolyzed polyacrylamide solution containing 2 g/2, of NaC1. Model parameters are: 7/0- 0.50 Pa-s, kc = 0.040 s -1, ~ = 0.016 s, d = 0.13, m - 0.95.

421

6.2. M u l t i s t e p Flows In section 6.1 we have considered the case of a constant shear rate being applied or removed at time t = 0. It is also possible to apply finite increments of shear rates and measure the corresponding stresses. Xu et al. [51, 52] have tested the De Kee-Carreau model for various multi-rate-step flows. Figure 12 shows the calculated and the measured shear stress for a PDMS sample (a viscoelastic standard material provided by Rheometrics) in a concave step flow. In this flow a constant shear rate ~ is applied at t = 0 and is maintained until the steady state is reached. The shear rate is then halved and is kept at this value (~/2) for a finite time after which it is increased to its original value (~5). When the steady state is reached, the shearing is stopped.

8

~6 '0 4

_

2

ol 0

o

1 20

o

I 40

o

o

o

?

o

I 60

I 80

I00

120

140

t(s)

Figure 12. Comparison of model predictions ~ : equations (55a to 56c) with shear stress data (o) for concave steps with different levels. Upper Curve" ~ - 0.5, 0.25, 0.5, 0.0 s-1. Middle Curve: ~ - 0.25, 0.125, 0.15, 0.0 s-1. Lower Curve: ~ - 0.125, 0.06, 0.125, 0.0 s-1. The calculation is based on the measured spectrum from dynamic data for the PDMS sample with c = 1.3 and f0 = 1.45 for all steps.

422 Figure 13 compares the theoretical and experimental values of "Cyx for the same material in a reverse shear rate flow. In this flow the fluid is subjected to a constant shear rate ~ at time t = 0 and after the steady state is attained the shear rate is reversed in direction but equal in magnitude (-~). This cycle is repeated.

10"

5 x 10 3

A o O. I

-5xlO

3

0

_

I 20

I 40

I 60

I 80

t

I00

120

I 140

(s)

Figure 13. Shear stress for reversed shear steps. Model predictions equations (55a to 56c). The model parameters are those of Figure 12. It can be seen in Figures 12 and 13 that the De Kee-Carreau model adequately predicts the response of the material. Details of calculations and experimental procedures are given in Xu et al. [52]. 7. D I S C U S S I O N In this chapter, constitutive equations have been derived in a fixed coordinate system. The constitutive equations can also be deduced in a convected (body) coordinate system and via an energy method. Lodge [ 16, 53] has discussed the merits of adopting the convected coordinate system in the formulation of constitutive equations.

423

Several constitutive equations based on structural kinetics have been shown to be related to the transient network theory. The Marrucci model mentioned in section 4.3 is one of them. The other models are discussed in De Kee [54]. These structural equations can be quite useful at describing the rheological properties of complex materials. De Kee and Chan Man Fong [55] have explored the capability of a structural equation in various flow situations. It is shown in sections 6.1 and 6.2 that the De Kee-Carreau model, a shear rate dependent model, can predict satisfactorily the rheological properties of polymeric systems. However, the shear rate dependent models have been objected to on the ground that they do not recover the linear viscoelastic behavior in small amplitude oscillatory flows. This objection can be refuted by noting that when the strain is small, the creation and loss of junctions are due to Brownian motion [6, 22, 31] and are independent of the imposed flow. In equations (56a, b), fp and gp are constants and we recover the Lodge model which reduces to linear viscoelasticity in small amplitude oscillatory flows. Ahn and Osaki [43] have examined the cases where fp and gp a r e in turn functions of strain rate, strain, chain length, and effective strain which is defined as the ratio of the primary normal stress difference to twice the shear stress. They found that the predictions obtained by assuming fp and gp t o be functions of the effective strain agreed best with their experimental data. Since both the primary normal stress difference and the shear stress are functions of the shear rate, we may conclude that fp and gp are functions of the shear rate. Hinch [56] in a computer simulation of the uncoiling of a polymer molecule in an elongational flow has found that the stress generated is strain rate dependent and not strain dependent. Further support for the strain rate dependent equation is given in Carreau et al. [4] and Macdonald and Carreau [57]. At present the model parameters of the network theory are not related to the molecular structure of the material and it is desirable to seek such a connection. Lodge et al. [34] have proposed to relate the rates of creation and loss to the molecular weight. In Figure 14 we have plotted tl and t2 of the De Kee-Carreau model against 7/0 M w for various polymer solutions. The curves are parallel straight lines implying that for all p, tp is proportional to

(770 Mw)~, where n is a constant depending on the polymer solutions. We also deduce that t2, t3 .... are multiples of tl and De Kee [23] found that the factor 0.1 fits the data (see Figure 6).

424

9

9

I 1 1800 I 0"s K)*

i iO?

_

sOe

i

, , ,,,l iO t

%Mw

Figure 14. Correlation between the parameters

tp

and the product 77o M

w .

The reptation model (section 3.1) has been improved by the des Cloizeaux [58, 59] double reptation model and he has been able to deduce that the viscosity 77 is proportional to M 3.4, where M is the molecular weight. In the double reptation model, the reptation of a stress point is considered in addition to the reptation inside the tube. A stress point is a point of entanglement of two polymers and if either of the two polymers reptates out of the stress point, the stress disappears. The stress point is the junction in the network theory and this prompts Mead [60] to state that "the evolution of molecular constitutive equations has gone full circle in the short span of 35 years." In the next circle, we need to examine the process of loss and creation of junctions and the deformation of the segments. The works of Wagner and Schaeffer [49] and des Cloizeaux [58, 59] which combine both the reptation and network models need further exploration.

425

REFERENCES

Oldroyd, J.G., Proc. Roy. Soc., A200 (1950) 523; A245 (1958) 278.

~

2.

Barnes, H.A., J.F. Hutton, and K. Walters, An Introduction to Rheology, Elsevier, New York, NY (1989). Bird, R.B., R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1 - Fluid Mech., Second Ed., Wiley, New York, NY (1987). Carreau, P.J., D. De Kee, and R.J. Chhabra, Polymer Rheology: Principles and Applications, Hanser, New York, NY (1997).

.

,

.

.

~

Larson, R.G., Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, MA (1988). Bird, R.B., C.F. Curtiss, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2 - Kinetic Theory, Second Ed., Wiley, New York, NY (1987). Rouse, P.E., J. Chem. Phys., 21 (1953) 1272.

11.

de Gennes, P.G., J. Chem. Phys., 55 (1971) 572. Doi, M. and S.F. Edwards, J. Chem. Soc. Faraday Trans. II, 74 (1978) 1789, 1802, 1818; 75 (1979) 38. Doi, M. and S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, New York, NY (1986). Curtiss, C.F. and R.B. Bird, J. Chem. Phys., 74 (1981) 2016, 2026.

12.

Green, M.S. and A.V. Tobolsky, J. Chem. Phys., 14 (1946) 80.

13.

Lodge, A.S., Trans. Faraday Soc., 52 (1956) 120.

14. 15.

Yamamoto, M., J. Phys. Soc. Japan, 11 (1956) 413; 12 (1957) 1148; 13 (1958) 1200. Chan Man Fong, C.F. and D. De Kee, Physica, A218 (1995) 56.

16.

Lodge, A.S., Rheol. Acta, 7 (1968) 379.

17.

Bogue, D.C., Ind. Eng. Chem. Fund., 5 (1966) 253.

18.

Bird, R.B. and P.J. Carreau, Chem. Eng. Sci., 23 (1968) 427.

19.

Meister, B.J., Trans. Soc. Rheol., 15 (1971) 63.

20.

Carreau, P.J., Trans. Soc. Rheol., 16 (1972) 99.

21.

De Kee, D. and P.J. Carreau, J. Non-Newtonian Fluid Mech., 6 (1979) 127.

8.

9. 10.

426

22.

Chan Man Fong, C.F. and D. De Kee, J. Non-Newtonian Fluid Mech., 57 (1995) 39.

23.

De Kee, D., Ph.D. Thesis, University of Montreal, Quebec (1977).

24.

Kaye, A., Brit. J. Appl. Phys., 17 (1966) 803.

25.

Phan-Thien, N. and R.I. Tanner, J. Non-Newtonian Fluid Mech., 2 (1977) 353.

26.

Phan-Thien, N., J. Rheol., 22 (1978) 259.

27. 28.

Tanner, R.I., J. Non-Newtonian Fluid Mech., 5 (1979) 103. Acierno, D., F.P. La Mantia, G. Marrucci, and G. Titomanlio, J. NonNewtonian Fluid Mech., 1 (1976) 125.

29.

Jongschaap, R.J.J., J. Non-Newtonian Fluid Mech., 8 (1981) 183.

30.

Wagner, M.H., Rheol. Acta, 18 (1979) 33.

31.

Wagner, M.H. and S.E. Stephenson, J. Rheol., 23 (1979) 491.

32.

Petruccione, F. and P. Biller, J. Chem. Phys., 89 (1988) 577.

33.

Herman, W. and F. Petruccione, J. Rheol., 36 (1992) 1461.

34.

Lodge, A.S., R.C. Armstrong, M.H. Wagner, and H.H. Winter, Pure & Appl. Chem., 54 (1982) 1349.

35.

Giesekus, H., Viscoelasticity and Rheology, edited by A.S. Lodge, M. Renardy, and J.A. Nohel, Academic Press, New York, NY (1985) 157.

36.

Johnson, C.W. and D. Segalman, J. Non-Newtonian Fluid Mech., 2 (1977) 255.

37.

Gordon, R.J. and W.R. Schowalter, Trans. Soc. Rheol., 16 (1972) 79.

38.

Liang, C.H. and M.E. Mackay, J. Rheol., 37 (1993) 149.

39.

De Kee, D. and C.F. Chan Man Fong, J. Non-Newtonian Fluid Mech., 62 (1996) 307.

40.

Gerhardt, L.J. and C.W. Manke, J. Rheol., 38 (1994) 1227.

41.

Manke, C.W. and M.C. Williams, J. Rheol., 36 (1992) 1261.

42.

Hua, C.C. and J.D. Schieber, J. Non-Newtonian Fluid Mech., 56 (1995) 307.

43.

Ahn, K.H. and K. Osaki, J. Non-Newtonian Fluid Mech., 55 (1994) 215.

44.

Tanaka, F. and S.F. Edwards, Macromolecules, 25 (1992) 1516.

45.

Wang, S.Q., Macromolecules, 25 (1992) 7003.

46.

van der Brule, B.H.A.A. and Hoogerbrugge, P.J., J. Non-Newtonian Fluid Mech., 60 (1995) 303.

427

47.

Vrahopoulou, E.P. and A.J. Mc Hugh, J. Rheol., 31 (1987) 371.

48.

Marrucci, G., S. Bhargava, and S.L. Cooper, Macromolecules, 26 (1993) 6483.

49.

Wagner, M.H. and J. Schaeffer, Rheol. Acta, 31 (1992) 22.

50.

Ait-Kadi, A., Ph.D. Thesis, University of Montreal, Quebec (1985).

51.

Xu, Y.Z., D. De Kee, and C.F. Chan Man Fong, J. Appl. Polym. Sci., 55 (1995) 779.

52.

Xu, Y.Z., C.F. Chan Man Fong, and D. De Kee, J. Appl. Polym. Sci., 59 (1996) 1099.

53.

Lodge, A.S., Body Tensor Fields in Continuum Mechanics, Academic Press, New York, NY (1974).

54.

De Kee, D., Recent Developments in Structured Continua, edited by D. De Kee and P.N. Kaloni, Longman Scientific and Technical, Harlow, U.K. (1986) 150.

55.

De Kee, D. and C.F. Chan Man Fong, Polym. Eng. & Sci., 34 (1994) 438; 35 (1995) 1031.

56.

Hinch, E.J., J. Non-Newtonian Fluid Mech., 54 (1994) 209.

57.

Macdonald, I.F. and P.J. Carreau, J. Rheol., 33 (1989) 367.

58.

des Cloizeaux, J., Europhys. Lett., 5 (1988) 437; 6 (1988) 475.

59.

des Cloizeaux, J., Macromolecules, 23 (1990) 4678.

60.

Mead, D.W., J. Rheol., 40 (1996) 633.

429

CONSTITUTIVE DERIVATIVES

BEHAVIOR

MODELING

AND

FRACTIONAL

Chr. Friedrich ~, H. Schiessel b'c and A. Blumen b

"Freiburg Materials' Research Center, Freiburg University, Stefan-Meier-Str. 21, 79104 Freiburg, Germany bTheoretical Polymer Physics, Freiburg University, Rheinstr. 12, 79104 Freiburg, Germany ~Materials Research Laboratory, University of California, Santa Barbara, CA 93106, USA

1. I N T R O D U C T I O N

The simplest decay behaviors are exponential, such as the dielectric relaxation associated with Debye and the mechanical relaxation named after Maxwell. Exponential decays depend on a single mode (or, equivalently, a single characteristic time). But most relaxation processes are governed by a large variety of characteristic times, see references [1-4] for reviews, and vast types of decay patterns follow, most popular being stretched exponentials (Kohlrausch-WilliamsWatts [2,5,6]) and power law behaviors. In this review we focus on the cases in which the decay function follows a power law for reasonably extended time or frequency intervals. Note that too short time or frequency windows do not allow to distinguish between different decay patterns [7]. The transition from the glassy relaxation zone to the transitional zone in the case of stress relaxation of a glassy polymer is an example for power-law relaxation. Consider polyisobutylene at the reference temperature To = 25~ In Figure l(a) we reproduce, following reference [8] its shear storage and loss moduli G' and G" as a ftmction of the frequency o~; in Figure 1(b) we display the corresponding shear relaxation modulus G and the shear creep compliance J as a function of time. As is evident by inspection, the high modulus plateau is followed by a power law that covers about four decades in time (Figure 1(b)) and frequency (Figure l(a)). Then a second plateau zone (called the entanglement

430

10

-4

10

G' .~

-5

8~

~.

O

-6 "7t~

8

~- 7

-7

o

6

-8

o

5

-9

O

i

i

i

i

-4

0

4

8

-12

log co/s-1 Figure l(a). Storage modulus G'(co) and loss modulus G"(co) for polyisobutylene as a function of frequency. The data are from Tobolsky and Catsiff [8].

~.~

l

i

i

-8

-4

0

e~ @ o

-10 4

l o g t/s

Figure l(b). Relaxation function G(t) and creep function J(t) for polyisobutylene (as in figure 1(a)) as a function of time.

5

~o 3

2 O

,--

*

G~- PB302

.ce,

1

~

-5

G'- PB304

l

1

1

I

I

I

I

I

I

-4

-3

-2

-1

0

1

2

3

4

5

log (aTCO) / rad s "l Figure 2. Storage modulus G' and loss modulus G" of unmodified (PB300) and urazole modified polybutadiene (PB302 and PB304) vs. the reduced frequency aTa~. The molecular weight of all samples is Mw - 31 kg/mol; the samples 302 and 304 correspond to the 2 mol % and to the 4 mol % modification, respectively.

431

plateau) shows up. This transition from one plateau to another, generally via a power law, is characteristic for amorphous polymers. Furthermore, power law behaviors also appear in the terminal relaxation zone of polymers. In the case of a single relaxation time one has a sharp transition from the entanglemem plateau to the flow zone, which obeys a typical liquid-like behavior, namely G'oc co2 and G"oc co. Physically or chemically cross-linked polymers [9], polymers with star-, H-, or comb-like topologies (see e.g. [10-19]) show a more general pattern, namely an intermediate power-law domain with G' oc co~ and G" oc coa , where cz and 13 lie between zero and one. In Figure 2 we contrast the behavior of neat, monodisperse polybutadiene to that obtained by attaching to its backbone active groups, which are able to form H-bonds. While for neat polybutadiene the shear storage and loss moduli obey G'oc o92 and G" oc co, the moduli of the modified polymers follow more general power laws. The reason for this behavior is that the active groups form temporary, random links between the polymers, fact which renders the relaxation process multimodal and cooperative; this leads here to power laws. Such laws are not the hallmark of polymers only; vast classes of substances, which range from inorganic glasses to proteins show such behaviors [1], and we like to recall the early works of Meinardi et al. [20-22] on power law relaxation in metals, in rocks and in glasses. In this chapter we focus on the possibility to portray such complex viscoelastic features of polymers by means of fractional calculus, a formalism which turns out to be exceedingly well-suited for this purpose. To this effect we start here by illustrating, using a simple example, how fractional calculus comes into play as a result of the superposition principle. We start with G(t), the shear relaxation modulus of a linear system. Now G(t) is defined as the response of the shear stress r(t) to a jump in the shear strata y(t)-yoO(t) where O(t) is the unit step function. We assume that G(t) obeys a power law, i.e.

G(t)

-

r ( 1 - p)

(1)

with 0 _ 13< 1. In equation (1) E and ~, are constants and I-'(x) denotes the Gamma function [23]; for convenience we chose in equation (1) the prefactors in such a way as to conform to the mare body of the Chapter. Due to the lmearity of the system, the response of the stress to a previous history of deformations y(t) is given by the superposition integral [24,25]"

432

r(t) - i dr' G(t- t') dy(t'___))

(2)

dr'

Inserting G(t) given by equation (1) into equation (2) we have t

Eft

r(t)- r ( 1 - p )

~dt' (t- t') -p dy(t') at'

(3)

Now equation (3) can be rewritten in the following compact form r ( t ) - E 2 p dP?'(t)

dtp

(4)

'

in which ~/dt ~ denotes the fractional derivative of order 13 [26,27] (see section 2 for details). Equation (4) with 0 0, is obtained by first picking an integer n, n > ct, then performing a fractional integration of order ct - n, followed by an ordinary differentiation of order n, i.e. [26,27]

d~ f (t) = d__"_( d~-" f (t)) dt ~ dt" k, ~-" J

(9)

Equation (9) defines fractional differentiation. Let us note that also another version of fractional calculus, the so-called Riemann-Liouville (RL)formalism [26,27], is of widespread use in rheology. In RL the lower limit of the integrals in equations (7) and (8) is set to 0. The RL formalism is particularly suitable for studying the transient response of materials after switching an external perturbation on, say at t = 0, so that v(t)= y ( t ) - 0 for t < 0. In this spirit the RL version can be understood as being the restriction of the Weyl formalism to a special class of initial value problems. In the following we will use the Weyl formalism; the translation into the RL version is straightforward.

434

Weyl's fractional calculus rams out to be algebraically very conveniem: the composition role for differentiation and integration obeys the simple form

d ~ d~f dV+~f = dt " dt ~ dt "+~

(10)

for arbitrary IXand v [27]. Furthermore, the Fourier transform oo

{f(t)} - f(o)) = I dt f(t)exp(-icot)

(11)

.-oo

turns the operation d~/d# into a simple multiplication [26,27]

{ d~f(t) } : (ico) ~f(co) dt ~

(12)

Let us illustrate the usefulness of these properties using the above-mentioned rheological example. A quick comparision of equation (3) with the Weyl integral, equation (8), leads immediately to the fractional stress-strata relation r(t)=EAp d ~-' dy(t)

dt a-~

dt

(13)

Furthermore using the composition rule, equation (10) with r = r - 1 and v = 1, equation (13) turns into equation (4), as stated in the previous section. The behavior of fractional derivatives under Fourier transformation is especially useful in determining the dynamical response functions. Let us consider the complex shear modulus

G* ( co) -icol;dzG(z)exp(-icoz )

(14)

which describes the response of the stress to a harmonic strain excitation y(t) =Yo exp(icot). From equation (2) it follows by the change of variables z - t - t' that 7(co) = G*(co) y(co). Using the multiplication rule, equation (12), one finds, say from equation (4)

435 G*(co) - 7(0)) / ~(co) - E(ico,;t)p .

(15)

From the complex modulus G*(co) follow the storage and the loss moduli, G'(co) = Re(G*(co) ) and G"(co)= Im(G*(co) ), the complex shear compliance J*(co)=l/G*(co) as well as storage and the loss compliances, J'(co) = Re(J*(co)) and J"(co) = -Im(J*(co)). Furthermore, we also consider the shear creep compliance J(t) (the response of the strain to a stress jump r(t) = r 0 tg(t)), which is given here by

J(t)-/-'(1+ fl) As a direct consequence of the multiplication relation, equation (12), we can easily derive the harmonic response functions G*(co) and J*(co) of a given fractional RCE. On the other hand, the analytical evaluation of the step response functions, namely of G(t) and J(t), turns out to be a hard task in many cases. Nevertheless these responses can be derived explicitly for a whole series of fractional RCEs of practical importance, cf. sections 5 and 6.

3. HISTORICAL SURVEY OF RCEs WITH FRACTIONAL DERIVATIVES

To our knowledge, the mathematically sound use of fractional differentiation to describe rheological properties of materials starts with Gemant [28,29]. He modified the Maxwell model by introducing the semiderivative of the stress (i.e. a = 1 and fl = 1/2 in equation (30), vide infra) in order to portray the properties of an 'elasto-viscous' fluid under oscillatory excitations. Nutting, on the other hand, pioneered power laws such as equation (1) to depict experimental results [30,31], although at that time the relation between power laws and fractional derivatives was not clear to the materials' science community. Other examples for the use of power laws are the works by P. Kobeko, E. Kuvshinskij and G. Gurevitch [32] (an expression used by them is equivalent to the Cole-Davidson function of dielectric relaxation) and by Alexandrov and Lazurkin [33]. There are even claims that stretched exponentials turn into power laws for exponents smaller than 0.4 [34], see, however also reference [7].

436

In rheology Scott-Blair et al. [35-38] made an extensive use of fractional integrodifferentiation to depict through power-laws the creep and relaxation in wide classes of materials. Their works rendered clear the intimate relationship between power laws and fractional calculus, and also introduced fractional generalizations of Newton's and Hooke's models, in which the fractional elements (FE) of this chapter were viewed as arising from 'quasi-properties', representing non-equilibrium states. Thus, in their notation, property X obeys generally an expression of the following type: X_

d'~r

(17)

Viewing z as stress and ~/as strain, the property X is an extension of the usual definition of viscosity, for which a = f l - 1 . Bosworth [39] made first considerations concerning the use of equation (17). After these basic pioneering works, rheology experienced a renewed surge of activity on fractional calculus starting at the end of the '60ies. Thus Slonimsky [40] applied the calculus to study rheological phenomena of polymers, a materials' class of growing importance. He described by a fractional relation the force acting on a polymer segment and the displacement experienced by it; in fact the operator used by him can be represented as an infinite series of simple fractional derivatives. Then Smit and deVries [41] investigated several material functions, such as the complex shear modulus of the fractional Kelvin-Voigt model (having a fractional derivative of the strain) and compared the results with experimental data. Next, Memardi and Caputo [20-22] developed the ideas of fractional calculus ft~her; they provided expressions for several material functions for the fractional standard solid model (see below), for which the order of fractional differentiation was the same for the stress and the strain. The evaluation of the relaxation in this case was an important step towards the understanding of the basic properties of the whole class of fractional models. These results were not widely noticed, and later rederived in rheology by Friedrich [11-13] and Nonnenmacher [42-44], who also solved the general case, in which the two fractional derivatives are of different order. Memardi and Caputo used their expressions [20-22] to describe relaxation measurements of rock materials, metals and glasses. A study of the fractional versions of the Maxwell and Kelvin-Voigt models, where only the derivative of the stress was replaced by a fractional counterpart was performed by Koeller [45]. He also considered the generalization to parallel

437

and serial arrangements, and also pointed out the close connection between the Rabotnov calculus [46] and fractional integrodifferentiation. The Rabotnov calculus, developed in the USSR in the '60ies, is based on the Rabotnov-operator 9t5~(f) (also called fractional-exponential operator), and was used widely for portraying relaxation phenomena in solid mechanics [46]. Friedrich and Hazanov poined out the relation between 9t-~(f) and fractional integrodifferentiation [47]. One has namely:

~ - ~ ( f ) = ~k=l( - f l ) * - '

d k(~

-')f dtk(a-1)

and

~ft~- a ( f ) -

d~-lf dtg-

(18)

1

w h e r e 0 _ a < l andfl>_0. More recently, one of the goals of research in our field was the detailed analysis of the fractional Maxwell and Kelvin-Voigt models. It was soon clear that the analytical determination of the response functions of these models is a mathematically difficult task (see below). Bagley and Torvik [48] connected the molecular theory of viscoelasticity to the fractional calculus, and showed that some aspects of the theory are mirrored by the fractional Kelvin-Voigt model. Friedrich and Heymann [15,16] pointed out the intimate relation between the order of differentiation in a fractional Kelvin-Voigt model and the degree of conversion for the sol-gel transition of a crosslinking polydimethylsiloxane. Later, in 1986, Bagley and Torvik [49] considered the fractional standard solid model in more detail. They established that this model, which contains two fractional derivatives- one of the stress and one of the strata- is compatible with thermodynamics if the order of both derivatives is identical. Thermodynamical admissibility of fractional order models was also analyzed by Friedrich for several models [12]. In 1983, Rogers [50] formulated general RCEs containing a large number of sums or products of fractional derivatives, both of the stress and also of the strain. The analysis of these models was restricted to those containing mainly two fractional derivatives, and dealt with (the more accessible) frequency domain. Another use of fractional derivatives was put forward by VanArsdale who generalized the Rivlin-Ericksen and White-Metzner tensors by including fractional orders of differentiation [51 ]. Only in 1991 did Friedrich [11] as well as G1Ockle and Nonnenmacher [42] succeed in obtaining G(t) and J(t) for the fractional Maxwell model containing two fractional derivatives of different order. They showed that this solution can be expressed through a special class of mathematical fimctions which will be

438

discussed later in this Chapter. The derivation in references [44,52] is based on an integral fractional representation of the standard solid model, whereas Friedrich's derivation uses the differential picture. Friedrich pointed out that the differential fractional version is thermodynamically admissible for wide ranges of parameters, whereas the parameter range of the fractional integral version is very restricted [17]. In subsequent works GlOckle and Nonnenmaeher clarified the mathematical basis of fractional calculus by pointing out its relation to the socalled Fox-functions [42]. Moreover, they explained the interrelation between fractional relaxation functions and the time-temperature superposition principle [52]. We note that these models were successfully applied to the description of the viscoelastic properties of filled polymers [53]. In general, much of the very recent work is characterized by the search for the physical background underlying fractional calculus. The relation of this calculus to the classical sprmg-dashpot representation of complex viscoelastic materials is a central aspect of the works by Schiessel and Blumen [54-58], as well as by Heymans and Bauwens [59-61]. These works showed how power law relaxation follows from exemplary arrangements of springs and dashpots. Schiessel and Blumen succeeded in explaining the process of polymer cross-linking on the basis of such mechanical networks; they also investigated terminated ladder arrangements which mimic pre- and postgel behavior [56]. Such mechanical analogs also showed the way how physically reasonable fractional RCEs can be constructed, aspects discussed by Schiessel, Metzler, Blumen and Nonnenmacher [58] and by Heymans [61]. By employing a formal analogy between linear viscoelasticity and diffusion in a disordered structure Giona, Cerbelli and Roman derived a fractional equation describing relaxation phenomena in complex viscoelastic materials. This analogy leads to a power law expression, which is in agreement with the experimental results [62]. At the moment we are still far from being able to relate in a reductionistic way the empirical models to a microscopic background. Nevertheless, as will become evident in section 4 of this Chapter, the representation of fractional derivatives through sprmg-dashpot analogs is helpful in understanding a series of phenomena which underlie the fractional calculus formalism. Another line of research, widely pursued nowadays, is the phenomenologically oriented, pragmatic modeling. An example for this is the work by Stastna, Zanzotto and Ho [63] who provided a relation between the Kobeko ftmction [32] and fractional calculus and used it to model rheological data of asphalts. Stastna et al. found a general inversion scheme which generalizes Rogers' results [50]; it allows the representation of the complex shear modulus in the form:

439

.oo I

+ o 1,tI

(19)

In equation (19) ~lk and ~2k are two sets of characteristic times. The authors of reference [63] succeeded in deriving the corresponding fractional RCE. Note that for m = 0 and n = 1 one recovers the Kobeko function [24]. We close this section by noting that nowadays fractional calculus is of widespread use in describing different rheological phenomena for wide classes of materials. As examples from the polymer literature we refer to [19,53,64-67]. In the following section we analyze under which circumstances spring and dashpot arrangements lead to power law relaxations.

4. THE REPRESENTATION OF RCE WITH FRACTIONAL DERIVATIVES BY MECHANICAL MODELS

Fractional RCEs may be formally derived from ordinary RCEs by replacing the first-order derivatives (d/dt) by fractional derivatives (~/dt ~) of non-integer order (0 < 13< 1). This formal procedure can, however, not assure a priori that the resulting expressions are always physically reasonable. This aspect was pointed out, for instance, in references [11,42,54]. Thus it is useful to have a procedure at hand that automatically guarantees mechanical and thermodynamical stability. As a first step Schiessel and Blumen [54-57] and Heymans and Bauwens [60] have demonstrated that the fractional relation, equation (4), can be realized physically through hierarchical arrangements of springs and dashpots, such as ladders, trees or fractal structures (which we will discuss in this section). The idea is that (disregarding for the moment the specific structure of the hierarchical constructions, these will be detailed later) equation (4) is obeyed by a fractional element (FE) which is specified by the triple (fl, E, 3,). We symbolize the FE by a triangle, as shown it Figure 3(c), where also its classical counterparts are depicted: the spring (cf. Figure 3(a)) obeying equation (5) and the dashpot (cf. Figure 3(b)) with the stress-strain relationship equation (6). Then, as a second step, more complicated RCEs can be constructed by combining two or more FEs in serial, parallel or more complex arrangements; this was proposed by Schiessel et al. [58] and by Heymans [61]. We will make use of this method in sections 5 and 6.

440

E (a)

__Lq (c)

(b)

Figure 3. Single elements: (a) elastic, (b) viscous and (c) fractional.

Eo E1 rio

E2 Eg/

r12 1//

Figure 4. A sequential spring-dashpot realization of the fractional element. Let us now consider different realizations of FEs. In reference [54] Schiessel and B lumen proposed a ladder-like structure with springs (having spring constants E0, El, E2,...) along one of the struts and dashpots (with viscosities r/0, ql, q2,-.-) on the rtmgs of the ladder (cf. Figure 4). As shown in reference [54] the complex modulus G*(co) admits a continued fraction expression, namely

G * ( c o ) - Eo (it~ -1~~ (it~ -~'~ ~(it~ -~ ~..., 1+ 1+ 1+ 1+

(20)

441 where we use a standard notation for continued fractions, [a/(b + ) ] f - a/(b + f ) , cf. reference (23). Choosing in equation (20) E o = E 1 - . . = E and r/0 - rh - . . . - r/it can be shown (by comparing terminating approximations of the continued fraction with the binomial series) that the complex modulus of the infinite arrangement is given by G*(co) - E (4(ic~ + 1)1/2 - 1 2(io92) -1

(21)

where we set )~-rl/E. For a)2 0 and v > 0.

The special case v - 1 yields the usual Mittag-Leffler function

(61)

E~,(z).

On the

basis of this definition, the generalized Mittag-Leffier functions for some special cases, in which la and v equal 0.5, 1 or 2, follow readily"

462

Eo.5,1(z ~

- e ~ erfc(_z ~ - - e ~ erfc(-z ~

1- 2 z

z El,l ( Z 1 ) - e z

(62) E1,2(zl) - l [e: - l] z

E2,1 (z 2 ) - c h ( z )

E2,2 (z2) _ l__sh(z) . z

All generalized Mittag-Leffier functions increase monotonically for z > 0. To obtain monotonically decreasing functions one goes to the domain of negative z. In the parameter range /~, v ~(0,1] the following asymptotic expansions for z >> 1 are of interest [26] la

z -2

for l a ~ l

(63)

V(1 1

~: F ( v - l a ) z

E,.~(-z)

-1

for v > la.

Consider now the relaxation function G(t) of the FMM, equation (32). Here the generalized Mittag-Leffier function has the parameters l a - a - 13 and v - 1-13. From definition (61) and the asymptotic expansion (63) one obtains for the two power-law regimes: G ( t ) ~: t -~

{

t -~

G ( t ) ~:

t

(a gps, orientational correlations are negligible [40]. The existence of a persistence length suggests that as far as the global properties of a flexible polymer chain are concerned, such as the distribution function for the end-to-end distance of the chain, the continuous chain could be replaced by a freely jointed chain made up of rigid links connected together at joints that are completely flexible, whose linear segments are each longer than the persistence length ~ps, and whose contour length is the same as that of the continuous chain. The freely jointed chain undergoing thermal motion is clearly analogous to a random-walk in space, with each random step in the walk representing a link in the chain assuming a random orientation. Thus all the statistical properties of a random-walk are, by analogy, also the statistical properties of the freely jointed chain. The equivalence of a polymer chain with a random-walk lies at the heart of a number of fundamental results in polymer physics.

3.1.1. Distribution ]unctions and averages In polymer kinetic theory, the freely jointed chain is assumed to have beads at the junction points betwen the links, and is referred to as the freely

478

jointed bead-rod chain [2]. The introduction of the beads is to account for the mass of the polymer molecule a n d the viscous drag experienced by the polymer molecule. While in reality the mass and drag are distributed continuously along the length of the chain, the model assumes that the total mass and drag may be distributed over a finite number of discrete beads. For a general chain model consisting of N beads, which have position vectors r~, u = 1, 2 , . . . , N, in a laboratory fixed coordinate system, the Hamiltonian is given by, 7{ --/C + r (rl, r 2 , . . . , r N )

(13)

where K: is the kinetic energy of the system and r is the potential energy. r depends on the location of all the particles. The center of mass r~ of the chain, and its velocity i% are given by 1

N

Er~ rc = N ~=1

;

1 N /'~= N E=/ ' ~ ~-1

(14)

where/-~ = dr~/dt. The location of a bead with respect to the center of mass is specified by the vector R~ = r~ - re. If Q1, Q2, ... Qd denote the generalised internal coordinates required to specify the configuration of the chain, then the kinetic energy of the chain in terms of the velocity of the center of mass and the generalised velocities Q ~ - dQ~/dt, is given by [2],

I c _ m N .2 1 2 rc + -2 V E g~t Qs Qt t

(15)

where the indices s and t vary from 1 to d, m is the mass of a bead, and g~t is the metric matrix, defined by, g~t - m E~ (OR~/OQ~). (OR~/OQt). In terms of the momentum of the center of mass, Pc = m N/%, and the generalised momenta P~, defined by, P~ - (OIC]OQ~), the kinetic energy has the form [2], 1

2

1

K: - 2m N p~ + 2 V y] G~t P~ Pt t

(16)

where, G st are the components of the matrix inverse to the metric matrix, F~t G~t gtu - 5s~, and 5~u is the Kronecker delta.

479 The probability, ~eqdrcdQ dp~dP, that an N-bead chain model has a configuration in the range dr~ dQ about r~, Q and momenta in the range dp~ dP about p~, P is given by, Peq (r~, Q, pc, P ) - Z -1 e -u/k"T

(17)

where Z is the partition function, defined by,

Z

I/f/

e -u/ksT drc dQ dp~ dP

(18)

The abbreviations, Q and dQ have been used to denote Q1, Q 2 , . . . , Qd and dQ1 dQ2 ... dQd, respectively, and a similar notation has been used for the momenta. The configurational distribution function for a general N-bead chain, Ceq ( Q ) d Q , which gives the probability that the internal configuration is in the range dQ about Q, is obtained by integrating Peq over all the momenta and over the coordinates of the center of mass, %, ( Q ) - Z -~ / / /

e -n/kBT drc dp~ dR

(19)

For an N-bead chain whose potential energy does not depend on the location of the center of mass, the following result is obtained by carrying out the integrations over Pc and P [2],

~)eq ( Q ) -

~/g(Q) e-r $ ~/-g(Q)e-r

dQ

(20)

where, g(Q) - det(gst) - 1/det(G,t). An expression that is slightly different from the random-walk distribution is obtained on evaluating the right hand side of equation (20) for a freely jointed bead-rod chain. Note that the random-walk distribution is obtained by assuming that each link in the chain is oriented independently of all the other links, and that all orientations of the link are equally likely. On the other hand, equation (20) suggests that the probability for the links in a freely jointed chain being perpendicular to each other, for a given solid angle, is slightly larger than the probability of being in the same direction. Inspite of this result, the configurational distribution function for a freely jointed bead-rod chain is almost always assumed to be given by the random-walk distribution [2]. Here afterwards in this chapter, we shall refer to a freely jointed bead-rod chain whose configurational distribution

480 function is assumed to be given by the random-walk distribution, as an ideal chain. For future reference, note that the random-walk distribution is given by,

( 1 ) N-'I N-1 Ceq (01,.--, 0N-l, (~1,--., (~N-1) -- ~ H sin Oi

(21)

i=l

where Oi and r are the polar angles for the ith link in the chain [2]. Since the polymer chain explores many states in the duration of an observation quantities observed on macroscopic length and time scales are averages of functions of the configurations and momenta of the polymer chain. A few definitions of averages are now introduced that are used frequently subsequently in the chapter. The average value of a function X (r~, Q, pc, P ), defined in the phase space of a polymer molecule is given by,

(X)e. -- f f f f

x ~:)eqdrc dQ dp~ dR

(22)

We often encounter quantities X that depend only on the internal configurations of the polymer chain and not on the center of mass coordinates or momenta. In addition, if the potential energy of the chain does not depend on the location of the center of mass, then it is straight forward to see that the equilibrium average of X is given by,

- f x r 3.1.2. The end-to-end vector

(23)

The end-to-end vector r of a general bead-rod chain can be found by summing the vectors that represent each link in the chain,

N-1 r -

y~ a ui

(24)

i=1

where a is the length of a rod, and ui is a unit vector in the direction of the ith link of the chain. Note that the components of the unit vectors ui, i = 1, 2 , . . . , N - 1, can be expressed in terms of the generalised coordinates

Q [2] The probability Peq(r)dr, that the end-to-end vector of a general beadrod chain is in the range dr about r can be found by suitably contracting the configurational distribution function Ceq ( Q ) [2],

Peq(r)- f 6 ( r -

Y~ a u i )r

i

(Q)dQ

(25)

481

where 5(.) represents a Dirac delta function. With Ceq ( Q ) given by the random-walk distribution (21), it can be shown that for large values of N and r - Irl < 0.5Na, the probability distribution for the end-to-end vector is a G aussian distribution, 3

3/2

1)a') ox. (.

_3r 2

1)a')

The distribution function for the end-to-end vector of an ideal chain with a large number of beads N is therefore given by the Gaussian distribution (26/. The mean square end-to-end distance, / r2 )eq, for an ideal chain can then be shown to be, (r2)eq - ( N - 1) a 2. This is the well known result that the root mean square of the end-to-end distance of a random-walk increases as the square root of the number of steps. In the context of the polymer chain, since the number of beads in the chain is directly proportional to the molecular weight, this result implies that R ~ M ~ We have seen earlier that this is exactly the scaling observed in theta solvents. Thus one can conclude that a polymer chain in a theta solvent behaves like an ideal chain.

3.1.3. The bead-springchain Consider an isothermal system consisting of a bead-rod chain with a constant end-to-end vector r, suspended in a bath of solvent molecules at temperature T. The partition function of such a constrained system can be found by contracting the partition function in the constraint-free case,

Z (r) - f f f f

5(r- ~ aui) e-~/kBTdrcdQdpcdP

(27)

i

For an N-bead chain whose potential energy does not depend on the location of the center of mass, the integrations over rc, Pc and P can be carried out to give,

Z(r)- C/5(r-

~

aui)r

(28)

i

Comparing this equation with the equation for the end-to-end vector (25), one can conclude that, Z (r) -- C Peq (r)

(29)

482

In other words, the partition function of a general bead-rod chain (except for a multiplicative factor independent of r) is given by Peq (r). This result is essential to establish the motivation for the introduction of the beadspring chain model. At constant temperature, the change in free energy accompanying a change in the end-to-end vector r of a bead-rod chain, by an infinitesimal amount dr, is equal to the work done in the process, ie., dA- F-dr, where F is the force required for the extension. The Helmholtz free energy of a general bead-rod chain with fixed end-to-end vector r can be found from equation (29), A(r) - - k B T In Z ( r ) -

A o - kBT In Peq (r)

(30)

where A0 is a constant independent of r. For an ideal chain, it follows from equations (26) and (30), that a change in the end-to-end vector by dr, leads to a change in the free energy dA, given by,

3kBT

dA(r) - (N - 1)a 2 r - d r Equation (31) implies that there is a ( 3 k B T / ( N - 1)a 2) r, which resists any thermore, this tension is proportional implies that the ideal chain acts like a stant H given by,

H-

3kBT 1)a2

(N-

(31)

tension F in the ideal chain, F attempt at chain extension. Furto the end-to-end vector r. This Hookean spring, with a spring con-

(32)

The equivalence of the behavior of an ideal chain to that of a Hookean spring is responsible for the introduction of the bead-spring chain model. Since long enough sub-chains within the ideal chain also have normally distributed end-to-end vectors, the entire ideal chain may be replaced by beads connected to each other by springs. Note that each bead in a beadspring chain represents the mass of a sub-chain of the ideal chain, while the spring imitates the behavior of the end-to-end vector of the sub-chain. The change in the Helmholtz free energy of an ideal chain due to a change in the end-to-end vector is purely due to entropic considerations. The internal energy, which has only the kinetic energy contribution, does not depend on the end-to-end vector. Increasing the end-to-end vector of

483

the chain decreases the number of allowed configurations, and this change is resisted by the chain. The entropic origin of the resistance is responsible for the use of the phrase entropic spring to describe the springs of the bead-spring chain model. The potential energy S, of a bead-spring chain due to the presence of Hookean springs is the sum of the potential energies of all the springs in the chain. For a bead-spring chain with N beads, this is given by, N-1

1H E qi" qi

(33)

where Qi = r i + l - ri is the bead connector vector between the beads i and i + 1. The configurational distribution function for a Hookean bead-spring chain may be found from equation (20) by substituting r - S, with the Cartesian components of the connector vectors chosen as the generalised coordinates Qs. The number of generalised coordinates is consequently, d - 3 N - 3, reflecting the lack of any constraints in the model. Since g(Q) is a constant independent of Q for the bead-spring chain model [2], one can show that, H Ceq ( Q 1 , . . . ,

Q N - 1 ) - rI. )

27~kBT

3/2

exp ( - H

Qj)

(34)

It is clear from equation (34) that the equilibrium distribution function for each connector vector in the bead-spring chain is a Gaussian distribution, and these distributions are independent of each other. From the property of Gaussian distributions, it follows that the vector connecting any two beads in a bead-spring chain at equilibrium also obeys a Gaussian distribution. The Hookean bead-spring chain model has the unrealistic feature that the magnitude of the end-to-end vector has no upper bound and can infact extend to infinity. On the other hand, the real polymer molecule has a finite fully extended length. This deficiency of the bead-spring chain model is not serious at equilibrium, but becomes important in strong flows where the polymer molecule is highly extended. Improved models seek to correct this deficiency by modifying the force law between the beads of the chain such that the chain stiffens as its extension increases. An example of such a nonlinear spring force law that is very commonly used in polymer literature is the finitely extensible nonlinear elastic (FENE) force law [2].

484

3.1.4. Excluded volume The universal behavior of polymers dissolved in theta solvents can be explained by recognising that all high molecular weight polymers dissolved in theta solvents behave like ideal chains. However, a polymer chain cannot be identical to an ideal chain since unlike the ideal chain, two parts of a polymer chain cannot occupy the same location at the same time. In the very special case of a theta solvent, the excluded volume force is just balanced by the repulsion of the solvent molcules. In the more commonly occuring case of good solvents, the excluded volume interaction acts between any two parts of the chain that are close to each other in space, irrespective of their distance from each other along the chain length, and leads to a swelling of the chain. This is a long range interaction, and as a result, it seriously alters the macroscopic properties of the chain. Indeed there is a qualitative difference, and this difference cannot be treated as a small perturbation from the behavior of an ideal chain [40]. Curiously enough however, all swollen chains behave similarly to each other, and modelling this universal behavior was historically one of the challenges of polymer physics [40,44,7,8,6]. Here, we very briefly mention the manner in which the problem is formulated in the case of bead-spring chains. The presence of excluded volume causes the polymer chain to swell. However, the swelling ceases when the entropic retractive force balances the excluded volume force. The retractive force arises due to the decreasing number of conformational states available to the polymer chain due to chain expansion. This picture of the microscopic phenomenon is captured by writing the potential energy of the bead-spring chain as a sum of the spring potential energy and the potential energy due to excluded volume interactions. The excluded volume potential energy is found by summing the interaction energy over all pairs of beads # and u, E - (1/2) E.,~=~ N E (r~ - r~), where E ( r ~ - r~) is a short-range function u u lly t

kr

E

-

-

k.T

- r.);

b

ing t h e exclua

a vol-

u m e parameter with dimensions of volume. The total potential energy of

a Hookean bead-spring chain with s is consequently, 1

N-1

r - -~ H ~ i--1

1

excluded volume interactions

N

Qi . Qi + -~v kBT Z ~,~=1 tt#u

5(r~-ru)

(35)

485

The equilibrium configurational distribution function of a polymer chain in the presence of Hookean springs and excluded volume can be found by substituting equation (35) into equation (20), and all average properties of the chain can be found by using equation (23). Solutions to these equations in the limit of long chains have been found by using a number of approximate schemes since an exact treatment is impossible. The most accurate scheme involves the use of field theoretic and renormalisation group methods [6]. The universal scaling of a number of equilibrium properties of dilute polymer solutions with good solvents are correctly predicted by this theory. For instance, the end-to-end distance is predicted to scale with molecular weight as, R ~ M ~ The spring potential in equation (35) has been derived by considering the Helmholtz free energy of an ideal chain, ie. under theta conditions. It seems reasonable to expect that a more accurate derivation of the retractive force in the chain due to entropic considerations would require the treatment of a polymer chain in a good solvent. This would lead to a nonHookean force law between the beads [7,29]. Such non-Hookean force laws have so far not been treated in non-equilibrium theories for dilute polymer solutions with good solvents. 3.2. N o n - e q u i l i b r i u m c o n f i g u r a t i o n s

Unlike in the case of equilibrium solutions it is not possible to derive the phase space distribution function for non-equilibrium solutions from very general arguments. As we shall see here it is only possible to derive a partial differential equation that governs the evolution of the configurational distribution function by considering the conservation of probability in phase space, and the equation of motion for the particular model chosen. The arguments relevent to a bead-spring chain are developed below. 3.2.1. D i s t r i b u t i o n f u n c t i o n s and averages

The phase space of a bead-spring chain with N beads can be chosen to be given by the 6N - 6 components of the bead position coordinates, and the bead velocities such that, ~O ( r l , . . . , rN, r l , . . . , rg, t ) d r 1 . . , drN d~l . . . drN

is the probability that the bead-spring chain has an instantaneous configuration in the range d r l , . . . , d r g about r l , . . . , r g , and the beads in the chain have velocities in the range d / h , . . . , drN about/'1,...,/~g.

486

The configurational distribution function ~, can be found by integrating /) over all the bead velocities, ( r l , . . . , rN, t ) -- f . . .f :P d f l . . . diCN The distribution of internal configurations r

(36) is given by,

r ( Q 1 , . . . , QN-1, t) - f ~I,'(r~, Q 1 , - . . , QN-1, t ) d r c

(37)

where, ~' - ~, as a result of the Jacobian relation for the configurational vectors [2], [ 0 ( r l , . . . , r N ) / 0 ( r c , Q 1 , . . . , QN-1)I -- 1. Note that the normalisation condition f r dQ1 dQ2 ... d Q N - 1 = 1 is satisfied by ~b. When the configurations of the bead-spring chain do not depend on the location of the center of mass, as in the case of homogeneous flows with no concentration gradients, ( l / V ) r = q~, where V is the volume of the solution. The velocity-space distribution function E is defined by, 7) (rl,...,rN, r l , - . - , I ' N , t ) -- ~ (38) Note that E satisfies the normalisation condition f . . . f E die1.., di'N = 1. Under certain circumstances that are discussed later, it is common to assume that the velocity-space distribution function is Maxwellian about the mass-average solution velocity, ---- ArM exp [

1

2 k B T [Trt(rl -- v) 2 -+- . . . nt- m ( r N

--

v)2]]

(39)

where ArM is the normalisation constant for the Maxwellian distribution. Making this assumption implies that one expects the time scales involved in equilibration processes in momentum space to be much smaller than the time scales governing relaxation processes in configuration space. Averages of quantities which are functions of the bead positions and bead velocities are defined analogously to the those in the previous section, namely, (X) - f...f

X 7) d r 1 . . , drN d ~ l . . , drN

(40)

is the the phase space average of X, while the velocity-space average is,

[[x ]] - f f z

dl'l

. . . dr'N

(41)

For quantities X that depend only on the internal configurations of the polymer chain and not on the center of mass coordinates or bead velocities, (X) - f X r d Q l d Q 2 . . , d Q N - 1

(42)

487

3.2.2. The equation of motion The equation of motion for a bead in a bead-spring chain is derived by considering the forces acting on it. The total force F~, on bead # is, F~ - Ei F(~), where the F(~), i - 1, 2,..., are the various intramolecular and solvent forces acting on the bead. The fundamental difference among the various molecular theories developed so far for the description of dilute polymer solutions lies in the kinds of forces F(~) that are assumed to be acting on the beads of the chain. In almost all these theories, the accelaration of the beads due to the force F~ is neglected. A bead-spring chain model incorporating bead inertia has shown that the neglect of bead inertia is justified in most practical situations [37]. The equation of motion is consequently obtained by setting F~ - 0. Here, we consider the following force balance on each bead #, F(h)+F(b)+F(r

(i~) -- 0

(#

1, 2 , . . . , N )

(43)

where, F(h) is the hydrodynamic drag force, F (b) is the Brownian force, F(r is the intramolecular force due to the potential energy of the chain, -# and -p(iv) is the force due to the presence of internal viscosity. These are the various forces that have been considered so far in the literature, which are believed to play a crucial role in determining the polymer solution's transport properties. The nature of each of these forces is discussed in greater detail below. Note that, as is common in most theories, external forces acting on the bead have been neglected. However, their inclusion is reasonably straight forward [2]. The hydrodynamic drag force F(h) is the force of resistance offered by the solvent to the motion of the bead #. It is assumed to be proportional to the difference between the velocity-averaged bead velocity [~i,u]] and the local velocity of the solution,

F.(h) = _r [

_ (v. + v,)]'

(44)

where ~ is bead friction coefficient. Note that for spherical beads with radius a, in a solvent with viscosity ~/,, the bead friction coefficient ~ is given by the Stokes expression: ~ = 67rr/sa. The velocity-average of the bead velocity is not carried out with the Maxwellian distribution since this is just the mass-average solution velocity. However, it turns out that an explicit evaluation of the velocity-average is unnecessary for the development of

488

the theory. Note that the velocity of the solution at bead # has two components, the imposed flow field vu - v0 + ~(t) 9r~, and the perturbation of the flow field v~t due to the motion of the other beads of the chain. This perturbation is called hydrodynamic interaction, and its incorporation in molecular theories has proved to be of utmost importance in the prediction of transport properties. The presence of hydrodynamic interaction couples the motion of one bead in the chain to all the other beads, regardless of the distance between the beads along the length of the chain. In this sense, hydrodynamic interaction is a long range phenomena. The perturbation to the flow field v~(r) at a point r due to the presence of a point force F ( r ~) at the point r ~, can be found by solving the linearised Navier-Stokes equation [1,8], v'(r) - ~ ( r -

r ' ) . F(r')

(45)

where ~ ( r ) , called the Oseen-Burgers tensor, is the Green's function of the linearised Navier-Stokes equation, n(r)-

rr ) 1 (1 + 87rq,r r-2

(46)

The effect of hydrodynamic interaction is taken into account in polymer kinetic theory by treating the beads in the bead-spring chain as point particles. As a result, in response to the hydrodynamic drag force acting on each bead, each bead exerts an equal and opposite force on the solvent at the point that defines its location. The disturbance to the velocity at the bead v is the sum of the disturbances caused by all the other beads in the chain, v" - - ~u ~ u ( r ~ - ru). F (h), where, ~ - ~v~ is given by, ~u~--

{ 87rrlsru~ 1 (1+ ru~ru~) r2 , ]zy

0

ru~--ru-r~,

for##v

(47)

for # - - u

The Brownian force F(~b), on a bead # is the result of the irregular collisions between the solvent molecules and the bead. Instead of representing the Brownian force by a randomly varying force, it is common in polymer kinetic theory to use an averaged Brownian force, F (b) --

-kBT ( 0 0r~ In

)

(48)

489

As mentioned earlier, the origin of this expression can be understood within the framework of the complete phase space theory [2,4]. Note that the Maxwellian distribution has been used to derive equation (48). The total potential energy r of the bead-spring chain is the sum of the potential energy S of the elastic springs, and the potential energy E due to the presence of excluded volume interactions between the beads. The force F(r on a bead # due to the intramolecular potential energy r is given by, F(~)_

0r

(49)

In addition to the various forces discussed above, the internal viscosity force F (i~) has received considerable attention in literature [3,38,43] though it appears not to have widespread acceptance. Various physical reasons have been cited as being responsible for the internal viscosity force. For instance, the hindrance to internal rotations due to the presence of energy barriers, the friction between two monomers on a chain that are close together in space and have a non-zero relative velocity, and so on. The simplest models for the internal viscosity force assume that it acts only between neighbouring beads in a bead-spring chain, and depends on the average relative velocities of these beads. Thus, for a bead # that is not at the chain ends,

(-

F ,(iv) = 99 ( r . + l - - r # ) ( r . + l l 2

rt~)).[~#+1__~#]]

[r.+l -- r. -- qO((r.- r._l)(r. [2rg_l)). [ [ / . _

1"#-1]]

(50)

I r u - ru_ 1 where ~p is the internal viscosity coefficient. A scaling theory for a more general model that accounts for internal friction between arbitrary pairs of monomers has also been developed [35]. The equation of motion for bead v can consequently be written as, - ~ [ ~-i-~]]- v0 - tr

A- ~ a u , . F (h) ] - kBT

0 ln~

#

Since F (h) - k , T (0 In ~ / 0 r u ) ranged to give, [[~]] - vo + tr

F(r

0r~

-4- F (~) + F (/') - 0(51)

--uF(i~), equation (51) can be rear-

1 0 ln~ + ~ Z "~,~" (--kB T~ u 0r~

+

+

(iv) )

(52)

490

where -),,~ is the dimensionless diffusion tensor [2], ~,~

5u~ 1 + ( f t ~

-

(53)

By manipulating equation (52), it is possible to rewrite the equation of motion in terms of the velocities of the center of mass r~ and the beadconnector vectors Qk, ~-/'c]] - v0 + ~ . r~

1 + 0Qk0r+ r"k ) N ( u,u,kE Bku "/t,u " (kBT OoQkln____~

1 [[Oj]] - ~" Q j - -~ Z Ajk" ( kBT 0 In 9

0r

f(iv)

(54) (55)

k

where, Bku is defined by, -Bk~ -- 5k+1,~- 5k~, the internal viscosity force, fk(i~), in the direction of the connector vector Qk is, Q'Q* I I

-

(56)

]]

and the tensor -Ajb which accounts for the presence of hydrodynamic interaction is defined by,

A.jk -- ~ B--j~,"ft,v-Bku -- Ajkl + ~(~j,k

+

~r~j+l,k+l --

~"~j,k+l - -

~-'~j+l,k)

(57)

L,, it

Here, Ajk is the Rouse matrix,

Ajk--

2 --1

for I J - k l - 0, forlj-kl-1,

(58)

0 otherwise In order to obtain the diffusion equation for a dilute solution of beadspring chains, the equation of motion derived here must be combined with an equation of continuity.

3.2.3. The diffusion equation The equation of Continuity or 'probability conservation', which states that a bead-spring chain that disappears from one configuration must appear in another, has the form [2],

O~ Ot

0

(59)

491

The independence of 9 from the location of the center of mass for homogeneous flows, and the result tr ~ - 0, for an incompressible fluid, can be shown to imply that the equation of continuity can be written in terms of internal coordinates alone as [2],

0r 0 0~ - - E o q / [ [ Qj] r

(60)

J

The general diffusion equation which governs the time evolution of the instantaneous configurational distribution function r in the presence of hydrodynamic interaction, arbitrary spring and excluded volume forces, and an internal viscosity force given by equation (56), is obtained by substituting the equation of motion for ~-(~j~J from equation (55) into equation (60). It has the form, 0r

:

-

j

0 0qj

1

qJ -

0r [0q

Ek

~(i~)

1)

0 kBT ~ OQj Aik" 0r -

+

~

j,k

(61)

OQk

Equations such as (61) are also referred to as Fokker-Planck or Smoluchowski equations in the literature. The diffusion equation (61) is the most fundamental equation of the kinetic theory of dilute polymer solutions since a knowledge of r for a flow field specified by to, would make it possible to evaluate averages of various configuration dependent quantities and thereby permit comparison of theoretical predictions with experimental observations. The diffusion equation can be used to derive the time evolution equation of the average of any arbitrary configuration dependent quantity, X( Q~, . . . , QN-~ ), by multiplying the left and right hand sides of equation (61) by X and integrating both sides over all possible configurations,

d (X) = E ( ~ ' Q j " dt j -

kBT E ( A j k " 0 l n r 9 OX ~ j,k OQk OQj

OX). OQj 0r

r

_ 1E- -~ ~m {( Qj

+ f2s) ]" qk> }

(65)

494

The second moment equation (65), which is an ordinary differential equation, is in general not a closed equation for ( QjQk }, since it invoves higher order moments on the right hand side. Within the context of the molecular theory developed thus far, it is clear that the prediction of the rheological properties of dilute polymer solutions with a bead-spring chain model usually requires the solution of the second moment equation (65). To date however, there are no solutions to the general second moment equation (65) which simultaneously incorporates the microscopic phenomena of hydrodynamic interaction, excluded volume, non-linear spring forces and internal viscosity. Attempts have so far been restricted to treating a smaller set of combinations of these phenomenon. The simplest molecular theory, based on a bead-spring chain model, for the prediction of the transport properties of dilute polymer solutions is the Rouse model. The Rouse model neglects all the microscopic phenomenon listed above, and consequently fails to predict many of the observed features of dilute solution behavior. In a certain sense, however, it provides the framework and motivation for all further improvements in the molecular theory. The Rouse model and its predictions are introduced below, while improvements in the treatment of hydrodynamic interactions alone are discussed subsequently. 3.3. T h e R o u s e m o d e l

The Rouse model assumes that the springs of the bead-spring chain are governed by a Hookean spring force law. The only solvent-polymer interactions treated are that of hydrodynamic drag and Brownian bombardment. The diffusion equation (61) with the effects of hydrodynamic interaction, excluded volume and internal viscosity neglected, and with a Hookean spring force law, has the form,

_ 0 H kBT 0 0r (66) 0 r _ _ ~ oqj" (~" Qj - --( ~-" Ajk Qk ) r + - - ~ ~ Ajk i)Qj'Oqk Ot

j

k

j,k

The diffusion equation (66) has an analytical solution since it is linear in the bead-connector vectors. It is satisfied by a Gaussian distribution,

1 ~ Qj. (er_l)jk " Qk] (Q1,..., QN-1) - Af(t) e x p [ - ~j,k

(67)

495 where Af(t) is the normalisation factor, and the tensor O)k which uniquely characterises the Gaussian distribution is identical to the second moment, o ) k - + (QjQk) . ~t + ......r H -~-~ [(QjQm-Amk> + 0,

F11Fzz>F12 z, (Fi=c3FIc3Ii,Fij=c3Filc3Ij)

F2>0,

(27)

imposed on the general form of potential F were also suggested [23,24,47]. The important implication of inequalities (27) and the proper use of this class of CEs are discussed in detail in [40]. In the simple case of bl=b2=l, with the neoHookean potential for F, it reduces to the simplest "Leonov model" which includes no nonlinear parameters.

2.1.5. Multi-mode Maxwell models. All the above single mode viscoelastic constitutive equations of Maxwell-type are usually extended to the multi-mode case. We briefly discuss below only models with potential stresses in the common incompressible case. The typical N-mode extension is as follows" N

F - y'Fk(T, ck), k=l

N

cy - ~ r ---ex

-=

(T,c ) --- ex,k

=k

(28)

~

Here Fk, Crex,kand Ck are the free energy, extra stress tensor and configuration tensor, respectively, in "k"th relaxation mode. Additionally, the general form of evolution equation presented in equation (18) holds for every configuration tensor s In the limit of very low Deborah number, this approach shows the linear viscoelastic behavior, with a discrete spectrum of relaxation times { Ok}. It means that every nonlinear relaxation mode is generated by the corresponding linear viscoelastic mode. The generalization to the multi-mode case can be justified only if the various relaxation modes are well separated, i.e. 01>>02>>...>>ON

(29)

In this case, it is reasonable to assume that the various relaxation modes act independently. All the known experimental datatestify in favor of inequalities (29). Additionally, the guess-independent Pade-Laplace method (see e.g. [57]) reveals the effective discrete linear relaxation spectrum in accord with (29).

2.2. Formulation of nonlinear single integral constitutive equations From a wide class of viscoelastic CEs of the integral type, only the single integral ones have been experimentally tested. In the common incompressible case, its general form is represented as [58]"

534 !

o%•= ~ [qg,(I,,I2,t- x)C- qg2(II,I2,t-

"r.)C-l]dx

(30)

--00

Here _O-ex is the extra stress tensor, C is the Finger total strain tensor for incompressible media, whose time evolution is described as follows" v

C-dC/dt-C.Vv-(Vv_) _ _

_._

_~_

r -C-O; ~

___

=

C[

=8 I =

T

(31)

- -

Here I l= trC, I2 = trC -1, and ~, and q32 are generally independent functions. We can also introduce many other measures of deformations. One of them, the Hencky measure, H = (l/2)lnC, will be used below. Experiments show that the simplified time-strain separable version of equation (31),

6k (Ii,/2,t-x) = m(t-x)q)k(I1,/2)

(k=l,2)

(32)

can be introduced. Here re(t-x)- dG(t-x)/dx, and G(t) is the relaxation modulus. Equations (30) and (31) are not the only single integral form of CEs. When, for example, the mixed convected time derivative is used, the CE can also be represented in an integral form (see, e.g., Ref [26]). Also, Kaye [57] and Bernstein et al. [60] proposed the potential form of equations (30) and (31):

(O, =(2p/G)OP/DI~,

~p2=(2,o/G)0~'/012;

(33)

or in the time-strain separable case"

q)~=(2p/a)3F/Ol~,

tp2=(2p/a)OF/OI2.

(34)

Here G is the elastic Hookean modulus. Equations (33) or (34) constitute the KBKZ class of single integral CEs. The potential F in equations (33) denotes the thermodynamic free energy with relaxation effects taken into account. For the time-strain separable viscoelastic CEs with potential F, the basic functionals such as the free energy W, the extra stress tensor o-e and the dissipation D are of the form [61 ]: ^

535

(35)

W = (p/G) iF ( I, ,I 2 )m(t - z)dx -oo

t

(36)

o-~ = (2p/G) IC. c3F/~C(I,,I 2 ) m ( t - , ) d , --O0

t

D - tr(crce)-dW/dt = Co~G) IF (I~,I 2 ) [ d m ( t - x) / dxld~

(37)

--oo

A comparison of equations (2) and (8) with equations (35) and (36) clearly shows that CEs of the differential type where the dissipation and free energy are generally independent, are more flexible for rheological modeling than CEs of the integral type where those quantities are roughly proportional. We now describe some particular CEs of separable single integral type by specifying q~, and q~2 in equation (30) or the potential F for the K-BKZ type in equations (34). Wagner et al. [62] proposed their first specification as: q~l=fexp(-nl 4I - 3 )+(1-J)exp(-n2 ~/I - 3 ),

I=flll+(1- /2;

r (38)

where f, nl, / 0

(55)

For the integral type constitutive equations, the condition (55) provides the stability during relaxation, which is included in the global stability requirement. On the other hand, the convexity of potential F, or the GCN + condition, can be represented as follows"

B ~jmn[30]3,,m > 0,

-02 F B ijmn -- Oh ~j0h,m

= 4Cmq

0c qn

eip

0C pj

Here hij is the Hencky strain measure, h__ = (1/2) I n c . To guarantee the thermodynamic stability, the inequality in (56) should be satisfied for any arbitrary symmetric tensor ,fl,j with the condition of incompressibility, trl~ = 0. In the potential (hyper-viscoelastic) case, the identity, Bijnm = B ijmn, holds. The comparison between the inequalities (55) and (56) shows that due to the symmetry of the tensor fl,j, the condition (56) is included in the inequality (55), i.e. the condition (56) imposes weaker stability constraints than the inequality

545

(55). It means that the conditions for the Hadamard stability are stronger than those of GCN +. Employing the algebraic procedure which has been used in hyper-elasticity [85], one can finally obtain the necessary and sufficient conditions for the global Hadamard stability as the set of algebraic constraints imposed on the functions q~k: (i) lai > O,

~i = ((Pl + (p2Ci)~k/CjCk

(i~j~k),

(ii) r q- 21-ti> O, q i =(Ii-Ci)(qOl+q02Ci)+2(Ii2-212-Ci 2-

(iii) [

,+21ai +

21

q>O,

s. )[q)llW(q)12+q)21)ci-l-q)22Ci 2] Ci

(57)

j + 2 ~ j ] 2 > g k - 2 ~ t k (iCj~k).

The additional constraint in (i), the positive definiteness of the tensor c, holds by definition for the integral CEs. It was also proved for the Maxwell-like CEs of differential type [36,102], however, only for the flow situations with a given history. The above approach to the global Hadamard stability has been recently extended by Kwon [93] on the compressible case. The new quality which occurs there is the possibility of longitudinal wave propagation. In the incompressible case, the speed of the longitudinal wave approaches infinity, whereas the speed of the transverse wave is finite. Hence, perturbation of basic solutions by the longitudinal wave was not considered in this stability analysis. The result was that the wave vector is always orthogonal to the vector of the main velocity field. However, in the compressible case, the speeds of both waves have finite values. Thus for stability, the initially infinitesimal amplitude of disturbing waves of either type (or a mixed type) should remain small all the time. It means that the conditions of Hadamard stability in compressible case are more rigid than those for incompressible one. It was demonstrated [89] on the simple example of Mooney-Rivlin potential with additional term dependent on density. Two sufficient conditions for the incompressible case have also been proposed: (1) The author's condition (27): the thermodynamic potential F for the author's class of viscoelastic CEs is a monotonously increasing convex function of invariants I1, and I2.

546 (2) Renardy's condition" the thermodynamic potential F for the K-BKZ class of CEs is a monotonously increasing convex function of ~ and Although Renardy's condition has been proved only for the K-BKZ class, it also holds for the Maxwell-like CEs with upper convected time derivatives [88]. Since the author's condition (1) is stronger than Renardy's (2), it also guarantees the global Hadamard stability for the K-BKZ class. The above sufficient conditions are much more easier to employ than the necessary and sufficient conditions for Hadamard stability (57). Therefore they are very useful for a brief evaluation of the stability for new formulations of CEs. For the compressible CEs, one sufficient condition for the global stability has also been suggested [89], but it is too complicated to use.

3.3 Dissipative stability criteria for viscoelastic constitutive equations As mentioned, there can be another source of instability originated from specification of dissipative terms in viscoelastic CEs. For viscoelastic CEs of differential type, this instability may happen due to an improper formulation of the dissipative term ~ (or ~b when ~ = 1) in equations (19), even for the Hadamard stable CEs with positively definite dissipation. For single integral CEs, the instability results from fading memory effects in equations (30) and (36). Although the global criteria for dissipative stability of viscoelastic CEs are far from being complete (if it is in general possible), we discuss in this Section two specific criteria that have been proven. In the case of compressible flow, no theorem on dissipative stability is known yet, but the following theorems are presumably valid also for the compressible CEs.

3.3.1 Criterion I of dissipative stability Theorem 1.1 (the case of CEs of the differential type [36]). Consider the set of upper convected Maxwell-like CEs (8) with the positive dissipation D = D(T, Ii, 12, 13) defined in equation (9). Let the free energy F be a non-decreasing smooth function of three invariants Ik. If for any positive number E, the asymptotic inequality

O > E'll ell

when Ilcll oo

(11 11- (trc2)m)

(58)

holds, then in any regular flow, the configuration tensor =e and the stress tensor ere are limited.

547

Theorem 1.2 (the case of single integral CEs [90]). In any regular flow, the functionals of free energy (35) and dissipation (37) are bounded, if (and only if) the thermodynamically or Hadamard stable potential function F(H1,H2,H3), expressed in terms of principal Hencky strains Hk, increases more slowly than exponentially. In the theorem 1.2, principal values of Hencky strain tensor and Finger tensor for the total deformation are related as" Hi =

(1/2)lnCi,

or

=/1= (1/2)lnC,

trH = 0.

(59)

Detailed proofs and definitions are given in the papers [36,90]. While the theorem 1.1 has been proved for differential CEs as a sufficient condition close to the necessary one, theorem 1.2 provides the necessary and sufficient condition for boundedness of single integral CEs. The above theorems were motivated by the fact that the globally Hadamard stable upper convected Maxwell model displays the unbounded growth of stress in simple extension when the elongation rate exceeds the half of the reciprocal relaxation time. As the consequences of the above theorems, (i) the upper convected Maxwell model which violates Criterion I, and (ii) the K-BKZ class with a potential F represented as an increasing rational polynomial function of basic invariants Iu, are dissipative unstable. Therefore, the Mooney and the neoHookean potentials as well as the potentials for the K-BKZ class of CEs which are subordinate to Renardy's sufficient evolution criterion also violate Criterion I of dissipative stability. Since the satisfaction of only Criterion I cannot prevent the viscoelastic CEs from severe dissipative instability, an additional criterion for dissipative stability has been introduced.

3.3.2 Criterion II of dissipative stability [88] For the stability of Maxwell-like and time-strain separable single integral CEs, it is necessary that both the steady flow curves in simple shear and in simple elongation have to be monotonously and unboundedly increasing with respect to the strain rate. It has been demonstrated [91] that the violation of Criterion II results in "blow-up" instability or even negative principal values of tensor __c in simple shear. Therefore the subordination to the combined criterion "I+II" was assumed in [88] to be presumably sufficient for the dissipative stability of both the differential Maxwell-like and the time-strain separable single integral CEs, at least in simple flows.

548 3.4. Application to viscoelastic CEs. Discussion Both the Hadamard and dissipative types of instability for such two broad classes of viscoelastic CEs have been discussed in this Section. These are the quasi-linear differential and factorable single integral models with instantaneous elasticity, which are the only ones in practical use today. The problem of global Hadamard stability for these two classes of CEs seems to be completely resolved in the isothermal, incompressible and compressible cases. This problem was reduced to that well known in the nonlinear elasticity, where the complete set of necessary and sufficient conditions of stability was formulated in algebraic form. It has been demonstrated that the proposed analysis of Hadamard stability for the two classes of viscoelastic liquids is reduced to the analysis of strong ellipticity. The physical sense of this is very evident: the studies of Hadamard stability involve very rapid disturbances which create only elastic response in viscoelastic liquids. In the case of dissipative stability, the global analysis is far from being completed, if it is generally possible. However, two distinct patterns of dissipative instability have been revealed, which are related to (i) the boundedness of stress, free energy and dissipation in a start-up flow problem under a given strain history (Criterion I), and (ii) the monotonously and unboundedly increasing steady flow curves in simple shear and simple elongation (Criterion II). Furthermore, it was assumed that the subordination of CEs to the combined criterion "I + II" is presumably sufficient for the dissipative stability in the simple flows. There is a tough problem as to how to distinguish the unstable behaviors caused by poor modeling of CEs and the observed physical instabilities which those equations should also describe. However, the long history of various branches of continuum mechanics and physics teaches us that the occurrence of either Hadamard instability or/and ill-posedness in ID situations without such important physical reasons as phase transitions, etc., is a distinct sign of inappropriateness in the CEs. Thus we can treat the instabilities demonstrated in this section as being associated not with the real instabilities observed in flows of polymer melts, but rather with the improper modeling of various terms in CEs. In numerical simulations of complex flows with unstable CEs, when the flow rate becomes high enough, the occurrence of various types of unphysical instabilities is inevitable. Even in the range of moderate Deborah numbers, the existence of singular points in flow geometry such as the comer singularity in die entrance region, is sufficient to spoil the entire numerical procedure.

549 All the results of the stability analyses found in various studies for popular viscoelastic CEs, are summarized in Table 1. An interested reader can find the details of calculations in references also provided in the Table 1. It is noteworthy that CEs derived from molecular approaches such as the Larson and the Currie models, exhibit the most unstable behavior. Surprisingly enough, none (to the authors' knowledge) of the time-strain separable single integral models are evolutionary. Appendix A represents the explanation of the reasons for that given by Simhambhatla [94]. He analyzed the time-strain separability concept for CEs and concluded that the Hadamard unstable CEs of time-strain separable type cannot properly describe the experimental data of stress relaxation after step-wise loading. The instabilities revealed in Ref. [94] exactly correspond to the results reviewed in this Section. It is astonishing that many CEs become Hadamard unstable even in viscometric flows. For the CEs of differential type, only three stable specifications exist. These are the FENE, the upper convected Phan-Thien-Tanner models, and the author class of CE's (8), (26) under convexity constraints (27). However, the FENE and the upper convected Phan-Thien-Tanner models predict zero value for the second normal stress difference in simple shear flow, which contradicts the experimental evidence for polymer melts and concentrated polymer solutions. It should be noted that all the necessary and sufficient conditions obtained for single-mode CEs become, strictly speaking, only sufficient for the multi-modal approach. Even though the necessity is not proved, it is thought that due to the inequalities (29), i.e. well separateness in the relaxation times, the exact conditions for Hadamard stability exposed above for a single mode CE, will be closed to necessary for multi-mode approach. It is also evident that the threshold of instability would only be delayed to some higher Deborah number region in the multi-modal approach, if any single-mode is unstable. For some viscoelastic CEs, regularization of ill-posedness may be achieved. E.g., it is well-known that adding a small Newtonian term to the stress stabilizes Hadamard unstable CEs. However, for complex flow simulations, this may not be enough to suppress numerical instability, and when the Newtonian term becomes larger, the description of the CE will deviate from the experimental data. In the case of Hadamard stable but dissipative unstable CEs which violate the Criterion II, one can also propose the more fundamental procedure of stabilization by changing the elastic potential. For example, the Giesekus model with or 0)

(30)

Here equation (28) describes the evolution equation for n-th mode, equation (29) is the formulation of the stress tensor in the N-mode model, with the artificial Newtonian term (with 'viscosity' rla) included, and equation (30) describes the momentum balance and continuity equations. In equations (28)(30), the subscript 'i' indicates the flow with the flow rate Qi and the superscript 'm', the number of iteration.

587 The above iterative scheme works as follows. For a given 'i', the "initial guess" for the virtual velocity field is obtained as" v ~

oo

oo

Xi vi_l,. Here vi_ 1, is

the solution of the problem with the flow rate Qi-1 and )v~ is a numerical parameter ()vi > 1). Since the vector field v ~ is still solenoidal, there is the relation: Qi = )vi Qi-1. Thus to guarantee that given flow rate values Qi-1, and Qi should be close enough, one should take the value of )vi just slightly above 1. Using the value _,v~, the strain rate and vorticity tensors for the initial guess, =,e.~ and mi~ are easily calculated. Then equations (28) should be solved with some proper (usually given upstream) conditions to find the space distribution of values for N six-component tensors c 1 ., and concomitantly, the extra stress =n,l

tensor, ~i e'~. The last step in the first iteration is the determining of the first iteration for the velocity field, vl, and the pressure p l at the first iteration. These are found as the solution of the linear non-homogeneous Stokes problem (30) under constraint Qi = const. Then one should repeat the iterative procedure till it converges, if it possible. Obviously, for any value 'i', the artificial viscous terms in equations (29) and (30) vanish at the convergence, m --+ oo. The described iterative scheme works well [ 16], if the CEs have some good stability properties. It allows to find effectively the velocity and stress distributions for very high level of Deborah numbers (flow rates Qmax). It is evident that the scaling approach can be applied to the computations at any step of iterative procedure, since the velocity field here has been found from the calculations on the previous step. To solve equations (28)-(30), some numerical schemes employing various discretization (e.g. pure FEM, or combined FEM and upstream) methods can be applied. These are not the subject of discussion in this paper. Obviously in this case, the highest computation burden is the numerical solution of the evolution equations (28) for the N tensors =n,i c m on any iterative step ' m ' with the given flow rate Qi. This is because for any given values 'n', 'm' and 'i', equations (28) is generally the set of six coupled nonlinear partial differential equations. Once again, it is wise to start solving equations (28) with the highest relaxation mode, n=l. Then using the scaling approach, one can find the distributed values of

tensors Cn, im and extra stress tensor iterative step 'm', as follows"

=13ie'm

for any level of flow rate Qi and the

588

m

m

__Cn,i - Cl,i

r

{a n -vm-1 }

N _ Z ans(r n=l

(an

--

0

n

/ 0 9 n1,

1,2,

"",

N)

mi ) Cn,

(31)

Here we need to compute equation (28) only once for n=l and then using an from interpolation procedure, obtain the values of the functionals =m cl,i {anvm-1} a search of already computed values. This is because in equations (31), an < an-1 (On < 0n-l). Since the flow rate Q is a linear functional of velocity v, the respective values of flow rates corresponding to the velocity a~y_iv-can be found as Qni = anQi. Since an 0,

(/)--71"

if

(7"1 -- 7"5)U' > 0,

(27)

essentially rejecting the alternatives on grounds of stability. Thus one can have uniform flow alignment tilted forward in the direction of shear or backwards, depending upon whether 75 is greater or less than I"1 [16,17]. The former case is consistent with one's intuition, and influences the material symmetry chosen, namely that our equations should be invariant to the simultaneous change of sign of the two directors. If one were to opt for invariance to the change of sign of a or e independently, no flow alignment would be predicted for this the simplest of shear flows involving smectic C liquid crystals.

598

Returning to the general formulation one quickly obtains from equations (22) ~I(T]I -~- ~2)U t - - Cl(~l -~- r/2 sin 2 r

c2r/2 sin r 1 6 2

,11(,/1 + r/2)v' -- c2(r/1 + y2cos 2 r

clr/2 sinr

(28) r

giving the velocity gradients as functions of the alignment. Also elimination of the velocity gradients from equation (24) leads to g~r

+ (rl - T5)(Cl sine- C2 COSr

-- 0,

(29)

which readily integrates to give K~r '2 + 2(T5 -- rl)(Cl cosr + c2 sin r

-- c3,

(30)

where c3 is a constant. The boundary conditions to be satisfied in general are u(d) -

V,

u(-d)

-

v(d) -

v(-d)

0, r

-

- r

, r

- r

(31)

the gapwidth being 2d with the origin chosen midway between the plates, and r and r two prescribed angles. It readily follows from equations (28) that 771(?']1 "~- ?~2)V -- c 1 f__ (Z]l -~- ?']2 J- d

0

C2 f__ (771 -~- 7]2

si"2r

- c2 r12

r

Cl ?']2

J- d

F sin r cos

Cdz ,

d

sin r

(32) Cdz ,

d

providing two equations for the constants Cl and c2. If the choice of angles r and r is such that r

- -r

(33)

and the angle r is an odd function, then the above implies that c2 is zero, and 771(~]1 -~- ~72)V -- 2Cl

(Z]l -~ Z]2 sin 2 r

(34)

so that Cl is clearly positive. While we know rather little concerning likely values for the various material parameters, they appear in the above equations simply as three combinations, the two viscosities rh and 7/2 and the group (T5 -rl)/K~r/1. Consequently a numerical integration of the equations is relatively straightforward. Gill and Leslie [16] give details for two cases,

599

with the angle r either symmetric or asymmetric. For the former the angle r tends to either zero or r in the centre of the cell, dependent upon whether T5 is greater or less than T~. For asymmetric solutions the results are more interesting with the twist in the centre of the cell unwinding as the shear rate increases, the flow causing the twist to concentrate near the plates with a growing region of uniform alignment in the middle of the cell. This result appears to be consistent with the early experiment by Pieranski, Guyon and Keller [18]. Note that for this problem the equations are equally valid for chiral or non-chiral materials, the difference between the two, if any, being in the initial equilibrium configuration. 3.2 P l a n a r layers p e r p e n d i c u l a r to t h e b o u n d i n g p l a t e s Our second example considers what is commonly called the bookshelf geometry with the layers planar but normal to the plates, and a shear imposed parallel to the layers would appear to be compatible with the smectic layering, as Carlsson, Leslie and Clark [17] discuss. In this case we examine solutions referred to Cartesian axes with the z-axis perpendicular to the plates of the form a-(0,1,0)

, c-

(cosr

,sine(z)), v-

(u(z),

v(z),

0),

(35)

this again satisfying the constraints (1), (2) and (3). Here, however, we must confine attention to non-chiral materials, since chiral materials would require the inclusion of a y dependence in the angle r and presumably also in the flow components, which would lead to an analysis beyond the scope of an initial investigation of the simpler options. As Gill and Leslie [16] discuss, the equations of linear momentum (16) with the above choice yield

t'~z - #1 (r

+ #2(r

t' z =

+

cosr

cl,

(36)

osr

where again Cl and c2 are constants, and there are no imposed pressure gradients or fields. The viscosity functions in the above equations are given by 2#1(r

= ~Uo+ ~ 4 ~- )k5 "4- 2A2 cos 2r + 2#3 sin 2 r cos 2 r

2~t2(r

-- tel -4- I"1 -4- "r2 -4- T5 -4- 2(n3 -4- 7"4)sin 2 r

2~t3(r

--/Zo 4-/z2 A- 2A1 + A4 + (~t4 -~- ~5 -- 2A2 + 2A3 d- A5 -I- )~6) sin 2 r

(37)

500

Here also one can solve the equations (36) to obtain the velocity gradients in terms of the angle r and the imposed shear stresses. The equations of angular momentum (15) reduce to f ( r 1 6 2 -4 21 dr(C) de r

_ (~5 + ~2 cos 2r

- (T1 -~- 7"5)v' c o s r

0,

(38)

where f(r

- g ~ + g~ cos 2 r + g~ sin 2 r

(39)

but this entails a choice of the vector/3 and the scalar 7 of the form t3-(~(z)y

, 0 , ~(z)y)

, 7=~(z)-y~(z),

(40)

the prime again denoting differentiation with respect to z. There are two respects in which this solution differs from the former, both giving rise to grounds for not pursuing it in any great detail at least for the present. One is that the material parameters appear in greater numbers, and our lack of knowledge of their relative magnitudes therefore inhibits progress. The second is that the surprising dependence of the vector/3 on the y coordinate means that the resulting torques can become large, and therefore the material may seek to relax to a less strained state, say by forming domain structures. Some guidance from related experimental studies would not go amiss in this respect. To close this section we consider the alignment dictated by shear flow in this geometry. Here the options are r162

rico

where r cos2r

, v'-0,

(41)

is the acute angle defined by - -~5/~2

, provided ~5 0,(43)

as Gill and Leslie [16] and Carlsson, Leslie and Clark [17] show. For such uniform alignment without transverse flow it is of interest to note from equations (36) and (37) that the transverse shear stress is in general non-zero, this an aspect perhaps more readily amenable to experimental observation.

601

4. F L O W I N S T A B I L I T I E S Continuum theory for nematic liquid crystals quickly gained credibility through its success in describing satisfactorily a number of instabilities that occur in these materials, when subject to external influences including flow. Here our aim is to show that similar flow induced instabilities are likely in smectic liquid crystals, being predicted by the present theory, and naturally one hopes to stimulate relevant experimental studies. In this section it suffices to give a simple illustration discussed by Leslie and Blake [19]. We return to the geometry of our first example in the previous section and thus consider smectic C layers confined between parallel plates, the layers parallel to the plates, and subjected to simple shear flow. Here, therefore, with the same choice of Cartesian axes, the z-axis normal to the plates and the x-axis parallel to the imposed flow, we again examine solutions of the form (21), and find as before that the governing equations are + (T1 -- rs)(u' sin r

g~r

v' cos C) - 0,

(r/~ + y2 cos 2 r

+ 7/2v' sin r cos r

c~,

(711 -~- 712 sin 2 r

+ ~/2u' sin r cos r - c2,

(44)

where v/1 and v/2 are again given by equations (23). Choosing the origin midway between the plates, a distance 2d apart, the relevant boundary conditions are u(d) - Y

,

u(-d)

= v(-4-d) - O ,

r162

(45)

where V is the velocity of the moving plate, and r is either zero or r. Straightforwardly the equations (44) have simple solutions subject to conditions (45) in which r162

,

t~z -

u-

Y(z + d)/2d

(~71 + r12)l//2d

,

,

(46)

v-O,

tyz - O.

As remarked above the solution anticipated is r

0

,

if r~ > T~ ,

r

,

if 7"5 < T1,

(47)

being unstable otherwise. Consider small perturbations to the solution (46) in which r

r

+ r

,

u - Y ( z + d ) / 2 d + ~t(z)

,

v - ~(z),

(48)

602

and one obtains the following equations for these perturbations g~$" + (~ - ~)~os r

~') - O,

(49)

(~1 + ~ ) ~ ' - a , ~1~' + ,7~V;~/2a - b, a and b denoting perturbations to the shear stresses. conditions for the perturbations are

The boundary

~(~d) = ~(+d) = r - O. (50) In view of the above the second of equations (49) at once implies that ~2 is zero, and elimination of ~3 between the remaining equations yields ~,, + V(7/a + rl2)(rl - r5)cos ~bo [ ~ _ 2bd/V(rll + r/2)] - 0.

(51)

2K~rl~ d If the product (rl - T5)cos Co is positive, the above equation has a solution subject to conditions (50) of the form -

2bd cos wz Y (7/1 + r/2 )(T1 - T5) COSr V(~l + ~ 2 ) ( 1 - coswd ) ' w 2 = , 2K~rlld

-

(52)

and the boundary conditions for ~ require that the last of equations (49) yield

712V / _ ~ Cdz - 2bd,

(53)

-~J d which in turn leads to

r/2 tan wd + rllWd - O,

(54)

and which straightforwardly yields positive values for w. However, if the product (rl - T 5 ) c o s r is negative, one finds that the relevant solution is 2bd cosh w z Y (r/1 + r/2)(r5 - T1) cos r Y(yl -~- Y2)(1 - coshwd ) ' w2 2K~rlld , (55) -

-

and equation (53) now yields r/2 tanh wd + rll wd - O.

(56)

As remarked earlier r/1 + 7/2 is necessarily positive, and consequently the above equation has no non-trivial solutions. As a consequence our analysis predicts an instability with respect to the particular perturbations considered for the unstable solutions at a critical velocity V~ given by

603 Vc - 2g~rll~2c/d(rll + r/2)(7-1 - 75)cos r

(57)

where ~ is the smallest positive root of ~1 x + v/2 tan x -- 0.

(58)

While other more general perturbations may lead to a lower threshold, it is perhaps premature to attempt further more complex analyses until experimental evidence becomes available, and also our knowledge of surface anchoring at a smectic-solid interface improves. With this latter aspect in mind, Blake and Leslie [20] discuss the above problem when a magnetic field is applied, this allowing consideration of the two limiting cases of strong anchoring and no anchoring of the c-director. 5. B A C K F L O W

EFFECTS

Given that the original motivation for the theory presented in section 2 is primarily to model behaviour in smectic devices, it is of more than passing interest to examine flow induced by switching of alignment in smectic cells, particularly so in view of experience of such effects in nematics [4-6]. Not surprisingly perhaps we turn first to the simplest geometry in which the layers are parallel to the bounding plates, with the alignment initially uniform due to strong anchoring at the surfaces. Application of a magnetic field parallel to the plates, but perpendicular to the initial alignment rotates the c director around the layer normal, the distortion symmetric about the plane midway between the plates. Below we present an analysis of the relaxation of this alignment when the field is removed. With a choice of Cartesian axes so that the z-axis is again normal to the plates with the origin midway between them, and the x-axis now coincident with the direction of the initial alignment dictated by the strong anchoring, it is natural to consider solutions of the form a

-

-

(0, 0, 1) , r - (cos r

t),

t ) , sin r

t), 0 ) ,

(59)

t), 0),

clearly consistent with the constraints (1), ( 2 ) a n d (3). With this choice equations (15) simply reduce to K~~z2 - 2)~5-~ + (7"1 - T5)(sin r

cos COz) -- 0,

plus expression for fl, 7, # and T, and equations (16) yield

(60)

604

0 [

C~U P Ot -

0---~ (rh + 7?2 cos 2 r

Ou

+ r/2 sin r cos r

Ov

+ ( ~ - ~-~)si~ r

Ov

0 [

P-~ - ~

Ov

~U t(r/~ + r/2 sin 2 r O--~z+ r/2 sin r cos r Oz

(61)

+ ( n - ~) r162 and an expression for the pressure p, the viscosities 771 and r/2 again given by equations (23). From the above equations it is at once evident that one must include both flow components in order to avoid an over-determinate system. The boundary and initial conditions are ~(+d) - ~(+d) - r177 u ( z , 0) - ~ ( z , 0) - 0 ,

r

- 0, 0) - r

the cell gap again 2d, and r

(62) ,

Izl < d,

a known even function of z.

Introducing the notation a - (7"5 - n ) sin r

, b - (7"1 - r5) cos r

m-r/l+r/2cos

, n - 7 ? l + r / 2 s i n 2r

2r

(63) , r-7?2sinCicosr

where r is a constant angle associated with the initial alignment, Leslie and Blake [19] replace the above equations by

02r

0r

Ou

Ov

g ~ -~z ~ - 2 ~ ~ -8-i - ~ -8-;z - b -8; - O ,

cgu

P~

Ov p-~

c92u

c92v

- m~z~ + ~ - ~

02u

02v

02r + a OzO----i '

(64)

02r

-- r -~z 2 q- n -~z 2 + b cgz cgt ,

whose solution presumably gives at least a good approximation to the initial relaxation. With the removal of a strong field, it would be appropriate to select r to be r / 2 . By introducing dimensionless variables given below, one finds that the inertial terms in the last two of equations (64) are negligible, and therefore it is reasonable to set p equal to zero in the above. As a consequence the last two equations quickly yield (mn-

c92u = ( r b - a n ) 02r ~)-~ ~ OzOt

'

(ran - r 2) c92v - (ra - bin) 02r (65) ~ o~Ot '

605 and hence with the first

02r

~162 (66) K~ Oz 3 = (OzOt' where a ( r b - a n ) + b ( r a - bin) _ 2)~5 - (75 - T1)2 - 2~ + . (67) m n - ?.2 rl1 By appeal to the viscous dissipation inequality one can show that the above parameter ~ is positive. Introducing non-dimensional variables as follows

K~ k-2,~5,

du

dv , ~--~-,

~-/~

z

kt r-d-- 7,

i=~,

(68)

the first of equations (64) and equations (65) and (66) can be rewritten 02r /)('2

0r OT

a 0~ 2A5 0~

b 0~ 2A5 0~

02~i

(ran - r~) ~ - ~ - ( r b - an) ~

-

(~

~" 0~

02r Or

- bin) ~ 1 6 2

(69)

O(Or'

03r 02r ( , 0 0 implies that - 1 < K < 0.5 and can establish that dv/dK > 0 under this condition. Hence, for all admissible values of 90, v(1; c~, go) < ~(1/a,, 0.5) (67) and we can calculate the fight-hand side of Inequality (67) numerically. For example, if (x = 1/19, ~,(19,0.5)= 20.6328. Finally a comparison theorem applied to Equations (59) and (61) shows that DR = u(1; c~,#,go) < v(1; c~,go) < g,(1/~,0.5) (68) which gives us an upper bound on DR for any chosen c~, irrespective of the values of # and go.

631

4.2. Multiple steady state solutions We consider uniform uniaxial extension, Equation (2), which starts at time t = 0. We will need to specify an initial configuration in order to obtain a specific solution to the evolution equations for the configuration tensor (and hence the stress). We consider the FENE-P model, Equations (45) and (46) and define the dimensionless configuration variables y=

A~ - A22 L2 ,

tr A All + 2A22 z = L2 = L2 .

(69)

The first of these, V, may be interpreted as describing the degree of alignment of the dumbbells with the direction of elongation while the second, z, gives the mean end-to-end length of the dumbbells in the flow (i.e. the extent to which the dumbbells are fully stretched). It is obvious that 0 < z < 1

(70)

and also fairly easy to establish that

-z/2

z to the region Y < z. When Y = z, ~, = y + c so the scalar product is n. v = (#+c)

- (#) = c > 0

(74)

and therefore solutions cannot leave the physically sensible region across the line y = z. The same argument may be applied to the line y = - z / 2 , (1/2, 1). Using this value of y gives il = s z - s z / 2 + z/2(1 - z) iz = - s z - z / ( 1 - z) + c and hence the scalar product n . v = ~ / 2 + fl = c / 2

> 0

with normal n = (75) (76) (77)

which shows that solution curves cross the line V = - z / 2 from the region y < - z / 2 into the region y > - z / 2 , as required. We may note, further, that for s > 0, i.e. for uniaxial extension (rather than for uniaxial compression or its equivalent, equibiaxial extension), # > 0 on y = 0 for 0 < z < 1 so that in extension alignment parallel to the flow is favoured (in the sense that if V(0) > 0 we have y(t) > 0 for all t > 0 while if y(0) < 0 we may expect v(t) > 0 at some subsequent time). Similarly if s < 0 alignment perpendicular to the flow is favoured, as we would expect in uniaxial compression. The same result may be proved for the FENE-CR model, Equations (47) and (48). The differences from the FENE-P model are only in the coefficient of the isotropic tensor, I, and the effect on the configuration evolution equations is that c is replaced by c/(1 - z) in Equation (73). Equation (72) remains unaltered and the arguments used above hold for the direction in which solution curves cross the lines Y = z and Y = - z / 2 . The behaviour as z ~ 1 is the same, provided that c < 1 and it is easy to see that this will be the case, since r > 1 would require b 3 (otherwise a 2 > L2). Hence the result that solutions which start off in the physically sensible region remain there for all time is established for this model too.

633

5. SOME OUTSTANDING P R O B L E M S IN EXTENSIONAL F L O W

There are many unsolved problems in extensional flow, and so plenty of work for rheologists to do. The choice of the best constitutive equation is still an open question- to which the safest answer is often that it depends on the purpose for which the constitutive equation is required. Good quantitative agreement with extensional flow data, and data on linear viscoelastic and viscometric properties, may usually be obtained only at the price of introducing a complicated equation which may present too great a challenge in the computational simulation of complex flows. There is, of course, the question of whether the need to choose a constitutive equation may be avoided by computational methods involving direct simulation of behaviour at a molecular or micro-structural level. My view on this is that we still have a need for constitutive equations, as an aid to our understanding, even if not for our use of computers to solve problems. A useful perspective on the importance of extensional flows may be gained from the list of key challenges in polymer processing given by Kurtz [46]. Of the five key challenges listed, two ("Blown film bubble stability" and "Draw resonance") quite clearly involve extensional flow. The other three ("Screw wear", "Sharkskin melt fracture" and "Scale up problems") may involve extensional flow as a more or less important feature. Some comments on this and other issues have recently appeared [47].

5.1. Interpretation of experimental results Two pieces of work may usefully be mentioned here. The first seemed at first sight to be a good idea, but probably will not stand up to scrutiny. That is the idea of a three-dimensional plot of transient extensional viscosity as a function of strain and time [48-50]. While this appears promising for limited data, its theoretical foundations are questionable [51 ] and the effort involved in preparing three-dimensional plots, let alone using them, does not seem to be justified by any increase in our understanding of extensional flows, or by any practical application of the three-dimensional plots. Certainly it is true that a transient extensional viscosity for a viscoelastic material should never be regarded as a function of instantaneous rate of strain alone and the emphasis of that point is a useful outcome of this discussion. It is doubtful whether, in general, one more parameter can carry all the information about the state of the material (as influenced by its flow history) that is needed to give a well-determined value for the transient extensional viscosity. The second set of results is more convincing, showing how results from the "Rheotens" apparatus (in effect a melt spinning device) may be presented in a set of "mastercurves" and "super-mastercurves" [52,53]. This presents a challenge to the theorist to explain why such behaviour is observed - does it tell us something about the family of experiments or about the appropriate constitutive

634

equation which we have not yet appreciated? A final comment on experimental methods is perhaps worth making. While the analysis of complex flows, such as converging flow or flow into a contraction, does not give a reliable extensional viscosity in the sense of a fundamental material property, such flows have a large component of extensional flow. Hence if the Trouton ratio of a fluid is large, the stresses in the fluid will be predominantly those associated with the extensional part of the rate of strain. A more practical point is that, if a consistent analysis can be made, measurements on such flows should give data which can be used for reliable predictions of stresses for similar flows in industrial processes. The test of a fluid property derived from converging flow is then not so much "Does it give a true extensional viscosity?" as "Does its use give reliable predictions in related practical flows?"

5.2. Stability of extensional flows This topic has considerable relevance to polymer processing, where the prediction of flow instability gives an understanding of limitations to production rates for artefacts made from polymeric materials. There are a variety of instabilities and failures in extensional flow which the author has discussed elsewhere [54]. One recurrent theme is the need for clear distinction between different instabilities such as "draw resonance" and the classical instability of a filament due to capillarity. Similar clarity of thought and of description is needed when filament rupture is considered [55]. Instabilities in extrusion ("melt fracture", "sharkskin") are not immediately associated with extensional flow, but there are strong links according to some of the mechanisms for these two distinct flow defects [46,56]. The topic of approximations for nearly extensional flows [3,4,57] has several connections both with stability and with the analysis of industrial processes like fibre spinning and film manufacture, both by tubular film blowing and by film casting. It is therefore not merely a topic of interest to theorists as a matter of scientific curiosity, but a topic which could shed much light on our analysis of a number of industrial processes.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

F.T. Trouton, Proc. Roy. Soc., A77 (1906) 426-440. J.M. Dealy, J. Rheol., 38 (1994) 179-191. C.J.S. Petrie, Elongational Flows, Pitman, London, 1979. C.J.S. Petrie, Rheol. Acta, 34 (1995) 12-26. J. Meissner, S.E. Stephenson, A. Demarmels and P. Portmann, J. NonNewtonian Fluid Mech., 11 (1982) 221-237. C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 14 (1984)189-202. C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 34 (1990) 37-62.

635

A. Demarmels and J. Meissner, Coll. Polym. Sci., 264 (1986) 829-846. 9. J. Meissner, Ann. Revs. Fluid Mech., 17 (1985) 45-64. 10. D.F. James and K. Waiters, A critical appraisal of available methods for the measurement of extensional properties of mobile systems, in Techniques of Rheological Measurement, Ed A.A. Collyer, Elsevier, New York, pp.33-53, 1994. 11. J.E. Matta and R.P. Tytus, J. Non-Newtonian Fluid Mech., 35 (1990) 215229. 12. T. S ridhar, Recent progress in the measurement of extensional properties of polymer solutions, in IUTAM Symposium on Rheology and ComputationAbstracts, Ed J.D. Atkinson, N. Phan-Thien and R.I. Tanner, University of Sydney, Australia, pp.29-30, 1997. 13. T. Sridhar, V. Tirtaatmadja, D.A. Nguyen DA and R.K. Gupta, J. NonNewtonian Fluid Mech., 40 (1991) 271-280. 14. V. Tirtaatmadja and T. Sridhar, J. Rheol., 37 (1993) 1081-1102. 15. J. Meissner, Trans. Soc. Rheol., 16 (1972) 405-420. 16. H.M. Laun and H. Mtinstedt, Rheol. Acta, 17 (1978) 415-425. 17. J.J. Linster and J. Meissner, Polymer Bulletin, 16 (1986) 187-194. 18. A. Demarmels and J. Meissner, Rheol. Acta, 24 (1985) 253-259. 19. J.J. Linster and J. Meissner, Makromol. Chem., 190 (1989) 599-611. 20. L. Berger and J. Meissner, Rheol. Acta, 31 (1992) 63-74. 21. J. Meissner and J. Hostettler, Rheol. Acta, 33 (1994) 1-21. 22. H. Gramespacher and J. Meissner, J. Rheol., 41 (1997) 27-44. 23. N.V. Orr and T. Sridhar, J. Non-Newtonian Fluid Mech., 67 (1996) 77-103. 24. S.H. Spiegelberg and G.H. McKinley, J. Non-Newtonian Fluid Mech., 67 (1996) 49-76. 25. J. van Nieuwkoop and M.M.O. Muller von Czemicki, J. Non-Newtonian Fluid Mech., 67 (1996) 105-123. 26. B. Ziille, J.J. Linster, J. Meissner and H.P. Htirlimann, J. Rheol., 31 (1987) 583-598. 27. T. Samurkas, R.G. Larson and J.M. Dealy, J. Rheol., 33 (1989) 559-578. 28. M.H. Wagner, J. Non-Newtonian Fluid Mech., 4 (1978) 39-55. 29. R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory, (Second Edition), Wiley, New York, 1987. 30. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1988. 31. Y. Kwon and A.I. Leonov, J. Non-Newtonian Fluid Mech., 58 (1995) 25-46. 32. M. Simhambhatla and A.I. Leonov, Rheol. Acta, 34 (1995) 259-273. 33. A.I. Leonov, Simple constitutive equations for viscoelastic liquids: formulations, analysis and comparison with data, in this volume, 1998. 34. C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 5 (1979) 147-176. 35. M.H. Wagner, T. Raible and J. Meissner, Rheol. Acta, 18 (1979) 427-428. o

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36. M.H. Wagner and H.M. Laun, Rheol. Acta, 17 (1978) 138-148. 37. A.C. Papanastasiou, L.E. Scriven and C.W. Macosko, J. Rheol., 27 (1983) 387-410. 38. M.H. Wagner and A. Demarmels, J. Rheol., 34 (1990) 943-958. 39. K. Feigl, H.C. 0ttinger and J. Meissner, Rheol. Acta, 32 (1993) 438-446. 40. M.H. Wagner and S.E. Stephenson, J. Rheol., 23 (1979) 489-504. 41. M.D. Chilcott and J.M. Rallison, J. Non-Newtonian Fluid Mech., 29 (1988) 381-432. 42. J.M. Rallison and E.J. Hinch, J. Non-Newtonian Fluid Mech., 29 (1988) 37-55. 43. L.E. Wedgewood, D.N. Ostrov and R.B. Bird, J. Non-Newtonian Fluid Mech., 40 (1991) 119-139. 44. R. Keunings, J. Non-Newtonian Fluid Mech., 68 (1997) 85-100. 45. C.JoS. Petrie, J. Non-Newtonian Fluid Mech., 54 (1994) 251-267. 46. S.J. Kurtz, Some key challenges in polymer processing technology, in Recent Advances in Non-Newtonian Flows, AMD-Vol 153/PED-Vol 14I, ASME, New York, pp. 1-13, 1992. 47. C.J.S. Petrie, Recent ideas in extensional rheology, in Polymer Processing Society Europe/Africa Regional Meeting - Extended Abstracts, Ed J. Becker, Chalmers University of Technology, Gothenburg, Sweden, KN 4:2, 1997. 48. J. Ferguson and N.E. Hudson, European Polym. J., 29 (1993) 141-147. 49. J. Ferguson and N.E. Hudson, J. Non-Newtonian Fluid Mech., 52 (1994) 121-135. 50. J. Ferguson, N.E. Hudson and M.A. Odriozola, J. Non-Newtonian Fluid Mech., 68 (1997) 241-257. 51. C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 70 (1997) 205-218. 52. M.H. Wagner, V. Schulze and A. G6ttfert, Polym. Eng. Sci., 36 (1996) 925-935. 53. M.H. Wagner, B. Collignon and J. Verbeke, Rheol. Acta, 35 (1996) 117-126. 54. C.J.S. Petrie, Prog. Trends Rheol., II (1988) 9-14. 55. A.Ya. Malkin and C.J.S. Petrie, J. Rheol., 41 (1997) 1-25. 56. C.J.S. Petrie and M.M. Denn, AIChE J., 22 (1976) 209-236. 57. C.J.S. Petrie, Predominantly extensional flows, in 2nd Pacific Rim Conference on Rheology - Abstracts, Ed C. Tiu, P.H.T. Uhlerr, Y.L. Yeow and R.J. Binnington, University of Melboume, Australia, pp.225-226, 1997.

637

MECHANICS

OF ELECTRORHEOLOGICAL

MATERIALS

K.R. Rajagopal Department of Mechanical Engineering Texas A &M University College Station, Texas 77802 1. INTRODUCTION Electrical and magnetic fields can significantly change the response characteristics of many materials and electrorheology is the name given to the branch of mechanics that is concerned with the flow of materials that are primarily affected by the action of electrical fields. Usually, electrorheological materials are dielectrics or semi-conductors in a non-conducting fluid, though recently Ferroelectrics have also been used. Winslow's study (see [1 ]) of non aqueous silica suspensions under the action of electrical fields seems to have been the first systematic analysis in electrorheology, though the effect of an electrical field on the viscosity of pure liquids was studied much earlier by Konig [2], Quinke [3] and Duff [4]. The work of Winslow has been followed by a great deal of work in the field, and much of this effort has been directed in fashioning such materials with a view towards producing a better fluid in virtue of the potential applications for such materials in shock absorbers, exercise equipment, valves, actuators and the like. However, for a variety of reasons such applications have not met the perceived potential for such materials. Initial attempts at manufacturing electrorheological materials were hampered by a lack of understanding of the role of water in such suspensions. Other stumbling blocks that have prevented the development of technological devices are the limited operational temperature range, the abrasive properties of the suspension that erode the devices that they flow through, the attrition of the particles, the stability of the suspension and the enormous voltage requirements that are necessary to produce the changes that are required. Much progress has been made to overcome these limitations. Polymer based particles that mitigate the problem have been developed

638

(see Bloodworth [5]), and stabilizers have been found that increase the structural stability of the suspension. Also, great strides have been made recently in decreasing the voltage requirement. A detailed account of the material science aspects of electrorheological fluids is discussed in detail in the review article by Zukoski [6], and an assessment of the technical applications of ER materials can be found in Krieger and Collins [7]. The reader is also referred to Deinega and Vinogradov [8] for a review of electrorheological materials. Here, we shall be primarily concerned with the mathematical modelling of electrorheological materials. Electrorheological fluids can be modeled starting at a microscopic level or within the context of continuum mechanics in a homogenized sense. Here, we shall restrict ourselves to continuum models; but even within the context of continuum mechanics thee are several ways to modeling electrorheological fluids. One approach is to treat the electrorheological fluid as a homogenized single constituent (see Atkin, Shi and Bullogh [9], Rajagopal and Wineman [10], Wineman and Rajagopal [11 ]). Another is to model it as a mixture (see Atkin and Craine [12], Bowen [13], Truesdell [14]) of a particulate medium and a fluid, each being treated as a single continuum (see Yalamanchili, Rajagopal and Wineman [ 15]) allowing for interactions between the two constituents. Here, we shall restrict ourselves to modeling the electrorheological suspension as a single continuum. Much of the modeling of the flows of electrorheological fluids have been restricted to one-dimension, though there has been some work on three dimensional models. Numerous three dimensional models can collapse to the same one dimensional model and at the present moment there is not a sufficient body of experimental evidence in general three dimensional flows which can be used to validate and select any one of these models as the one that is best suited. In view of this, we shall restrict our discussion to a reasonably general class of models. Experimental evidence suggests that electrorheological fluids thicken significantly on the application of the electrical field, respond in a Bingham like fashion with the yield depending on the applied field, develop normal stress differences and stress relax (see Gamota and Filisko [ 16], Yen and Achom [17], Gamota, Wineman and Filisko [18], Jordan and Shaw [ 19], Jordan, Shaw and McLeish [20]). The response of electrorheological materials is also significantly affected by the thermal conditions (Conrad, Sprecher, Choi and Chen [21 ], Jordan and Shaw [22]). Here, we shall not consider thermal effects but restrict ourselves to an isothermal analysis.

639 The fiber like structures that are formed on the application of the electrical field suggest that such fluids ought to be modeled as anisotropic fluids. This leads to an additional level of complexity which should be introduced after a better understanding of such materials is achieved. However, this aspect of the modelling is crucial and has to be reckoned with if one is to capture the behavior of electrorheological fluids. For instance, the perceived viscoelasticity of the fluid that is characterized by a time constant for the response could be due to the response time associated with the alignment of the particles on the application of the field. The electrorheological response of liquid crystals have also been studied (Carlsson and Skarp [23], Yang and Shine [24]) where the material is anisotropic even before the application of the electric field. We shall not consider these issues here, suffice it is to say that the new framework that has been developed recently that allows for variations in the synunetry of the body with various configurations that are natural to the body can be used successfully to model such materials (see Rajagopal [25]). Such an approach has been used to model crystal plasticity (Rajagopal and Srinivasa [26]), twinning and solid to solid phase transition (Rajagopal and Srinivasa [27], [28]), multi-network theory for polymers (Rajagopal and Wineman [29]) and anisotropic viscoelastic fluids (Rajagopal and Srinivasa [30]). 2. K I N E M A T I C S AND BALANCE LAWS Let f2 denote the reference configuration of a body B. By the motion of a body we mean a one-to-one mapping ~ that assigns to each point Xc f] a point x belonging to a three dimensional Euclidean space, at each instant of time t, i.e., x=x(X,O.

(2.1)

The image of f] under X, denoted as ~"~t, is the configuration occupied by the body at time t. We shall assume that X is sufficiently smooth to render all the following operations meaningful. The velocity v and acceleration a are defined through 1Various properties ~ associated with a material point at different instants of time can be defined through d~=~(X,t)=~(x,O. We denote

r ax

d~ _ 0~ a~ _ O~,v~= 0__~and grad dt a t ' a t at ox

Also, div ~ denotes tr [ grad ~] and Div ~ denotes tr [V ~].

640

O~ dx ,

(2.2)

a - 0 2 X - ~d 2.x

(2.3)

v-

Ot dt

and

Ot 2

dt 2

The deformation gradient F and the velocity gradient L are given respectively through

F: OX:Vx,

OX

(2.4)

and L - d_~v: grad v. dx

(2.5)

The symmetric and skew part of L are denoted by D and W, respectively, i.e.,

D:~(L +Lr),W:I(L-L r).

(2.6)

We shall keep our kinematical definitions and the documentation of the basic equations to a minimum while ensuring that the treatment be self-contained. A complete and proper flame-work for the study of electrorheological fluids would require the laws of electromagnetism in addition to the usual laws of thermomechanics. There are many ways of expressing the equations of electromagnetics and we shall use the Minkowskian formulation. A detailed discussion of the basic laws of field dependant materials within the context of continuum mechanics can be found in Truesdell and Toupin [31] and Eringen and Maugin [32]. We shall use the dipole-current loop model (see Pao [33]).

641

The conservation of mass is given by 2

0p +div (pv)=0,

(2.7)

Ot

where p is the density. The balance of linear momentum takes the form

(2.s)

divTr+pf+f~=P dt"

where T is the Cauchy stress 3, f is the external mechanical body force, and fo the electromagnetic force density given by

f e : : q E + l j x B +10P xB +l div[(PxB)~v] +[grad B] r M+[grad E]P; (2.9) c c Ot C qo is the electric charge density, E is the electric field, J the conduction current, B the magnetic flux, P the electric polarization, and M is defined through

(2.10)

M=M +lvxP c

where M is the magnetic polarization. The balance of angular momentum takes the form

div(x• T) +xx pf+Q = x x p

dv

dt'

(2.11)

where ~o is the electromagnetic angular momentum density given through

2

A documentation of the governing equations can be found in Rajagopal and Ruzicka [34]

3While the phenomenological continuum models assume that the stress is symmetric, some models based on particle dynamics lead to expressions for the stress that are not symmetric. We shall assume here that the stress is symmetric. A discussion of the asymmetry of stresses of electrorheological materials can be found in Rosensweig [35]

642

~e" =xxf e +Px~+MxB,

(2.12)

where g" is the electromotive force intensity given through

g'=E+lvxB.

(2.13)

c

The balance of energy takes the form

1 2) pd(e+-~lvl

+div q -div(Tv) +Of-V+pr+We,

(2.14)

where e is the specific internal energy, q the heat flux vector, r the radiant heating and w~ is the energy production density given by

where J is given through J - J - q~v.

(2.16)

Even in classical continuum thermomechanics, the specific formulation of the second law of thermodynamics is an object of much contention. Thus, the exact form of the second law in electromagnetics is far from settled. Here, we record the second law in the form of the Clausius-Duhem inequality, though alternate interpretations of the second law are possible:

p drl r dt +div (O)+p -~>_0,

(2.17)

where 11 is the entropy and 0 the absolute temperature. The Clausius-Duhem inequality places restrictions on the allowable forms of the constitutive expressions.

The above balance laws are the usual laws of thermomechanics modified to account

643

for the effects of the electrical and magnetic fields. We have to augment the above equations with Maxwell's equations in the Minkowskian form. The conservation of electric charge is

J=0.

Oqe +div

(2.18)

0t Gauss' law is given by (2.19)

div I)e=qe , where D~ is the electric displacement field given through

(2.20)

De=P+E. Faraday's law is given by

1 0B curl E +-=0.

(2.21)

c Ot

The conservation of magnetic flux is given by div B=0,

(2.22)

and Ampere's law takes the form

curl H-

1 ODe c

Ot

1

+•

(2.23)

c

where H is defined through

H: =B-M.

(2.24)

644 The system of equations (2.18) - (2.24) can be manipulated to obtain other equations that can prove more amenable to use. We shall not get into this here but refer the reader to [36] for a discussion of the same. While the equations of thermomechanics are invariant under Galilean transformations, Maxwell's equations are invariant under Lorentz transformations, and Galilean transformations are not uniform approximations of Lorentz transformation (see [37]). In general, to solve flow problems involving electrorheological fluids, it would be daunting to use the full system of equations (2.7) - (2.24) except in the simplest of problems. Thus, we need to simplify the system of equations. A gross simplification is to ignore Maxwell's equations, as well as effects due to the fields in the thermomechanical equations as field variables but treat the electric field as a parameter. Much, though not all, of the modeling in electrorheology is in this spirit. We shall discuss such an approach in some detail later. A less drastic simplification is that based on the fact that the fluid is nonconducting and that we are dealing with a dielectric, i.e., J=O,

(2.25)

and M=O.

(2.26)

It follows that (see Rajagopal and Ruzicka [34])

d9 +9 div v=0, dt

(2.27)

dv div T+pf+f = p ~ dt

(2.28)

~

de -kA0 =T.L + dP "g'+(P-g0div v,

dt

(2.29)

645

T+O 2

1 "L+

>_0,

0

(2.30)

div(E +P) =qe,

(2.31)

1 aB

curl E + - - ~ = 0

(2.32)

c at

divB=0,

(2.33)

curl B+lcurl(v• c

dqe ~+qe dt

div v=0.

1a

1

= c - ~ (E +P) ---qeV'c

(2.34)

(2.35)

In the above equations we have assumed that the heat flux is given through Fourier's law and ~ is the specific Helmholtz potential. The above equations can be further simplified on the basis of dimensional arguments for problems of interest in electrorheology. A detailed treatment of the same can be found in [36]. In general we will have to solve the coupled partial differential equations (2.27)(2.35) which is tantamount to fourteen partial differential equations and one constraint inequality for the appropriate variables, a most arduous task even under idealized conditions. Since in most problems of practical relevance, the gaps are exceedingly small, it may be reasonable to assume the electric field to be a constant (of course, this assumption could be totally inappropriate as the electric field could vary tremendously within the short gap). Such an assumption is often made in electrorheology and the electric field then plays the role of a parameter in the problem. We shall further simplify the analysis by restricting ourselves to isothermal processes and essentially ignoring the thermal variables. Even such a simplified situation serves to highlight certain interesting features concerning the flows of electrorheological fluids.

646

It is possible that there are some electrorheological fluids that undergo only isochoric motions when the electric field E is held a constant, however the density changing with the electric field. This situation is similar to fluids which can undergo isochoric flows in isothermal processes while their densities can change with temperature. Such an assumption is the starting point for the celebrated Oberbeck-Boussinesq approximations in fluid mechanics. The fact that the density can change with the electric field, but is a constant in all processes in which the electric field is a constant can be expressed through det F =f(E). A detail discussion of the consequence of the above constraint can be found in Rajagopal and Ruzicka [36]. 3. S I M P L I F I E D M O D E L S B A S E D O N T H E E L E C T R I C A PARAMETER: DIFFERENTIAL TYPE MODELS

F I E L D AS

We shall restrict our analysis to an incompressible electrorheological fluid. Further, we shall assume that the Cauchy stress is given through

T - -pl § f(D,e),

(3.1)

where -pl is the indeterminate part of the stress due to the constraint of incompressibility and D is the stretching tensor defined through (2.6). We shall assume that the material is isotropic. The response of materials of the form (3.1) has been studied in some detail by Rajagopal and Wineman [29]. It follows from flame-indifference and isotropy that f must satisfy f(QDQr QE)=Qf(D,E)Q r

VQr

(3.2)

where O is the orthogonal group. It follows from standard representation theorems that T can be expressed as (see Spencer [38]):

647 T= -pl +aIE~)E+a2D+a3D2+a4(DE~)E+E~)DE) +as(D2E|

(3.3)

+E|

where a~, I=l, ...,5 are scalar functions that depend on the following invariants Ii:tr (E@E),/2 =trD2,/3 :tr(DE| (3.4) I4=tr(D 2E| On the other hand if we require that (3.2) holds for proper orthogonal transformations O +, i.e., invariance under rigid body motions, then T has the representation T : -pl +&IE~E +tx2D+~3D2 +t~4(DE| +6cs(D2E|

+E|

+E|

+&6(MD +DM r) +&7(MD2+D2M 7,)

+&8(DMD2+D2M rD) +&9(E(~)MA+MA|

+&lo(E|

(3.5)

+MB|

where A, B and M are defined through A=DE,

B--DE

and

M=cE,

(3.6)

where e is the alternator tensor and the &; , i=l,... 10 are functions of the invariants I 1=tr(E |

=trD 2,I3: tr(D E |

trD3, (3.7)

I s =tr(D2E@E),/6 =tr[D2M TA|

rD]I 1.

Invariance under O or O § are both assumptions, whichever choice is made. We note that even in the simpler case corresponding to invariance under the full orthogonal group, where temperature effects are ignored, and the electric field is treated as a parameter, there are five arbitrary material functions that appear in the representation, and in general it would be difficult to devise a reasonable experimental protocol to

648 determine these material functions. In order to illustrate some interesting interactions between the electric field and the deformation, we shall simplify the model further. However, we shall see even in this simplified model, the presence of normal stress differences can be induced in simple shear flows due to the applied electric field. Also, the fluid can thicken, i.e., the viscosity can increase considerably with the electric field as is to be expected if the model is to describe the response of real electrorheological fluids. 3.2 Simple Shear Flow

Here, we shall essentially outline the analysis of Wineman and Rajagopal [29] to illustrate that the model (3.3) can describe normal stress differences, thickening and other behavior characteristic of electrorheological fluids. Let us consider the simple shear flow of an electrorheological fluid modeled by (3.3) with an electric field applied transverse to the direction of flow, i.e., v =u(y)i,

E=E2j +E3k

(3.8)

A straightforward calculation yields (see Rajagopal and Wineman [29])

T11_ _p+_~

~2

T22--p+ot2Ef+-~IX4 ~2 +_~ y2E2,2 T33= -p +o~2E2, +~

T12:(-~ T

(3.9) 2)Y,

~5

TI3 =--~-YE2E3, T23=o~2E2E3+~.~6]t2E2E3. where

649 Y:u'0')

(3.10)

First, suppose that E3=0. We notice that the shear stress T12 can be expressed as T12: [itt(u

2)]y,

(3.1 1)

where the generalized viscosity ~t is dependent on both the shear rate and the electric field. Thus, for appropriate forms of ~t, the model can describe the increase in viscosity that is observed in electrorheological fluids due to the application of an electric field. Depending on the nature of the fluid, it can shear thin or shear thicken and this can either enhance or ameliorate the thickening due to the electric field. It is also worth noting that in general the normal stresses Tll , T22 and T33 would be different and thus normal stress differences that are characteristics of non-Newtonian fluids are also induced by the electric field. We note that

r, 1

-TY

2,

tX4- 2_~2E32, T11-T33=TY

T22-T33=o 2(E2-E2)

(3.12)

y2 + Ix6

and thus in general the normal stress differences are distinct. We also notice that there is a contribution to the normal stress differences due to the sheafing as well as the electrical field. Moreover, we recognize that there is a coupling effect between the mechanical and electrical fields that arises in the normal stress differences Tll-T22 and T22- T33. Next, we shall illustrate the effect of the electrical field in a simple flow. 3.3 Flow Between Parallel Plates

Consider an electrorheological fluid modeled by (3) flowing between two infinite

650

parallel plates along the x-axis due to an applied pressure gradient with the electric field applied along the y-axis. It follows from the balance of linear momenttLm, in the absence of body force fields,

To,=-Cy,

(3.13)

where C . . . .Op-constant > 0 . For the problem under consideration substituting

Ox

(3.13) into (3.11 ) leads to

[g(u',E2)lu'=Cy,

(3.14)

and the above equation is solved subject to the boundary conditions

u(-h)=0,

u(h)=0.

(3.15)

It has been observed that in the presence of an electric field the fluid flows only after a critical value is reached for the shear stress, i.e., it exhibits "Bingham type" behavior. Of course, it is possible that the fluid flows ever so slowly even below the "yield stress" in that its flow is imperceptible within the time frame of the observation. To simplify the problem, we shall assume a Bingham type behavior. Thus, we shall consider a representation for Txy of the form

Oo(E)+v(E)v, T=

0,

y=0

with the assumption that Oo(0)=0.

(3.16)

651

We could assume more complicated responses wherein la depends on both y and E, i.e. ~t= la(y,E). For a fixed value of E, the maximum value of the magnitude of the shear stress occurs at y=• and its value is Ch. If Ch < Oo(E), then the maximum value of the stress is less than the yield stress and there will be no flow. Let C be such that Ch > Oo(E). Then, there is some y* such that Cy *= Oo(E). It then follows that - Oo(E)=~t(E)u '- -Ay,

y* tl when the response is fluid-like

(4.6)

655 t

Tnuid = -pl +y~E|

+f Ml(t-s)Ct(s)ds t1

t

f

+: :M2(t-Sl,t-s2)[~'t(Sl)C-~t(s2)+f-,,(~2)C-,t(Sl)]dSldS2 tl t1 t +f M3(t-s)[Ct(s)E| +E|

(4.7)

t1 t

t

+ff t1

t(s2)E~)E+E~)(~t(s,)Ct(s2)]Edsids2,

tI

where I~:B t (t)-I and Ct(s):Ct(s)-I . The scalars 13i,I= 1,--7, and Yi, I=1, 2 are constants. Also, Li, i--1, - - 6 and M., i-1, --4 satisfy certain continuity and monotonicity properties. The model outlined above is too general to be useful and it is unlikely that an experimental protocol can be devised where all the arbitrary material functions can be determined. The purpose of documenting the model is to merely show general, principles of continuum mechanics can be used to develop general enough models can describe the response characteristics of electrorheological fluids. Of course, the above model could be further simplified with the multiple integral terms ignored which would then lead to models similar to those used in viscoelasticity to describe the time dependant behavior of such materials. For instance, there has been considerable interest in the response of electrorheological materials when subject to oscillatory shears (see McLeish, Jordan and Shaw [42], Gamota and Filisko [16], Otsubo, Sekine and Katayama [45]). Wineman and Rajagopal [11] use the above model to study small simple shear and oscillatory shear in electrorheological fluids. In the case of the oscillatory shear problem the stress oscillates with the frequency to of the oscillatory shear, and G~ and G~ the storage and loss modulii of linear viscoelasticity depend on the electric field and the phase angle depends on both the frequency to and the electric field E. They are able to correlate and explain phenomena observed in the oscillatory response of electrorheological fluids by Gamota, Wineman and Filisko [16], and Gamota and Filisko [46], [47].

656

General three dimensional continnum models that can describe the response of electrorheological materials such as thickening, stress-relaxation, normal stress differences, yield etc., have been presented. The simple shear flow of such fluids has been analysed and even in such a simple flow it is found that the models are capable of describing the characteristics exhibited by real electrorheological materials such as thickening due to the applied electric field, normal stress differences and stress relaxation.

657

BIBLIOGRAPHY

1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

W.M. Winslow, 1949, J. Applied Physics, 20:1137. W. Konig, 1985, Ann. Phys., 25:618. G. Quinke, 1897, Ann. Phys., 62: 1. A.W. Duff, 1896, Phys. Rev., 4: 23. R. Bloodworth, 1995, Electrorheological Fluids Based on Polyurethane Dispersions, Bayer Corporation Report. C.F. Zukoski, 1993, Material properties and the electrorheological response, Annual Reviews in Material Science, 23: 45. I. Krieger and E.A. Collins, 1992, Electrorheological fluids, A Research Needs Assessment, Washington, DC: US Dept. Energy, Office Energy Res. Program Analysis. Y.F.Deinega and G.V. Vinogradov, 1984, Rheol. Acta, 23: 636. R.J. Atkin, X. Shi and W.A. Bullogh, 1991, Journal of Rheology, 35" 1441. K.R. Rajagopal, R.C. Yalamanchili and A.S. Wineman, 1994, International Journal of Engineering Science, 32" 481. A.S. Wineman and K.R. Rajagopal, 1995, On Constitutive Equations for Electrorheological Materials, Continuum Mechanics and Thermodynamics, 7: 1. R.J. Atkin and R.E. Crame, 1976, Q.J. Mech. Appl. Mathematics, 17: 209. R.M. Bowen, 1976, Theory of Mixtures in Continuum Physics Vol. 3., ed. A.C. Eringen, Academic Press. C. Truesdell, 1984, Rational Thermodynamics, Springer-Verlag. K.R. Rajagopal, R.C. Yalamanchili and A.S. Wineman, 1994, International Journal of Engineering Science, 32" 481. D.R. Gamota, A.S. Wineman and F.E. Filisko, 1991, Journal of Rheology, 35- 399. W.S. Yen and P.J. Achorn, 1991, Journal of Rheology, 35: 1375. T.C. Halsey, J.E. Martin and D. Adolf, 1992, Phys. Rev. Letters, 68" 1519. T.C. Jordan and M.T. Shaw, 1991, Electrorheology, MRS Bulletin, 38. T.C.Jordan, M.T. Shaw, and T.C.B. McLeish, 1992, Journal of Rheology, 36: 441. H. Conrad, A.F. Srecher, Y. Choi and Y. Chen, 1991, Journal of Rheology, 35: 1393. T.C. Jordan, M.T. Shaw, and T.C.B. McLeish, 1992, Journal of Rheology, 36: 441. T.Carlsson, K. Skarp, 1981,Mol. Cryst. Liq. Cryst., 78: 157. I. Yang and D. Shine, 1992,Journal of Rheology, 36: 1079.

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25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

K.R. Rajagopal, 1995, Constitutive Relations and Material Modeling, Univ. Of Pittsburgh Report. K. R. Rajagopal and A. Srinivasa, Mechanics of the inelastic behavior of Materials, Parts I and II, to appear in International Journal of Plasticity. K.R. Rajagopal and A. Srinivasa, 1995, International Journal of Plasticity, Vol. II: 653. K.R. Rajagopal and A. Srinivasa, 1997, International Journal of Plasticity, Vol. 13: 1. K.R. Rajagopal and A. S. Wineman, 1990, Archives of Mechanics, Vol. 42: 53. K.R. Rajagopal and A. Srinivasa, Viscoelastic Anisotropic Fluids, in preparation. C. Truesdell and R. Toupin, 1960, The Classical Field Theories of Mechanics, Handbuch der Physik, Vol. 3, Springer-Verlag, Berlin. A.C. Eringen and G. Maugin, 1989, Electrodynamics of Continua, Vols. I and II, Springer, New York. Y.H. Pao, 1978, Electromagnetic Forces in Deformable Continua, Mechanics today (New York) (S. Nemat-Nasser, ed.), Pergamon Press, 209. K.R. Rajagopal and M. Ruzicka, 1996, Mechanics Research Communications, 23: 401-407. E. Rosensweig, 1992, Magnetic Fluids, Cambridge University Press. K.R. Rajagopal and M. Ruzicka, Mathematical Modeling of Electrorheological Materials, submitted for publication. J.M. Levy-Leblond, 1977, Rivista del Nuovo Cimento, 7:187-214. A.J.M. Spencer, 1995, Theory of Invariants, Continuum Physics (New York) (A.C. Eringen, ed.), Vol. 3, Academic Press. Y.Z. Xu and R.F. Liang, 1991, Journal ofRheology, Vol. 35:1355. G. B. Thurston and E.B. Gaetner, 1991, Journal of Rheology, Vol. 3: 1327. T.C.B. McLeish, T.C. Jordan and M.T. Shaw, 1991, Journal of Rheology, 35: 427. G. V. Vinogradov, P. Shulman, Yu. G. Yanovskii, U.V. Barancheeka, E.V. Korobko and I.V. Bukovich, 1986, Inzh.-Fiz, Zh., 50: 605. W. Noll, 1972, Arch. Rational Mechanics and Analysis, Vol. 48: 1. C. Tmesdell and W. Noll, 1992, The Non-Linear Field Theories of Mechanics, Springer-Verlag, Berlin. Y. Otsubo, M. Sekine and S. Katayama, 1992, Journal of Rheology, 36: 479. D.R. Gamota and F.E. Filisko, 1991,Journal of Rheology, 35: 399. D.R. Gamota and F.E. Filisko, 1991, Journal of Rheology, 35: 399.

659

CONSTITUTIVE EQUATIONS FOR ELECTRORHEOLOGICAL FLUIDS BASED ON MOLECULAR DYNAMICS R. Tao

Department of Physics, Southern Illinois University at Carbondale, Carbondale, IL62901, USA 1. INTRODUCTION Electrorheological (ER) fluids are a class of materials whose theological characteristics are controllable through the application of an electric field. A typical ER fluid consists of colloidal dispersions of dielectric particles in a liquid of low dielectric constant. When an electric field is applied, the effective viscosity of the ER fluid increases dramatically. If the field exceeds a critical value, the ER fluid turns into a solid whose shear stress continues to increase as the field is further strengthened [1-5]. The above novel properties are the result of a structure change in ER fluids under an electric field. It is known that upon application of an electric field the dielectric particles in ER fluids form chains spanning between the electrodes. The chain will aggregate to form thick columns. The body-centered tetragonal (bct) lattice was predicted theoretically as the ground state structure for an ER fluid and has been verified experimentally [4,7]. The microstructure of ER fluids is a fundamental issue. The viscosity increase and solidification of ER fluids are all related to this microstructure. The mechanical and physical properties of ER fluids also strongly depend on the induced structure [8]. Recently, the ER effect has also been used to produce new composite materials [9,10]. Therefore, information about the microstructure of ER fluids under various conditions is very important for the growing number of applications. In this paper, we will start from the first physics principle to derive constitute equations for ER fluids based on molecular dynamics. Then we will apply computer simulations to investigate the structure of ER fluids. Our ER system is confined between two electrodes located at z = 0 and z = L respectively, upon which a voltage may be applied. The system consists of spherical dielectric particles of diameter a and dielectric constant ep suspended in a nonconducting liquid. The liquid has a dielectric constant of e / a n d viscosity r/. In an electric field, each particle has an in-

660

duced dipole moment of p - o~ef(a/2)3Eto~ where a - ( e p - e l ) / ( e p + 2el) and Eto~ is the local field. The formation of stru&ure is driven by dipolar interactions, viscous drag forces, and Brownian motions. Molecular simulations on this model have shown that the ER fluid can readily form the bct lattice under a strong electric field [11]. Several other computer simulations on a similar model have also been reported [12-16]. In our extensive simulations, we will examine the detail of the induced ER structure under various conditions. Our results indicate that ER fluids under an electric field may develop into five different structures under different conditions. In a weak electric field, ER fluids can move from a liquid state to a nematic liquid crystal state which only has ordering in the field direction, but no ordering in other directions [17]. This ordering indicates chain formation along the field direction. In both liquid and liquid crystal states, the columnar particle density remains uniform. When the electric field is strong and the thermal fluctuation is weak or moderate, ER fluids develop into a body-centered tetragonal (bct) lattice. In a very strong field without or with very weak thermal fluctuations, ER fluids may develop into a polycrystalline structure consisting of many small bct lattice grains. If both the electric field and thermal fluctuation are strong, ER fluids develop into a glass-like structure in which the particles are aggregated together to form thick columns, but the structure has no appreciable ordering. In all of the last three structures, the columnar particle density peaks in a small region, indicating a thick-column structure. Our simulation also provides information about the solidification and the chain formation time in ER fluids. The chain formation time is much shorter than the solidification time. Both of them depend mainly on the ratio of the Reynolds number to the Mason number, especially in the over-damped case. Here the Reynolds number is the ratio of the inertial force to the viscous force while the Mason number is the ratio of the viscous force to the dipolar force. However, when the viscosity is not too strong, the ratio of the Brownian force to the dipolar force also plays a role in these two time scales. 2. CONSTITUTIVE EQUATIONS We use the Langevin Equation to describe the motion of particle i,

d2ri _ dri m--~- - F i - gTrarl-~+Ri(t ).

(1)

Here Fi includes all electric forces on particle i, 37rorlvi is the Stokes' drag force, and Ri(t) is a Brownian force. Fi is given by the following

661

expression . _.-~.[fiJ

Fi --

+fidrep ]+fi self +fi wall

jr

(2)

where fij is the force acting on particle i by particle j and all of j's images, firfp is a short-range repulsive force to prevent particles i and j from overlapping, f~lf is the force on particle i due to all its own images, and f~an is a short-range repulsive force to prevent particle i from penetrating the two electrodes. The dipolar force exerted by particle j on particle i is given by 3/) 2

(3)

(efr~j) [e~(1-3 cos 20ij)-eo sin 20ij]

where rij - r i - rj and Oij is the angle between the z direction and the joint line of the two dipoles. When a dipole is placed inside a capacitor at r i - (xi, yi, zi), an infinite number of images are produced at (xi, yi,-zi) and (xi, yi, 2Lk 4-zi)for k - -4-1, +2,--.. The force that the j t h particle and its images exert on the ith particle is given by

fij,,

_

p2 oo 4s 3r 3 ST~fli j STrZ~' STrZj efL----4 y~ (xi -- Xj) f(1 ( L ) cos( L .....) cos( L ) Pij ,

s=l

f ij,y

_

p2

~

ef L4 ~

s=l

p2

fij~z

4

8371_3

(Yi - Yj)1(1(

sTrPij,)

L "cos(

Pij

co

ef L~ y ~ 4s3 7r3I(o( STcPij ~-"

sTrzi

8~zi~

"L "

cos(

8~zj,)

L "

(4)

87FZj

" L )Sin( L,, )cos( n )

s--1

where Pii - X/(xi - Xd)2 + (Yi yj)2 and Ko and Kt are modified Bessel functions. The force on a particle due to its own images is in the z-direction and given by

f self3p2[1 i,z

8E f

~176 -- 7Z -}- y ~

s=l

1

1

]

(Zi -- 8 L ) 4 - (zi q- 8 L ) 4

(5) "

To introduce hard spheres and hard walls into the simulation, we use a

662

short-range repulsion between two particles ,~p 3p2e, e x p [ _ l O O ( r o / a _ l ) ] fij ~- e_/a 4

(6)

and the short range repulsion between a particle and the electrodes as fwall

3p2ez -- e - ~ { e x p [ - l O O ( z i / a - O . 5 ) ] - e x p [ - l O O ( ( L - z i l / a - 0 . 5 ) ) } .

(7)

The random force Ri(t) has a white-noise distribution,

(Ri.~) - O.

(Ri..(O)Ri.~(t)) - 67rkBTa~6.a6(t)

(8)

where kB is Boltzmann's constant and T is the temperature. We introduce a subinterval T which is shorter than the time steps used in the integration of Eq.(1) but much longer than the molecular collision time 1 [.t+rp. [18]. The average of Ri(t) over T, Ri,~(t, T) -- -~a, - . , , ( t ' ) d t ' has a Gaussian distribution

W(R,..(t..))

-

1

(9)

where ~t - V / 6 r k B T a ~ / ' r . For a time step 6t > T, we can divide it into many subintervals of duration T, in which all quantities except Ri(t) can be treated as constants. Then, for any smooth function r X , = ft ~+6~r has a probability distribution

W(X,)

- (Trq)-'/2 e x p ( - X 2 / q )

where q 12~vkBTa~ (~)d~ and is independent of T. Although the value of T is not very uniquely defined, Eq.(10) implies that our results do not depend on a specific choice of T. The intrinsic time scale in Eq.(1) is to - m / ( 3 r a ~ ) . We re-scale the variables, t - tot*, F i - FoF* where F o - 3p2/(ela4), R i - ~R*(t), and ri - ar* in Eq.(1). The scaling produces a dimensionless constitutive equation ri''* + ri ' * - A(F* + BR*)

(11)

where A - Foto/(3rrla ) and B - gt/Fo. It is interesting to note that A is the ratio of the Reynolds number to the Mason number. The

663

Reynolds number R is the ratio of the inertial force m v 2 / a t o the viscous force 37raCy where v is the speed of dielectric particles. Hence R = mv/(37rrla 2) = vto/a. The Mason number M n is the ratio of the viscous force to the dipolar force, M n = 3raTlv/Fo. Then, it is clear that A - R / M n . In our problem, dielectric particles have negligible speed before the electric field is applied or after the solid structure is formed. During the process, the particles have steady or typical speed, either. As a result, there is no typical Reynolds number or Mason number in our problem. However, their ratio, A, is independent of the particle speed and hence a good parameter for our simulation. We also note that parameter B is the ratio of the Brownian force to the dipolar force. In addition, 1 / ( A B 2) - (1.5T/to))~ where A = ( p 2 / E l a 3 ) / k u T , a crucial parameter [19] in study of ER system in equilibrium process [17]. However, we must remember that our electric-field induced solidification is a non-equilibrium process, during which the temperature is not uniform throughout the system. 3. MOLECULAR DYNAMICS SIMULATIONS

In our simulation we use 122 particles confined in a space of dimensions L~ - Ly - 5a and L~ - 14a. This corresponds to a volume fraction of r - 0.183. There are periodic boundary conditions in the x and y directions. We probe the structure at each step using the following three order parameters, N

1 pj - [ ~ ~

(12)

exp(ibj- ri)[

i=1

where the bj are the reciprocal lattice vectors of the bct lattice bl-

(27cIo)(2etzlv~-ez), b 2 - (2~lo')(2etylV/-6-ez), b3

- 47re~/a. (13)

Of the three unit vectors, ez is along the field direction and e~,ey~ are along the intrinsic axes of the bct lattice. In the calculation of the order parameters, we must rotate the coordinate system around the z axis to find the intrinsic axes of the structure which maximizes pip2. The order parameter P3 characterizes the structure along the z direction while pl and p2 characterize structure in the x-y plane. In the simulation, we assume that an electric field is applied at t - 0 instantaneously. Then we apply an adaptive step-size control RungeKutta method to integrate the equation of motion. It is clear from

664 Eq.(11) that the final structure depends on the two parameters A and B. We examine the dynamic process by monitoring the particles' positions, velocities, and the structure's order parameters. In most cases, after application of the field, the particles quickly move to form chains, then the chains aggregate to form a thick structure. Afterwards, the particles usually fluctuate slightly around their positions in the structure and the order parameters of the system change very little. Hence we are able to determine the solidification time and analyze the final structure. The other interesting quantity is the chain formation time which is shorter than the solidification time. As seen from equations (6) and (7), collisions may interrupt the structure formation if some particle's position change in the integration is too big. Therefore, we specify a value 5r~. If the largest position change among all the particles' motion during one time step 5t is greater than 5r~, the next time step is reduced. Otherwise, 5t will be increased. Our method speeds up the integration and effectively preVents possible problems from the collisions because we control the position change by selecting a proper 5r~. Since the time step changes at every step with this method, we must employ Eq.(10) to handle the Brownian force. There are some special situations which need to be discussed. In the over-damped case, the viscous force becomes so strong that we have i~* - -i~*+A(F*+BR*) ~ 0.

(14)

The over-damping condition requires IA(F*+BR*)I __0.9.

Varying the two parameters A and B in the simulation can be realized by changing the applied electric field and temperature in the experiment. We have clearly seen five different structures of E R fluids in a wide range of A and B" liquid, nematic liquid crystal, glass-like structure, poly-crystalline structure, and bct lattice. Figure 1 depicts the final structures the system evolves over a wide range of A and B. In order to make a comparison, we derive all these final structures in our simulation from an initial randomly distributed state (Fig.2). In Fig.la, we present the result with axes A and B. In Fig.lb, we present the same result with axes A and )~. The boundary between the liquid and the nematic

667

liquid crystal and the boundary between the bct lattice structure and the nematic liquid crystal structure seem to have A = 6.7 and A = 167 respectively. Although this is expected for an equilibrium statistical physics, we should note that the final structure in our simulation may not be an equilibrium state. Therefore, Fig.1 may differ from a conventional phase diagram. 4.1 BCT Lattice and Poly-crystalline Structure As seen from Fig.l, the ordered state, a bct lattice, has been obtained in quite a wide range of A and B. In Fig. 3, we plot a typical bct lattice structure with order parameters p], p~ ~ 0.8 and p3 >_ 0.9. The projection of tile three-dimensional structure onto the x-y plane shows a centered square lattice (Fig.4), a typical characterization of the bct lattice [4]; the marked square also has its side ~ lv/~.5.ha; and the characterization of these chains is also correct for the bct lattice [4].

"1 .--/

~ 3

.'7~

/

"

/.

~ 2 9

....

0

l

I ' ' ' ' 1 ' - ' ' " 1

1

2

....

3

I ....

4

I

5

x (~) Fig.4. Projection of the bct lattice on the x-y plane. The marked square has its side ,-~ x/1--5a. We also note that there is a small region inside this ordered region where the system may likely develop into a poly-crystalline structure (Fig.5). This small region has a very small B and a big A (___ 0.1). Examination of this structure reveals that the system has thick columns consisting of several bct lattice grains. However, these grains do not form a single crystal. There is some mismatch, mainly caused by their rotation around the z axis by slightly different angles. In Fig.6, we plot a part of a thick column of poly-crystalline structure which clearly shows a twist of bct lattice grains. Since these rotations do not affect P3 very much but reduces p] and p~, P3 remains ,-, 0.9 while Pl and p2 are reduced to ,-, 0.5. The poly-crystalline structure is a produ~:t of fast solidification.

668

Because of very small B in this region, the E R system may not be able to relax into a good crystal. It is thus easy to understand that in this region the final structure is somehow sensitive to the initial random state. The computer simulation confirms the conclusion, too: in this region from some r a n d o m initial state the E R system may develop into a good bct lattice, while from some other random initial state the system ends up in a poly-crystalline structure. To further understand the issue, we have paid special attention to the situation of B = 0 which can be realized at zero temperature. If A _ 0.1 and B - 0, the final structure developed from the initial state in Fig.1 has three order parameters around 0.5 while P3 is close to 0.6. However, the final structure at A _> 0.1 and B - 0 is now very sensitive to the initial random state. For example, after changing the initial state at A - 0.1 and B = 0, we have ended up with a good bct lattice" pl and p2 are close to 0.84 and p3 is close to 0.92. This also implies that the ER system at B = 0 can be easily trapped in a local energy minimum. The final structures derived at a moderate B is not sensitive to the initial random state because the ER system can get out from a local minimum energy state and develop into the global energy minimum state with the help of the thermal fluctuations. z 14.00

14.0

11.67

11.6

9.J3

9.3

7.0

7.00

4.6

4.67

2.3

2.33 0.0

3.62 48 ~~

,r e

0.00 4. o.83

Fig.7. In a nematic liquid crystal state, the particles do not aggregate together. D(x, y) is quite uniform over most of the region.

Fig.8. In a nematic liquid crystal state, the system has ordering in the z direction but almost no ordering in the x-y directions.

We have compared the final structures in the poly-crystalline region

670

when we fix A and increase B. For example, at A = 1.0 and B = 0, the final structure derived from the initial state in Fig. 1 has Pl and p2 around 0.5-0.6 and pa abot/t 0.83. When we increase B, although we continue to start the system from the same initial state, the final structure improves. For example at A = 1.0 and B = 0.1, the final structure has Pl and p2 around 0.8 and P3 around 0.87. This implies that a moderate B can help the system relax into a global energy minimum state. z

(~,)

14.00

14.00 11.67 11.67

9. ,1,1

9.,13

7.00

1.00

4.67 4.67

2. ,I,1

2.33

o.o

,1.92

2.30 .x,

0.00 !

,68 ] ....

z.~l

1.15

x(~) Fig.lO. In a glass-like structure In a glass-like structure, tile ER fluids form thick columns in a small columbut t h e t h r e e order p a r a m e t e r s ar~ nar region, a main difference bealmost the same as that of tile ne tween a glass-like structure and matic liquid crystal state. liquid. 4.2 Nematic Liquid Crystal and Glass-like Structure W h e n we increase B, equivalent to raising the t e m p e r a t u r e , we come from the region of the bct lattice to a region of final s t r u c t u r e with significant P3, but small pl and p2. Typically, pl and p2 are around 0.3 while P3 >_ 0.6. This implies t h a t the system has ordering in the z direction with weak or no ordering in the x-y directions. From the definition of a particle density Fig.9.

D(x, y) is peaked

671

N

D(r) - E

5(r-rj)

(18)

j=l

where rj is the position of the j t h particle center, we define a columnar density, L

D(x, y) - / D(r)dz.

(19)

0

After analyzing the columnar density of these structures, we find that within this region, there are two slightly different structures. When A is relatively small and B is relatively strong, D(x, y) is quite uniform, as in Fig.7. This implies that in these structures, the particles do not aggregate together to form thick columns though there is some ordering in the field direction. The system remains in a liquid state, but similar to a nematic liquid crystal structure (Fig.8). When A is relatively strong and B is relatively small, the ER fluids form thick columns. As indicated by D(x, y) in Fig.9, the particles are concentrated in a small region, a main difference distinguishing this structure from a nematic liquid crystal structure. The three order parameters of this structure are not too much different from that of a nematic liquid crystal structure. Although there is some ordering in the fielddirection, there is no significant lateral ordering. Therefore, this is a glass-like structure (Fig.10). In this region, the strong electric field forces the particles to aggregate to form thick columns, but the thermal fluctuations prevent the system from forming a crystalline structure. 4.3 Liquid A further increase of B leads to a region which has the final structure in a liquid state (Fig.l 1). All three order parameters are very small for these structures. The particles are randomly and quite uniformly distributed in the space, as seen from D(x,y) in Fig.12. In this region, the random Brownian force is too strong to prevent formation of any ordered structures.

4.4 Non-equilibrium Process and Boundaries. Our simulation shows a dynamic process. The poly-crystalline structure is a product of non-equilibrium processes. The difference between the glass-like structure and liquid crystal is only in the columnar density, not in the ordering: In a glass-like structure, the particles aggregate together while they do not in a nematic liquid crystal (see Fig.7 and

672 Fig.9). Therefore, poly-crystalline and glass-like structures may not be closely related to the equilibrium state. On the other hand, both the boundaries between the liquid and nematic liquid crystal and the bct lattice and liquid crystal structure seem to be related to the equilibrium state. Although they are not exact, these two boundaries both have A B 2 roughly as a constant. Since the parameter A is proportional to 1/(AB2), these two boundaries are roughly along the lines of a constant A [17]. The boundary between the liquid and the nematic liquid crystal has A B 2 close to 0.25, corresponding to A ,.- 6.7, The boundary between the bct lattice and liquid crystal state has A B 2 close to 10 -2, corresponding to ,.- 167. z (,)

DCx,v)

14.00

14.00

11.57

11.67 [~

9.33

7.00

9.,33 I li~ ~ 7.O0 f-~ II "

4.67

4.57 ~t II

2..13 0.00

4. .f" (~j

4.20 45

0.75

0.0~

~"~

Fig.ll. In a liquid state, three order parameters are vanishingly small and the particles are randomly distributed in the space.

Fig.12. In the liquid state, D(x, y) is quite uniform over the whole region.

5. RESPONSE TIME OF ER FLUIDS

ER fluids are marked for their fast response to an electric field. A number of experiments established that a typical response time of ER fluids is of the order of milliseconds. This response time is Usually defined as the time needed for ER fluids to have a significant viscosity increase

673

immediately after an electric field is applied. In our simulation, we define the solidification time as the time interval between the application of an electric field and the establishment of a final structure [20]. It is clear that our solidification time should be longer than the response time since ER fluids deliver a significant increase of viscosity before they reach their final structure. However, these two time scales are closely related and our solidification time has clear physical meaning and is important for applications as well. The relationship between the solidification time and the parameters A and B is in Fig 13. We note that at a fixed B, the solidification time is getting longer as A gets smaller. In the over-damped case, Eq.(17) indicates that the solidification time is inversely proportional to ' A. Our simulation verifies this conclusion,

t~o,id 60/A.

(20)

-

This relationship holds up to A ,,~ 10 -2. In real time, for example, at A - 10 -3, this solidification time is of the order of a second. As the value of A increases, the viscosity reduces. When the system is not overdamped, the solidification time further decreases as A increases, but this reduction is slower than 1/A. For example, as A increases from 10 -2 to 10 -1 , the solidification time only slightly reduces. 109

lOa

o

B=I B=0.1 B=O.01

o

k: o'3

"

107

E 0

m

.+J .p..i

10e

= 10s o~.,4

104

103

.,,l

1 0-e

!

t

i |ll|ll

1 0-s

i

i

| tJl|ll

__l

1 0-4

!

Ill|l|l

1 0-3

I

!

* |nlJll

10-2

I

I

I |Ill|

1 0-f

A

Fig.13. The relationship between the solidification time and the parameters A and B.

674

At a fixed A, the solidification time increases with B. This is due to high fluctuations of the Brownian motion. However, the effect of B is significant only when B is large enough. For example, if B < 10 -2, the solidification time is almost unaffected by B. If B > 10 -2, the thermal fluctuations delay the solidification process. For example, at A = 10 -~ and B = 10 -2, the formation of a bct lattice structure takes about 1.738 • 105t0 while at A = 10 -3 and B = 10 -1 the solidification of a similar bct lattice takes 2.81 x 105t0. In our simulation, we also determine the chain formation time by examining the order parameter p3. From Fig.14, it is clear that the chain formation is much faster than the formation of a final structure. This again implies that the particles in ER fluids form chains first, then chains aggregate together to form thick columns. Typically, the chain formation time is about one third of the solidification time or shorter. In the overdamping case, the chain formation time is also proportional to 1/A. We also notice that in real time, the chain formation time is of the order of milliseconds, the same order as the response time found in engineering applications.

'~ I%

lOe

%,.

I-

\\

o

B=I

o ,

s=o.1 s=o.ol

lO7

=

1 oe "~

lOS

104

10a 1 o-e

10-S

10-4

A

10-a

10-2

10-I

Fig.14. The relationship between the chain formation time and the parameters A and B. 6. DISCUSSIONS In this section, we want to compare our simulation results based

675

on the constitutive equations with experiments. For a real E R system, such as dielectric particles in petroleum oil, e / ~ 2, % >> 1, ~ ~ 0.2 poise, a ~ 10#m, and the mass density of the particle p ,.., 3 g / c m 3. We estimate to ~ 8.33 x 10 -7 s. If we choose the subinterval 7 = 0.4t0, then as E0 varies from 0 to 4 K V / m m at T=300 K, A changes from 0 to 10 -2 and B reduces from c~ to 10 -3. When A - 10 -2 and B - 10 -2, for example, our simulation finds the chain formation time is about four milliseconds while the bct lattice and the solidification time is less than one second. In the experiment, the chain formation takes milliseconds to complete, but the formation of bct lattice is slower than that in our computer simulation

[7].

As the particle size becomes big, the inertial time to and A increase. For example, if the above ER fluid has everything the same except a ,.~ 100#m instead of 10#m, then we have A ,,~ 1 at E - 4kV/mm. Hence, from Fig.l, we notice that ER fluids with large particles are easy to develop into a poly-crystalline structure in the non-equilibrium process. This interesting result is useful in production of composite materials by the ER effect [9,10]. Our results at B = 0 are interesting enough to warrant some experimental investigation. The fact that the final structure at B = 0 is sensitive to the initial state indicates that the Brownian force plays an important role in driving the ER system from a'local energy-minimum state into a global energy-minimum state. On the other hand, if B is too strong, the thermal fluctuations prevent the system from forming a good bct lattice. Therefore, an experimental determination of this range of B will be very interesting. This goal may be achieved by examination of ER fluids at cryogenic temperatures. Our simulation also reveals that the response time defined in ER engineering applications is related to the chain formation time. We have also found a relationship between the solidification time and the viscosity, temperature, and electric field. It will be very interesting to see if this relationship holds in experiments. A CKN OWLED G E M ENTS

This research is supported by a grant from National Science Foundation DMR-9622525. REFERENCES

1. Electrorheological Fluids, edited by R. Tao and G.D. Roy (World Scientific Publishing Comp., Singapore, 1994). 2. L. C. Davis, J. Appl. Phys., 72 (1992), 1334; 73 (1993), 6so.

676

3. H. Block and J.P. Kelly, US Patent No. 4,687,589 (1987). 4. F.E. Filisko and W.E. Armstrong, US Patent No. 4,744,914 (1988). 5. R. Tao and J. M. Sun, Phys. Rev. Lett., 67 (1991), 398; Phys. Rev. A, 44 (1991), R6181. 6. J. E. Martin, J. Odinek, and T. C. Halsey, Phys. Rev. Lett., 69 (1992), 1524. 7. R. Tao, J.T. Woestman, and N.K. Jaggi, Appl. Phys. Lett., 55 (1989), 1844. 8. T. J. Chen, R. N. Zitter, and R. Tao, Phys. Rev. Lett., 68 (1992), 2555. 9. G. L. Gulley and R. Tao, Phys. Rev. E, 48 (1993), 2744. 10. X. Wu. X. Zhang, R. Tao, and R. P. Reitz, Bull. of Amer. Phys. Soc., 41 (1996), N.1, 191. 11. C. A. Randal, C. P. Bowen, T. R. Shrout, G. L. Messing, and R. E. Newnham, in ref. 1, p516. 12. R. Tao and Q. Jiang, Phys. Rev. Lett., 73 (1994), 205. 13. D. J. Klingenberg, F. van Swol, and C. F. Zukoski, J. Chem. Phys. 91 (1989), 7888; 94 (1991), 6170. 14. N. K. Jaggi, J. Stat. Phys. 64 (1991), 1093; W. Toor, J. of Colloid and Interface Science 156 (1993), 335. 15. K. C. Hass, Phys. Rev. E, 47 (1993), 3362. 16. R. T. Bonnecaze and J. F. Brady, J. of Chem. Phys.,96 (1992), 2183. 17. H. X. Guo, Z. H. Mai, and H. H. Tian, Phys. Rev. E, 53 (1996), 3823. 18. R. Tao, Phys. Rev. E, 47 (1993), 423. 19. For example, see, S. Chandrasekhar, Rev. Mod. Phys. 15 (1943), 1; R. Reif, Fundamental of Statistical and Thermal Physics (McGrawHill, New York, 1965), 560-562. 20. P. M. Adraini and A. P. Gast, Phys. Fluids, 31 (1988), 2757. 21. H. See and M. Doi, J. of Phys. Soc. of Japan, 60 (1991), N.8, 2278.

677

ELECTRO-MAGNETO-HYDRODYNAMICS AND SOLIDIFICATION

G. S. Dulikravich

Aerospace Engineering Department, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

1. INTRODUCTION Fluid flow influenced by electric and magnetic fields has classically been divided into two separate fields of study" electro-hydrodynamics (EHD) studying fluid flows containing electric charges under the influence of an electric field and no magnetic field, and magneto-hydrodynamics (MHD) studying fluid flows containing no free electric charges under the influence of a magnetic field and no electric field. Traditionally, this division was necessary to reduce the extreme complexity of the coupled system of Navier-Stokes, Maxwell's and constitutive equations describing combined electro-magnetohydrodynamic flows. Recent advances in numerical techniques and computing technology, as well as fully rigorous theoretical treatments, have made analysis of combined electro-magneto-hydrodynamic flows well within reach. A survey of electro-magnetics and the theory describing combined electro-magnetohydrodynamic (EMHD) flows is presented with an emphasis on describing the intricacies of the mathematical models and the corresponding boundary conditions for fluid flows involving linear polarization and linear magnetization. This survey concludes with a presentation of EHD and MHD flow models involving solidification. NOMENCLATURE b - electric charge mobility coefficient, kg A s2 B_B_ = magnetic flux density vector, kg A 1 s2

678

(

d = Vv_ + Vv

= a v e r a g e rate o f d e f o r m a t i o n t e n s o r , s -~

Do

= e l e c t r i c c h a r g e d i f f u s i o n c o e f f i c i e n t , m 2 sl

D=eo_E+P

= e l e c t r i c d i s p l a c e m e n t field v e c t o r , A s m -2

e : c~T + (v._v)/2

= total e n e r g y p e r u n i t m a s s , m 2 s ~

E

= e l e c t r i c f i e l d v e c t o r , k g rn s 3 A -~, or V m ~ = e l e c t r o m o t i v e i n t e n s i t y v e c t o r , k g m s 3 A -1

E=E+vxB

f g

= mechanical body force vector per unit mass, m s2 = a c c e l e r a t i o n d u e to g r a v i t y , rn s -2 = h e a t s o u r c e or s i n k p e r u n i t m a s s , m 2 s -3

h

a_=B_/ J= J,

to-M + Jd

= m a g n e t i c f i e l d i n t e n s i t y v e c t o r , A m -1 = electric current density vector, A m 2 = electric conduction current vector, A

m 2

J d = V_qo

= e l e c t r i c drift c u r r e n t v e c t o r , A m -2

M

= total m a g n e t i z a t i o n v e c t o r p e r u n i t v o l u m e , A m ~

M=M+vxP

= m a g n e t o m o t i v e i n t e n s i t y v e c t o r p e r u n i t v o l u m e , A m -1

P

= p r e s s u r e , k g m -~ s -2

P qo

= total p o l a r i z a t i o n v e c t o r p e r u n i t v o l u m e , A s m -2 = total or f r e e e l e c t r i c c h a r g e p e r u n i t v o l u m e , A s m -3

q

= h e a t flux v e c t o r , k g s -3

S

= e n t r o p y p e r u n i t m a s s , m 2 kg-' K" s -2

T

= absolute temperature, K

V

= fluid v e l o c i t y v e c t o r , m s -~

GREEK SYMBOLS = v o l u m e t r i c t h e r m a l e x p a n s i o n c o e f f i c i e n t , K -1 E

= C h o r i n ' s ( 1 9 6 7 ) artificial c o m p r e s s i b i l i t y c o e f f i c i e n t = d i e l e c t r i c c o n s t a n t ( e l e c t r i c p e r m i t t i v i t y ) , k g -~ m -3 s 4 A 2

ro = 8 . 8 5 4 x 10 -12

= v a c u u m e l e c t r i c p e r m i t t i v i t y , k g ~ m 3 s4 A 2

8r = S / F , o

= relative electric permittivity

K

= thermal conductivity coefficient, kg m s3 K l = e l e c t r i c c o n d u c t i v i t y c o e f f i c i e n t , k g -1 m -3 s 3 A 2

P

= fluid d e n s i t y , k g m -3

_x_v = 2 g v d + g v E I ( V 9v ) - N e w t o n i a n v i s c o u s stress t e n s o r , k g m -1 s -2 x

EM

= e l e c t r o m a g n e t i c stress t e n s o r , k g m -1 s -2

679

v

EM

x__=x__ + x__

= stress tensor (viscous plus electromagnetic), kg m -~ s -2

~t

= magnetic permeability coefficient, kg m A 2 s -2

~to = 4~ x 10 - 7

=

ktr

relative magnetic permeability = shear coefficient of viscosity, kg m ~ s -~ = second coefficient of viscosity, kg m ~ s -~

= ~t [ k t o

~tv ~tv2 E = er _ 1 M = [t r _

1

= x__~'d

magnetic permeability of vacuum, kg m A -2 s 2

=

= electric susceptibility = magnetic susceptibility = electric potential, V = viscous dissipation function, kg

m ~ s -3

2. BACKGROUND

The scientific field of study that analyzes the ability of electro-magnetic fields to influence fluid flow-field and heat transfer has been investigated for decades. The equations that are most often used to model this phenomena consist of the system of Navier-Stokes equations for fluid motion coupled with Maxwell's equations of electro-magnetics augmented with the material constitutive relations. The field studying these flows is often called electromagneto-dynamics of fluids [ 1], electro-magneto-fluid dynamics (EMFD) [2-5], electro-magneto-hydrodynamics [6], magneto-gas-dynamics and plasma dynamics [7], or the electro-dynamics of continua [8-10]. The full system of governing equations has, until recently, been far too difficult to solve because Navier-Stokes system becomes very complex when modeling flows involving turbulence, chemical reactions, multiple phases, non-Newtonian effects, etc. When coupled with Maxwell's equations, the complexity of the combined EMHD system is raised by orders of magnitude. To reduce this complexity, the analytical modeling has traditionally been divided [11] into flows influenced only by externally applied electric fields acting upon electrically charged particles in the fluid, and flows influenced only by externally applied magnetic fields without electric charges in the fluid. The former are called ElectroHydrodynamic (EHD) flows [12] and the latter Magneto-Hydrodynamic (MHD) flows [13]. More recently, rigorous continuum mechanics treatments of EHD [14] and unified EMHD flows [9,10] have been developed. These continuum mechanics approaches are limited to non-relativistic, quasi-static or relatively low frequency phenomenon [ 15-17].

680 This chapter should provide an introductory survey of the background theory to allow implementation of numerical analysis of unified EMHD flows and of classical MHD and EHD flows with addition of liquid/solid phase change. An overview of electro-magnetic theory with concentrated effort placed on descriptions of the electric and magnetic fields and electric charges and currents will be made to provide a physical understanding of the field-material interactions causing polarization and magnetization effects. The system of equations governing the unified EMHD theory and the corresponding boundary conditions will be presented together with its fully conservative form that is ready for numerical discretization.

3. POLARIZATION AND GAUSS' LAW Charge polarization is created when electric charges of opposite signs are separated by a distance. Although many references define several sources of polarization [ 18], there are essentially two main sources of polarization: natural and induced [13]. Natural polarization arises from natural dipoles and charged particles. An example of a natural dipole is a water molecule which has a geometry such that the centers of positive charges and negative charges do not coincide. Since the molecules are allowed to move freely and orient randomly, water will not have polarization on a continuum level. Now consider the fluid water as it is frozen with an applied electric field. An induced polarization will be created by the electric field by inducing an initial charge separation in neutral particles [19], by causing greater charge separation within the molecules, and by causing molecular alignment with the applied electric field in case of natural dipoles [19]. Once locked in the ice crystal structure, the water molecules will no longer be able to change their position or orientation. Consequently, even after the electric field is removed, the ice will still have polarization on a continuum level since the polarization caused by the electric field aligning the water molecules was literally frozen into the ice. From this example it may seem that there is no reason, when dealing with fluids, to consider natural polarization. This, however, would be an erroneous assumption. Though the natural polarization may show no continuum effects without the presence of an electric field, in an electric field the total polarization, _P, combines both the induced polarization due to the electric field and the natural polarization of the molecules which are now aligned by the electric field [ 13, p.22].

681

If polarization is assumed to be a linear function of the steady or relatively low frequency electric field, then it can be defined as P__- GoXE(E__4- v x B)-- gp(E q- v x B ) = GpE

(1)

The electric displacement vector then becomes [19, p.164] [9, p.178]. __D-- 8oE_E_+ P__= 8o(1 + xE)E + CoXE~ x B -- go8r___E+ 8p V X B - ~___+ 8p V X B

(2)

where the material property, ZE, is the dielectric susceptibility. It is typically obtained experimentally [20, p.86] and could be a function of frequency. Electric charges come in two types: free and bound. Free charges arise from electrons in the outer or free atomic shells and from ions. Bound charges are those arising from the molecular geometry and displacement of atomic inner electron shells [13, p.21]. Gauss's law for a linearly polarizable medium then becomes [ 13, p.22]

V.D_=qo

(3)

or

V. (ao___+ P_)= V .(~_ + epVX B_B_)= qo

(4)

At this point it is important to note that qo multiplied with the charged particle drift velocity, v d , creates the convection or drift electric current, Jd [ 13, p.67], while polarization current, J_p, is defined as the variation of the total polarization with respect to time [ 19, p. 121 and p. 147].

4. M A G N E T I Z A T I O N AND A M P E R E - M A X W E L L ' S L A W If the material in question may be considered linear, that is, if the magnetization is a function of one material property and the strength and direction of the applied magnetic field, then the magnetization is defined as [9, p.178] [19, p.164] [20, p.92-96] [21, p.371-377] ~M

Po ~1+

~M)

682 In addition to the electric currents arising from magnetization and direct charge motion, other phenomenological currents have been observed and must be taken into account when defining the total current, J [9, p.162-163]. Introducing the effects of magnetization and polarization and rearranging constants, the Ampere-Maxwell's law of electrodynamics may be rewritten as [ 13, p.30]

VxB__--po VxM-k-Jd +~p -k-

(6)

Magnetization and magnetic field vectors are often combined to form the magnetic field strength vector, H__,defined as H - B _ M

(7)

~to The total current, J, is defined as the sum of the apparent magnetization current, V x M, charge drift current, Jd, and phenomenological polarization currents, J, [13, p.26] since the contribution to the magnetization current by intrinsic magnetization is zero. The Ampere-Maxwell's law for polarizable, magetizable media can therefore be written as [ 19, p.132] 0D -

VxI-I=-j

(8)

0t Detailed descriptions of these equations can be found in any number of texts [19,20,21].

5.

A MODEL OF UNIFIED ELECTRO-MAGNETO-GASDYNAMICS (EMGD)

The full system of equations governing unified EMGD flows consists of the Maxwell's equations governing electro-magnetism, the Navier-Stokes equations governing compressible fluid flow, and constitutive equations describing material behavior. Assuming a single-phase fluid and only one type of charged particles in the fluid, this set has a minimum of 12 partial differential equations that contains 13 unknowns: p, qo, T, p, and the three vector components of _v, E__, and B, respectively. The thirteenth equation is the equation of state for the

683

fluid. The foundations of the electro-magneto-gasdynamic (EMGD) theory were formulated by Eringen and Maugin [9,10] and are based on continuum mechanics [22-25]. The rigor with which the constitutive, force, and energy terms were derived leads to a model more complete and robust than any of those found in classical literature [8,7,1,11-13,18-21 ]. Dulikravich and Jing [6,26] have shown that a compact vector form of the unified EMGD system can be written as a combination of the Maxwell's electro-magnetic subsystem and the Navier-Stokes fluid flow subsystem. The Maxwell's subsystem (consisting of seven PDE's) is composed of Ampere-Maxwell's law for polarizable and magnetizable medium 8D -

-

Ot

VxH=-J

(9)

that can also be written as OE 8t

Vx--=--l+ eo eo

-

(10)

Faraday's law 0B -

~VxE=0

(11)

8t and conservation of electric charges 8qo ~+ 8t

V._J_ = 0

(12)

that is a combination of Gauss' law V.DD_ = qo

(13)

and the Ampere-Maxwell's law. Conservation of magnetic flux V.B =0

is also a part of the Maxwell's subsystem, but is not solved for explicitly.

(14)

684

The second part of the unified EMGD is the viscous, compressible flow Navier-Stokes subsystem consisting of five PDE's and an equation of state of a perfect gas. It is composed of conservation of mass equation OP + V.(p_v)- 0 0t

(15)

and a conservation of linear momentum (including electromagnetic effects) 0p_v

+V (v_gv+pI-~) 9

_

--

V (v(P 9

__

B/)

X

--

V ((B_B_M II+(___ P)I) 9

9

=

"

-"

P

S

V

(16)

0t

Here, I is the identity (unity) tensor and S v is a vector of source terms. following dyadic identifies were used in equation (16)

The

(va). M_= v. ((B. M _ ) I ) - ( ) . B_

(17)

(vE). e = v. ((E. e)i)-(ve)._E

(18)

Conservation of energy equation is also a part of the Navier-Stokes subsystem

0(Pe----~)-t- V- (pev + (pI-__x)-v)+ V. ~ 1 - p h - 9 E . ~ Ot

-

-

-

+ Iql. D B - J c .1:: = 0 Dt

--

Dt

(19)

-

It can be replaced by the entropy generation equation [9,2,4]

9

Ds _ oh +_______~@V. _ 9DtT

-T5

J-

Dt

-+&.E

-- Dt T

(20)

The viscous stress tensor for a non-linear fluid is given as v

1; - 21avd + ).tv2I(V. _v) + oqd 2

(21)

685 In the case of a media with non-linear physical properties, the unified EMGD formulation for the electric conduction current and the heat flux can be expressed as [9, p. 161-162].

Ic -- O'1~q- G2~'--~ -t- ~3 d2 "E + G4VT 4- ~5 d" VT + 0"6d2" VT + o7E x B (22)

+ cY8(tl-(E x B ) - (d- E) x B__)+ cY9VT xB + o'lo(d_. (VW x B ) - ( d " VT) x BB)+ (Yll(B 9E)B__+ o'12(B_-VT)B__ q = K1E -I'-K2d" ~ q- K3d2 "E q- K4VT + tcsd-VT + K6d2. VT + K:vE_x B

(23)

+ ~a(a=. (g • B_)- (a_. E) • B_)+,,:9VT • B_ + Klo(d" (VT x B ) - ( d - V T ) x B)+ Kll(B'E)B + KI2(B 9VT)B

The electro-magneto-thermal stress tensor for a non-linear fluid can be expressed as [9, p. 177-178]

I;

EM

= ~2~ (~)E + ot3B (~)B + o~4VT (~)VT + or5(_E| d. E)s + 0~,6 (_E~)d 2 -~__.1+o~7(VT | d-VT)s+ ot8@T |

2- VT~

-ko~9(_d.'W-W-d)q-O,,loW.d-W -+-Otll(d=2 . W - W . d

- w 2 . . w),- 0,3( |

T/s +

2)

E | E-W/s

+ O~15(___.~) ~ W" F)S -t- O~16~" ~(~)W2 9E_~ + oqv~W-(E| VT- VT | + o~,18d-(E (~ V T - VT |

(24)

s

0~,18(E 1~)VT - VT |

where ~____W=Wij=gijkBk, while the subscript s indicates symmetrization. Expressions for total polarization, P, and magnetization, M, of non-linear media can be modeled with expressions of similar complexity [9, p.175]. In these formulas, c~i, cE and Ki are the physical properties of the media. Most of these coefficients are still unknown although their exploitation can offer potentially significant benefits in applications involving interacting electric, magnetic, thermal, and stress fields. This theory is valid for the frequencies of the electric and the magnetic fields that are less than approximately 1 kHz and for fluid speeds considerably less than the speed of light [14-17]. For higher frequencies, certain physical properties become functions of the frequencies. For higher speeds, relativistic effects will have to be taken into account.

686

6. CONSERVATIVE FORMS OF E L E C T R O - M A G N E T O HYDRODYNAMIC (EMHD) SYSTEM A necessary condition that an iterative numerical solution of the EMGD system will converge to the exact solution of the analytical EMGD system as the computational grid is infinitely refined, requires that the EMGD system must be rewritten in a fully conservative (divergence-free) form. This is especially needed if strong gradients of dependent variables are expected to exist in the solution domain. The fully conservative forms can then be used directly in the finite difference, finite volume, or finite element discretization of the EMGD system and its iterative integration process. In the following derivations, it will be assumed that the fluid is incompressible, homocompositional, that it has linear polarization and linear magnetization properties, and that the frequencies of the applied electric and magnetic fields are less than approximately 1000 Hz for this mathematical model to be realistic. These are the only assumptions to be used in this model which will be referred to as a unified electro-magneto-hydrodynamics (EMHD). A fully conservative EMHD system in a vector operator form is given as [6]

OE_E_

Vx

8B 8t Oqo

-H-

=

8o

Ot

(25)

S E -

+ VxE = 0

(26)

+v.J_

= 0

(27)

V.v

= 0

(28)

ot

Ov - + v . l v v + - (1p i 8t

-8e - + - V1 & p

P

)) - 1 v . I v ( l , • B)+(B.M)I+(_EE.p)!]=S v P

.

.

.

.

.

-=

(29)

-

.(pe_v+ ( p I - x=)-_v + _q ) = S e

For simplicity of notation we can define the following terms as [6,26]

(30)

687 1

F=/.to (r

1

M) =--~t

(31)

1 =-

(32)

ep =8oZ E --e--e o

(33)

=

A=

1

Co(1+ Z E)

(1 p +

zEI+Pe p B . B

(34)

8 = pf_ + qo_E + J_• a + (v_). P + (va). M + v . (v(_P • ~_))- v . (~_v + p i - a)(35)

_Dt=V•

IB /

(36)

- -M_M_ - J _ = V x H - J _

go

Pt = A D t +A(VxE_)xx 80 + p(1 + Z E

Dt +

(37) x

A[Rt 9 - P_P_x (V x E__)]x B

If we now assume that the fluid which is subjected to applied electric and magnetic fields is of Newtonian type and if we allow only for linear polarization (equation 1) and linear magnetization (equation 5), the constitutive relations for the electric conduction current and for the heat flux vector become [9, p.173-174]

lie ~ ~I(E~ -b X x B)-4- t34VT -t- o'7(]E -+-v x B__)x B -!-(Y9VT • B q- 0"1,(B 9(~E + Ir • B__))B-b o12(B- VT)B

(38)

688 _q- K1(_E + v x B_)+ ~:4VT + K:7(_E+ _v x B_) x B (39)

+ K:9VT x B_B+ KI, (B_-(__~+ v x B))B + KI2(B- VT)B_

Then, the EMHD source terms can be given in a compact vector form [6,26] as

S_E =

1 (j_.+ Pt )

(40)

~o s_V _ [ + _ h o E + l 9

(V x _ ) x ___-(VM). B_B- (VP__)-E+ (J + P--t)x B_B_]

S e --h +!(EwvxB__.). [(v-v)e+ J_c + ~t]--!~Z M B. ((v-V)B-V x F~) 9

(41)

(42)

9

Notice that these source terms have been formulated in such a way as not to contain explicit time derivatives [6,26].

6.1 Fully conservative Cartesian form of the EMHD system The EMHD system of equations (equations 25-30) can now be written in a general conservative form in terms of (x,y,z) orthogonal coordinate system as cqQ ~+ &

c3E

0F 3G +~+~=S 0x 0y 0z

~

(43)

Here, the solution vector of unknown quantities is given as

{

Q=

E x,

Ey, E z,

B x,

By, B z,

qo, p Vx Vy v z e

}"

(44)

where the asterisk symbol designates transpose of a vector. The vector of source terms (those terms that do not contain divergence operator) is given as

- { x S , Sy, S z, 0, 0, 0, 0, 0,Sx,v Sy,v Sz,vse} S=

(45)

689 In equation (44), Chorin's [27] artificial compressibility coefficient, 13, was used to create the unsteady term in the mass conservation since physical unsteady term does not exist in the mass conservation for incompressible fluids. By combining equations (5), (7), (1), and (31), the Cartesian components of the magnetic field intensity vector can be defined [6,26] as H x = gB x + t~pVy(E z + vxBy - VyB x ) - 8pV z (Ey + vzB x - vxB z) Hy = ~By + t3pVz (E x + vyB z - VzBy) - 13pVx (E z + vxBy - vyB x )

(46)

Hz = g B z + gpvx(Ey + vzB x - v x B z ) - gpvy(Ex + vyB z - vzBy) The flux vectors in equation (43) can then be defined as

Hz/eo

/~o

-Hy/eo 0

-Ez

-Ex

Ey

Jy ~=Vy

Jx

E= Vx

2 l

Vx

Evvx-txy+

EP

[V~y--}-~~)--'lTyy-- N~BP-VyN~yB)

VxVy--?

Ev,vzt z+vz y t (47)

(48)

690

Hy/eo

-nx/~o 0

0 Jz G=,Vz 1

VzVx-~ (~xz+ Vx~ ~)

~z +!(p-~zz-~.~-Vz~z ~) p-

(49)

eVz§ Here, we have written components of (__Px B_B_) as NxPB= PyBz _ PzBy

(50)

PB

Ny =PzBx - P x B z N PB = PxBy - PyBx

In addition, we have defined the terms NEPBM-- EX Px + EyPy + EzP z + B x ( Bx - H x ) + By ( B y _ Hy ) + B z ( Bz - H z)

go

0T

(51)

go

(52)

Nap = BxP x + ByPy + BzP z

0T

go

0T

N ~ - ax ~ + ar ~ + az--~z

(53)

691 NV~ _ v x ( - p + Xxx)+ Vy'l~xy+ Vz'ISxz

v,~ Ny - v x'l;xy + Vy ( - p + "l;yy) + Vz'l;yz

(54)

NV~ - VxXxz + Vy'l;yz + Vz ( - P + Zzz)

Components of the electric current vector, J, were defined as

(~1 ~p

~ 0"7 PB ~Z 0T +~N x +cy9(B zB ) ~XX {~p ~-~Z y

Jx=vxqo+~Px+~4

+

O'11 NBpB x + O'lzNBTBx ~;p ~1

= Vyqo + - - e y ep +

+0" 4

OT c~

+--N

Sp

Y

+(I 9

~Z B x - ~ - B z )

{3"11 - NBpBy + o'12NBTBy E;p

O"1 Jz - Vzqo + ~ P z

+ 0"4

{~p

OT

+

G7

~ZZ ~p

PB OT OT Nz + (I9 By B x)

(~-

~-

O'11.NBpB z + (I12NBTBz 8p

(55)

and heat flux vector c o m p o n e n t s were defined as

qx = ~~(~px + K4 ~~

_ KI

~

1(7 PB + K9( ~ ,Bz - o~F B ) + ~KI~ +~Nx N B p B x + K:12NBTBx 8p -~C~Z Y 8p

1(7

PB

K:7

Pa

{ly -8pPy 1" K4 ~0y' 1 " ~8pN y -

Clz - }c--LPz+ K:4

~p

0T

~Z

+~Nz

8p

0T

0T

NBpBy + K12NBTBy (56) + K9(-~-ZBx - ~ - Bz)+ 1(1! 8p

~

+ 1(9(

aT

By - - ~ - B x ) +

Kll

- N B p B z + K12NBTB z

8p

692 7. CHARACTERISTIC-BASED I N F L O W AND O U T F L O W BOUNDARY CONDITIONS For most boundary value problems of electro-magneto dynamics, jump conditions are exclusively used [9,28] to formulate solid wall boundary conditions where a discontinuity occurs. At the inflow and outflow boundaries where no surface or line discontinuities exist, an alternative approach based on conservation law for continuous surfaces or lines become necessary. Characteristic boundary condition formulation [29,30], which starts from a characteristic form of the EMHD system, will be sketched here since it leads to non-reflecting boundary condition formulation [31-36,26]. To find the characteristic boundary conditions, it is first necessary to determine analytical expressions for all eigenvalues of the characteristic system. The most common approach is to use one of the symbolic programming languages software (LISP, MACSIMA) in order to determine analytical expressions for each eigenvalue. Since these software packages cannot be used for systems that have more than five coupled partial differential equations, in the case of a complete EMHD system which has twelve coupled partial differential equations, it is impossible to find the eigenvalues using available symbolic programming software. Consequently, we will use an alternative approach in which we will divide the unified EMHD system into a Maxwell's subsystem and the Navier-Stokes subsystem [33]. Each of these two subsystems will then be analyzed separately by finding the analytical expressions for its eigenvalues by hand. 7.1 C h a r a c t e r i s t i c - b a s e d b o u n d a r y conditions for M a x w e l l ' s s u b s y s t e m

For example, characteristic treatment of the Maxwell's subsystem can be formulated by rewriting the fully conservative Maxwell's subsystem

at

+~

Ox

+

Oy

+~

SEM

az

(57)

in a non-conservative (characteristic) form as

a0EM+ AEM 0EM + BEM 0t

-

0x

0y

+ CEM --

0EM -- SEM 0z

(58)

693

In order to perform characteristic analysis for Maxwell's subsystem, care must be exercised to ensure that all the terms appearing in the fluxes EEM, FEM,GEM are expressed as functions of the primitive variables

-

{

QEM--Ex,

E z,

Ey,

B x,

B z, qo

By,

}*

(59)

For illustration, the flux vector EEM can be extracted from equation (47) as 0 Hz/eo -Hy/8 o (60)

0

EEM --

-E z Ey Jx

For fluids with linear polarization and magnetization, Hz and Hy are the same as in equations (46), while Jx is given in equation (55). The flux vector Jacobian matrix A___EMis obtained as

A EM -- 0EEM = -OQEM

0

0

0

0

0

0

0

azl

a22

0

a24

a25

a26

0

a31 0

0 0

a33 0

a34 0

a35 0

a36 0

0 0

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

_a71

a72

a73

a74

a75

a76

Vx_

(61)

where the coefficients are a21 = --~EVy

a31 =--~Ev z

(62)

a22 = zEv x

a33 = ~Ev x

(63)

694 a24 = ~Ev x Vz

a34 = --~Ev x Vy

(64)

a25 - ~EVyVz

a36

= -Z E

(65)

a26-

-

+Vy

VyVz

a35 = - ~ + Z ~go

ktgo a71 = 0-1 + ~llB2x

+Vz

a72 = cY7Bz + o l l B x B y

a74 = t37(VyBy + VzBz) + Oll(ExB x +

- 0 " 1v z -

O"7(Pz + epvxB

(67) (68)

a73 = -(YTBy + O'llBxB z

a75 =

(66)

NBp )

+ O'I2(NBT +B x

)--0"9 0T +~11Ey B +

0"7 ~p

0T vy

a76 =O,Vy + - s - ( P y - g p v x B z ) + 0 9 ~ +

/9"I'

(69)

Bx 0I'

(70)

0T 0z

(71)

Ol,EzBx +Ol2B x

Matrices BEM and CEM may be obtained in the same fashion as equation (61). After tedious algebraic manipulations [26], the vector of eigenvalues of the flux vector Jacobian matrix A s s is found as

-

{

k~M = 0, Zk, k~, 0, Z~,, k~,, Vx

}"

(72)

This means that the eigenvalues ~1 "- ~4 - - 0 , while ~'7 -- Vx" The remaining four eigenvalues can be obtained from the fourth order algebraic equation ~4 "t- (/,EM ~3 + VEM~,2 + ~EM ~ + 6EM -" 0

(73)

where the coefficients in the fourth order characteristic polynomial are ~EM ------a22 -- a33

(74)

695

VEM "- a22a33 - a26 + a35

(75)

~EM -- a26a33 - a22a35

(76)

~EM -- a25a36 -- a35a26

(77)

The four eigenvalues are the analytical roots given as +

10

~E=--4 "

~'E=--~

+

19

fil

1 (II/EM-I-nEM~

(78)

"i-~O2EMI--~(II/EM'F~-)EMo)

(79)

EMI+-vi-602MI---2

~/'1

EMI--

1

~:~ - -~-r

i

~1

1

- -~(v~,,, - a~.,,,o)

(80)

jm~ 0 2 M 2 -- -~1 (ll/EM -- ~"~EMo)

(81)

+ igc, G

~'B -- -- ~ (I)EM2 --

1

Here, different terms are defined as 2 _ 4 V E M + 4~EM = O~EM + (I)EMo (I)EMI -- (gEM + ~/ (I'EM

(82)

2 (g EM - 4 v EM + 4~4/EM = (X,EM --OEM o

(83)

(I) EM2 = ~ EM --

(I)EMo = 4Ot,2M --4VEM + 4q/EM

(84)

~"~EMo = ((~EMVEM -- 2~EM)/OEMo

(85)

3

ZEM =

3(aEMYEM_ 48EM) _ VEM 2

(86)

(87)

696 [

3 YEM = VEM (4~EM -- (XEM~tEM) I VEM

xl4VEM6EM-- ~tEM 2 -- (XEM~EM) 2

6

2

27

(88)

For illustrative purposes, the following are the eigenvalues in the case of onedimensional EMHD flow where Vy - v , - 0 and a22 = a33 and a25 = 0. Hence

~-~-

~E--~B--~

;[E IE2 / 1 x Vx+ x Vx+4 ~;o~to(1 +

" X

Vx--

2)]

X M ) -- ~EVx

xE2Vx +4

--Z Vx eo~to(1 +

(89)

(90)

)

1 Since /~o~-~o equals the speed of light in vacuum, it seems that for most

practical applications the incoming and the outgoing electromagnetic waves will not be influenced by the fluid except in the situations where the fluid is very highly ionized or when the fluid moves with a speed comparable to the speed of light. In the case of a pure electro-magnetics without any fluid motion, polarization, magnetization, or electric charges (v = __P= M =qo =0), these eigenvalues reduce to the eigenvalues of Maxwell's equations for electromagnetic fields in vacuum [35] 1 -

0, 4

1 ' -

1 S'

0, 4 '

1 -

}*

(91)

After introducing the similarity transformation matrix S__EMof the flux vector Jacobian matrix ~ r M ' the eigenmatrix ~EM corresponding to ~EM becomes ~"

+

~EM diag[ 0, ~E, LE,0, L+B,L-B, v x ] -

-

(92)

where )~E, )~E, )~a, )~a are given by equations (78-81). For locally one-dimensional problems, wave propagation direction is well defined. For multi-dimensional problems, there is no unique direction of

697

propagation, because the flux vector Jacobian matrices AEM , BEM , CEM cannot be simultaneously diagonalized. Therefore, characteristic boundary condition analysis allows that only one of these matrices (relating to only one coordinate direction) can be diagonalized at a time. In the case that the x-coordinate is in the main flow direction, premultiplying the equation (58) with the inverse of the similarity matrix, SEI~, gives

SE1M~QEM

~-t-

Ot

'~ -1 ~QEM ~EMSEM ~-t-

=

c~x

-I ~' SEMHEM -- 0

(93)

Here, vector HEM is given as

H r u = BEta 0QEra + CwM aQEM SEM --~ = aZ

(94)

For the hyperbolic system, time dependent boundary conditions could be derived based on the principle that outgoing waves are described by characteristic equations, while the incoming waves may often be specified by a non-reflecting boundary condition [31,32,36]. Following this approach, the characteristic and non-reflecting boundary conditions at the inlet boundary x = a and at the outlet boundary x = b can be given by the i-th equation of the system -1

(93). Here, the left eigenvector Si,EMis the i-th row of S ~

S-1 C3QEM I=i,EM -O~ + Li,EM + Si,EMHEM) x=a,b - 0

(95)

where L~,EM- 0 for incoming waves, while for outgoing waves (96) ,

63x

698

7.2 Characteristic-based boundary conditions for Navier-Stokes subsystem Similar derivations can be used to determine analytical expressions for the eigenvalues and the non-reflecting boundary conditions of the Navier-Stokes subsystem of the unified EMHD as shown by Dulikravich and Jing [26]. Characteristic treatment of the Navier-Stokes subsystem of the unified EMHD system can be performed by converting its conservative form 63QNs + aENs - -b aFN S -1 63GNs = SNS

at

ax

as

az

(97)

into its non-conservative (characteristic) form g31~NS t- ANS 691~NS I- BNS C3QN----------~s + CNS g31~N---------~s = SNS & 0x 0y -0z

(98)

where the solution vector of unknowns is given as

-

{

QNS= P/~,

Vx, Vy, Vz,

e

}*

(99)

From equation (47) it can be seen that flux vector ENs becomes

Vx 2 Vx +

/

P ENS=

~,~1

p % __I'~BM Vx~x / P

VxVy VxVz

P PB p 'l~xy VyNx ~xz

ev_~ P

Vz x

N_~ '~ P

|

/

(100)

!|

J

Terms related to d,d_ 2 and VT will not be considered in the evaluation of coefficients of the flux vector Jacobian matrix ANS since they are associated with first derivatives of velocity, v, or temperature, T. The flux vector Jacobian matrix ANS = c3ENs/0QNs then becomes

699

0

1

0

0

0

~[p

a22

a23

a24

0

0

a32

a33

a34

0

0

a42

a43

a44

0

_Vx[5/9

a52

a53

a54

Vx_

ANS -

(101)

The coefficients in this matrix are given in detail by Dulikravich and Jing [26]. Eigenvalue vector of the flux vector Jacobian matrix ANs is

-

{

~,NS = Vx, Uu, Z+v, Uw, Ue

}"

(102)

which can be written as a diagonal eigenvalue matrix ~'Ns== diag[Vx' Z+~' Uv' Uw' U~]

(103)

The eigenvalues Uu, X~, Uw, Ue are obtained analytically by solving a fourth order characteristic polynomial (similar to equation 73) where

~ VNS

= - ( a ~ +a3~ +a,4)

-

a22a33 + a22a44 + a33a44 - a34a43 -- a24a42 -- a23a32

-

(104)

(105)

)'NS -- a34a43a22 - a22a33a44 - a24a32a43 + a24a33a42 + a23a32a44 -

a23aa4a42 + (a33 + a44)13 P

~NS -- (a34a43

_

a33a44

)_~

(106)

(107)

9

so that the four eigenvalues are

~+-"

4

NS1 -I-

ONS1 ---2 "(ll/NS q- ~'~NS~)

(108)

700

~/NS q- ~'-~NSo)

(109)

~+w -- -- 4 (I) Ns2 q" i--~(I)Ns2 -- 2 (V NS -- ~'~NSo)

(110)

_lt9 ~1 2 )~+e- 4 NS2-- ~ ~Ns2 --2(~I/NS- ~'-~NSo)

(111)

--'4"(I)Ns1 --

1

NSI --

~/1

2

1

with the coefficients given by equations of the type similar to equations (82-88). Characteristic waves defined by the Navier-Stokes equations in the EMHD system have a great dependency on both fluid dynamics and electro-magnetodynamics, in particular, the electro-magnetic properties of the media and electro-magnetic field quantities. When electric and magnetic fields are absent, these eigenvalues reduce to the well-known eigenvalues of a classical NavierStokes system for Newtonian, incompressible flows. These eigenvalues are {Vx,V~VxV x + c, Vx - c }. Here, the equivalent local speed of sound is defined as C - - 4 V 2 + (~/p) X

"

Following Thompson's approach [30,31], non-reflecting boundary conditions for the Navier-Stokes subsystem are hence formulated as follows. The characteristic form of Navier-Stokes subsystem influenced by the electromagnetic effects is possible to write as

SN s r

&

~ aQNs + ~NsSNIs ~ + = 0x

-1 ~ S__NsHNs=0

(112)

where the i-th equation is S_ 1 ~QN_______SS =,,NS & + NSSN s OQNs & + Si,NsHNs _ 0

(1 13)

and the new source vector is HNs - BNS aQN-----~s+ CNS 0QN------As- SNs Oy = Oz

(114)

701

Here, the left eigenvector S--i,NS "1 is the i-th row of S~ls

--1--/I

Si.NS O0NS + Li,NS + Si,NsHNs Ot

-- 0

(115)

x=a,b

where Li,NS --0 for incoming waves, while for outgoing waves

,NS

0X

(116)

Practical implementation of Thompson-type [31-33,36,26] non-reflecting boundary conditions deserves further comments. The essence of his approach is that one-dimensional characteristic analysis can be performed by considering the transverse terms as a constant source term. In order to provide well-posed non-reflecting boundary conditions in multi-dimensional cases, substantial modifications may be required to take into account the transverse terms at the boundaries [37,38]. It should be emphasized that physically there are cases where flow information propagates back from the outside of the domain into the inside through the boundaries by the incoming waves [39]. This fact makes it possible that building a perfectly non-reflecting (absorbing) boundary condition [40] might lead to an ill-posed problem. Under these circumstances, corrections may be needed to make them partially non-reflecting.

7.3 Numerical integration of EMHD system It is often highly desirable to have a time-accurate unsteady solution to the governing EMHD equations. One numerical integration algorithm that could be used is an advanced form of the dual time-stepping technique, also called an iterative-implicit technique, originally developed by Jameson [41 ]. To create an instantaneous picture of the solution of the entire EMHD system at a given physical time, equation (43) must be driven to zero in its entirety, not, as is commonly done in time-marching techniques by driving only the physical time-dependent term to zero. To this end, a pseudo-time derivative is added to the EMHD system (equation 43) which can be rewritten as c3Q 0E 0F 0G -00 + ~ + ~ + ~ + ~ _ S c3z 0 t 0x 0y 0z

(117)

702

or as

aQ=

_

aQ &

(118)

where ~ is a composite of the spatial and source terms and is called the residual. Thus, given a physical time step the governing equations are time marched in pseudo time, x. Upon convergence, the fight-hand side of equation (118) becomes zero and the solution at the desired physical time level, t, is obtained. Note that the pseudo-time dependent variable vector, 0 , does not have to be the same as the physical time dependent variable vector, r An additional concern of great importance is that the system of equations develops zero terms in the pseudo-time dependent variable vector, 0 , for incompressible fluids, fluids without electric charges, or systems in which the electric and magnetic fields are non-interacting. This poses significant problems for time-marching numerical solutions. This problem may be alleviated, however, by proper selection of pseudo-time dependent variable vector, 0 , and through the use of matrix preconditioning. By premultiplying I~ with a properly selected matrix, it is possible to directly control the system eigenvalues. This prevents development of zeros in the pseudo-time dependent variable vector, 0 , and vastly improves iterative convergence rates over a wide variety of flow regimes (low and high Mach and Reynolds number combinations). The preconditioning matrix, F'(0), for the EMHD system could be based on one developed by Merkle and Choi [42] for the Navier-Stokes system. The preconditioned EMHD system may be written as

r, aQ_ aQ ax -&

=

(119)

Equation (119) can be transformed to a body-conforming non-orthogonal curvilinear time-dependent (~, 1-1, ~; t) coordinate system. A high order of accuracy is desired to properly resolve unsteady motions. A finite difference scheme using fourth order accurate spatial differencing and second order accurate physical time differencing could be used while the solution is advanced in pseudo-time using a four-stage Runge-Kutta scheme which is second order accurate for non-linear problems. Fourth order accuracy should be selected for

703

the spatial derivatives based on extensive research completed by Carpenter et al. [30] which found that a Runge-Kutta advanced fourth order accurate scheme provided the best convergence and stability of higher order schemes at reasonable computational cost. Second order accurate differencing in physical time could be selected based on stability and convergence studies performed by Melson et al. [43] who found that for a Runge-Kutta advanced dual timestepping scheme second order backward differencing provided the most stable physical time discretization while providing excellent resolution. The new physical time step could be treated implicitly in pseudo-time, while all old physical time steps and spatial derivatives could be treated explicitly. This is unlike Jameson's early method [41] that treats both the physical time and the spatial derivative explicitly and causes a restriction on the maximum physical time step allowed. The discretized preconditioned system may be written as 0 ~ =(~" F'F-I+ a i -~

(120) A'I~ "~/~ i _

-

~ z- ~- r i m + l ,

i-1

_4, m2,,t (~ n-,-1= 0 4

i=O

(121)

(122)

where m=1,2,3,.., represents the physical time step, n=1,2,3,.., represents the pseudo-time step, and i=1,2,3,4 is the Runge-Kutta stage number. Also, F = 0 0 / / ~ and c~i are the Runge-Kutta coefficients. Note that the physical time-dependent term on the right hand side of equation (121) is held constant for all four Runge-Kutta stages.

8. S U B M O D E L S

OF EMHD

Until now, the numerical solutions of the unsteady three-dimensional EMHD flows that have been reported in the open literature [34-36] did not account for polarization or magnetization effects and did not involve charge density transport equation. The reason is that the complete unified EMHD system is very large having extremely complicated source terms and two extremely

704 different time scales for the electro-magnetic fields and the flow-field. Consequently, a number of simplified versions of the EMHD system have been traditionally used in practical applications. These submodels can be grouped in two general categories: EHD models and MHD models [ 11-13,44]. From the unified EMHD model, it can be seen that the electromagnetic field is not the only cause of electric current and that the temperature gradient is not the only source of heat conduction as is commonly assumed. The electric field, magnetic field, heat conduction, and deformation (strain) may couple to produce charge motion and heat transfer. These couplings are called phenomenological cross effects and may be placed in four general categories: 1) thermoelectric, 2) galvanomagnetic, 3) thermomagnetic, and 4) second order effects [9, p.161163]. These categories are based on the source of the effect and each will be described in turn, as will be a comparison between classical EHD and MHD models and the unified EMHD theory. The comparison concentrates on similarities and differences between electro-magnetic force and electric current and heat conduction terms in the EHD, MHD, and EMHD models. The inadequacies of simple superpositioning of classical simplified models to fully describe the unified EMHD flows are also noted. Couplings between the temperature gradient and the electric field cause thermoelectric effects so that a temperature gradient in the material produces an electric current (Thompson effect), while applied electric field produces heat conduction in the material (Peltier effect). These two effects together are known as the Seebeck effect and form the basis for thermocouples. Also note that the or, term in the electric conduction current (equation 22) and the K4 term in the heat conduction (equation 23) are the ohmic charge conduction and Fourier heat transfer, respectively. When the electric and magnetic fields are simultaneously applied but are not parallel, electric current (Hall effect) and heat conduction (Ettingshausen effect) perpendicular to the plane containing the electric and the magnetic fields are induced in the media. These effects are termed galvanomagnetic [9, p. 161-163]. When the temperature gradient and the magnetic field are simultaneously applied but are not parallel, electric current (Nernst effect) and heat conduction (Righi-LeDuc effect) perpendicular to the plane containing the temperature gradient and the magnetic field are induced in the material. These effects are termed thermomagnetic. It should be noticed (equation 22) that the interaction of the average rate of deformation tensor and the electric field can also create the electric current, while the interaction of the material deformation tensor and the electric field can create the temperature gradient (equation 23). These piezo-electric and piezomagnetic effects can further be enhanced if the material is non-isotropic.

705

8.1 Classical e l e c t r o - h y d r o d y n a m i c s ( E H D )

As mentioned previously, EHD flows are those in which magnetic effects may be neglected and charged particles are present, while only a quasi-static electric field is applied so that the magnetic field, both applied and induced, may be neglected [ 11 ]. One of the implied assumptions is that the flows are at non-relativistic speeds, although in astrophysical flows this assumption cannot be made [1]. Atten and Moreau [44] present a detailed coverage of classical EHD modeling and discuss the relative importance of terms in the force and electric current through stability analysis. With these assumptions, the Maxwell' s system reduces to [ 11 ] V . D - V.(~E_)- qo

(123)

c3q~ ~ - V . , J - 0 &

(124)

With classical EHD assumptions, the electro-magnetic force in the unified EMHD theory reduces to" (125)

f_.v_,M_ q~E + (V_E)-P = qo_E+ (VE). e,pE

This is not the form of the electro-magnetic force usually seen in classical EHD formulations [11]. Through the use of thermodynamics and the material constitutive equation of state, the electric force per unit volume in EHD is most often used in the following equivalent forms [10, p.505-507][8, p.59-63] f EM

:

qo_E- E . E Ve + - V

-

--2-

-

2

2

E-E p -

p=const

(126) T=const

constl

(127)

The three terms in the equation are the electrophoretic, dielectrophoretic and electrostrictive terms, respectively. The electrophoretic force or Coulomb force is caused by the electric field acting on free charges in the fluid. It is an irrotational force except when charge gradients are present [45].

706

The dielectrophoretic force is also a translational force, but is caused by polarization of the fluid and particles in the fluid. The dielectrophoretic force will occur where high gradients of electric permittivity are present. This condition will be true in high temperature gradient flows, multi-constituent flows, particulate flows [ 18] or any time the electric field must pass through two contacting media of different permittivities [46]. Grassi and DiMarco [47] treat the dielectrophoretic force as it applies to bubbly flows and heat transfer. Poulter and Allen [45] note that the dielectrophoretic force produces greatest circulation when the dielectric permittivity is inhomogeneous and non-parallel with the applied electric field. The last force, the electrostrictive force, is a distortive force (as opposed to the previous translational forces) associated with fluid compression and shear. The electrostrictive force is usually smaller than the -phoretic forces. It is present in high pressure gradient flows, compressible flows, and flows with a non-uniform applied electric field. Pohl [18] describes this phenomenon in greater detail. Classical EHD modeling derives directly from the unified EMHD theory. Thus, the electric current density, using EHD assumptions, reduces to J = qo_V+ OlE + o-47T

(128)

However, this is not the form seen in classical EHD models [ 11 ] which typically define the conduction electric current as only the first term of equation (22). However, more advanced classical EHD models define the current as [9, p.562] J- = qo_v + J_c = qo_v + qob E_- DoVqo

(129)

The last two equations imply that the temperature gradient is directly related to the electric charge gradient. This may be shown to be true based on the Einstein-Fokker relationships, derived from studies of Brownian motion [25, p.264-273], which relate any concentration gradient to a charge mobility and a diffusion. Newman [48] also provides a detailed discussion of the concepts of diffusion and mobility. The electric charge diffusion term is often neglected where only limited amount of free charges are available [49]. By introducing classical EHD assumptions in the unified EMHD theory, the equation (23) for heat flux reduces to CI = KI__E+ K4VT

(130)

707

The classical EHD models neglect the contribution to heat transfer from the electric field so that equation (130) reduces to Fourier's law of heat conduction. Cl = -~:VT

(131)

Although classical EHD modeling seems to neglects heat transfer induced by the electric field and electric current, Joule heating effect ( - I , . _ E term from EMHD equation 19) is usually included in the EHD computations [50,51 ]. 8.2 Classical m a g n e t o - h y d r o d y n a m i c s (MHD) The classical modeling of MHD assumes non-relativistic and quasimagnetostatic conditions. It implies that electric current comes primarily from conductive means and that there are no free electric charges in the fluid [11 ]. With these assumptions Maxwell's system becomes

V.B_.= 0 VxE= -

(132) 0B -0t

(133)

Vxl-l= /

(134)

V-J -0

(135)

The modifications to the Navier-Stokes relations come from the electromagnetic force on the fluid from which all induced electric field terms have been neglected. Using the MHD assumptions, the electro-magnetic force per unit volume in the unified EMHD theory becomes [ 11 ] [EM = j x B_B+_ ( V B ) . M___

(136)

The second term, source of dimagnetophoretic and magnetostrictive forces, is typically neglected in classical MHD [10, p.508]. Thus, the electro-magnetic force per unit volume in the classical MHD is modeled as [ 11 ] frM = j x B

(137)

708

By making MHD assumptions, the conduction current in the EMHD can be expressed with equation (38). However, classical MHD theory usually defines the electric conduction current as [ 10, p.510] Jc = cYlE + o4VT = cYlE + O'l(V x B) + cY4VT

(138)

Here, (5'4 is the Seebeck coefficient [9, p.174] which in some classical MHD formulations is not used [11 ]. Clearly, the classical MHD formulations neglect a significant number of physical effects [52,53]. Similarly, in classical MHD modeling, Joule heating is often included in the energy relation, but the heat transfer constitutive relation remains the same as in equation (131). In comparison, the unified EMHD model for the heat flux with classical MHD assumptions can be expressed with equation (39). It could be concluded that classical EHD models include many important effects and correspond to the unified EMHD theory well, while classical MHD formulations need improvements in the force, current, and heat transfer terms. As in classical EHD modeling, it is important to be aware of the fact that many force, current and heat transfer terms can be written in several different forms, each of which is equivalent. It is, therefore, important to recognize the potential danger of simply adding terms from different EHD and MHD models.

9. S O L I D I F I C A T I O N W I T H E L E C T R O - M A G N E T I C FIELDS During solidification from a melt, if the control of melt motion is performed exclusively via an externally applied variable temperature field, it will take quite a long time for the thermal front to propagate throughout the melt thus eventually causing local melt density variations and altering the thermal buoyancy forces. It has been well known that an externally applied steady magnetic or electric field can, practically instantaneously, influence the flowfield vorticity and change the flow pattern in an electrically conducting fluid [51-59,33]. Similarly, it is well-known that applying an electric potential difference to a flow-field of a homogeneous mixture will cause fractionation or separation of the homogeneous mixture into regions having high concentration of the constituents. This phenomena, known as flee-flow electrophoresis, has been extensively studied experimentally and, to a lesser extent, numerically [50] using classical EHD modeling. Nevertheless, there are no publications yet on actual algorithms for determining the proper variation of intensity and orientation of the externally applied magnetic and electric fields. This is not a

709

trivial problem because we are dealing with a moving electrically conducting fluid within which an electric current is induced as the fluid cuts through the externally applied magnetic field lines [11]. This induced electric current generates heat (Joule effect) as it passes through the fluid that has a finite electrical resistivity. In the case of solidification, the amount of heat generated through the Joule effect due to the externally applied magnetic field is often neglected compared to the latent heat of solidification and the amount of heat transferred in the melt by thermal conduction. The latent heat released or absorbed per unit mass of mushy region (where Tliquidus > T > Tsolidu s ) is proportional to the local volumetric liquid/(liquid + solid) ratio often modeled [59] as

f =

Ve

=

v,+v,

(

T - Tsolidus

/n

= 0n

(139)

Wliquidus -- Wsolidus

Here, 0 is the non-dimensional temperature, the exponent n is typically 0.2 < n < 5, subscripts g and s designate liquid and solid phases, respectively, while f 1 for T > Tliquidusand f 0 for T >Ot, Equation (7) gives ~c=l, in which case the particle/liquid conductivity mismatch dominates the ER response. The conductivities of the particles and the host liquid are assumed to be ohmic in the complex polarization models presented in [ 11-17]. These models still predict that the attractive force between particles and the resulting shear yield stress are proportional to the square of the applied field. Although the models include the conductivity in the ER response, they can only give an estimate of the shear yield stress for two limiting cases: (a) dc or low frequency ac field and (b) very high frequency ac. For example, Davis [17] provided a simplified analysis for a suspension of oxidized metal-particles in a weakly conducting oil. His prediction of the yield stress with ac electric field is in good agreement with the experimental results by Inoue [18], but one order of magnitude lower than those measured with dc electric field. Although one frequently obtains reasonable agreement between the measured rheology of ER suspensions and that predicted by the complex polarization model using the complex permittivities [11-15,17,19-20],

720

the model does not account for the fact that the electric field dependence of the flow stress is frequently less than quadratic, as observed in some experiments at dc field or low frequency ac field [ 14,18,21-24]. Also the model does not explicitly explain that the flow stress generally correlates with the associated current density [14,25]. These discrepancies pertaining to the polarization models were first addressed by Foulc et al [26,27] for the case when the conductivity of the particles is much greater than that of the host oil. A new concept in their study was that the host liquid exhibits non-ohmic (non-linear) behavior. They proposed that the effect of the field on the conductivity of such oils was given by a simplified expression of Onsager's theory [28] of(E)- j/E = of(0) [ (l-A) + Aexp~E/Ec],

(8)

where of(0) is the conductivity at low electric fields and A and Ec are constants which depend on the oil. In their model the conductivity mismatch F=Op/Of(0) between the particles and the host oil is the important electrical parameter. Further, to make the necessary calculations they defined two regions in the contact zone between spherical particles: (a) for a distance from the center line x>6 the sphere surface is equipotential and the electric current leaving the sphere is negligible and (b) for x6). The distance 6 is obtained by setting the conductance of the solid sphere equal to the conductance of the host oil. This gives

724

(a/5)ln(a/5)- F/~,

(14)

where a is the sphere radius and F=~p/C~f(0) the ratio of the conductivity of the sphere to that of the oil at low field. They then calculated the total axial attractive force between the two spheres for low applied electric field to be fa = 4ua2~fF2Eo2/[uln(a/5]-

(15)

Equation (15) gives a quadratic dependence of fa on F and E o. At high electric field when x_~5the enhancement of the conductivity of the host oil becomes significant because of the non-linear electric field dependence of the oil conductivity; see Equation (8). The axial attractive force in this case becomes

fa-- 2ua2efEcEo{ln[(10F/u)(~]2 Eo/Ec ]}2.

(16)

Equation (16) gives an approximately linear dependence of the force on the applied field and only a weak dependence on the conductivity ratio F. In a more simplified estimate they give the attractive force fa = 2ua2~f'EcEm,

(17)

where E m is the saturation field in the host oil due to the strong nonohmic behavior of the oil. They estimate that E m ranges from 30kV/mm to 40kV/mm for the mineral oil they used. Thus, they concluded that for two nearly-touching spheres, under low dc applied field, the attractive force is proportional to Eo2, while under very high dc field the force is proportional to the applied field Eo. Their analysis gave qualitative agreement with force measurements on two nearly-touching, large scale, semiconducting half-spheres (a=7mm). It should be pointed out that although their model and experiment show that when Eo>lkV/mm the dependence of the attractive force on the electric field is linear, the shear yield stress does not have a linear dependence on the field. This is because when a chain is sheared the particles in the chain will separate and thus are no longer nearly-touching (see section 2.3). The saturation field E m in the host liquid is an important parameter in the non-ohmic conduction model. Wu and Conrad [30] give the following empirical equation as an estimate of the saturation field

725 F o Ec E ~ / E o - 30(~) "1(~0-0)09

(18)

and Davis and Ginder [32] give for the radius of the saturation region of two nearly-touching spheres 5/a = ~]2Eo/Em

(19)

Felici, Foulc and Atten [27] gave the following expression for the relationship between the axial attractive force fa and the current I passing through the spherical particle:

fa = 4~;a2~p2Vo 4/I2,

(20)

where Vo is the potential difference between the two adjacent halfspheres. Qualitative agreement with Equation (20) was obtained for experiments on polyamide half-spheres (a=7mm) in mineral. Although the conduction model for two nearly-touching spheres presented a new concept for ER response, it could not predict the shear behavior (i.e., the shear yield stress and the shear modulus) of a general ER suspension, which is of interest in engineering applications. This question and the approach will be addressed below

2.3 Conduction model for separated spheres The strength of ER suspensions can be understood in terms of the static electric interaction between the particles. A single-row chain of the particles is the most basic and the simplest structure in ER fluids. Hence, most of the theoretical studies on ER response [6-12,29-37] are based on an analysis of the mechanical and electrical properties of a single chain. Two geometric arrangements for the single chain are used: (a) the chain is parallel to the applied field and (b) it makes an angle with respect to the applied field. Klingenberg and Zukoski [7] used the geometric arrangement (b) and gave an exact analysis for the restoring force of the two-sphere system. Most studies, especially in the nonohmic conduction models, consider the arrangement (a) to obtain first the distribution of the local field and then to estimate the shear modulus and the shear yield stress when the chain is sheared. Method (a) is simpler than (b) and has an acceptable accuracy [42]. All of the work on the conduction model covered in this section uses this method.

726

Assume that with application of a dc field E 0 the chains formed in an ER suspension consist of spherical particles with radius a and are distributed uniformly between the two electrodes. Due to the symmetry we need consider only two half-sphere neighboring particles as shown in Figure 1. The following two approximate equations then apply [29,30,34]: OpEp(x)= of(E)E(x)

(21a)

[R-h(x)]Ep(x) + h(x)E(x) - RE o

(21b)

where R=2a+s is the distance between the two half-spheres' centers, h(x) the gap between the two half-spheres at any location x, Ep(x) and E(x) the local field in the particles and in the host oil, respectively. Equation (21a) is the continuity condition of current density and Equation (21b) that of the potential. Wu and Conrad [36] have shown that Equations (21a-b) give a good estimate of the local field distribution.

of Ef

- -~'I h(x) _Io I

(a)

eo

(b)

Figure 1. Schematic of the conductivity model for two separated spheres [29,30]: (a) The geometry and symbols employed; (b) an area element in the horizontal plane, of is the conductivity of the host liquid, Op the conductivity of the particles, Ef the dielectric permittivity of the host liquid, epthe dielectric permittivity of the particles, a the radius of the particles, s the separation of the particles, and h(x) the gap between the two particles at location x.

727

It was found that most of the current passes through the contact zone, which dominates the main behavior of an ER fluid. Good agreement occurred between the predicted and experimental data [21] for the effect of applied electric field on the quasi-static yield stress XE of a zeolite/silicone oil suspension. Based on the work of Felici, Foulc and Atten [27], Davis and Ginder [32] predicted the shear modulus of an ER suspension with non-ohmic conductivity at dc field to be G= 3EoKf0EoE m

(22)

and the shear yield stress 4

1

XE = ~EoKfOE ~ . 5 ~

(23)

where ~ is the volume fraction of the particles in the suspension. Wu and Conrad [30] derived the saturation field Em in the "contact zone" to be that given in Equation (18) (note: they used the conductivity definition o=j/E, which will be used in the remainder of this chapter except where noted otherwise). Outside the "contact zone" they used the equipotential assumption. Figure 2 shows the maximum local field in the liquid layer Ef=E(x=0) versus the separation of the particles. It is clearly seen how 104 ~u~otential: 103 ~-

E/Eo:I+I/S

~

A=0.007

Ec:O.ZikV/mm

=. o

r.l.l 10 2

101

; Eo(kV/mm)= ~ :

10o ' 10-4

:

;4

-'

r=lo 7 " '

. . . . . . . . .

~

,

,A,,,,I

,

10-3

,,

,,,,,I

,

10-2

,,

,,,,,I

10-1

,

,,

,,,,

10 0

S=s/2a Figure 2. The ratio of the maximum local field in the fluid layer to the applied field v s the normalized separation for different applied fields [30].

728

the saturation field varies with the normalized separation S and the applied field E o. Figure 3 shows the normalized attractive force F (f~=na2~oK~o2F) between the two particles. If Em/EoI+I/S , a saturation field occurs in the liquid layer and the normalized attractive force is F m -- 66(F/A)~ l(Ec/Eo )n

(25)

in the range ofEo=l-10kV/mm , Ec=0.001-1kV/mm , A=0.0005-0.5 and F=103-109. We have n=0.92178Ec ~176 when Ec=0.01-1kV/mm and n = l when Ec=0.1-0.3kV/mm. Therefore, the general form of the normalized force can be written as F = min.{Fm, 0.955/S}.

(26)

The shear stress of an ER suspension is given by 3 (~oK~o2F(?) ~

lO~

(27)

~ " ' '

'''"'1

' '

]o3 ~102

:: -_

101

-

10 0

I/

10-4

'''"'!

'

'

....

"1

' '

'"'~"_~

F=]O 7 A=0.007 ~ Ec=0.21kV/mm ~ j F=fa/[:n;eoKfEo 2a21 1 . "~ =0.955/S

,

,

lO

, ~,~,,I

10 .3

..... ,

n ......

I

10 -2

,

,

, , ,,,,!

10-1

. . . . .

]00

S=s/2a

Figure 3. Normalized attractive force between two particles normalized separation for different applied fields [30].

US

729

The shear yield stress can then be obtained by maximizing Equation (27). Wu and Conrad [30] give the following estimate for the relationship between the saturation field and the saturation radius for any separation of the spheres: Em/E o - (l+S)/[S+1_41_(5/a) 2 ].

(28)

When 5/all0~ In the contrast, the particles may loose some of their water content at high temperatures (i.e., >100~ giving a decrease in their conductivity opposite to the temperature effect on the oil conductivity. The combined effects of temperature and water content in both the particles and the host oil give the highest conductivity mismatch for the ER suspension at about 100-110~ It should be pointed out, the above non-ohmic conduction models predict that the interaction force between the particles increases continuously with the conductivity ratio of F-Op/Of(0). However, 800. . . . .

I ....

I ....

Zeolite/Silicone 0il ~=0.28

600 " W=6wt.%

400-/ / ~ 9 / ~

-

0-,

~1~ ~ ) l,

0

2kV

//"

" : 200 " -

t .... Open" Exp. Filled: Pred.

,

I , , , ,

50

./A

A

"'A A

A

\ .--'~

~, "~

~

Eo=lkV/mm u"O.. I , , ,

100 T (~

,

I

150

I

I

,

,

200

Figure 6. Comparison of the measured shear yield stress dependence on temperature for a zeolite/silicone oil suspension and that predicted [30].

733

experiments [44] show that the interaction force first increases, reaches a maximum and then decreases with F. Boissy, Atten and Foulc [45] and Wu and Conrad [46] found that when F is small the ER response is negligible and when F>Of, giving the conductivity polarizability at dc field f~= 1. Figure 10 shows that the non-ohmic conduction models predict the shear yield stress better than the polarization models. 100

_

,.

i i[lll i ! . . . . . . .

i

i

!

i

:

,

[

i

1

,

1

Experiments.: -

,

.

1

o ~ A

~_,1~ 0

_T

T

i

i.t

t

,

|

3 Spheres ~,~[ 4 Spheres ............................... ~ ~ ~ 5 Spheres ~ ~ ! -

! i ii I '!-...........J..........i-~-i-i~--i--i-'-'~-i -!....!---i......Theoreticati _ T h e o r e t i .......... .c...a tii-....... i i--i I 01

0.20.3

i ~. ~ i i

i

~

~

..... Tangetal[49] -----Wu & Conrad 13(

:

'

0.5 E o

2

'

'

3 45

(kV/mm)

Figure 8. Comparison of the measured current density of a static single chain under de field [50,53] with that predicted [30].

736 51_

. . . .

I

. . . . . . . .

~.~_. ~"

4 1: ~

~ ................i............J 0 \ I

1

. . . .

- .....

o

-I

3 Spheres ]-]

i

iA

i ......; ................................

:

~.,...................... !...................... i ....... o .........

o ?ii i,, i, :i i i i i 0

I ....

A 4Spheres/:t 0 5Spheres1

~~i

,~ 2 *'~

I

_Expefimen.tal_ : :._ _~

0.1

0.2

,,1

0.3

0.4

0.5

u Figure 9. Comparison of the measured current density of a sheared single chain of humidified glass beads v s shear strain with that predicted by non-ohmic conduction models under dc field" 9 Tang, Wu and Conrad [49]; Wu and Conrad [30]. I

L

:

1

1

!i

J

9 Experiment [51,55] J /, ~ ~ ......... Wu & Conrad [30] J ~Z'.,~";" E] " Tang etai [49] [IIIIZ]III]I]I::]I::I~]:I]L~~I!!!-_ [i:J - Davis & Ginger [32] J:::::::::::::::::::::..~ ~:::::::::[---J -- ---Klingenberget al [7] [........i - ~ .....!.............Z k if!___.~..- .- --Conrad et al [54] ..........!2~iiiiiiiiiiiiiii~iiiiiii--

.......i:i_..:i:!i:i::::ii:ii~ii!:i::::::i~iiii:iiliZ.ii~:i:2:1111!iiiiii:i~ii:ii:ii _:_:.i_i 0.1

0.6

~

,J'2" 0.8

i

1

2 E

o

3

4

( k W m m )

Figure 10. Comparison of the measured shear yield stress of a single chain of humidified glass beads [51,55] with that predicted by the nonohmic conduction models: Tang, Wu and Conrad [49], ..... Wu and Conrad [ 3 0 ] , - - - - D a v i s and Ginder [32]; and the polarization models: Klingenberg and Zukoski [7], Conrad, Chen and Sprecher [54]. In the predictions by the polarization models it is assumed that *= 13c=1, i.e., Op>>Of

737

2.4.2 Highly conducting particles with a low conductivity film For highly conducting particles with a weakly conducting surface film, the current first passes across the surface film and then continues in the highly conducting bulk (core). Wu and Conrad [34] developed a model for such particles. They define a combined conduction parameter A

a OI

= 6~f(0) ' where 6 is the thickness of the surface film, oI the

conductivity of the surface film and gf(0) the conductivity of the oil at low field. Taking silicone oil as the host liquid (with of(0)=2.4x1012S/m, Ec=0.21kV/mm, A=0.007), they found that increasing A (i.e., increasing o I or decreasing 5) increases the ER response under dc applied field. If the surface film is an ideal insulator and thick (i.e., ai=0), there is no ER effect under dc field. If 5 is too thin, electric breakdown will occur in the surface film when the applied field exceeds some value, which was observed in the experiments [18,34]. Therefore, a reasonable thickness of the surface film is desired to avoid breakdown. The shear yield stress, shear modulus and the current density of an ER suspension with a volume fraction ~ of particles suspended in silicone oil were determined to be given by the following empirical equations: T E (Pa)

= 19r

G ( P a ) - 55r J(pA/cm z) - 6xl 07r

(35) nG 1 22c~f(0)AnJ

(36) (37)

for the range 100), an equation similar to Equations (42a-b) can be used to estimate the local electric field distribution between the two adjacent particles. However, a and s in Equations (4247) should be replaced by a'=a~/1-(x/a) 2 and s'=s+2a[1-~/i-(x/a)2]. Further, the normalized separation S should be replaced by S'=(S+I)A/1-(x/a) 2 -1. This simplified analysis gives an estimate of the local field with good accuracy [36]. Having obtained the distribution El* (x) of the complex amplitude of the local electric field, we can get the complex amplitude of the current density 9

$

$

$

(49)

J(x) =(:If (x)Ef (x).

The current density at location x at any time t is j(x,t) - ~/2 j(x)cos(~ot + 0j(x))

(50)

where j(x)= Ij(x)* I / ~ is the rms value of j(x,t) and

5-

-lr Illl(j (X)*).

(51)

is the phase angle of the current density shift from the applied electric field, which is a function of the location x. The rms value of the average current density along a chain is 2~ r

j = {~_

[ ~:a12

~x~j(x)eos(mt+~(x))dx]2dt}l/2

(52)

747

The attractive force between two particles is given by a

fa(S,t ) = ~soKf 1 f 2 ~ IEl(X)* 12cos2(eot+0E(x))xdx = e 0 g a 2 I ~ ~ 2 F(S,t),

(53)

where the normalized force F(S,t) is 1

F(S,t) = [ 2(E(~)/Erm~)2 COS2(cot+0E(~))~d~.

(54)

Here ~=x/a and E ~ is the rms value of the applied electric field. The rms value of the normalized attractive force is tl

O

Fr~s =

co ~ F(S,t)2dt

(55)

and the mean value of F(S,t) is 2n O)

Fmean

(9

2~

J F(S,t)dt

(56)

The shear yield stress is given by

Tnm=SfE,.m~ 2 max. (F,.m~(~,)q:+?2)

(57a)

or

TEmean--E:Ferms 2 m a x .

(Fmean(Y)q;~2)

(57b)

In most experiments, the measuring meters give the rms value of the ac signal; but some meters may give the mean value.

748

In a real ER fluid, many chains span the two electrodes, Equations (57a-b) should then be multiplied by the factor 3(]), where ~ is the volume fraction of the particles in the ER suspension. Conrad, Chen and Sprecher [54] suggested that a structure parameter As should also be included if the particles are not sphere-like and the structure consists of clusters of chains. If the chains of spherical particles in an ER suspension are considered to be ideal, single-row chains and to act independently, we can take A~=I. Calculation of the current density of an ER suspension differs slightly with ac field from that with dc field. With dc field the current passing directly through the liquid phase between two electrodes is negligible compared with that passing along the chains. However, at high frequency ac field this current can not be neglected. If the currents passing through both the chains and liquid phases are considered, the current density of the suspension is: 2n 0)

J-(~-

[ 0

I ~

nOx~j (x)cos(~ot+Oj(x))dx a0~3

3 ) ~ j (a)cos(mt+Oj (a))]2 dt}l/2 + (1-~r

(58)

or it can be roughly estimated by omitting the phase angle effects of the current passing through the chains and the pure liquid, giving 2yt 03

j _ ~{ ~ _ 3co

[ a21- u x ~ j (x)cos(o)t+Oj(x))dx]2dt}~/2+ (1_2~)j(a)

(59)

q where j(a) is the current density (rms value) of the pure host liquid under the applied electric field and is given by j (a) = ErmsqOf2+(2nf~:f)2' 9

(6O)

749

3.2.2 Ohmic conductivity of the host oil Figure 16 shows how the ER response changes with the normalized frequency g~ of the applied field for a single chain. If Fa=F, the frequency has no effect on ER strength. If F > F , the conductivity mismatch dominates the ER strength at low frequency of applied field, and if F~104 (b) Thickness 5= 10 -3--l~um, or 5/a< hz + < he +, e

(65)

where e is the (unknown) thermal penetration length (Figure 8). The required boundary condition is established by means of the Galerkin method, using linear test functions and taking into account the flux conditions 9 - k s -a-T~ ( z = h ) = q w ,

-

- - ~ ( z - h + e) = 0

.

(66)

After some calculations, this yields the pair of equations e 0"7"w 3 k s ( T . PsCs-+2 e ~ w - Te)-qw 40t

-0

0

(67)

PsCs -~((Tw - Te )e 2) = 6ks (T w - Te ) ,

whichgovem T w, qw and e as long as e < e c. When e reaches single condition coupling qw and T w must be applied :

e c, a

963 -e~0Tw + ~ksl~Tw-Te)Z PsCs 3 0 t e

~ _ qw

=

0

.

(68)

The thermal behavior of the upper and lower walls is complicated by the fountain flow, which generates a thermal shock between the core polymer and the walls, with a resulting singularity on the front-wall intersection lines [28a]. This effect can be analyzed by means of the thermal shock theory between two semiinfinite rigid bodies of initial constant temperatures Te and TZ o [44], since the fluid motion and the non-uniform temperature field play no role close to the singularity during the first instants following the shock. This theory shows that Tw remains constant after the shock. In a similar way, when a thermal shock is considered between a semi-infinite body (the fluid) and a model body (the steel wall) where conditions (67) hold, Tw again remains constant after the shock [26] 9 T w - (or T e + [3 Tfo

)/(o~ +/~) ,

,

(69)

with

-(3S s sl8)

-(pc.kl

)

and where

Tfo

denotes the temperature of the fluid making contact with the

wall at time t f .

After the shock, the temperature profile inside the "semi-infi-

,

nite" fluid is easily calculated in terms of the complementary error function, while the thermal penetration length is

e -(6ks(t-

tf)/PsCs) 1/2 .

(71)

3.2.3 Bifurcations and abrupt changes of thickness The theory developed to build up a simplified model of fountain flow can be easily extended to the so-called singular regions, where thickness variations are abrupt or very steep and take place in a region of the same order of magnitude as the part thickness. Singular regions include edges, abrupt changes of thickness and bifurcations as in the case of T shapes (Figure 9). Inside the singular regions, the same scaling (56) is introduced and a fixed Eulerian reference frame is used in each "infinitely small and rapidly fading" inner zone, where asymptotic analysis shows that the scaled flow is essentially 2D and quasi-steady, while pressure is almost constant, and heat conduction and viscous heating are negligible. According to Couniot et al. [17] and Dheur [16], the following jump conditions must therefore be imposed" 9 Each singular region is infinitely thin, consists of straight segments normal to

964

(a)

11111/I////

(b)

I//I/111/I

/////

~/i//

I

I//11111/11111

(d)

(c)

/

f / / / / / /

///I/IS

//////,

IIII/

,,,..._ ,,....-

/ / / / / / / / 7 / / / / / / / /

7 ~ 7 / / / / / / I / / / / / /

Figure 9. Singular regions in Hele Shaw flow" (a)edge; (b)abrupt change of thickness; (c) bifurcation with separation; (d) bifurcation with junction. the midsurface(s), and is represented by a line on the midsurface. 9 Pressure is govemed only by the parabolic conservation equation (42) and is continuous across the singular line, without singular losses of head. 9 The temperature of a material point crossing the singular region remains constant (Figure 10). This condition has the same form (62) as in fountain flow modelling. The pairs of levels associated through condition (62) are determined by mass conservation. In the case of the separating flow in a bifurcation of upstream and downstream half-thicknesses h-, h( and h~, the levels z 1 and z 2 are linked by the condition ,~22 p (v, p) - 1 / 2

in the case of a salient comer). The thickness is uni-

form. Define the velocity potential r (with ~-~ - ~9r f = ~0+ i ~

and the stream function

- e~/~ o~/0X/~ ), in such a way that the complex potential

is an analytical function of the complex variable ~" = x 1 + ix z , with on F(t).

the front boundary condition r

by expressing ~" as a function ( ( f )

The solution can be found easily

since the functional domain of ~ is the

fixed region (r < 0 , V > 0). The complex velocity ~ = V1 + i V2 is given by the relation (77) Self-similitude of the solution at different times imposes that ~ be orthogonal to

(~/t - ~) along the front, or equivalently that

Re 0~-

-1

-0

0r

- 2t ,

if

t>_O, ~ - 0

.

(78)

The other boundary conditions are : Im(r

: o,

if

-

Im(~e-mn/) : 0,

if

lit-O,

if

Ifl

d~/df

~ eim~r/Vo,

-Kt O) .

[l~o~l[-1 i~ continuous

(91) (92) across the singular

line, as is II0p/0xall2 So 2 h -2 v # (from (83)), and thus"

2 since

-I1 / 11 s;

h+v;f / S ; - h-v2f / S ;

,

(94)

in view of the pressure continuity gradient. Therefore, comparing (91) and (93)

973 shows that

H n (h [lOp/ oaxaIi/*0) is continuous across the singular line, since 7/0

and "ro are functions only of p. Hence, h IlOP/Oqxall/'r0 is itself continuous since H n is strictly increasing, and thus from (93) : Sp+ // h +3 - S p / h -3 , which also involves from (94) the continuity of

(95)

v2f/h 2

(equation (85)).

Further developments proceed as with a power-law viscosity. Similarly, it can be shown [23] that, when an unbalanced bifurcation is fed with incompressible isothermal fluid through the upstream branch, and the incident velocity is uniform and normal to the singular line, the motion of the fronts downstream of the bifurcation depends only upon the thickness ratio, but not on the viscosity law. These considerations have far reaching consequences and explain why the filling kinematics are, in general, only weakly dependent upon shear thinning effects and dominated by thermal effects. 3.4 Viscoelastic effects Since the aim of this chapter is not to analyze in detail the role of viscoelasticity in injection and compression molding, this section is devoted to a review of the existing literature. Viscoelastic effects are significant in critical regions, mainly the front zone, the edges, abrupt changes of thickness and bifurcations of the part, the gates, and the branchings of the delivery system. They also influence the flow stresses (via normal stress differences) and the cooling stage [118], and are directly connected to the generation of frozen-in orientation. Nevertheless, the GN model generally provides accurate results since the flow is dominated by shear, and also thanks to the rather low sensitivity of the gap-averaged kinematics upon shear thinning effects as shown in the previous section. Early studies (see the review of Grmela [47]) were limited by the tremendous difficulty of the viscoelastic problem in normal processing conditions [107]. Tadmor [27] and Dietz et al. [29] developed simplified models to evaluate fountain flow extensional effects and the resulting frozen orientation. Kamal and coworkers (see e.g. [70,122]) investigated 2D cross-sectional flows in the cavity with a White-Metzner rheology, but this approach is limited by the poor properties of this model [ 131 ]. Most of the effort bore upon using the Leonov constitutive equation (see Section 2), since its material constants can be determined from standard experiments. The 1D or 2D incompressible multi-mode model [108111] was used by Isayev and Hieber [30], followed by several authors [53,54,65, 67,112], to analyze the effect of the processing conditions on the flow and

974

N1 (MPa) 0.1

Direct -

Indirect

6 0.06

0.04

tO

0.02

0 -'

-0.5

0

0.5

1

Distance midplane (mm)

Figure 13. Predicted first normal stress difference after 4.7 s in the filling of a 80 • 50 • 2 mm PC rectangular plate (vf = 120 mm/s, T/ = 320~ Left" direct simulation; right" comparison between direct (solid line) and indirect (dotted lines) calculations 8 mm from the linear gate. (From [57]). residual stress distributions in junctures or in the front region. Experimental comparison by means of birefringence measurements showed reasonably good agreement, such as in the subsequent investigations performed by Haman [60], using the 2D compressible Leonov model [57], and by Chang and Chiou [63], using the K-BKZ model. In these latter studies, such as in the packing stage analysis of Nguyen and Kamal [61] and in [65], an attempt is made to extend the lubrication approximation to viscoelastic rheology by assuming a priori equal scales for the extra-stress components (without considering the less degeneracy principle), while simplifications are introduced more carefully in [60a]. The scaling problem still needs to be solved for viscoelastic flows in thin parts. An important step was made when Baaijens [57] demonstrated that simulating GN kinematics, and re-calculating the stresses from a viscoelastic rheology and this pre-calculated velocity field usually provides a fairly accurate estimate of the solution in shear dominated flows. This indirect method, which was further used by Douven et al. [64], Caspers [132] and Kabanemi et al. [62] to simulate the molding of Leonov and Wagner fluids, represents a key step forward since it is much easier to solve the indirect than the direct (coupled) problem. Figure 13 depicts a typical result from [57], where the agreement between indirect and direct calculations is obviously excellent.

975 4.

N U M E R I C A L SOLUTIONS

4.1 Review of available methods Beside pioneer studies performed by means of the finite difference (FD) method in order to solve the 189 problem [31,35a,40b], most of the effort in injection molding modelling has borne upon developing mixed algorithms, where the 2D pressure field is evaluated on the midsurface, while the temperature and other 3D fields are integrated by means of a hybrid scheme over the entire filled region. These techniques, which are detailed in the literature below and will not be discussed, will be classified according to the solution of the free boundary problem since the main difficulty to be overcome in the simulation of molding processes arises from the fact that the filled domain evolves considerably with time. The particular method developed at UCL is described in the next section. Two general approaches were followed, the 1 st consisting in using control volumes on a fixed mesh covering the entire midsurface, while the 2 nd class was based on tracking the flow front(s) and adapting the mesh (or grid) at each time step in order to cover only the filled domain. The l~t technique, which is very robust and by far the easiest to implement, does not involve a discrete front representation. Front phenomena are thus more difficult to model and strong mesh refinement is required in the front zone. This approach, which was 1~t used by Lord and Williams [40b] and Broyer et al. [41] to model 2D non-isothermal crosssectional, or 2D isothermal in-plane flows, has been the object of intense investigations from the Cornell group [56a,71,72,78]. It was further used to simulate compression molding [95] and extended to the K-BKZ model [63], see also [75, 76,79,80]. Following Chiang et al. [56a,78], this technique attaches to each node of the 2D mesh (surrounded by a given control volume) a variable which represents the volume fraction of the injected polymer within the control volume. Solving alternatively for this variable and the pressure field, and for the temperature field (using subvolumes in the gap and a streamline-upwinding technique) provides a reliable and efficient way to simulate the cavity filling, with an easy representation of all the geometrical details and the delivery system. Normally, at most one element should be filled per time step, but the VOF method developed by Voller and Peng [81 ] to simulate RTM shows great promise as to the removal of this limitation. Finally, the Taylor-Galerkin method developed recently by Pichelin and Coupez [82] to simulate the 3D free surface flow of a GN fluid represents a very important step forward obtained by means of an approach which can be associated with the control volume technique. The front tracking-mesh adaptation methods are based on mesh deformation or remeshing. In the former case, the 2D filled zone is mapped onto a fixed domain

976

over which the problem is discretized. There is no longer any free boundary, with the drawback that it becomes very difficult to address the complexity of typical industrial parts. Whereas conformal mapping was used in early studies [44,128a], the Thompson algorithm seems to behave better, see the results obtained by Gtigeri and co-workers [74,77,98,133]. Particular methods are developed in [77,98] to couple the 2lAD Hele Shaw model with full 3D calculations in the singular flow regions (front zone and junctures). On the other hand, the 1st remeshing algorithms were based on the generation of successive element layers according to the 2D front progression [52,66,73a], but this technique can hardly be extended to general shapes. The best approach, as used in the MOLDSYS software (see the next section) and in [80,104,132,134], consists in tracking the front(s), recovering as many elements as possible from a fixed mesh covering the entire midsurface, and generating new elements in the remaining portion of the filled area. To close this review, it is worth recalling the various techniques elaborated to simulate the packing stage [14,56a,60a,61,64,68,70,78,104,132] and to understand the fountain flow goveming mechanisms [48,50,51,54,55,70,77,122,123].

4.2 Description of the MOLDSYS approach The MOLDSYS software developed by Dupret and co-workers at UCL [9-23, 25,26] (see also Chapter FS) solves the free boundary problem on the basis of the following concepts : (i)front tracking and automatic remeshing, in order to capture the shape of the moving free boundary while providing a systematic method of adapting to geometrical events such as weldline formation or front split after collision with a side wall; (ii) Eulerian integration of the temperature or any other 3D field (such as crystallization degree, reaction degree, fiber orientation); (iii) extrapolation of these fields in the region located between the successive flow fronts, using an extrapolation mesh and taking the fountain flow conditions into account; (iv) 2D implicit FEM pressure calculation. The front region plays a critical role in molding processes in view of the key effect of fountain flow, which causes fast out-of-plane advection and material deformation in the front zone. As far as a general purpose algorithm is desired, in order to simulate injection and compression molding, RIM, RTM .... and predict crystallization, curing, fiber orientation and their effects on the flow, it is essential to accurately track the fronts. A remeshing strategy was selected instead of deforming meshes in order to adapt easily to the topological changes of the filled domain. Moreover, as in-plane diffusion is neglected in the lubrication approximation, the simplified energy equation (46) is such that, during any given time interval, the fountain flow thermal condition (62) has no influence on the tempera-

977

rare evolution upstream of the initial front of the time step, which can be treated as an outlet section (without thermal boundary condition). The characteristics of (46) are indeed leaving the domain along this "initial" flow front. On the contrary, the effect of fountain flow must be taken into account in the region located between the initial and final fronts, since characteristics are leaving or entering this region across the moving front depending on whether the material points move faster or slower than the front. The extrapolation mesh is therefore built on in the inter-front region and equation (62) is considered for the fluid layers that re-enter the domain after having experienced fountain flow. This method is general and can be applied to any field for which in-plane diffusion is neglected. To describe the time-dependent algorithm, it is sufficient to consider a single time step. Let M n, F n, ~ n denote the mesh, front(s) and discrete domain at time t,, and [Phi, [T~] the associated column vectors of nodal pressures and temperatures, while [Vn] and [Wn] stand for the discrete in-plane and out-ofplane velocity components along the front(s) and inside the domain at the integration nodes. Additional fields [ A n ] represent the nodal crystallization or curing degrees, or orientations ... if need be. While the flow fronts move from time t n to tn+ 1 , they scan the "inter-front region" An+1 . This area is discretized as a quadrilateral "extrapolation" mesh denoted by En+ 1. T h e sequence of operations needed to calculate these quantities at time tn+ 1 is organized as follows : 9 Front displacement so as to obtain the extrapolation mesh En+ 1 together with ^

provisional representations of the flow front(s) Fn+ 1 . 9 Tracking of front-front and front-side wall meetings, and front conditioning to generate the definite front(s) Fn+ 1 . Remeshing to generate Mn+ 1 on ~n+l" 9 Eulerian time integration of the temperature and other 3D fields on the previous domain soastoobtainprovisionalvalues

[7~n+1], ['4n+1] on M n. Extrapo-

lation of [7~n+l], [/~n+l] on the inter-front zone

An+1 , taking fountain flow

conditions into account. Correction of the temperature and other fields so as to obtain I T n +l ] , [an+l] on M n +l . 9 Implicit pressure time integration so as to obtain

[Pn+l]" Calculation of in-

plane and out-of-plane velocities [Vn+1] and [Wn+1]. 4.2.1

D e t e r m i n a t i o n o f the n e x t c o m p u t a t i o n a l d o m a i n

Front velocities are oriented along the boundary for vertices lying on the side walls, and along the bisectors at the front for other vertices. The fronts are

978

I

|

Figure 14. Extrapolation mesh in the presence of a bifurcation. moved using the Euler scheme [13]. A more sophisticated technique would be very difficult to implement since, during the time step, not only can the shape of the domain evolve, but so can its topology as flow fronts merge, divide, or meet the boundaries. Also, it was shown by Couniot [13] that the present remeshing algorithm is of a very low order and that there is no point in devising sophisticated techniques. The Euler scheme has the benefit that the area scanned by a front segment is represented by a quadrilateral, possibly degenerating into a triangle. The union of these elements defines the extrapolation mesh. In the case of thickness discontinuities or bifurcations, as velocities undergo a transformation when the material points cross the singularity (see Section 3.3.3), each quadrilateral originating from an upstream facet (relative to the singularity) yields one transformed quadrilateral per downstream facet (Figure 14). Managing the extrapolation mesh then becomes relatively easy [ 17]. The situation is more complex regarding front junctions. Computing the new position of the fronts features the difficulties of computational geometry [ 13], these stemming from the discrepancy between pure geometry and its numerical transposition. As a consequence, geometrical algorithms involve tolerances, which means that the transitivity of geometrical properties is lost (e.g., two straight lines which are parallel to a third line to a given tolerance are not necessarily parallel to each other to the same tolerance). The complexity of these algorithms is therefore often dramatic, particularly in the simulation of molding processes when a front tracking strategy is selected, since the size of the element sides must be controlled in order to obtain well-behaved meshes. To overcome these problems, the solution must be implemented as a mix of numerical and logical operations. In particular, the present front motion algorithm is

979

Completely filled

[

/t//tltl

I

Partially filled

Completely empty

Figure 15. Partition of the elements of the fixed mesh. based on the concepts of "virtual front nodes" and "events" [13], and uses as reference the fixed mesh. Basically, a virtual node corresponds to the intersection of one front segment with one side of a mesh element. As the front moves, the virtual node moves along the element side until it reaches a mesh vertex, which constitutes a first type of event. Other event types include the crossing of a mesh boundary or a re-entrant comer along the boundary. All the events are sure to be detected and simultaneous events can be processed asynchronously (which consists in removing or inserting virtual nodes in the front description). Hence, operations are mostly logical and not numerical.

4.2.2 Generation of the temporary mesh used for the next iteration The discrete front description provides a partition of the elements of the fixed mesh into three classes, viz. the elements which are (i) completely filled, (ii) completely empty, and (iii) partially filled (Figure 15). A simple approach would be to re-use as such the elements of class (i) and to generate additional elements in the remaining portion of the filled area ~ the so-called generation zone. This approach is workable with some enhancements. First, the discrete front description must be modified in order to handle front-side wall and front-front meetings, and filtered to ensure that the local front discretization scale corresponds to the local fixed mesh scale. Such conditioning can be achieved by redistributing the front vertices so as to preserve mass balance. As front collisions are treated by accepting a slight overlap between the colliding fronts, or a slight crossing of the side walls, it is easy to detect and remove the dead front segments. Front filtering must also remove the unfilled areas which are regarded as negligible and increase the filled areas which are still negligible (removing matter can lead to a halt or a non-natural motion of the fronts). The simulation time is finally adjusted on the

980

basis of a comparison between the total mass integrated at the nozzle and the mass currently present in the mold. After front filtering, the temporary mesh is generated. First, the fronts are located with respect to the fixed mesh and, if strictions are detected, some totally filled elements are considered as partially filled. Generation is then performed. As a 1 st step, triangles are built using a Delaunay triangulation that is constrained to use the boundary vertices of the generation zone. This method, which leads to optimal triangulations with respect to some criteria, can be implemented quite efficiently and extended to curved surfaces. Preserving imposed boundaries can be achieved in several ways [13]. As a 2 "d step, as many triangles as possible are merged two by two into quadrilaterals. The temporary mesh Mn+ 1 is built by merging the filled elements and the generated ones.

4.2.3 Integration of temperature and additional fields The temperature field is discretized by means of fimte elements along the midsurface and cubic splines in the gap [25,26] (full polynomials as in [9,11] are less stable):

T --- ~, T~i) (z,t) r

) ,

(96)

i

where the shape functions r

are linear or bilinear on the parent element,

while T~i) (z,t) is the temperature profile approximation at node x~ ) . Space discretization of the energy equation is performed by means of a hybrid scheme combining a S UPG technique along the midsurface [ 11,25] and collocation in the gap. Weighted residuals of equation (46) are l~t integrated over f~n, and are further considered for each node

z (k) in the gap. The original SUPG

method [135] consisted in defining the test functions ~i)(xa ) as" ~(i) -- ~)(i) +

where

A'i" v a o3~)(i) /oaxot ,

(97)

A~" = (l~'~lA~ +lVo[Ao)/(-~/~Va 2) is an element characteristic time (~

and 7"/ denote local coordinates, with - 1 < ~,7/< 1 for quadrilaterals and 0 < ~ , 0 , ~ + 7/1 ,

else D C-aw = O, Dt

~ c) j Dav2 Dt

The parameters t oi represent the isothermal induction times, while nonisothermal induction times are calculated from the accumulated factors 0i [159], which reach 1 when induction is complete and the associated mechanism starts. Equations (105) refer to the model of [157,158] : the time derivative of the specific volume VI of the fictive nuclei is the product of their specific surface S 1 by the growth rate G, while DS1/Dt is 4~r times the product of the fictive specific diameter 2R 1 by G, and DR 1/Dt is the product of the specific number of fictive nuclei N 1 by G; the evolution of N 1 is governed by the nucleation rate /~ (Figure 23). As the fictive specific volume (V1) is also the fictive crystallinity degree, the Avrami postulate provides the relative crystallinity degree

C Ctvl

of the first mechanism. C

The relative degree for secondary

crystallization av2 is calculated by means of the differential form of the Avrami equation, with k 2 and n 2 as kinetic parameter and Avrami index. The real degree of crystallinity is calculated by combining the 1st and 2 nd mechanisms

997 (with the weight W 1 ), taking into account the asymptotic degree of crystallinity oo

tzv . The model of [157,158] was not used for the secondary mechanism, whose physical behavior is totally different. It is assumed that nuclei pre-exist in the melt, and that, in static conditions, the number of activated nuclei /V1 is independent of time (with instantaneous nucleation). Hence, /V1 is a given function of T, ~' and p, which can be determined experimentally. When the material undergoes a complex thermo-mechanical history, /V is obtained from the relation /V - max (D1Vm(t)/Dt, 0), in order to avoid any decrease of N 1 when the polymer is heated. This expression should be improved. This model can predict PET crystallization in typical laboratory apparatus (such as DSC or cone and plate rheometer) in both isothermal and non-isothermal conditions [20,93]. In principle, the dependence of PET material properties on txv should be specified. Detailed material functions are given in [20]. However, these effects may be neglected in PET injection molding since crystallization develops mainly during the packing and cooling stages.

6.2.2 Experimental and numerical results Injection molding experiments were performed using the equipment described in Section 5.2. The half-thickness of the plate was 3 ram. Processing conditions are detailed in [20]. The simulation was peformed by imposing a gate flow rate of 60 cm3/s during filling, while the experimental pressure evolution was prescribed during packing and cooling until the gate was frozen. A non-uniform wall temperature (from 293 K to 303 K) was imposed, corresponding to wall sensor measurements performed at the beginning of the filling stage. The results obtained by prescribing a uniform inlet temperature of 568 K are depicted in Figures 24(a) and 24(b), where the predicted and experimental crystallinity profiles are represented at two locations in the plate, together with the profiles of N 1 . The results obtained by taking only thermal effects into account, and only thermal and shear effects, are shown for comparison purposes. In all cases, a zero a v is predicted along the walls, indicating a quench of the polymer when it meets the walls, while a uniform crystalline core is obtained only if pressure-induced crystallization is taken into account (when only the thermal effect is considered, the core crystallinity reaches 3 %, which is far from the experimental 28%; when the shear stress effect is added, no significant difference is observed). However, the predicted and experimental crystallinity profiles do not fit well in Figure 24(a). As explained in Section 5.2, this results from the important temperature increase observed in an annular region located at the inlet of the gate and

998 0.6 i

(a)

II

0.4 "~

-~

~

i

"'

~ .

.

I

.

I

.

.

.

.

.

'"

.

I

.

~

cr

k1~6l ~O: ,

0

.............................. ..-.................. x .............. :.....-.........::...........................

rf/

0.3

.....

1020

i

-"-~176 -""--~

\~1014

0.2 r,.)

~.

0.1

I

0.0

__.~ _=_- : ...............k ........... : - ~----'~""i'~"-'" I I I -2 -1 0 1

-3

~176 ~

....- ,L. ! % ..

0.6

%e

.

.

~

"] 1012

.

.

.

1020

.

0.5 "~ o ,,,,,~

o.2

b ro

1018 101s

-/"

0.1 -

~,,~.

I" o~

.--

-.:.-:..:...-...~...(....-..-'=.....':...=...--"

?~1 \ v4

-

i

i

i

...............t..':..___.. L/

-1

0

1

2

3

1012 c_.._l

101~

0.6

102o

0.5

10 TM

(c)

0.4

, ,...r

r~

b

0.2

1016

t

0.1 L _ ~ ~ r 0.0 I -3

~

O

1014

0.0 t.7 .... _:..r ............ _1 -2

exl)

z cur

0.4 0.3

:tt:

~..o %LA .......~ - . . ~ . % - ~ l 10 lo 2 3

(b)

"9

~

i

% -

I

-2

-1

......

I.....

I

0

1

-

Z

0

1014 c~ i,,,~ 9

1012 L--..-I

101~

~

2

3

z [mm]

Figure 24. Gapwise crystallinity profiles at distances of 55 mm (a and c) and 310 mm (b) from the gate in the symmetry plane. Comparison between experiments (e) and numerical predictions with (a and b) uniform and (c) improved inlet temperature profiles. Dotted lines" only thermal effects considered; dashed lines" only thermal and shear effects; solid lines : all effects. (From [20]).

999 due to viscous heating in the nmner. Since nuclei are generated mainly upstream from the delivery system (i.e. where p is the highest), they are partially destroyed in the nmner under the action of the high peripheral temperature. Hence, their inlet distribution is not uniform, and this strongly affects crystallization. Also, the non-uniform inlet temperature profile again had a major impact on the front shape. A simple model was developed to evaluate the distribution of nuclei downstream from the gate [20]. The resulting profiles are shown in Figure 24(c), which exhibits a much better agreement between calculation and experiments.

6.3 Filling of fiber mats SRIM and RTM differ fundamentally from classical injection molding. Usually, in these processes, a thermosetting resin is injected into a cavity where a fiber reinforcement has been placed beforehand [7]. Using SRIM or RTM, it is possible to achieve, in one shot, complex composites with a high ratio of continuous fibers and excellent mechanical properties, while the low resin viscosity (< 0.1 Pa.s typically) allows the manufacturing of large parts (of several m 2) with low cost tools. In addition, good orientation control is obtained. Although the reinforcement considerably modifies the physics, the same simulation tool developed for TIM and RIM may be used to simulate SRIM and RTM provided some particularities are taken into account [21,25,80,96,97,100,133, 160-163]. The equations must be established via a multiphase analysis, in which the liquid flows through a solid porous medium (cf. Tucker and Dessenberger [8] and the references therein). For an incompressible Newtonian resin, stationary fiber reinforcement, negligible inertia and long range viscous forces, the resin flow is governed by the Darcy law, which relates the intrinsic fluid velocity v i to the pressure gradient O~/OqXi ,

v i = - / ~ ~ Kij Op/Oxj

,

(107)

where q~ is the porosity (i.e. the liquid volume fraction), 7"/ is the resin viscosity and Kij the permeability tensor. The Darcy law is useless in describing how the resin penetrates the bundles and wets each individual fiber. As in TIM and RIM, most SRIM or RTM parts are thin (see Section 3.1). Dimensional analysis thus leads to combining the mass and Darcy equations as

c~xa

1000

h

where

S ao -

I / ~ KaO dz .

(109)

-h

Greek indices mean that the quantity is limited to its in-plane components. Equation (109) has the same form as (42), the fluidity factor S being replaced by a fluidity tensor Sag. The procedure outlined in Section 4.2 is thus applicable, provided minor modifications of the front tracking algorithm in singular regions are introduced [21,25]. The in-plane permeability Ka# is generally determined by eigenvector-eigenvalue measurements in simple flows [7,97]. In addition, progress has been made in relating the reinforcement geometry and draping with Ka~ (taking into account in some cases the elasticity of the preform) [164-170]. To illustrate the effect of permeability, the injection of an automobile front hood has been simulated using two different kinds of reinforcement (Figure 25). The filling time is 141 s. Crosses indicate where vents should be placed. This result shows that permeability is a key factor in determining the geometrical and operating parameters for a given part. It should be noted that, as perfect draping along the borders or the junctions is difficult, high permeability zones are often found, resulting in easy flow channels or race tracking [162,171-173]. In SRIM and RTM, solidification generally results from a chemical reaction initiated by heat or mixing during filling. In order to reduce the cycle time, the mold walls are set to a high temperature so as to enhance this mechanism; hence, thermal effects must be taken into account in the model [ 100,102]. However, the heat transfer in the porous medium is much more complex than in Section 3.1, since hydrodynamic dispersion must be considered due to the tortuous nature of the flow paths. Assuming local thermal equilibrium between the fiber bundles and the resin, the energy equation [99,101] reads as :

(op, c,

p,c, v, O [Kij+Kii~o~I"

1 ap

@

(11o)

where P ph and Cph denote the specific mass and specific heat (with subscripts l and s referring to the liquid and the solid), Ki~ and Kia are the thermal conductivity and mechanical dispersion tensors, and ~ is the specific heat source due to chemical reactions. In (1 lO), K~ is a function of the conductivity

1001

(a)

(b)

(c) (d) Figure 25. Isothermal simulation of the filling of an automobile front hood" (a) finite element mesh; (b) example of temporary mesh; (c and d) evolutions of the flow front for a non-isotropic (Kxx/Kyy - 3 ) and an isotropic reinforcement. of the two phases and the reinforcement geometry [21,174], while

KiJ, which

formally accounts for the mixing convection heat transfer, is often considered to be proportional to Mal et al. [21] propose the equation"

Uv,II

Kid where

=" ~Pl Cl IIVml1-1 Zijkl Vk Vl ,

Lijkt

(111)

is a fourth order tensor. This relation can induce different mixing

effects parallel or perpendicular to the flow, in agreement with experimental observations [175]. Also, it is observed that hydrodynamic dispersion is invariant with respect to time scale changes, corresponding to a purely geometrical mixing model. As for Kij, the symmetries required for Lijkl are not clear. A complex dimensional analysis shows that, in SRIM and RTM, gapwise mixing is

1002

t = 70s (end)

31530~ 32.5~ 335~/'~ (a)

t = 70s (end~ " "

305 ,,,-~ 315 325/'~~ (b)

3

t = 141s

(end)

305

,,,,315~-,,7"~

33~.~@r" 3 4 ~

4

t = 141s

(c)

(end)

~

...315~

305

33~~, / 345~~~~,,~

5>

(d)

Figure 26. Simulation of the non-isothermal filling of an automobile front hood. Average temperatures (in K) at end of filling. Top : filling time = 70 s; bottom: filling time = 141 s; (a, c) no dispersion; (b, d) Lzz = 0.0017 mm. (From [21]). predominant when compared with in-plane contributions, and

tijkl

V k. v t

thus

has a single non-negligible component Lzzal3 v a v/3. ff it is further assumed that Lzzoc~

is transverse isotropic [21] ( L z z a o - L z z tSaO), the equations involve a

single mixing coefficient Lzz and are equivalent to the model of Tucker and Dessenberger [8]. Thanks to the material data provided in [99], realistic numerical experiments may be carded out. In particular, the non-isothermal filling of the automobile front hood of Figure 25 has been simulated by Mal et al. [21 ] in order to compare

1003

the results obtained with and without neglecting mechanical dispersion (Figure 26). Even though Lzz is small, its effect on the average temperature distribution leads to a difference of up to 10 K in the entry zone when filling is faster. It is essentially in the immediate vicinity of the gate that its effect is important, because mechanical mixing is very effective in the regions where thermal gradients and velocities are high. A complex balance between dispersive and conductive heat transfer takes place during filling and, the higher the flow rate, the less effective conductive heat transfer is (while mixing acts on higher thermal gradients). In conclusion, the global heat transfer is increased in the vicinity of the gate, while it is reduced in regions of lower velocities. Two closing remarks are in order. First, chemical reactions can be modelled using the methodology of Sections 4.2 and 6.1, as long as the kinetics and material data are known. The evolution equation of species concentration is similar to (110), with an additional term relating to curing kinetics, while molecular diffusion can generally be neglected and mechanical dispersion is governed by the tensor Lijkt. Secondly, it is easy to extend to SRIM and RTM the theory of abrupt changes of thickness and bifurcations developed in Sections 3.2.3 and 3.3.3. In the case of isothermal flow, tmiform material properties and a transverse isotropic permeability, mass conservation exactly provides (81), while pressure continuity across the singular line forces the continuity of the tangent velocity

(v2f +- v2f-),

instead of (82) (see Figure 12). At the front, equation

(83) remains valid and thus, after a few calculations, (81) and (83) show that, when h + ~ h - , the normal velocity component vanishes at the flow front +

(Vlf -Vlf

-0).

This simple effect, which holds true in the case of bifurca-

tions, can be observed in practice. purposes. 7.

It can also be exploited for computational

CONCLUDING REMARKS AND A C K N O W L E D G M E N T S

This chapter summarizes the models and numerical techniques that were elaborated to develop the MOLDSYS simulation software. Combining sound physical knowledge with rigorous mathematical approximations and accurate algorithms proves efficient in addressing the main issues of process modelling. Various polymer molding techniques, all based on the thin cavity assumption, have been implemented in the simulation program. The numerical method, which is based on front tracking, automatic remeshing and extrapolation (taking fountain flow into account), has been successfully extended to the filling of complex parts.

1004

Results prove the validity of this approach. Also, it is shown that the simulation tool can be used to better understand the tmderlying physics when complex materials are molded. Besides the authors of this chapter, several people participated in this research, including Marcel Crochet, Luc Dheur, Olivier Hansen, Kali Kabanemi, Natasha Van Rutten and Vincent Verleye, whom the authors wish to thank here for their contributions. The work was carded out within the framework of the European BRITE project RI1B-0087-F(CD), the "Multimat6riaux" project of the Walloon Region of Belgium, and the program of Interuniversity Attraction Poles of the Belgian state, and in collaboration with the Shell Research and Technology Center in Amsterdam (the Netherlands) and the "P61e Europ6en de Plasturgie" of Oyonnax (France). The experimental work was possible thanks to Rapha61 Favier's permission. Grants from the IRSIA (Belgium) and the FRIA (French Commtmity of Belgium) are acknowledged. The authors wish to thank JeanPierre Gazonnet, G6rard Dechavanne and Virginie Durand for their friendly help and advice in performing the molding experiments. The efficient page setting work of Victor Vermeulen was also appreciated. REFERENCES

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F l o w o f P o l y m e r i c Melts in Channels with M o v i n g B o u n d a r i e s A.I. Isayev a, C. Z o o k and Y . Z h a n g alnstitute of Polymer Engineering, The University of Akron Akron, OH 44325-0301, U.S.A.

1. I N T R O D U C T I O N

1.1 Significance of the Problem The development of numerical techniques to accurately approximate the flow of polymer melts in channels with moving boundaries is of paramount importance in polymer processing. In polymer processing, such a flow situation occurs in injection molding, extrusion and simultaneous injection/compression molding. In the case of injection molding, such a flow situation occurs in the non-return valve located on the front of the machine screw. During the injection stage of the molding process, the valve is required to close to stop the flow of polymer melt back into the screw region. Typically, a ring, ball or piston is utilized to close the flow passage into the screw to facilitate this shut-off. In extrusion, varying the geometry of the die by using choker bars or deformable lips allows the control of melt flow to obtain products according to desired specifications. In simultaneous injection/compression molding, polymer enters a mold and is compressed by a moving boundary that is perpendicular to the flow direction. In all of the examples, the boundary moves perpendicular to the dominant flow direction. Current research in our laboratory has focused on the simulation of the nonreturn valve during the molding process. Figure 1 shows a typical non-return valve used in an injection molding machine. This valves has a cylindrical ring which closes during the injection stage to close the passage into the screw. This closure stops polymer melt from flowing back into the screw. During the recovery stage, this ring opens to allow melt to accumulate in front of the valve/screw assembly. This melt will be injected into the mold during the next injection step.

1012

Valve

Ring o.~tflo~.~,( \

Outflow /

_Inflow

Inflow

Figure 1. Sketch of typical ring type non-return valve for the injection molding of thermoplastics. A plot of the pressure traces in front of the valve on the barrel wall (downstream) and behind the valve (upstream) in the screw section and screw displacement during the injection and recovery stages is shown in Figure 2. At the start of injection both pressures increase rapidly. The upstream pressure reaches a steady value, while the downstream pressure in the screw metering section decreases drastically. This decrease in pressure indicates the ring closing the flow passage into the screw. Thus, the valve is closed. The closing time of the valve can then be determined. For the recovery cycle, an oscillating pressure is observed in the upstream pressure transducer. This is caused by the movement of the flights of the rotating screw over the stationary pressure transducer. The downstream pressure measurement has a steady pressure due to the accumulation of polymer melt in the large reservoir (shot size) in front of the valve for the next injection stage. The pressure drop across the valve/screw determines the amount of resistance during the recovery stage, which affects the length of the recovery time. The ability to simulate this process of the valve opening during the recovery stage and closing during the injection stage will determine the time for the valve to open, close and the forces acting on the ring, which will help in the development of more efficient valves [1 ]. In order to understand the phenomena that takes place during the polymer processing with moving boundaries, an experimental slit die with a moving boundary has been designed (Figure 3). The polymer flows through the slit die as a wall closes the flow passage. The transient pressure drop across this wall is measured. To numerically simulate the transient channel flow with a moving boundary, computational code based on the f'mite element method has been developed using the Giesekus viscoelastic model. In addition, a commercial computational package is utilized to understand the transient polymer flow using a quasi-steady approach. The pressure drop from the die experiments and numerical simulations will be compared in an attempt to validate the numerical technique. The contribution of the elasticity of the polymer will be investigated

1013

30

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Figure 2. Pressure measurements across valve and screw displacement versus time during the injection molding of HDPE with an injection speed of 2.54 cm/s and a melt temperature of 204.4~ by comparing viscoelastic simulations with generalized Newtonian-inelastic simulations using the Cross model. Changes in the velocity and stress components will be discussed to indicate the effect the moving boundary has on these variables. The usefulness of these methods in polymer processing will then be discussed. V,

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of viscoelastic simulations

In the past two decades, polymer melt flow has been extensively researched. This research has led to an understanding of the basic flow of viscoelastic fluids in many different polymer processing applications. The advancement of this knowledge base has been expanded mainly through the implementation of computational simulations, which can only approximate realistic polymer

i

1014

processes. In particular, the simulations is carried out for two or three dimensional flow domains, complex geometry such as abrupt and tapered contraction flows, and flow with transient or unsteady boundary conditions. Nonisothermal boundary conditions have also had some limited development in extrusion, but not in the area of abrupt or complex geometry. Advanced numerical techniques available to simulate the complex flow of viscoelastic fluids include the fufite difference and the fimte element techniques. The books of Crochet [2] and Tucker [3] give a good review of these techniques. To properly model viscoelastic flow, one needs to understand numerical simulation techniques, the non-linear viscoelastic constitutive equations and the mathematical characterization of these systems of equations. Techniques used in past research include analytical solutions, decoupled fimte difference, decoupled f'mite element, and coupled/mixed f'mite element methods. In a decoupled approach, the goveming equations and comtitutive model are solved at different steps of the solution process. In a mixed/coupled approach, the goveming equations and constitutive equation are solved simultaneously. The choice of governing equations (energy, momentum and continuity) and the assumptiom which constrain the applicability of these equations, as well as the choice of constitutive equation, determine the type of viscoelastic flow a numerical technique can realistically simulate. Together, the goveming equations and constitutive equation form a set of nonlinear differential equations. This nonlinear set of differential equations is descritized into an algebraic system of equations, which can be solved with numerical techniques. The flow domain is divided up into elements or nodes and solved locally for each set of algebraic equations. The solutions from these local elements are then related globally to the entire flow domain to reach an accurate approximation. One technique to solve this system of equations is the finite difference method, although, much of the recent interest in numerically modeling viscoelastic flow has been concerned with the finite element method. In this type of approximation, the flow domain is discretized into a set of nodal points. The governing equations are then approximated at each node point using a Taylor series approximation. The constitutive equation is then solved separately using an iterative approach. Past work in this area has been conducted by many authors including Perera et al [4,5,6], Davies et al [7,8], Gatski and Lumley [9], Townsend [10,11], Tiefenbruck and Leal [12], Yoo et al [13,14], Phelan et al [ 15] and Yuan et al [ 16]. The method of choice for the simulation of viscoelastic flow is the finite element method. In this method, the flow domain is descretized into triangular or quadrilateral elements. The system of equations is approximated using the

1015 Galerkin method or method of virtual work. The equations are then solved by standard elimination techniques such as Gauss or Gauss-Siedel elimination. The solution of these equations gives approximations of the pressure, velocity components, extra stress components, etc. The two techniques to solve these equations are the decoupled and mixed/coupled methods. In the decoupled method, the continuity and momentum equations are solved for the Newtonian flow field. Then, the constitutive equation is solved separately using some type of streamwise integration. Usually, a Picard iteration, which is also called successive substitution, is applied to the constitutive equation to attain convergence. The extra stress contribution from the constitutive equation is then incorporated into the momentmn equations and then a new solution is found for the momentum and continuity equation. This iterative procedure is repeated until the convergence criteria is attained. Researchers to have used this method include Upadhyay and Isayev [17,18], Isayev and Huang [19], Bush et al [20], Lou and Tanner [21,22,23], Lou and Mitsoulis [24], and Hulsen and Van der Zander [25,26]. Since the streamlines start as flow boundary, the recirculation zones in contraction flows are solved by a numerical technique found in Upadhyay and Isayev [17,18]. The use of decoupled techniques has also been utilized for integral constitutive equations by Viriyayuthakom and Caswell [27], and Dupont et al [28] using particle tracking and Lou [29] using the control volume approach. Other methods include the Choleski decomposition method with a Picard iteration by Mitsoulis et al [30,31 ], streamline upwinding (SU) and Lesaint-Raviart methods by Fortin and Fortin [32], streamline-upwinding/PetrovGalerkin (SUPG) by Lou and Tanner [22], Barakos and Mitsoulis [33], and Lou [34], the adaptive viscoelastic stress splitting/streamline integration (AVSS/SI) scheme and adaptive viscoelastic stress splitting/streamline-upwind PetrovGalerkin (AVSS/SUPG) by Sun et al [35] and botmdary integral methods by Bush and Phan-Thien [36,37]. The use of mixed methods to simulate viscoelastic flow has received a large amount of attention due to the inability of these schemes to converge at high Deborah numbers. Early work on these techniques employed the method proposed by Kawahara and Takeuchi [38], where the Galerkin method and a Newton-Raphson solver converged only at low Deborah numbers. Researhers to have used this method include Crochet and Bezy [39], Crochet and Keunings [40,41,42], Davies et al [43], Dupret et al [44], Debbaut and Crochet [45], Keunings [46], Musarra and Keunings [47], Mendelson et al [48] and Kajiwara et al [49]. Other earlier research of Yeh et al [50] applied numerical methods designated as quadratic stress interpolation (QLL) and stress interpolation using bilinear polynomials (QQL) with Newtons method to solve the system of

1016

equations. The QQL method was found to converge at higher Deborah numbers and have a more stable convergence. This QQL method was also used by Brown et al [51]. Later, Marchal and Crochet [52] developed a more advanced mixed method where each element is divided into several bilinear sub-elements for stress to improve the algorithms convergence ability. In addition, some authors used the SU and SUPG methods introduced by Hughes and Brooks [53,54] to improve convergence. Other authors to have used this method include Debbaut et al [55], Pumode and Crochet [56], Hartt and Baird [57] and Yunm and Crochet [58]. Methods to improve the convergence of mixed numerical schemes have been researched by Brown, Armstrong and coworkers [59,60,61,62]. King et al [59] implemented a numerical method into the momenaun equations which makes the elliptic property of this equation explicit called the Explicitly Elliptic Momentum Equation (EEME). SUPG is also applied to this numerical scheme. This type of method has also been utilized by Burdette et al [60] and Coates et al [61]. One other formulation developed by Rajagopalan et al [62] is the elastic viscous split stress (EVSS) where the viscoelastic stress is split into the viscous and elastic stresses defined by x= _~ + ft. In addition, Sasmal [63] used this method. Other methods applied to the mixed solution technique include the spectral/fmite-element method by Beds et al [64], the timediscontinuous/Galerkin least-squares (TD/GLS) method by Baaijens [65,66], the Taylor-streamline upwind-Petrov-Galerkin/pressure correction formulation by Townsend and Webster [67,68], and the modified EVSS by Geunette and Fortin [69] and Azaiez [70]. Integral models have also been solved by mixed methods. An example of this is the paper by Papanastasiou et al [71 ]. All of these numerical techniques incur some difficulties in the approximation of viscoelastic flow. One problem studied by Holstein and Paddon [72] and Lipscomb et al [73] is the singularities which occur at the re-entrant comer of contraction flows. The extra stress terms can reach physically unrealistic values since the flow behavior of the fluid at these comers is not well understood. The gradients of the stresses at the boundaries can become excessively high. To overcome this problem, either the mesh size is increased in the region of the reentrant comer or a slip condition is applied at the boundary. Another source of instabilities is inherent in the constitutive equations themselves. Kwon and Leonov [74,75] have investigated the Hadamard stability and the positive def'miteness of the configuration tensor of different constitutive equations. Models proposed by Larson, Leonov and Giesekus, which are known to accurately predict viscoelastic flow, experience blow-up instabilities. These instabilities can be overcome by properly modeling the dissipative terms, adding a small

1017

Newtonian viscosity or using a multi-mode approach with respect to the relaxation spectrtma [17]. Van der Zanden and Hulsen [26] have also reported on these types of instabilities in the simulation viscoelastic flow. Joseph [76,77] investigated instabilities during the flow of viscoelastic fluids. He determined that hyperbolicity changes the system of equations from an elliptic to a hyperbolic type, thus making numerical solutions difficult. The simulation of the moving boundary requires the use of a transient numerical scheme. The basic understanding of the unsteady flow of constitutive equations with a time dependent history of the fluid deformation is well documented. The nonlinear response of viscoelastic fluid is different from the response typically seen for a Newtonian fluid. To more accurtely simulate this fluid, a viscoelastic model is applied in conjunction with a multi-mode approach to the deformation history. One type of flow situation, the sudden imposition of flow, occurs in many polymer process applications. The types of unsteady flow include a suddenly imposed velocity gradient, constant pressure gradient, a periodic pressure gradient, start-up flow with a wall moving parallel to flow, etc. In unsteady flows, the elasticity does not effect the f'mal velocity, but at the imposition of the velocity or pressure gradient a nonlinear response occurs in the velocities and stresses of the fluid. For a suddenly imposed velocity gradient, an overshoot in the pressure occurs. For the suddenly imposed pressure gradiem, a velocity overshoot is observed and is followed by a velocity minimum before a steady value is attained. Researchers to have investigated this phenomena include Fielder and Thomas [78], Waters and King [79], Chong and Franks [80], Chong and Vezzi [81], Townsend [82], Akay [83], Duffy [84,85], Balmer and Fiorina [86], Ryan and Dutta [87], Upadhyay et al [88], Isayev [89] and Kolkka and Ierley [90]. The solution of time dependent problems for the full set of governing equations has held much interest in the modeling of viscoelastic fluids. Along with the inclusion of a time variable, numerical stability and convergence difficulties are already confronted when trying to model the steady flow of a viscoelastic fluid. In addition, time dependent flows add an increased amount of computational time to solve problems. The numerical scheme typically used to approximate the time variable is a fmite difference approximation. Past publications applying transient simulations include the research of Townsend [10] using a De-Fort Frankel scheme, Keunings [91], Northey et al [92] using fully implicit integrator and semi-implicit integration, Keiller [93], Olsson [94] using a predictor-corrector method for time stepping, Rasmussen and Hassager [95] using a Lagrangian integral method and Baloch et al [68]. Results indicated that stable time dependent solutions were attainable, but flow occurring at high Deborah numbers

1018

were not presented. In order to understand the behavior of a viscoelasstic fluid, the experimental techniques that have been employed include streak photography, stress birefringence studies, multiple flash technique and laser doppled velocimetry. An excellent review of this research has been published by White et al [96] and lsayev and Upadhyay [97]. These methods work well when clear polymers are used in the experimental research. Unfortunately, fluids such as rubber compounds which cannot be seen through or polymers which can only be processed with industrial processing machinery cannot take advantage of these experimental methods. The only other experimental variable to measure is the pressure at various locations in the flow channel. Experimental work on pressure prediction to numerical simulations has been conducted by Isayev and Upadhyay [97] and Huang et al [98,19]. Results though indicate that the prediction of pressure is not as accurate as the prediction of the other field variables. In addition, this error can become significantly high for higher flow rates and in an abrupt contraction or expansion. In the present research, a decoupled finite element method previously used in two dimensional unsteady viscoelastic modeling is applied to the solution of polymer flow in channels with a moving boundary. The momentum and continuity equations are solved using the method of virtual work. The constitutive equation is solved by a Picard iteration technique. This formulation has been shown stability up to a Deborah number of 50 and a Weisenberg number of 4.2 by Upadhyay and Isayev [17], a Deborah number of 270 by Isayev and Upadhyay [98], and recently to a Deborah number of 845 by Isayev and Huang [19]. The only limitation in achieving a higher value of the Deborah number is the density of the mesh. A higher density mesh, hence, allows convergence at higher Deborah numbers. Added to this past f'mite element code is the simulation of transient flow with a moving boundary, which is closing the flow domain. This boundary moves in the direction perpendicular to the main direction of flow. In addition, the computational package FLUENT has been utilized to simulate the flow of a generalized Newtonian fluid based on the Cross model. The simulation using this package applies a quasi-steady approach with a velocity imposed at the moving boundary where the wall is closing the channel. Through these two paths of numerical simulations, the understanding of the effect the transient nature of flow with a moving boundary has on viscoelastic flow will be developed and understood.

1019

2. THEORETICAL The modeling of polymer processing can be grouped into three particular areas of research. One is based on the generalized Newtonian theory, which takes into account only the nonlinear shear rate dependent viscosity of the material. The other is based on the viscoelastic theory of polymer melts, which has many different forms, but is mainly concerned with incorporating the non-linear stress terms into some type of constitutive equation along with the nonlinear viscosity. To complete the theory of viscoelastic modeling, the viscoelastic-plastic fluid theory is also covered. 2.1 Generalized Newtonian Fluid

For most types of fluids, the viscosity of the material does not change as the shear rate on the material increases. For an incompressible Newtonian fluid the shear stress is related to the strain rate by the equation r~ = -ja~

(1)

where ~t is the Newtonian viscosity which is a constant for a given temperature, pressure and composition. Early rheologists used the empiricism that the viscosity could be a function of shear rate to model the shear rate dependence of the viscosity of suspensions, pastes, polymer melts and solutions. Thus, they incorporated the dependence into the previous equation as

(2) where r/(~;) is the apparent viscosity, which is a function of shear rate 2, pressure and temperature. Many generalized Newtonian equations have been postulated since the realization of this type of relation. The types of available models are the CrossArrhenius, Power Law, Klein, Carreau-Yasuda, Spriggs tnmcated power law, Eyring, Powell-Eyring, Sutterby, Ellis, and Bingham. Bird gives a good review of many of these models in chapter 4 of reference [99]. Two very useful generalized Newtonian relations, are the power law and the Cross-Arrhenius equations. The power law equation of Oswald and de Wade [ 100,101 ] is written as -n--1

r/=mg

(3)

1020

where m is a constant and n is the power-law index. This equation can only fit the high shear rate range of the viscosity curve called the power law region. This equation is useful since many analytical solutions are available, which can be easily understood. Many of these solutions can be found in the book of Bird [99]. In addition, this equation is capable of being implemented into numerical solutions easily. An example of this is a paper by Hieber and Shen [ 102] and also in Hieber [103] in which a finite element/finite difference is used to simulate the injection-molding filling process. Hieber [104] and Isayev and Upadyhay [97] also used the power law equation along with the finite element method to estimate the pressure drop and extra entrance length across planar and axisymmetric contractions. The main drawback of this equation is the inability to accurately model the Newtonian or low shear rate range of viscosity. The model by Cross [ 105, 106] is a three parameter equation of the form 7/ n(Y) =

o

i-n

(4)

where rio is the zero shear viscosity and ~* is the shear-stress level at which rl is in transition between the Newtonian limit rio and the power law region. The effect of temperature and pressure on the melt viscosity can be taken into account through an Arrhenius type dependence. Hieber gives a thorough description of these types of equations in chapter 1 of reference [ 103]. The modeling of the injection molding process by the Cross-Arrhenius equation has been well documented since the early 1980's. Hieber et al [107] used the Cross-Arrhenius type equation to model material viscosity to understand the effect of juncture losses resulting from the elasticity of the fluid. By comparing the numerical viscous simulation and experimental data, they were able to deduce the amount of juncture pressure from the elastic effects. Sobhanie, Deng and Isayev [ 108,109,110,111 ] have extensively used the Cross-Arrhenius equation for the injection molding of robber compotmds. An additional parameter was added to this simulation to account for the cure kinetics of the rubber compounds during injection molding. The numerical pressure predictions showed qualitative agreement with the experimental pressure data. Other researchers who have used generalized Newtonian models include Duda and Vrentras [112]. These authors used the Powell-Eyring equation to understand the effect nonlinear viscosity has on the amount of extra pressure loss

1021

in juncture regions. Kim-E, Brown and Armstrong [113] used a Carreau type viscosity equation to simulate the flow through an axisymmetric contraction. Results indicate steep velocity gradients occurring near the wall caused high shear rates and a shear thinning viscosity. These steep gradients effect the accuracy of the finite element approximations.

2.2 Viscoelastic Fluids The limiting factor in the ability of generalized Newtonian theory to predict polymer melt flow is the inadequacy of the equations to predict the elastic effects in a flowing polymer. These effects are extremely important in flows of polymers through sudden contractions and expansions. In addition, nonlinear stresses arising from polymer melts contribute to large pressure drops and steep stress gradients. Also, the elastic effects can dominate in start-up or transient flows of polymers. Thus, these elastic effects have a significant impact on the flow of polymer melts in many areas of processing. The modeling of viscoelastic phenomena employs two main types of constitutive equations to model polymer flow. The two types are the integral and differential type equations [99]. In the present paper, we will consider the differential models only. To characterize the flow of viscoelastic fluids, the dimensionless Deborah number defines the rate of straining on a fluid as De = .3U ~,

(5)

where 3U/b is the characteristic shear rate with U being an average velocity downstream of the contraction and ~, is the mean relaxation time defined as N

~rlkkk k = x_-,

(6) N

with TIk being defined as 770= 77~+~ = k, TI~ is a parameter similar to the lower Newtonian viscosity, and TIk, and Ek are the viscosity and the relaxation time in the k th mode. Most differential models are only an extension of the Maxwell [114,99,115] type equations from the 1860's. The Maxwell model assumes a linear relation for the viscous and elastic responses of the fluid. This relation can be written as

1022

Xxy+ G /)t - -tt~xy

(7)

where 17xy is the shear stress and G is the shear modulus. For steady-state motions, the equation simplifies to the Newtonian fluid. This equation shows time-dependent response upon an imposition of flow. In addition, other types of constitutve equations can be explained depending on the type of time derivative utilized for equation (7). For sudden changes with time, the equation simplifies to a Hookean solid. One can introduce the time derivative 5/8t in the convected coordinate system such as V

~-,__=a=$-Vv.a-~=.Vv

T

(8)

where ~ is the substantial time derivative, which translates with the material particle, Vv is the velocity gradient with Vv~ denoting the transpose. This system is called the upper convected coordinate system where the base vectors are parallel to material lines. Therefore, the vectors are stretched and rotated with the material lines, t, the time derivative, is written as D a ~ = b--Tx== ~-x=+ v__.Vx__

(9)

Limitations for this model exist which exclude it from properly predicting realistic polymer flow at high Deborah numbers. For instance, the model does not include any type of shear rate dependent viscosity. The first normal stress coefficient is not shear rate dependent either. The elongational viscosity becomes infinite at f'mite strains and moreover, the recoverable strain is over-predicted at high strain rates. In addition, the model cannot predict polymer behavior after the imposition displacements strains. In later developments, Oldroyd [116] proposed a quasi-linear differential model which was frame invariant called the Oldroyd-B model or the convected Jeffrey's model [117]. This model included a time dependent deformation tensor and is written as

~.,~-~+~ = 2no D+ X.,~D

(lo)

1023

For this equation, one needs to provide three parameters where rio is the zero shear rate viscosity, D is the deformation tensor, s is the relaxation time, and is the retardation time. This equation is capable of describing time-dependent flows, but is still unable to correctly predict rheological behavior. Equations of this type are limited to polymer flows with small deformation rates or low Deborah numbers. To account for these deficiencies more elaborate differential models have been developed such as the Maxwell corotational, White-Metzner, Gordon-Schowalter, Johnson-Segalman and Oldroyd-8 constant models which appear in Bird [99] and Larson [115]. These models were capable of describing more of the rheological properties of melts, but were still unable to describe all of the material properties correctly. In the mid-1960's, models were proposed that could describe polymer flows more accurately. All of these equations contain nonlinear stress terms. With the incorporation of this term, the constitutive equations are more suitable in predicting stresses in shear flow. One model which includes the quadratic stress term is the nonlinear viscoelastic equations developed by Giesekus [118-122]. This constitutive equation is based on the concept of a deformation-dependent tensorial mobility or drag. The equation is derived from the theory for concentrated solutions and melts using the dumbbell theory for dilute solutions. The assumption is made that the mobility is not dependent upon the individual configuration of each polymer segment. Instead, an average configuration of all the segments is used to relate the mobility tensor to the configuration tensors. This averaging of the polymer segment configuration bridges the gap between the molecular ideas from which the constitutive equation comes from and the treating of the polymer chains as a continuum of polymer melt or solution. This equation replaces the scalar mobility constants Bk with a non-isotropic mobility symmetric second order tensor 13to give an equation of the form ~.a+~-~-~= = 2rid

(11)

where $ is the elastic stress term, I"1 is the shear rate dependent viscosity, G is the shear modulus and D is the rate-of-deformation tensor. To achieve realistic predictions, two assumptions are made on the dependencies of $ and l~ on the elastic strain tensor C. First, the relation [G-5] of _ to ~ is of the neo-Hookean type dependence and is written as

1024

where ~ is the unit tensor. The shear modulus, G, is related to nkT, where n is the number of beads per unit volume, T is the temperature, and k, is Boltzmann's constant. The second assumption [120] is that a linear relation exists between 13 to C and is written as = ~ + oc(C- ~)= (1- ct)8 + txC

(13)

In this equation tx is an empirical constant of proportionality (mobility factor) and is related to the compressibility of the material, oc must satisfy the condition of 0 < cz < 1. The most general form of the constitutive equation can be written as

a

2

3

+ ~-~ + ~,~-~ = 2r/D

(14)

When oc equals O, equation (14) reduces to an isotropic mobility tensor and the UC Maxwell model is retained. When oc equals 1, the anisotropy is at its maximum and equation (14) produces results in shear and extension similar to those for the corotational Maxwell model. For the research presented in this paper, the assumption of incompressibility of the polymer is assumed and oc is equal to 1A. In addition, a multi-model approach is applied to equation (14) resulting in 1

l: + ~ ~ k

2

3

+ 2,k-~_~ = 2 r/kD

(15)

where k denotes the k th mode. The basic understanding of this model has been well documemed for many types of simple flows. Giesekus [120,121] published results on the predictuon for simple shear and simple extension which predict shear thinning and a nonvanishing first and second normal stress difference. In addition, this research has defined where the Giesekus model predicts real solutions by Yoo and Choi [123] and viscoelastic instabilities in shear by Oztekin et al [124]. Other f'mdings include the Giesekus and other models ability to predict Poiseuille flows by Schleiniger and Weinacht [ 125], sinusoidally undulating channel flow and TaylorCouette flow instabilities by Beds [126], shear flow and experimental comparisons by Vlassopoulos and Hatzikiriakos [127], exponential shear stress coefficient and elongational flow by Schieber and Weist [128], shear flow, and

1025

uniaxial and biaxial extension by Khan and Larson [129], steady and transient shear flow by Quinzani et al [130], and tmiaxial, biaxial and elongational flow by Isake et al [131]. Larson [132] has compared several different models including the Giesekus model for steady-state flows, start-up steady straining, stress relaxation following cessation of steady straining, single and double step strains, and elastic recovery, in sheafing, or uiaxial, biaxial and planar extension. Other work has discussed the compatibility of equations with equilibrium thermodynamics by Grmela and Carreau [133] and the restriction of the extra stress tensor and requirements that this tensor must be positive def'mite by Hulsen [134,135]. Other differential equations which are often used in the modeling of viscoelastic fluids are the Leonov [136,137] equations, the Phan-Thien Tanner (PTT) model [138,139],the f'mite extensible nonlinear elastic (FENE) model [140,141] and the Doi-Edwards [142] model. For practical purposes, constitutive equations quadratic in stress are the most popular for today researchers. More complex models can be derived, but the implementation of these models would probably be to cumbersome due to the complexity of the mathematical equations. In addition, to many parameters would need to be specified in order to implement the models. Excellent reviews of constitutive equations and the relations between the different models is available in the books by Bird et al [99] and Larson [115].

2.3 Viscoelastic-plastic fluids In the modeling of polymeric fluids, one other area of interest is the modeling of filled polymer systems. The inclusion of the interaction between polymer and filler adds additional assumptions to the constitutive model. These filled systems are characterized by the rheological behavior of the matrix, the particle characteristics, the dispersed state, the interaction between particles, and the interaction between the filler and the polymer matrix. In addition, the particle size effects the rheological properties of the filled polymer melt. For large particles, an increase in shear and elongational viscosity if observed. In systems with small particles, yield values in shear and elongational flow occur, as well as, other changes in the rheological properties. The approach to modeling these filled systems is either a continuum approach or a micro-mechanics approach. The first to propose a stress-deformation rate equation for a fluid with yield stress was Bingham [143]. This was subsequently followed by the equations of Prager and Hohenemser [144]. These authors proposed a constitutive model for the flow of an incompressible viscoplastic material as D = 0 for J2 - t85~

Vw -- Vclosing for 0< t < t85~

Calculations are conducted for the quasi-steady steps of the fully open position and 85% closed position for a velocity of zero and with the specified closing speed velocity. Thus, an instantaneous starting and stopping velocity is assumed. The second type of analysis employed the actual velocity by determining the local slope at each quasi-steady position. This variation of wall velocity during the

1052

closing of the slit channel is shown in Figure 1 la, 12a and 13a. Results of the simulations of pressure traces with the ideal velocity are shown in Figures 1 lb, 12b and 13b for the flow rates of 20, 40 and 70 g/min, respectively. These figures indicate the pressure at each position during the closing of the moving boundary. In all of the graphs, as the flow rate increases for a given boundary velocity, the pressure increases, but not by a significant amount. As the boundary velocity increases for a given flow rate, the pressure increase and peak pressure increases drastically. This indicates that for a viscous quasi-steady simulation the velocity of the moving boundary has a dominant effect on the pressure as the moving boundary closes the slit channel. For the start-up and stopping of the moving boundary, an abrupt change is seen in the value of the pressure. Results for viscous simulations with the actual wall velocity are shown in Figures 14 and 15 for the various closing speeds and the flow rates of 20, 40 and 70 g/min, respectively. The simulated pressures show a sharp increase at the start of the closing of the slit channel. This increase in pressure is not as drastic in the case of the ideal velocity simulations since an instantaneous velocity is not assumed. As the wall closed fiulher, the predicted pressure increases until the end of the closing of the channel is reached. At this point, the pressure reaches a maximum value. The location of this peak depends on the value of the wall velocity at each quasi-steady position. Near the end of the closing simulations, the actual velocity of the wall decreases rapidly. Thus, lower predicted pressures are found when compared with the simulations which use the ideal velocity at the end of the closing of the channel. 5.3 Viscoelastic Numerical Results Viscoelastic numerical results are presented for the transient flow of a polymer in a channel with a moving boundary. The simulations use in-house code which is transient and accounts for the moving boundary. The Giesekus constitutive equation is applied with the fully transient momentum equations. The simulations include the elasticity of the polymer, the transient flow of the polymer and the movement of the wall boundary into the flow domain. The flow rate, zero velocity of the stationary boundaries, velocity of the moving boundary, steady viscoelastic entrance and exit conditions and flow exit into a zero pressure region are applied for the simulations along with the appropriate material parameters. To account for the moving boundary, the viscoelastic simulations employ the actual velocity of wall. Results of the viscoelastic simulations, which use the actual velocity are shown in Figures 14 and 15 for the flow rates of 40, 20 and 70 g/min, respectively. A time step of 0.005s was chosen in the simulations. This

1053

le+7 I -0.317 ~ r .

-0.154 .~

-0.071

t~

le+6 @

,

j

- 4 - - Viscous w/Vw= -Vw(t) Viscoelastic w/Vw= -Vw(t) Experimental Data

7 le+5

-

I

I

I

I

I

I

I

i

3

4

5

6

7

8

9

10

Log t, s Figure 14. Experimental and predicted pressure as a function of time for a flow rate of 40g/min and various wall velocities. time step is capable of capturing the time-dependent response of the polymer, which occurs for the multi-mode Giesekus model. The HDPE used in the experiments exhibits a low relaxation time compared to the external time of straining based on the flow rates and closing speeds. Thus, only the steady state response and not the time dependent response of the second and third modes will contribute to the predictions of the field variables. In addition, if the second and third relaxation modes were included in the simulations, the computational time would become excessively large. The viscoelastic simulations that use the actual wall velocity are shown in Figures 14 and 15. This velocity is similar to the velocity used in the viscous simulations, except that a smoother pressure development occurs since a greater number of time steps are included in the calculations. The simulations predict a sharp increase in the pressure at the start of the closing of the wall. The pressure gradually increases, until a peak pressure is reached near the end of the simulation. However, the peak pressure is lower than that for the viscous

1054

le+7 I --~'- Viscous w/Vw= -Vw(t) Viscoelastic w/Vw= -Vw(0 Experimental Data

,A

le+6 tm O a

le+5 !

I

I

I

I

3

4

5

6

7

Log t, s le+7 -

- Viscous w/Vw= -Vw(t) Viscoelastic w/Vw= -Vw(t)

_.l(~*,~ . . . .

9

r,

9

le+6

o -1 le+5

b

--

I

I

I

I

I

3

4

5

6

7

8

Log t, s Figure 15. Experimental and predicted pressure as a function of time for a flow rate of 20g/min (a) and 70/min (b) for the actual wall velocity with a set velocity of -0.317 cm/s. simulations that use the ideal and actual velocity. This pressure then reduces to a steady value. After the wall has stopped moving the simulated pressure has not reached the steady value as shown in Figures 14 and 15a. This would indicate that a small amount of relaxation occurs in the viscoelastic simulations. In Figure

1055

15b, a large peak in the pressure development does not occur. The pressure development for the viscoelastic simulation almost attain the steady pressure with no overshoot.

6. COMPARISON OF NUMERICAL AND EXPERIMENTAL RESULTS

Figures 11, 12 and 13 are the comparison of simulated and experimental pressure traces for the viscous simulations with the ideal velocity. Figures 11a, 12a and 13a are the experimental velocity of the wall during the experiments. Figures l lb, 12b and 13b are the comparison between the simulated and experimental pressure during the closing of the die channel. For the predicted pressure drops using the Cross model with a quasi-steady type simulation, the simulated pressures do not predict the experimental pressure very accurately at the beginning or end of the transient flow experiments. In most cases, for the flow rates in Figure 11, 12 and 13 the simulated pressures are higher than the experimental pressure at the beginning of the closing of slit die. At the start of the closing of the slit die, an initial jump is observed in the experimental pressure. This experimental pressure jump occurs for the three flow rates and the five closing speeds. The viscous simulation results predict a faster pressure jump than the experimental pressure. This is true for all simulations conducted. Then, the simulated pressure shows a much higher overshoot than the overshoot of the experimental pressure. The simulated pressure instantaneously goes to the steady pressure when the wall stops moving. However, the experimental data indicates that the pressure does not reach a steady value instantaneously, but after a short period of time reaches the steady value. The lower peak pressure and longer time to reach a steady value after the wall has stopped moving indicates the viscoelastic response of the HDPE. The lower peak pressure indicates a delay in the development of pressure due to the viscoelastic effect. The slow recovery to a steady value of pressure after the wall has stopped moving indicates the importance of the relaxation of the polymer. This is very interesting since the polymer has a relatively low relaxation time, but the time dependent effect of the moving boundary results in the viscoelastic effect becoming more pronounced. For the flow rate of 20g/min in Figure 11, the predicted and experimental pressure traces show overshoot at all closing speeds. For the flow rate of 40g/min in Figure 12, some more interesting fluid behavior is seen. For the wall closing speeds of-0.0287 and -0.0711 cm/s, no overshoot is seen in the experimental data, although, an overshoot is numerically predicted for all of the

1056 different closing speeds due to the quasi-steady approach taken and the elastic effects are ignored in these simulations. When no overshoot occurs, the pressure development is delayed and the steady value of experimental pressure is not reached until after the wall has stopped moving. This indicates the viscoelastic effects of the polymer are relevant in the time dependent experiments, which the viscous simulations cannot predict. For the other three closing speeds, an overshoot is seen in the experimental pressure, but the numerical predictions for the peak pressure are much higher. For the flow rate of 70g/min in Figure 13, the experimental pressure for the four closing speeds do not indicate any overshoot. In addition, when the pressure development is delayed, the steady experimental pressure is not attained until after the wall has stopped closing the slit channel. In addition, the time to reach the steady pressure value has a longer time span for the experiments with a slower wall velocity. Part of this delay observed in the experimental data could come from the slowing of the moving wall during the end of each experiment. In addition, the ideal velocity used in the simulations does not take into account the slowing of the moving wall. The simulated pressures predict an overshoot as in the previous numerical results. This reinforces the fact that the viscous simulations cannot predict the pressure accurately for the flow of a viscoelastic fluid, even though the relaxation modes of the polymer are relatively small. Thus, the viscoelastic effects appear to be extremely important in the flow of a polymer through a slit channel with a moving boundary. For the viscous simulations where the actual velocity of the moving boundary is applied, the predicted pressures indicate different results when compared with the experimental values as shown in Figures 14 and 15. The prediction of the pressure during start-up compares well with the experimental pressure since enough quasi-steady increments were chosen to approximate this region. The simulated pressures show good agreement with the experimental pressure after the start-up region except at the end of the simulations. As with the viscous simulations with an ideal velocity, the simulations predict a higher pressure even thought the actual velocity is employed in the simulations. Thus, the simulations based on the quasi-steady analysis cannot predict the experimental pressure at the end of the closing. For the viscoelastic simulations where the actual velocity of the moving boundary is applied, the predicted pressure shows different results. The predicted pressure compares well with the experimental pressure at the start of the closing of the channel. Then, the pressure gradually increases and reaches a maximum and after that reduces to the steady value of the pressure. The experimental peak pressure is not predicted accurately, although, the viscoelastic

1057

2 0.6

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--

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l cm/

~ 0.30.20.10.0

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Figure 16. Predicted velocity profiles at five different cross sectional areas of the flow channel at Q=40g/min and Vw= -0.317 cm/s for viscous (solid lines) and viscoelastic (dashed lines) simulations. simulations that use the actual wall velocity predict a much lower pressure than the other simulations. The predicted peak pressure is always higher than the experimental pressure except in some instances after the wall has stopped closing the channel. The pressure jump at the start of the closing of the channel is predicted well since the actual velocity of the moving boundary is employed and a higher amount of time steps are used in this analysis when compared to the viscous simulations which used the actual velocity. The viscoelastic simulations offer the closest prediction to the experimental data. When an overshoot occurs in the experimental pressure as seen in Figures 14 and 15, an overshoot occurs in the viscoelastic predicted pressure. In addition, the relaxation of the pressure due to viscoelasticity of the polymer occurs in this simulation. When a suppression of the pressure development occurs in the experimental pressure, very little overshoot occurs in the predicted pressure. Unforttmately, the simulations were unable to predict this suppression of the pressure development. The predictions of pressure can be improved by using a smaller time step. This would allow the simulation to include all three relaxation modes of the Giesekus model. The velocity vector profiles of the simulations are shown in Figure 16. The v and u velocity components are shown in Figure 17a and 17b, respectively, for a flow rate of 40 g/min, and the second fastest closing speed of-0.317 cm/s. The moving wall is eighty percent closed for these figures. Figure 16 indicates that

lO58

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-0.25

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Figure 17. Predicted profiles of v- (a) and u- (b) velocity component at different cross sectional areas of the channel at Q - 40g/min and Vw- -0.317cm/s for viscous (solid lines) and viscoelastic (dashed lines) simulations. the variation in the velocity magnitudes across the gap are close for positions 1 and 2 and slight differences occur at positiom 3, 4 and 5 between the viscous and viscoelastic simulations. It should be noted that the velocity magnitude at position three in Figure 16 has a value greater than zero at the wall due to the velocity contribution from the moving boundary. The velocity components in Figure 17 indicate a good corr~adson between the viscous and viscoelastic simulations for u-velocity components. However, some differences in the vvelocity component are shown. This good agreement would indicate that the

1059

elastic effects of the polymer have little effect on the velocity profile of the flowing polymer during the closing of the slit channel. Thus, the viscoelastic simulations that use the actual moving boundary velocity predict pressure better than the other simulations. The viscoelastic simulations predict a lower pressure which is much closer to the experimental results. When a suppression of the pressure occurs as seen in Figure 15b, the viscoelastic simulation predicts a minimal amount of overshoot. When an overshoot occurs, the viscoelastic simulations predict a lower pressure than the viscous simulations. For the viscous simulations which implement the ideal velocity of the moving boundary, the pressure development is poorly predicted. Thus, the variation in the wall velocity has an important impact on the proper prediction of pressure development. In addition, the predicted pressure overshoot is drastically higher than the experimental overshoot of pressure. For the simulations where actual velocity was employed, the viscoelastic simulations predict a lower pressure than the viscous simulations. In both simulations, predicted pressures are lower than in the simulations where the ideal wall velocity is used. The inclusion of the elasticity of the polymer allows for a much better prediction of the pressure development. The simulation predicts a pressure much closer to the experimental data.

7. CONCLUSIONS In the present study, experiments were conducted to measure the pressure drop during the flow of a polymeric fluid through a slit die with a boundary moving at a specified velocity, which is perpendicular to the direction of flow. The polymer flows through the die as the wall closes the flow passage. The polymer used in the experiments was HDPE. Experiments were conducted for three different flow rates and five different moving wall velocities. A data acquisition system was used to record the transient pressure drop as the wall closes the die channel. The flow rate was held constant for each experiment. The velocity of the moving wall was held constant, but was found to actually vary as the wall closes. Experimental results indicate that for the flow rate of 20g/min an overshoot occurs in the pressure as the wall closes the channel. For the flow rate of 40g/min, an overshoot does not occur for all of the closing speeds. This overshoot occurs only for the two fastest closing speeds. For the flow rate of 70g/min no overshoot occurred for any of the wall closing speeds. When no overshoot occurs, a delay is present in the development of the experimental pressure. This delay was found to be a combination of the change in the velocity

1060

of the wall and a delay in the development of the pressure due to viscoelastic effects. The prediction of the experimental pressure was simulated using three different types of numerical simulations. These simulations included viscous and viscoelastic simulations that employed an ideal and actual wall velocity. The viscous simulations used the generalized Newtonian fluid model according to the Cross model. The viscoelastic simulations used the Giesekus model with a timedependent stress evolution. The viscous simulations were conducted with the computational code FLUENT and used a quasi-steady approximation for the wall movement. The viscoelastic simulations were conducted with in-house FEM code, which is transient and accounts for the moving wall boundary. The simulations that assumed an ideal wall velocity predicted a sharper increase in the pressure development at the start-up of the wall velocity. In addition, these simulations predicted a peak pressure which was drastically higher than the experimental pressures. All simulations with the ideal wall velocity predicted an overshoot in the pressure at the end of the closing of the channel. The simulations which employed the actual wall velocity gave better predictions of the experimental pressure. The viscous and viscoelastic simulations predicted the start-up pressure very well. The prediction of the pressure overshoot for the viscous case was much higher that the viscoelastic predictions. For the simulations that used the actual wall velocity, a prediction of the pressure during the start-up of the closing of the channel by the moving wall was found to be accurate. The simulations, though, did not predict the delay in the development of pressure. A small amount of overshoot in the pressure was predicted even in the cases when no overshoot occurs in the experimental pressure traces. The use of the actual velocity improves the prediction of the peak pressure. The viscoelastic case was found to predict the transient pressure development better than the other numerical simulations. The viscoelastic simulations which used the actual velocity predict a lower pressure than the viscous simulations. This indicates that the elastic nature of the polymer as described by the Giesekus model helps in the prediction of pressure. These viscoelastic effects occur even though the HDPE melt would seem to not exhibit a significant amount of elasticity. The transient nature of the experiments make this elasticity become more pronounced. For a better prediction of the pressure smaller time steps must be implemented in the viscoelastic simulations. The comparison of the velocity magnitudes and components across the gap height for several different positions in the die at 80% closed indicate little difference. Thus, the elasticity does not effect the velocity profiles. For the simulations conducted, the viscoelastic FEM code can offer significant insight

1061

into the prediction of the experimental pressure. For polymers with higher amounts of relaxation, this code could possibly help in the prediction of the suppression of the pressure development. In these instances, a delay in pressure could possibly be predicted. Research is currently being conducted on the closing in the absence of flow through the slit or squeeze flow experiments. In addition, polymers which have much higher relaxation times will be incorporated into the current research on the channel flow of polymer with moving boundaries.

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126. A.N. Beris, M. Avgousti and A. Souvaliotis, J. Non-Newtonian Fluid Mech., 44 (1992) 197. 127. D. Vlassopoulos and S.G. Hatzikiriakos, J. Non-Newtonian Fluid Mech., 57 (1995) 119. 128. J.D. Schieber and J.M. Wiest, J. Rheol., 33:6 (1989) 979. 129. S.A. Khan and R.G. Larson, J. Rheol., 31:3 (1994) 207. 130. L.M. Quinzani, G.H. McKinley, R.A. Bronw and R.C. Armstrong, J. Rheol., 34:5 (1990) 705. 131. T. Isake, M. Takahashi, T. Takigawa and T. Masuda, Rheol. Acta, 30 (1991) 530. 132. R.G. Larson, J. Non-Newtonian Fluid Mech., 23 (1987) 249. 133. M. Grmela and P.J. Carreau, J. Non-Newtonian Fluid Mech., 23 (1987) 249. 134. M.A. Hulsen, J. Non-Newtonian Fluid Mech., 30:1 (1988) 85. 135. M.A. Hulsen, J. Non-Newtonian Fluid Mech., 38:1 (1990) 93. 136. A.I. Leonov, Rheol. Acta, 15:2 (1976) 6. 137. A.I. Leonov, J. Non-Newtonian Fluid Mech., 25 (1987) 1. 138. N. Phan-Thien and R.J. Tanner, J. Non-Newtonian Fluid Mech., 2 (1977) 353. 139. N. Phan-Thien, J. Rheol., 22:3 (1978) 259. 140. R.B. Bird and R.J. DeAguiar, J. Non-Newtonian Fluid Mech., 12 (1983) 149. 141. J.R. DeAguiar, J. Non-Newtonian Fluid Mech., 13 (1983) 161. 142. M. Doi and S.F. Edwards, J. Chem. Soc. Faraday Trans. II, 74 (1978) 1789. 143. E.C. Bingham, Fluidity and Plasticity, McGraw Hill Book Company, New York, N.Y. (1922). 144. W. Prager and K. Hohenemser, Uber die Ansatze der Mechanik der Continua Zeitschrit~ fur Angewandte Mathematik und Mechanik 12 (1932) 216. 145. A. Slibar and P.R. Parsley, J. Appl Mech., 29 (1959) 107. 146. A. Slibar and P.R. Parsley, in Elasticity, Plasticity and Fluid Dynamics, Haifa ( 1962). 147. J. Harris, Rheol. Acta, 6 (1967) 6. 148. T. Schwedoff, Congres de Physique, 1 (1900) 478. 149. J.F. Hutton, Rheol. Acta, 14 (1975) 979. 150. J.L. White, J. Non-Newtonian Fluid Mech., 5 (1979) 177. 151. J.L. White, J. Non-Newtonian Fluid Mech., 8 (1981) 195. 152. Y. Suetsugu and J.L. White, J. Non-Newtonian Fluid Mech., 14 (1984) 121. 153. S. Montes and J.L. White, J. Non-Newtonian Fluid Mech., 49 (1993) 277. 154. A.I. Isayev and X. Fan, J. Rheol., 34 (1994) 35.

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155. D. C-H Cheng and F. Evans, Br J. Applied Physics, 16 (1965) 1599. 156. D. C-H Cheng, Br J. Applied Physics, 17 (1966) 253. 157. Z. Kemblowski and J. Perera, Rheol. Acta, 18 (1979) 702. 158. A.I. Leonov, J. Rheol., 34 (1990) 1039. 159. M. Simhambhatla, PhD Dissertation, The University of Akron (1994). 160. P. Coussot, A.I. Leonov and J.M. Piau, J. Non-Newtonian Fluid Mech., 46 (1993) 179. 161. M. Sobhanie, A.I. Isayev. and Y. Fan, Rheol. Acta, 36 (1997) 66. 162. FLUENT User's Guide, Fluent Inc., Lebanon, NH, V4.2 (1994) 854. 163. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, HcGraw-Hill, New York (1980). 164. J.P. Maruszewski, Fluent Inc. Technical Memo, TM-049, 1991. 165. B.P. Leonard, Draft for First National Fluid Dynamics Conference, Cincinnati, Ohio, July, 1988. 166. S.F. Shen, Int. J. Num. Meth. Fluids, 4 (1984) 171.

1069

FREE SURFACE VISCOELASTIC AND LIQUID CRYSTALLINE POLYMER FIBERS AND JETS S t e p h e n E. B e c h t e P , M. Gregory Forest b, Qi W a n g r and H o n g Zhou b

Department of Aerospace Engineering, Applied Mechanics, and Aviation, The Ohio State University, Columbus, Ohio ~3210 b Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250 r Department of Mathematical Sciences, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana ~6202 1. I N T R O D U C T I O N

We provide a cursory history of thin filament modeling of fibers and fiber manufacturing processes (referred to as fiber spinning in the industry); extensive references can be found in our published articles. Our goal here is to identify those aspects of fibers and filaments which are accessible from slender I-D models, and which have been addressed in our own work. As such, this chapter will be a biased account of viscoelastic and liquid crystalline polymer fiber developments. An important issue is a unified derivation of I-D fiber models. Different industrial processes vary widely in the dominant competing physical effects. For example, [1] analyzed the competition between inertia and Newtonian viscosity; [2] considered viscosity and elastic relaxation. It is a fact of life that one does not know a priori what the dominant physics in the slender flow is, and moreover, it is highly likely that a slight change in the physical competition may be either beneficial or detrimental. For this reason, we have focused on a unified derivation together with a comparison of all the various leading order physical regimes; a similar point of view has been taken in [3] for Newtonian fibers where the number of possible dominant balances is manageable. For a wide range of inviscid [4], Newtonian viscous [5, 6, 7, 8, 9, 10],

1070

viscoelastic [11, 12, 13], and liquid crystalline polymer [14, 15] constitutive relations, together with surface tension, gravity, and inertia, we have developed a slender fiber perturbation algorithm that produces all onedimensional slender, axisymmetric, isothermal and thermal models in the literature, as well as several relevant new models. This approach posits a separable radial and axial structure at every order in perturbation, and therefore has one major advantage and disadvantage. The advantage is that the method is algorithmic, can be implemented with symbolic manipulations, and works whenever there exist separable slender flows. The disadvantage is that the method does not address whether the separable flows it describes are uniquely specified by the slender longwave approximation. For Newtonian flows this issue is resolved (private notes). But for more complex constitutive laws of the types considered here, it is an open problem to construct slender fiber flows which are not captured by our posited radial/asymptotic expansions. We have also addressed how to select the specific regimes and corresponding model which best fits a particular experiment or process. This modeling feature is essential both for direct simulations of a given experiment/process, and for inverse material characterization [16, 17, 18]. We refer to our original published work for details of these developments over the past decade. For this volume we survey a selection of leading order models for increasingly complex physical regimes. Similar to work by [19] and [20, 21], we have exploited the mathematical structure of transient models to deduce which quantities can, or must, be specified at upstream and downstream boundaries of the fiber spinline. In viscoelastic [13, 22] and liquid crystalline polymer [14] fiber models we have discovered a novel space-time change-of-type in which the governing equations for the fiber flow may experience a transition from a well-posed hyperbolic structure to a catastrophic elliptic structure. We have also identified this transient changeof-type with predictions for the onset of polymeric fiber failure [23]. The mathematical structure of the I-D models for our selection of physical regimes will further be shown below to reflect on the stability of two very different classes of steady states" cylindrical jets and noncylindrical fibers under tension.

1071 2. S L E N D E R F I B E R M O D E L S F O R I N C O M P R E S S I B L E I N V I S C I D , N E W T O N I A N , V I S C O E L A S T I C , A N D LIQUID C R Y S T A L L I N E P O L Y M E R F I B E R FLOWS

We refer the reader to our journal articles for the 3-D, incompressible axisymmetric free surface formulation and slender fiber perturbation theory. The fluid rheology is modeled in the 3-D equations by the constitutive (i.e. stress vs. rate-of-strain) relation one posits. The I-D models we survey below arise from the following constitutive assumptions for the stress t e n s o r T: Inviscid Fluids: T -- - p I ,

(1)

where p is the scalar pressure maintaining the incompressibility constraint, I is the identity tensor. Newtonian Viscous Fluids: T = - p I + 2~7D,

(2)

where r/is the zero strain rate viscosity, D is the rate-of-strain tensor given by

[Vv +

(3)

with V denoting the Eulerian gradient and v the fluid velocity. Viscoelastic Fluids: T - - p I + q?,

(4)

where

+A~DT=2y z)t

D+A2

.

(5)

z)t

Here '~1 is the relaxation time, A2 is the retardation-time constant, which is also called "polymeric viscosity", and - (~-~ + v . V ) ( . ) + Ft(.) - (.)f~ - a ( D ( - ) + ( . ) D ) ,

(6)

1072

where a is a slip parameter ( - 1 _< a 0, indicating well-posed evolution. In particular, the system (16) is: 9 Hyperbolic type (3, 1) (meaning 3 positive and 1 negative characteristics) if and only if A > (Ov) 2, where s'l, s2, s3 > 0; 5"4 ~ 0. In this case, there are exactly 3 upstream boundary conditions and 1 downstream boundary condition. Therefore, the system is well-posed for modeling fiber spinning processes as long as this inequality on the solution is maintained. Moreover, the Riemann variables prescribe what process quantities may be imposed upstream and downstream. 9 Hyperbolic type (4, 0) if and only if 0 < A < (~v)2 where all four characteristic speeds sj are positive. In this case all information moves downstream and one can only impose upstream boundary conditions. This same system therefore describes extrusion flows as long as the above constraints are satisfied. 9 Mixed elliptic/hyperbolic type if and only if A < 0, indicating illposedness. The equations are not applicable for dynamical evolution in this regime. The boundary between well-posed hyperbolic evolution and ill-posed mixed-type behavior is A = 0, which is not an invariant condition (i.e. the sign of A may change along exact solutions). [22] studies families of exact model solutions which yield all varieties of hyperbolic to elliptic/hyperbolic transitions. In Figure 1 we show one possible space-time boundary of this well-posed/ill-posed transition, corresponding to the exact model solution: (~ - - r

V --

VO~

Tzz(t, z ) - e-t/AiAo(z- vot),

(24)

T~(t, z) -- e-t/a~[Ao(z -- vot) + A~ where A ~ qSo, vo are constants and the function Ao(s) is arbitrary except that Ao(s) =/=O , - A ~

1077

10-

.

61

n

2

4

6

8

10

Figure 1" The regions in the (z,t) plane of hyperbolic and mixed elliptic/hyperbolic type which evolve from the Cauchy d a t a with the discriminant A - A ( e c ) + e-88 v0t), where v0 - A1 - 1, A(z) - 2 + sin(z) _> 0, ~ >- 0, z x ( ~ ) - 2wA~ ~o ( ~- A o 1) _ - 0 . 5 , so t h a t the system is hyperbolic at time t - 0 for all z _> 0. A(ec) - - 0 . 5 implies t h a t the system becomes mixed elliptic/hyperbolic as t --+ oc for all z > t. From [22].

1078

2.4 Model 4" Viscoelastic fibers dominated by surface tension, inertia, viscosity, relaxation and retardation (a Johnson-Segalman fluid) When an elastic retardation effect is included, the extra stress can be decomposed into two parts, A2 rr _ Tp + 2r/~ll D

(25)

where Tp is the polymer part of the stress, A2 is the retardation time and 2 q ~ D is the Newtonian part of the stress. Substituting the decomposition into the generalized Oldroyd Fluid-B model, the constitutive equation (5) becomes a Maxwell-type equation for Tp, rrp 4- )kl - ~

= 27/(1 - ~11 D.

(26)

This is the Johnson-Segalman formulation for viscoelastic flows. From this formulation, the leading order model is a 4 x 4 degenerate parabolic system, involving the same unknowns as in Model 3: ut + C(u)u~ = f(u) + Duzz,

(27)

where u -- (4), v, 7"zz, ~,.,.)t, v

~2 1

+

0

0

-B

B

Tzz)

C(u) -

(28)

0

-2a~z

0

Z(1-A1) a Z , r + " A1

0 0

0 0

O0

0 0

0 0

0

0

V

'

A,, T1 + 6 BZ-~11dz Vz f(u)

O0

v

0

0 3 B Z ~A1 0 0 D

- 2Z(1-~)_A~

-

.

_T_~ A1

T~ A1

(29)

1079

All the physical parameters are the same as before. Due to the decomposition of the extra stress, the range of A2 is restricted to 0 _< A2/A1 _< 1. 2.5 M o d e l 5: L C P inviscid fibers d o m i n a t e d by i n e r t i a , s u r f a c e t e n s i o n , gravity, a n d m o l e c u l a r - s c a l e o r i e n t a t i o n effects" p o l y m e r k i n e t i c e n e r g y , an e x c l u d e d volume potential, and anisotropic drag

From [14, 15] the uniform leading order behavior in the LCP fiber crosssection is given by the special uniaxial representation for the orientation tensor Q: 1 3'

Q-sdiag[

1 2 3' 3 ].

(30)

where s is the scalar order parameter, which is restricted to the range 1 < s < 1 From this representation, the slender fiber equations for this regime take the form [14]: 2

~

~

"

ut + C ( u ) U z - f,

where u -

(31)

v, s) t,, f - (0, 1/F, -ad/AU(s)) t, and

(0, v

6"~.(U ~ __ '~.w't

1

we ~

]

2aU(s)

o

r

r2 V

0

(32)

-~U'(~)

-h(~)

v

The eigenvalues of the coefficient matrix, which determine the characteristic speeds, are ~1

m

V~

~,3 - v + qA(~, r ~, w, m), A(,. r ~. w. -

- ~[h(.)U'(.) ~[h(~)U'(~)- u ( ~ ) - c_~], ~ 0 the quasilinear system is strictly hyperbolic. W h e n / k - 0 the system is degenerate, with three identical characteristics and only two independent eigenvectors. When A < 0, then the system is of mixed elliptic-hyperbolic type, and is therefore ill-posed as an evolutionary system. In the context of general solutions, the sign of A varies through a competition of the solution variables s, ~; at particular values of s, ~ the hyperbolic versus elliptic behavior is governed by the parameters N and Cal~p. 2.6 M o d e l 6: T h e highly a n i s o t r o p i c d r a g limit of M o d e l 5

This model is formally obtained fi'om Model 5 by letting ad = 0 in the leading order equations; refer to [14]. The nonlinear classification in Model 5 is unchanged, since the terms proportional to ad only contribute to the nonlinear lower order terms in the equation for s. Thus, the criteria for hyperbolic versus mixed hyperbolic-elliptic type are as given in Model 5. In this model one loses the selection of discrete branches of equilibria determined from the zeroes of U(s); the intermolecular potential is effectively constant in this limit so all values of s are allowable equilibria. The parameter values for which A(s0) < 0 correspond once again to ill-posed equilibria, as discussed in Model 5.

1081 2.7 Model 7" Viscous generalization of Model 6

This model couples viscous terms, both Newtonian and orientationdependent, to Model 6, retaining its highly anisotropic drag limit (ad = 0) [14] The leading order system is given by r + V~z + ~Vz/2 = o,

(02~), + (02~)z

_

1(~2

~

1

+ ~r

(38)

+ ((,7~sf0 i i

~ U ~ 0. In [15] we have also studied how the steady spinning solution responds to variations in Reynolds number (R), Weber number (W), and Froude number (F). Our numerical results indicate negligible quantitative changes in the free surface r axial velocity v, and uniaxial order parameter s due to variations in W and F, and still very small quantitative changes due to variations in R. Sensitivity to these hydrodynamic parameters becomes evident, however, when we analyze stability of these steady states [15]. For further studies on the quantitative influence of LCP parameters (e.g., anisotropic drag coefficient, LCP kinetic energy relative to inertial energy, and dimensionless relaxation time), refer to [15]. In Figure 11 we present linearized stability results of the steady isothermal LCP spinlines of Figures 9 and 10. Figure 11a displays the maximum growth rate curve for the s0-parametrized family of Figure 9, from which we conclude that the entire Figure 9 family of steady states is linearly stable. Figure l l a further indicates that a weaker upstream degree of orientation leads to more stable steady states. The maximum linearized growth rates as a function of Dr, for the steady state family of Figure 10 is plotted in Figure llb, from which we deduce that faster processes are less stable in this region of parameter space. This might seem intuitively obvious, but we refer to a study [5] of slender Maxwell fluid models where the maximum growth rate curve oscillates about zero with varying Dr, creating alternating windows of stable and unstable draw speeds. Figure 11c provides the full two-parameter neutral stability curve: the critical draw ratio, Dr*(so) for each so, above which the steady state is linearly unstable. We deduce that the critical draw ratio for all steady states of Figure 9 is between Dr - 27.5 and Dr - 30.5. Most notable, however, is the shape of this neutral stability curve, indicating there is preferred initial degree of orientation (around so = 0.25) associated with

1094

the maximum stable drawing speed (around Dr - 30.4) for this twoparameter family of steady states. Now we present steady fiber spinning solutions to Model 9. Consistent with our nondimensionalization, the upstream conditions on fiber radius, velocity, and temperature are fixed:

r

1, v ( 0 ) - 1,0(0)- 1.

(79)

The downstream boundary condition is selected by the assumption that axial thermal conduction is negligible downstream, i.e.,

0(r Oz

(1) - 0 .

(80)

The remaining boundary conditions are free processing parameters to be specified/varied in the simulations below" s(0), v(1).

(81)

The upstream degree of orientation (s(0)) is a function of spinneret design, whereas the take-up speed (v(1) - draw ratio - Dr) is a measure of process speed and throughput. In Figure 12 we present typical steady state solutions of Model 9 due to changes of A0. Here a two-phase model has been posited [26], where the governing equations for the temperature above the glass transition temperature (0 > 0g) are given in Model 9, and in the solid phase (0 _ 0g) we employ a rigid cooling LCP fiber model so the velocity is constant (fixed by the take-up speed) whereas the orientation and energy equations are maintained. (The parameters used here are for academic illustrations, and are not taken from experiments.) Figure 13 depicts the critical draw ratio Dr* as a function of the nonisothermal parameter A0. Three different forms of N have been used, the constant value N - 4 and then two temperature-dependent forms [26]. These forms are chosen so that the intermolecular potential f U(s)ds has only the isotropic critical point for sufficiently high temperature, then passes to a double-well potential for lower temperature. As shown in Figure 13, the critical draw ratio grows with A0. As A0 increases from 0 to 0.4, there is 9.3% gain in Dr* for N c~ 9.7% gain for N linear, and 9.5% gain for N quadratic. These predictions clearly indicate that the cooling process increases stable spinning speeds. Note that for A0 - 0, N c~ -" 4, N linear : 2.1, and N quadratic - - 2.52, the qualitative differences in these isothermal steady states are negligible,

1095

"'111111/I/1[[

1,...

.

12 .

.

.

~

0.8

0.6

10

>

6

0.4

0.,~ .....................

1 ..,

0.8

0.{

0.2 ~

0.4

z

9

0.6

0.8

o

,

1 (3.,'}

%

II iii|I |1|

Ill

0.2

0.4

0.6

0.8

o:6

o'.8

12

. ."""/. ////////,//////////,,,,:i~

O.z

1

0.~

......

11111111111111|I

..............

CO

1

i

0.2

0.4

!

i

I

0.6

0.8

1

z

(b) ~

o'.2

o:4

z

12

/

10

0.!

/-.:y

0.6

if:: > 6

.../.! i /

0.4

........;:;r i ....

0.2 ........... .,.,.,.,~.,~,-r~ . . . . .

Oo

0.2

0.4

0.6 z

0.8

1

o -0

0.2

0.4

0.6

0.8

z

Figure 6: The variation of fiber radius 4) and axial velocity v due to the changes of: (a) l / F , where 1IF goes from 0 to 5 in increments of 0.5, with 1 / W - 1 / R - 1.0; (b) l / W , where 1/W goes from 0 to 5 in increments of 0.5, with 1 / F - 1 / R - 1.0; (c) l / R , where 1 / R goes from 1.0 to 5 in increments of 0.5, with 1/F - 1/W - 1.0.

1096

20 0.8

15 >,, ,,,,,,,,,,

0

~50.6

o10

L

>

0.4

0.2 0

!

0 0

,,,

0.5 axial coordinate

I

0.5 axial coordinate

1

0.5 axial coordinate

1

30 25 03

-0.5

20 -I

9-~ X 15 L

-I .5

10 !

0

0.5 axial coordinate

1

-2

0

F i g u r e 7: A t y p i c a l s t e a d y s t a t e profile for a M a x w e l l l i q u i d f i l a m e n t in a fiber spinning process.

The parameter

1, A1 - 0.05, 0 ( 0 ) - v(0) -

values are B -

1, T,.,.(0) - 0, D r - 20.

W -

F -

Z - a -

1097

20 0.8

|

15 >.,

to,)

,,,..,.,

rj

~0.6

o10 (1) >

0.4 0.2 0

0 0

I

0.5 axial coordinate

I

0.5 axial coordinate

300

r,/)

200

(l)

L

~-4

,,,,,... ~

-2

..,,,

•

"0

100

L

-6 0

~

0

~

0.5 axial coordinate

--

--8

1

"'

0

~ '"

0.5 axial coordinate

Figure 8" A typical steady state profile for a Johnson-Segalman liquid filament in a fiber spinning process.

1

1098

Figure 9: Steady solutions of LCP fiber spinlines reflecting variations due to changes in the initial degree of orientation, s(0). We vary s(0) = so from .05 to .95 in increments of .05. Arrows indicate the direction of increasing so. Figures 9a,b,c display the family of solutions, fiber radius r so), axial velocity v ( z ; so), orientation s ( z ; so), respectively. All parameter values are fixed at order one values" 1 / R - c~ - 5, 1 / W - 1, 1 I F - 1 , N - 4, A 1, a = 0.5. Boundary conditions are: 4)(0) = v(0) = 1, s(0) = .5, D r = 10 = v(1).

1099

Figure 10: Variations in the steady spinning state due to changes in draw ratio, Dr. All other parameter values and initial data are the same as Figure 9. D r - v(1) is varied from 2 to 40, in unit increments.

1100

//i /

-1.6 //

-1.7 /

-1.8

/

/

/

t

/

/

/

J

1. / i

/

/

l/max

//

-1.9

/J

/

/

-2.1

/

(b) 0.4

i

I

l 9

0.6

I

i i l I l

......................

0.2

/

/

/

//

/

/

/

// j/

/

/

j/

l/max

I

//

/

,

,

,

~

i0

0.8

.

.

.

.

:

D?

80

.

2O

.

.

.

:

3O

~

.

40

.

/ - ....... \

3o /

/'

\,

/

\ \

:

\

/

\ ',

29.5

'\

Dr* 29

\

28.5

(c) 0

0.2

0.4

0.6

0.8

80

Figure 11: Linearized stability information. Figure l l a concerns the s0parametrized family of Figure 9; Figure 11b concerns the Dr-parametrized family of Figure 10; Figure 11c summarizes the neutral stability boundary for a full two-parameter variation of the spinning steady state. Each data point in Figures l la,b depicts the maximum real part, Uma~ (vs. so in Figure 11a, vs. D r in Figure l lb), of all linearized growth rates for that respective steady state; a negative value of U,ia~ indicates linearized stability.

1101

1

10

: ""--:----:; ....-.....:...... -.... ;

0.9

B

0.8

J/#I/Y

....... ................

0.7 0.6 :i!~i

i

0.5

, ~ :i!;i:i:!:~:~:~''

~) ....

0.4

0-30

0:2

....0:4

0:6

~

0:8

0'2

o14

Z

z

0'6

0'8

1

(b) 0.8 0.6

..,~iii~!~

o:i~~~,,,~ ~ii!i!:~il!i,i:i~:~ .,

~ii~

0.4 0.2 0.65 O0

012

014

z

016

018

1

f

06;

0:2

0:4

z

0:6

. . . 11

0'8

(d) Figure 12" Steady state variations with respect to the nonisothermal parameter A0, 0

2

0

I

I

[

50

100

150

200

Axial distance (cm) Figure6. Velocity distribution along the spinline at various power-law indices.

1136

3.2. Attenuation along the Spinline Of all factors affecting fiber dimension and properties, such spinning conditions as take-up velocity, quench air velocity and flow rate seem to be the most important. Figure 7 shows that fiber diameter attenuates slowly at such low take-up speed as 4 km/min. However, as one increases the spinning speed, it decreases rapidly to its final value, as if a neck-like deformation occurs in the spinline. This is because the crystallization proceeds rapidly under the high speed and the extrudate solidifies quickly. Hence, the deformation is concentrated on the narrow region in the amorphous state. Figure 8 demonstrates the effect of quench air speed on the fiber diameter at the spinning speed of 6 km/min. From this result, we can draw a conclusion that the quench air speed does not have a significant influence on the fiber diameter. Effect of mass flow rate on the fiber diameter is illustrated in Figure 9. It is evident that the fiber diameter, and hence, the distance of the onset point of crystallization from the spinneret increases with the increase in the flow rate. 0.1 V ai r --'- 2 0

m / m i n

W = 1.5 g/min, hole

0.01

~

4 km/min

9 v,,,,q

"Q~

6

\ ~ ~ _ 0.001

I

0

50

:__,__:,

.

.

.

.

.

.

.

.

I. . . . .

100

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

,

I

150

200

Axial distance (cm) Figure 7. Diameter distribution along the spinline at various take-up speeds (symbols are for experimental results). [] ~ 4 km/min A ... 5 km/min o--- 6 km/min

1137

0.1

v = 6 km/min W = 1.5 g/min, hole

0

~--

0 m/min 20 3O

o 0.01 cO

?5

0.001 0

50

100

150

200

Axial distance (cm) Figure 8. Diameter distribution along the spinline at various quench air speeds (symbols are for experimental results). [] 0 m/min 9 20m/min o--- 30 m/min 0.1 v = 6 km/min Uair = 20 m/min r

(D r

\~"~'~ ',,\ "\

2.0 g/rain, hole ~_1.5 1.0

\

0.01

?5

.o _ -----

0.001 0

.

.

.

.

.

.

.

.

.

.

.

.

.

_ . . . . . . . . . . . . . . . . . .

I

I

I

50

100

150

200

Axial distance (cm) Figure9. Diameter distribution along the spinline at various mass flow rates(symbols are for experimental results). n 1.0 g/rain 9 hole A ... 1.5 g/min, hole o--- 2.0 g/min 9 hole

1138

In Figures 10-12, we can observe the temperature profile along the spinline as a function of take-up speed, quench air velocity and mass flow rate, respectively. The temperature decreases slowly (slow in a sense not of time but of distance), as the spinning speed increases. Similarly to the previous result on the fiber diameter, the change of quench air speed rarely alters the temperature profile as shown in Figure 11. We can examine the significant effect of flow rate in Figure 12. As the flow rate increases, cooling becomes very slow due to the increase of fiber diameter, and therefore quick temperature variation is suppressed. Figures 13-15 are the results for crystallinity variation. Figure 13 clearly shows that the rate of crystallization and the final value of crystallinity increase as the spinning speed increases. Again, the effect of quench air speed on the crystallinity may be regarded as negligible. Concerning the effect of flow rate, even though the crystallization starts later, the equilibrium value of crystallinity increases with its increase.

300

-••

v air = 20 m/min ', \

~

W = 1.5 g/min, hole

200

"- x\ "",, \ \ / / - 100

shift(cm) 4km/min 0

5

10

" ~,l-v/ 0 o 50 m >' 10" Ld V) "I" 12. I (r

**,/- OATA THEORY,

rl# + Tleon

kZ w

THEORY,J ' ' " ' " ' ' " -

""-.

rl# ONLY


- V O 0

,

(19)

1171

here c is the concentration of the bubble trains, 1 is the length of the train, and p is a phenomenological constant. In equation (19), the vector u(s) denotes a vector which joins two adjacent macrolamellae, and arclength s is measured along the train. The angle brackets denote an average over the orientations of the macrobubbles which link the macrolamellae. It is convenient to rewrite equation (19) in the following form

/,

(20)

J = J o - cp jd,sS. V P , 0

Jo = - c - ~ - V P , S=


,

I ij =6ij , S ' u = S i j u j , S : u = S i j u i u j ,

where tensor S is the tensor of the order parameter. One can see that the first term on the right hand side of equation (20) may be included into the ordinary Darcy law associated with the flow of flee gas. Thus, equation (20) is written in a form in which it is convenient to treat the foam motion in a porous medium as a flow of a "solution' of bubble chains (Doi and Edwards [57]). For such a hypothetical system, the detailed derivation of the constitutive equations has been presented by Kornev and Kurdyumov [50]. The system of these equations contains a generalized Darcy law for gas flow in the presence of a foam

v = - k f VP

-L,

L =J-

Jo '

(21)

and a kinetic equation which describes an evolution of the "blocking force' L due to the action of the velocity gradients. In a one-dimensional case, e.g., for a foam motion in a porous tube, the model has the following form Ov

(22)

Ox DL

DL

~ - l - V

.

Dt

v =-k

.

Dx 3t:'

L .

~-L f 3x

(23)

.

r .

(24)

Here v is the gas velocity, v is a relaxation time, and k~ is an effective seepage coefficient. The system of equations (22)-(24) may also be easily elucidated in

1172

the terms of the so-called lamella "break-and-reform' mechanism of foam motion (Holm [58]). Namely, equation (24) expresses the balance of forces for the moving foam. The viscous force on the left hand side of the equation is balanced by the pressure gradient and a blocking force. The latter arises by vim~e of a blockage of the gas channels by 'valve' lamellae. In the first approximation, the blocking force is directly proportional to the number of 'valve' lamellae. We thus can express the lamella density through the function L and, ignoring the generation rate of lamellae, we may treat equation (23) as a kinetic equation for the lamella population balance in a moving foam. The parameter r should thus be treated as a lifetime of a 'valve' lamella, and it may be represented as 1

1

1

r

rt

rh

where r, is a 'thermodynamic' lifetime of a lamella, and r, is its hydrodynamic traveling time. So, each 'valve' lamella blocks a gas path until it rips due to inherent thermodynamic or hydrodynamic instability. Within the framework of the above formulated bubble-train-flow-mechanism, the relaxation time can be obtained by analyzing a random walk of a bubble train as a whole. To describe this random walk, the following h y p o t h e s i s has been used: once formed, the system of active channels cannot be destroyed. Then the picture of a random walk resembles a reptation of a polymer molecule through the obstacles (de Gennes [59]; Doi and Edwards [57]; Kornev and Kurdyumov [50]). A bubble train can change its active channel only by moving through a network of active channels in a worm-like manner. In fact, only its ends participate in movement, but the rest of the macrolamellae are effectively trapped within the existing (at the given moment) active channel. The model in equations (22)-(24) reproduces the main features of the flow of a foam through a porous medium (Figures 10-11). The one-dimensional flow is governed by the parameters r and b = Lok I r ! H , where Lo is a boundary value of the blocking force L, and H is the length of the sample. These parameters can be extracted from the experimental data by using a piecewise-linear approximation (Figure 10) as follows AP kf H = ( l + b ) v , v --->0, AP H kf H =v+ b, v ~o

(25) (26)

1173

where AP is the applied pressure drop. The point at which the straight lines intersect has the coordinates

vr/H

=1, APrkf I H 2 = l + b ( 1 - e - 1 ) = l + b

(27)

.

W 5~

11 2/ 31

4

2

0

2

4

6

8

10

12

14

16

AP Figure 10. The velocity as a function of the pressure drop. 1-b--1, 3-b=5,

2 - b = 3,

4-b=10.

The coordinates of the imersection poim can be used as the fitting parameters. An additional parameter needed to specify the model is the breakthrough time T* = t*r . At this instant, a from of the foam first reaches the exiting end of the sample. The dimensionless parameter t* can be found from the following transcendental equation

1174

1=

f

t

H 2

b

1 - exp(-(1

1 + b + 1 +-~"

+

1+ b

b)t*) "

(28)

The system of equations (25)-(28) is closed so that the parameters b , r and ke can be found in each individual experiment. Then, these parameters can be plotted as functions of the degree of water saturation, properties of the foaming agent, etc. Despite the fact that the model contains a small number of the physical constants, the theory cannot be directly spread onto the two- or three-dimensional flows (because the parameter b depends upon the boundary value of the order parameter S). The similar problem occurs in the theory of polymer viscoelasticity (Bird, et al [60]) and also remains unresolved.

10-

t* u

I

I

m

0

1

2

"

II

4

1

6

l

b

8

1

AP Figure 11. The breakthrough time as a function of the pressure drop. 1 - b = 1, 2-b=3,

3-b-5,

4-b=10.

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6. STRONG FOAMS As follows from the previous sections, the main feature of foam rheology concems its 'pseudoplasticity'. The foam pseudoplasticity is very sensitive to the foam texture inside a porous medium and to a fraction of the pores blocked by lamellae. The physical reasons of the blocking action of weak and strong foams are different. For a weak foam, even in the absence of a start-up pressure gradient, an enhanced friction of bubble chains will result in an apparent pseudoplasticity of the gas-foam system. But for a strong foam, when the bubble trains flow through a small fraction of the pore channels, the blocking effect is originated from at least two mechanisms. The first is caused by trapped lamellae which do not participate in the movement. Such a blockage leads to a permeability that is orders of magnitude lower than that for the single phase flow. The second contribution to a reduction of the gas flux in the presence of a strong foam concerns an enhanced friction of the bubble trains. Thus, the total gas mobility, i.e., the coefficient in the generalized Darcy law, is the product of permeability divided by apparent viscosity. (If the liquid also participates in the movement so that its saturation changes during the process and alters the fraction of the trapped pores, then, instead of the absolute permeability, the relative permeability to gas has to be considered). In addition to a nonlinearity of the gas mobility, a start-up pressure gradient plays the key role in the treatment of the hydrodynamic effects in strong foams. This is the pressure gradient which is required to depin some bubble trains and to create a network of active channels (Cottrell [61]; Read [62]; Hirth and Lothe [63]; Suzuki, et al [64], Komev [30], Dautov, et al [32]). Thus, the term plasticity, in its own physical meaning, must be attributed to strong foams. Both, the bubble trains and the corresponding network of active channels play the same role as the one prescribed to the dislocations and the network of dislocations in solid state physics. In fact, studies have shown that the trapped gas saturation of strong foams in porous media can be as high as 80% (Rossen [14]). Therefore, the trapped foam forms an elastic field around the network of active channels. In a short period of time, the trapped foam resides as effectively motionless lamellae. However, the diffusion processes need to be considered to characterize the long term behavior of a foam (Falls, et al [65]; Cohen, et al [66]). In particular, a mechanism like the Nabarro-Herring-Lifshitz mechanism [67-69] of a diffusion-induced plasticity might play an important role in the motion of trapped foam. Indeed, under a pressure gradient, the gas will diffuse through the lamellae so that the gas pressure within the bubbles will change with time. Because of a difference in the Laplacian and gas pressures, the

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trapped lamellae will creep as well. A sophisticated analysis of this effect is highly desirable. There is an important evidence of self-organized criticality in foam flows through porous media. Namely, the typical manifold reduction of the gas mobility speaks in favor of the fact that, during the flow, the network of the active channels remains nearby in the same state, as it would be at the percolation threshold (Rossen and Gauglitz [70]). Scaling estimates of the gas mobility reduction by foams have been obtained by operating the ordinary percolation theory (de Gennes [71]; Rossen and Gauglitz [70]; Entov and Musin [72]). However, the critical behavior of the network of active channels has been assumed ad-hoc. Though most researches believe that the transport phenomena in strong foams obey the percolation laws, the physics of the hydrodynamic processes remains unclear and puzzling. 7. FOAMS IN FIBER SYSTEMS

Foams have been found various applications in industrial technologies dealing with fiber structures and fibrous materials, such as dyeing, printing, mercerizing, and finishing of textile fabrics, paper coating, resin-impregnation of fibrous mats and fabrics [73-78]. In these processes, the usage of foams instead of bulk liquids, as a vehicle for delivering small amounts of liquid solutions to fiber surfaces, leads to substantial energy savings because of small amounts of residual solvent to be removed at the drying step. Another class of processes, involving bubble generation in fiber structures, is fabricating fiber reinforced composites. Herewith, bubble formation during the stage of liquid resin impregnation causes a negative effect of non-uniform polymer distribution in the products (Judd and Wright [79]), and the mechanisms of air entrapment and bubble interactions within a fiber network have been studied aiming to diminish this phenomenon at technological conditions (Mahale, et al [80, 81 ]). Interactions of bubbles with fibers and fiber networks have certain specifics compared with capillaries and pore networks in solid-wall materials. The major difference is that in contract with granular materials, where the sizes of pores and grains are commensurable, the typical diameter of fibers, which constitute a skeleton of a pore structure, is commonly smaller than the typical diameter of voids/pores between the fibers. In fibrous materials, it is not easy to identify single pores, their shapes and dimensions. The definition of pore sizes and their distribution in fibrous materials is a matter of convention. The most rational way to introduce the pore dimensions in a real fiber system is to consider a model system of "effective" pores, in which some characteristic processes would occur

1177

obeying the same peculiarities as in the fiber system under consideration. For example, the effective pore sizes are estimated from the experiments with capillary equilibrium of immissible fluids (commonly, wetting liquid and gas) and with steady or quasi-steady forced flow of a non-wetting fluid (commonly, gas) within and/or through a fibrous sample. An advanced technique of liquid porosimetry has been developed by Miller and Tyomkin [82] for determining pore volume distributions in fiber systems and other materials. The method is based on the consecutive, quasi-equilibrium wetting fluid - gas displacement under precisely controlled pressure at isothermal environment. Therewith, the effective radius of a pore, where the liquid and gas phases coexist at given pressure, is operationally related to the mean radius of curvature of the equilibrium meniscus between the phases through the Laplace equation. During the process of gas-liquid displacement in fibrous materials, bubble formation occurs mostly due to a hydrodynamic instability of the wetting film clinging to the fibers. This is the common mechanism of lamella formation in any porous media [27]. However, the fiber structure causes some peculiarities. It is likely that in case of most of fibrous materials, we deal with strong foams with the bubbles commensurate to the pores. This conclusion follows from the experiments of Gido, et al [83], who examined the flow of foams through fibrous mats by characterizing the bubble sizes before and after injection. The authors observed that the size of the output bubbles exiting the fiber system was independent of the bubble size of the input bulk foam. For fibrous mats with different pore structure, the output bubble sizes were found correlated with the pore size distributions measured by the liquid porosimetry [82]. This result can be interpreted assuming that the foam bubbles, residing within the fiber network during the foam flow through this network, are grouped basically into two configurations: a system of immobilized bubbles strongly pinned to fibers and system of unpinned bubbles, which are formed in the bubble trains sliding along the active channels confined by the immobilized bubbles. The bubbles are immobilized when they are transpired by several fibers which intersect or are not collinear. These bubbles are crucified at fiber crossing and/or stretched by differently oriented fibers. The mobile bubbles are commensurate with the pore constrictions in order to pass through them without essential deformation. The movement of lamellae in the active channels within a fibrous structure should be like the lamella stick-slip motion within porous solids described in detail above. However, foam flow through a fiber mat is more difficult to formalize than the flow in a solid-wall porous body. At present, no quantitative approach exists to describe foam flow in fibrous materials. This problem is still awaiting its solution.

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While considering equilibrium distribution of bubbles and foam flow in fiber networks, we have to account for a specific behavior of wetting films on fiber surfaces. In particular, the lamellae (bubble films) should coexist with the wetting films and the Plateau borders at the intersections of lamellae and/or lamellae and wetting films. The equilibrium configurations of such a complex system are determined by capillary and surface forces (the latter is expressed via an additional so-called Derjaguin's disjoining pressure [84], [15], [16]). The fiber surfaces are commonly convex. This means that the capillary pressure acting on the liquid-gas interface of the wetting film covering the fiber is positive and tends to squeeze liquid out of film into the regions of fiber crossings. This tendency leads to a reduced mobility of wetting films in fiber systems compared with ordinary capillary systems. These effects have been considered in literature as related to the liquid spreading and drop residence on fibers first by Carroll [85, 86], who accounted for the capillary forces only, and then in great detail by French researchers from the de Gennes group [87-91 ]. Brochard was the first to emphasize a central role of long-range intennolectdar forces in residence, stability, and spreading of films and drops on fiber surfaces [87, 88]. Di Meglio [89] experimentally observed wetting films stabilized by the Van der Wa,~s forces, and proved that mass transfer between the drops residing on a fiber occurs through these films. Similar effects should be important in phenomena involving the bubbles on fibers and in fiber networks, however their description is lacking in literature. A theory of foams in fiber systems cannot be advanced without a solution of a chain of particular problems: equilibrium shape of a bubble residing on a fiber, transition zone between the bubble lamella and wetting film coating the fiber, slippage of a bubble along a fiber, bubble crucifixion at a fiber crossing, motion of a system of contacting bubbles transpired by a fiber ("bubbles on a spit"), etc. 8. CONCLUSIONS The motion of foams through porous media is a challenging problem in physicochemical hydrodynamics. The basic mechanisms of foam transport reviewed in this article contain some, but certainly not all, of the relevant physics of foam flow in porous media. Foam flow in porous media is a multifaceted process in which, on one hand, foam texture strongly governs foam rheology, and on the other hand, foam texture is in turn regulated by the porous medium through the capillary pressure. We have analyzed the main features of this process on examples of foam motion in model pore channels. The modem theories of the foam lamella transport in pore channels of varying cross-section and the models

1179

of the "weak foam" flow are discussed in detail. Careful analyses of the flow on the scale of individual pores or channels are useful in exposing effects of various physical parameters on foam motion and in identifying flow-induced patterns. In addition, the basic physical mechanisms of foam microhydrodynamics tmderlie a variety of technological processes in oil recovery, groundwater/soil remediation, textile manufacturing, etc. REFERENCES

1. C.V. Boys, Soap Bubbles and the Forces Which Mould them. Soc.for Promoting Christian Knowledge, E. and J.B.Yotmg, London,1890.Reprinted in Doubleday Anchor Books, New York, 1959. 2. K.J. Mysels, K. Shinoda and S. Frankel, Soap Films, Studies of their Thinning and a Bibliography, Pergamon Press, New York, 1959. 3. J.J. Bikerman, Foams, Springer-Verlag, New York, 1973. 4. I.B. Ivanov (ed.), Thin Liquid Films: Fundamentals and Applications, Marcel Dekker, New York, 1988. 5. A.M. Kraynik, Ann.Rev.Fluid Mech., 20 (1988) 325. 6. A. Wilson (ed.), Foams: Physics, Chemistry and Structure, Springer-Vedag, New York, 1989. 7. P.M. Kruglyakov and D.R. Exerowa, Foam and Foam Films, Khimia, Moscow, 1990. 8. J. Stavans, Rep.Prog.Phys., 56 (1993) 733. 9. R.K. Prud'homme and S.A. Khan (eds.), Foams:Fundamentals and Applications, Marcel Dekker, New York, 1995. 10. S.H. Raza, Soc.Petr.Eng.J., 10 (1970) 328. 11. S.S. Marsden, Foams in Porous Media - SUPRI TR-49, US DOE, 1986. 12. J.P. Heller and M.S. K u n t a m ~ l a , Ind.Eng.Chem.Res., 26 (1987) 318. 13. L.L. Schramm (ed.), Foams: Fundamentals and Applications in the Petroleum Industry, Advances in Chemistry Series 242, 1994. 14. W.R. Rossen, in [9], p.413. 15. L.I. Kheifetz and A.V.Neimark, Multiphase processes in porous media. Khimia, Moscow, 1982. 16. B.V. Derjaguin and N.V. Churaev and V.M. Muller, Surface Forces, Nauka, Moscow, Nauka, 1985; Surface Forces, Consultants Bureau, New York, 1987. 17. A.V. Neimark and M. Vignes-Adler, Phys. Rev. E, 51 (1995) 788. 18. P.G. de Gennes, Rev.Mod.Phys., 57 (1985) 827.

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19. A.H. Falls, G.J. Hirasaki, T.W. Patzek, P.A. Gauglitz, D.D. Miller and Y.Ratulowski, SPE Res. Eng., 3 (1988) 884. 20. A.N. Fried, The Foam-Drive Process for Increasing the Recovery of Oil, Report US Bureau of Mines R.I.5866 (1961). 21. G.J. Hirasaki and J. Lawson, Soc.Petr.Eng.J., 25 (1985) 176. 22. C.W. Nutt and R.W. Burley, in [6], p. 105. 23. A.V. Bazilevsky, K.Komev and A. Rozl~ov, in Proceedings of the ASME Symposium on Rheology & Fluid Mechanics of Nonlinear Materials, Atlanta, November 17-22, 1996. 24. O.S. Owete and V.E. Brigham, SPE Res.Eng., 2 (1987) 315. 25. K.T.Chambers and C.J. Radke, in N.Morrow (ed.), Interfacial Phenomena in Petroleum Recovery, Marcel Dekker, New York, 1990, p. 191. 26. R.A. Ettinger and C.J. Radke, SPE Res.Eng., 7 (1992) 83. 27. A.R. Kovscek and C.J. Radke, in [ 13], p. 113. 28. E. Rabinowicz, Friction and Wear of Materials, Wiley, New York., 1965. 29. F.P. Bowden and D.T. Tabor, Friction and Lubrication of Solids, Claredon Press, Oxford, 1986. 30. K.G. Kornev, JETP, 80 (1995) 1049. 31. K.G. Kornev, V. Mourzenko, and R. Dautov, in E.P. Zhidkov (ed.), Proceedings of the Conference on Computational Modelling and Computing in Physics, JINR, Dubna, September 16-21, 1996. 32. R. Dautov, K. Kornev, and V. Mourzenko, To appear in Phys.Rev. E. 33. V.L. Pokrovsky and A.L. Talapov, Zh.Eksp.Teor.Fiz., 78 (1980) 269. 34. P. Bak, Rep.Prog.Phys., 45 (1982) 587. 35. I. Frenkel' and T.A.Kontorova, Zh.Eksp.Teor.Fiz., 8 (1938) 1340. 36. B.D. Josephson, Adv. in Phys., 14 (1965) 419. 37. I.O. Kulik, Zh.Eksp. Teor.Fiz., 51 (1966) 1952. 38. A. Seeger and P. Schiller, in W.P.Mason and R.N.Thurston (eds.), Physical Acoustics, Academic, New York, 1966, Vol. IliA, p.361. 39. A. Barone and G. Paterno, Physics and Applications of the Josephson Effect, Wiley, 1982. 40. A.J. Lichtenberg and M.A. Lieberman, Regular and Chaotic Dynamics, 2end ed., Springer-Verlag, New York, 1992. 41. L.D. Landau and E.M. Lifshitz, Mechanics, Nauka, Moscow, 1965. 42. G.M. Zaslavsky and R.Z. Sagdeev, An Introduction to Nonlinear Physics: From Pendulum to Turbulence and Chaos, Nauka, Moscow, 1988. 43. L.G. Aslamasov and A.I. Larkin, Zh.Eksp.Teor.Fiz.Pis'ma, 9 (1969) 150. 44. H.A. Barnes, J.F. HuRon, and K. Waiters, An Introduction to Rheology. Second ed., Elsevier, 1993.

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45. F.P. Bretherton, J.Fluid Mech., 10 (1961) 166. 46. A.H. Falls, J.J. Musters, and J. Ratulowski, SPE Res.Eng., 4 (1989) 55. 47. J.E. Hanssen and M. Dalland, in [ 13], 319. 48. L.W. Schwartz, H.M. Princen, and A.D. Kiss, J.Fluid Mech., 172 (1986) 259 49. W.L. Olbricht, Ann.Rev.Fluid Mech., 28 (1996) 187. 50. K.G. Komev and V.N. Kurdyumov, JETP, 79 (1994) 252. 51. C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1956. 52. K. Lonngren and A. Scott, Solitons in Action, Academic Press. N.Y., 1978. 53. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, 1984. 54. J.S. Langer, in G. Grinstein and G. Mazenko (eds.), Directions in Condensed Matter Physics, World Scientific, Singapore, 1986, p. 165. 55. Y. Braiman, F. Family, and H.G.E. Hentschel, Phys.Rev.E., 53 (1996) R3005. 56. R. Musin, Ph.D. Thesis, Moscow, 1997. 57. M. Doi and S.F. Edwards, The Theory of Polymer Dynamics. Claredon Press, Oxford, 1986. 58. L.W. Holm, Soc.Petr.Eng.J.,.8 (1968) 359. 59. P.G. de Gennes, J.Chem.Phys., 55 (1971) 572. 60. R.B. Bird, R.C. Armstrong, O. Hassager, and C.F. Curtis, Dynamics of Polymeric Liquids, Wiley, New York, 1977, Vols. 1, 2. 61. A.H. Cottrell, Dislocations and Plastic Flow in Crystals. Oxford University Press, London, 1953 62. W.T. Read, Dislocations in Crystals, McGraw-Hill, New York, 1953. 63. J.P. Hirth. and J. Lothe, Theory of Dislocations, Wiley, New York, 1968. 64. T. Suzuki, H. Yoshinaga. and S. Takeuchi, Dynamics of Dislocations and Plasticity. Mir, Moscow, 1989. 65. A.H. Falls, J.B. Lawson, and G.J. Hirasaki, JPT, Jan (1988) 95. 66. D. Cohen, T.W. Patzek, and C.J. Radke, J.Colloid Interface Sci., 179 (1996) 357. 67. R.N. Nabarro. Report of the Conference of the Strength of Solids, Phys.Soc.London, London, 1948, p. 75. 68. C.J. Heri~g, J.Appl.Phys., 21 (1950) 5. 69. I.M. Lifshitz, Zh.Eksp.Teor.Fiz., 44 (1963) 1349; The Collected Works of II'ya Lifshitz, Nauka, Moscow,1987. Vol.1. 70. W.R. Rossen and P.A. Gauglitz, AIChE J., 36 (1990) 1176. 71. P. G. de Gennes, Revue De L'lnstitut Francais Du Petrole, 47 (1992) 249. 72. V.M. Entov and R.M. Musin, Preprint IPM RAN, Moscow, No 560 (1996).

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73. T.F. Cooke and D.E. Hirt, in Foams, ed. R.K. Prud'homme and S.A. Khan, Marcel Dekker, NY, 1995, p.339. 74. C.C. Namboodfi and M.W. Duke, Textile Res. J., 49 (1979) 156. 75. Gregorian, R.C., Text. Chem. Color., 19 (1987) 13. 76. D.E. Hirt, R.K. Prud'homme, L. Rebenfeld, AIChE J., 34 (1988) 326. 77. D.E.Hirt, R.K. Prud'homme, L. Rebenfeld, Textile Research J., 61 (1991) 47. 78. E.L. Wright, Textile Research J., 51(1981) 251. 79. N.C. Judd and W.W. Wright, SAMPE, 14 (1978) 10. 80. A.D. Mahale, R.K. Prud'homme, and L. Rebenfeld, Polymer Eng. & Sci., 32 (1992)319. 81. A.D. Mahale, R.K. Prud'homme, and L. Rebenfeld, 4 (1993) 199. 82. B. Miller and I. Tyomkin, J. Colloid & Interface Sci., 162 (1994) 163. 83. S.P. Gido, D.E. Hirt, S.M. Montgomery, R.K. Prud'homme, and L. Rebenfeld, J. Dispersion Sci. & Tech., 10 (1989) 785. 84. B.V. Derjagum, Kolloid Zeits, 69 (1934) 155. 85. B.J. Carroll, J. Colloid & Interface Sci., 57 (1976) 488. 86. B.J. Carroll, Langmuir, 2 (1986) 248. 87. F. Brochard, J. Chem. Phys., 84 (1986)4664. 88. F. Brochard-Wyart, C.R. Acad. Sc. Paris, Serie II, 303 (1986) 1077. 89. J.-M. di Meglio, C.R. Acad. Sc. Paris, Serie II, 303 (1986)437. 90. F. Brochard-Wyart, J.-M. di Meglio, and D. Quere, C.R. Acad. Sc. Paris, Serie II, 304 (1987) 553. 91. D. Quere, J.-M. di Meglio, and F. Brochard-Wyart, Revue Phys. Appl., 23 (1988) 1023.

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FLOW OF NON-NEWTONIAN FLUIDS IN POROUS MEDIA Shapour Vossoughi University of Kansas Department of Chemical and Petroleum Engineering Lawrence, KS., USA 1. INTRODUCTION Non-Newtonian fluid flow through porous media has become increasingly important in a wide range of disciplines and industrial segments. This is the result of availability of a wide variety of polymers that have interesting fluid flow properties, and there is a growing demand for their industrial use. Catalytic polymerization process, the injection of polymer and surfactant solutions into petroleum reservoirs to enhance oil recovery, food processing, and fluid flow through riving tissues are examples of the vitality of understanding the nonNewtonian fluid flow through porous structures. This section deals with the different aspects of non-Newtonian fluid flow through porous media and will bring together the different treatments of the subject matter commonly practiced in different disciplines. It will cover the complexity of both non-Newtonian fluid flow behavior and flow through porous media. Anomalous behavior of non-Newtonian fluid flow through porous media could be due to the fluid, the nature of porous media, or the interaction of fluid and porous media. Therefore, flow study should be carefully designed to distinguish between these effects or at least to acquire knowledge of which aspect of the non-Newtonian fluid flow is being studied. In this chapter, the nature of non-Newtonian fluids that are commonly employed will be studied, followed by a look at the nature of the idealized porous media and the geometrical complexity of true porous media. Fluid and porous media interaction, such as adsorption, mechanical entrapment, and inaccessible pore volumes, will have direct effect on the flow and will be analyzed and quantified. Microscopic and macroscopic view of the flow will be studied next; and, finally, the predictive models presently available for the study of the non-Newtonian fluid flow through porous media will be investigated. This will cover models based on hydraulic radius concept, friction factor/Reynolds number relationship, and empirical methods.

1184

2. NATURE OF FLUIDS Non-Newtonian fluid flow through porous media is not limited to the polymer solutions. There are a wide variety of non-Newtonian fluids that could be of interest. Following are samples that have already established their association with porous media.

2.1 Polymer Melts and Polymer Solutions Polymers come in a variety of forms with respect to their average molecular size and the geometrical shape of their molecules. This leads to a wide variety of rheological behavior termed as "non-Newtonian". To illustrate the variety of viscoelastic responses, Ferry [1 ] sampled polymer from seven different groups; four of them were uncross-linked polymers and the other three were cross-linked. Among the uncross-linked ones, he picked up amorphous polymers of low and high molecular weight, amorphous polymers of high molecular weight with long side groups, and amorphous polymers of high molecular weight below its glass transition temperature. Among the cross-linked polymers, he selected a lightly cross-linked amorphous polymer, a dilute cross-linked gel, and a highly crystalline polymer. He showed that each of these seven structural types has a characteristic viscoelastic behavior. He then concluded that there is a strong correlation between the viscoelastic behavior and the molecular structure of the polymer. A macromolecule chain in a solution is capable of assuming a variety of configurations by rotating around its chemical bonds. This makes the polymer solutions behave differently from their parent polymer gels. The polymer solutions might behave significantly different depending on their level of polymer concentrations, and interaction between polymer molecules increases as polymer concentration increases. The effect of the polymer molecules entanglement on the solution viscosity may become highly significant in the case of a concentrated polymer solution. The polymer solution viscosity becomes an increasingly nonlinear function of the polymer concentration for the concentrated polymer solutions. Injection of polymer solution into oil reservoirs to enhance oil recovery has become a common practice. Oil recovery during the primary stage is due to the depletion of the initial energy stored in the reservoir. Over seventy percent of the oil is still trapped in the rock pores after the primary recovery is depleted. During the secondary recovery stage, a fluid, such as water or gas, is injected into the reservoir to sweep the oil out and push the oil bank toward the production well. Waterflooding is a common secondary oil recovery technique [2]. Water, because of its low viscosity, tends to finger through the oil zone and creates early water breakthrough, but when a small amount of polymer is added to the water to increase its viscosity it creates a more stable front. Synthetic polymer, such as partially hydrolyzed polyacryalmide, and biopolymer, such as Xanthan, are the two types of

1185

polymers most commonly used for this purpose. Understanding the flow of aqueous polymer solution through porous media is essential for producing a reliable model for reservoir simulation.

2.2 Boger Fluids Constant viscosity fluids that are highly viscous and highly elastic were first introduced by Boger [3], hence, they are called Boger fluids. The first report of the fluids which exhibited elastic properties but remained Newtonian in viscous behavior was made by Giesekus [4]. The Boger fluid is prepared by dissolving a small amount of polymer in a highly viscous solvent. These fluids can be divided into two general groups; aqueous solutions consisting of a small amount of polyacrylamide dissolved in corn syrup [5,6]; and an organic-based Boger fluid consisting of a small amount of polyisobutylene dissolved in a mixture of kerosene oil and polybutene [7-9], or high molecular weight polystyrene dissolved in a solvent composed of low molecular weight polystyrene in dioctylphthalate [9]. The two classes of Boger fluids seem to have some differences in their rheological behavior. For example, Kemielewski, et al. [10], observed significant differences in the drag ratio measured for the two classes of Boger fluids over the same Weissenberg number range. The general reported characteristics of these fluids are; high viscosity, approximately non-shear thinning; high relaxation time; and optically clear. The Boger fluids are ideal model fluids for studying viscoelastic behavior in the absence of shear thinning. 2.3 Micro and Macro Emulsions Micro and macro emulsions are mixtures of two immiscible liquids in the form of small droplets of one phase into the other. The size of the droplets in the microemulsion solution is much smaller than those in the macroemulsion. Macroemulsions are turbid and thermodynamically unstable. The two phases will eventually separate into the original two immiscible liquids.On the other hand, microemulsions are translucent and thermodynamically stable. These fluids have been known for many years, and a wealth of literature is available on their properties and on their production techniques [ 11-13]. Micro and macro emulsions and their flow through porous media are frequently encountered in the oil industry. The produced crude oil is often in the form of emulsion with water, and the emulsion is broken to separate the oil from the water before it is shipped. Therefore, the flow of macroemulsion through porous media is an important aspect of the fluid flow near the production well. On the other hand, injection of microemulsion into the oil reservoir has been practiced to enhance oil recovery [ 14]. Microemulsions, sometimes called micellar solutions, are formulated for a specific crude-oil/reservoir-brine system to achieve ultra low interfacial tension of less than 10.3 dynes/cm. These microemulsions consist of hydrocarbons,

1186

water, surfactant, cosurfactant (such as alcohol or another surfactant), and electrolytes [ 15]. In many cases, polymer is also added to increase the viscosity of the micellar solution, since higher viscosity is desired to achieve a stable front and minimize viscous fingering effect. Micellar solutions are generally injected in various slug sizes to economize the chemical flood process. The micellar slug is then chased by a polymer solution to maintain its mobility control. Loss of surfactant due to the adsorption to the rock surface and dispersion at the front and at the back of the slug are the main reasons for the eventual breakdown of the slug effectiveness to mobilize the residual oil. Microemulsion can also be generated in situ as observed in alkaline flooding of oil reservoirs. It has been established [16-18] that some components of crude oil, such as organic acids, asphaltenes, and resins, react with alkaline solutions and form micellar-type structures. Rheological properties of an emulsion depend strongly on its composition. Its viscosity could change an order of magnitude in a narrow range of its concentration change. This is believed to be caused by the structural change of the microemulsion. The presence of electrolytes, such as salt, enhances the non-Newtonian behavior of the emulsions, and the effect of polymer addition on the viscosity of the microemulsions can be quite significant [19]. This enhanced viscosity is shear sensitive, and the viscosity recovery after the removal of the high shear is extremely slow. In general, suspensions and emulsions do not exhibit the same level of viscoelastic behavior as polymer solutions and polymer melts. However, emulsions of gel-like structure may exhibit marked viscoelastic behavior [13].

2.4 Suspensions Suspensions, sometimes called slurries, differ from emulsions in that one of the two phases is solid. The dispersed phase, which is a solid, is finely ground and mixed into a liquid. If the dispersion lasts, the mixture is called a suspension. Surfactants are usually added to make the suspensions more stable by preventing agglomeration of the solid particles. Suspensions at high volume concentrations are affected by many factors, such as hydrodynamic interaction between particles, doublet and high-order agglomerate formations, ultimately mechanical interference, and surface chemical effects between the particles as packed bed concentrations are approached. There is a critical concentration, after which, the viscosity increases sharply [20,21]. The abrupt increase in the viscosity levels may be caused by the strong inter-particle forces between the solid particles. Vossoughi and AI-Husaini [22] studied rheological behavior of the coal, oil, water slurries, and the effect of polymer as an additive. All the systems they studied were pseudoplastic and showed shear thinning behavior. Slurries made of finer coal particles were more viscous than those made of coarser grains. They studied the dynamic properties of coal slurries with and without polymer, using a small

1187

amplitude oscillatory test. In general, for all slurries tested, the dynamic viscosity was observed to be a decreasing function with frequency, and the modulus of rigidity was found to be an increasing function with frequency. Their study was limited to the frequency range of 0.24 to 6.0 cps. Flow of suspensions and slurries through porous media involves many challenges unique to this class of non-Newtonian fluids, and they need to be met. For example, in the filtration of suspensions or drilling mud infiltration near the wellbore in the petroleum drilling industry. 2.5 Gels Placement of gel in petroleum reservoirs to improve oil recovery has become an accepted practice. The technology is known as permeability modification or profile modification. Water, or any other fluids that are injected into the reservoir to displace oil, tends to pass through the more permeable zones leaving behind a significant amount of oil in the reservoir. Gelling solutions are injected following the waterflood to plug the already swept zones. Resumption of waterflood, forces the water to find a new path which leads to additional oil recovery. Gelling solutions are typically an aqueous polymer solution and some kind of heavy metal ions, such as chromium or aluminum, as cross linkers. Gels are highly nonNewtonian and their rheological properties are unique for each gel system [23]. A somewhat different process for profile modification, known as combination process, consists of the sequential injection of polymer solution and aluminum citrate solution [24]. The permeability reduction is believed to be due to the formation of layers of polymer/aluminum ion structure onto the wall surface of the pores. The adsorbed polymer molecules from the first polymer treatment acts as a base for the buildup of the structure. To enhance creation of the base and to increase adsorption of the polymer molecules to the rock, the rock was first treated by cationic polymer before the injection of the first anionic polymer cycle [25]. In-depth Permeability modification for the oil reservoirs with high permeability variation has also been achieved by injecting colloidal dispersion gels into the reservoir [26-28]. This consists of polymer with aluminum citrate crosslinker injected as a homogeneous solution. Above examples clearly reveal the complexity of the non-Newtonian fluid flow through porous media associated with the gel treatment. The fluid flow behavior is not simply a function of the rheological properties of the fluids involved, but the interaction between the porous media and the fluids plays an important role. 2.6 Foam Foams are a dispersion of a gas phase into a liquid phase, are unstable, and break easily. However, addition of surfactants can increase their stability and prolong the life of the foams almost indefinitely. Foams are injected into the petroleum

1188

reservoirs for the same reasons as gels. The profile modification for the case of foam injection is aimed toward the displacement processes where gas is being used as the displacing phase such as carbon dioxide flood, steam flood, or any other gas injection schemes [29,30]. Flow of foams through porous media has been studied in the literature [31,32]. The apparent viscosity of the foams flowing through sandpacks was measured [33] and was found to be significantly higher than the viscosities of the constituent fluids. This is one major factor for its effectiveness in profile modification. In some studies the permeability reduction for foam flow was found to be an order of magnitude larger than the permeability reduction predicted just based on the gas/brine mixture viscosity [34]. Foam rheology is crucial in designing an effective permeability modification for a given reservoir. Foams are non-Newtonian pseudoplastic fluid. Viscosity measurement in a capillary viscometer revealed that the data fit the Ostwald and de Waele (power law) relationship [35]. It was noticed that the bubbles do not behave as rigid particles but flow and slip at the same time. Therefore, a fixed-slip velocity could not be determined from the data. 3. N A T U R E OF POROUS MEDIA

Any solid body containing space to hold a fluid can be considered a porous medium. This can be as simple as a pipe or as complex as riving tissue. Only the interconnected pores, which could contribute to the flow, is of interest to the study of the fluid flow through porous media, but not all the dead pores should be ignored. Those dead pores, which have connection to the main flow path, could act as sink or source for the flow. Experimental investigation of the fluid flow through porous media is frequently carried out on idealized porous medium to avoid the complexity of the true nature of porous medium. 3.1 Idealized Porous Media Idealization, or simplification, of porous media is aimed toward studying a particular aspect of fluid flow through porous media. For example, capillary tube models are ideal to study viscous behavior of the fluid flow through porous media, and the Hele-Shaw model is ideal for studying interface instability based on perturbation theory. One should be careful in selection of idealized physical model and its capability to reflect the physical phenomenon of interest. A bundle of capillary tubes will not be able to reflect the viscoelastic nature of a fluid and, similarly, in studying two-phase flow through porous media in the absence of capillary effect, one should pick up a physical model with large pores to allow neglecting capillary forces. Following are a few examples of idealized porous media studied in the literature.

1189

3.1.1 Hele-Shaw Model One hundred years ago, Hele-Shaw [36] observed that streamlines in an inviscid flow can be visualized by making the gap of the two parallel plates small enough so that a sheet of water as thin as the boundary layer could only flow through. He added localized color to visualize streamlines. His remarkable observation put him, as an experimentalist, ahead of the mathematicians to experimentally generate streamlines for the geometries that fluid flow equations had not been solved yet. To show his remarkable observation, two of his photographs generated for the flow of water through a sudden enlargement are scanned and presented here. Figure 1 is when the gap between the two plates is large, i.e. a thick sheet of water is flowing

Figure 2. Sudden enlargement (thin sheet) [36] through the gap, and Figure 2 is when the gap is very small. This clearly shows that by reducing the gap, a truly two-dimensional inviscid flow can be produced. A Hele-Shaw model can be simply constructed by placing two parallel plates very close together. Incompressible fluid flow through porous media and Hele-Shaw model becomes analogous [37] and the average velocity components will be identical if permeability in the Hele-Shaw model is defined as,

1190

h2

(i)

12

where, k is permeability and h is the spacing between the two plates of the HeleShaw model. Hele-Shaw cell has been used to study interface instability related to the viscous fingering in porous media [38-45]. For example, Wiggert and Maxworthy [45] injected air into a Hele-Shaw cell saturated with silicon oil. The unstable immiscible interface resulted in a series of viscous fingering patterns that were photographed and digitized. Their Hele-Shaw cell consisted of two glass plates separated by a gap of 0.21 cm. Viscous fingering is an interface instability which occurs when the displacing phase is less viscous than the displaced phase. The creation of viscous fingers and their growth have been visualized in Hele-Shaw models as well as sand packs. Figure 3 is an example of the existence of the fingers even in laboratory systems where porous media are much more uniform and much more homogeneous than the actual reservoirs. Figure 3 is reproduction of one of the photographs taken by van Meurs [46] in his transparent three-dimensional glass beads model where oil was displaced by water.

Figure 3. Viscous fingering [46] Addition of a small amount of polymer to the displacing water has been shown to have significant effect on the stability of the front [47] and, consequently, produces much higher recovery efficiency. 3.1.2 Porous Media Made out of Glass Rods A two-dimensional flow that more closely mimics the actual reservoirs can be physically modeled by assembling glass rods parallel to each other and having a flow perpendicular to the axis of the rods. Vossoughi [48] used such a model to

1191

visualize the flow path in porous media and to study the non-Newtonian flow distribution in a porous bed. He used a regularly spaced matrix of 6 mm diameter glass rods in a triangular arrangement. Figure 4 is a schematic top view of the bed geometry used in this study.

@

d

~

X

0 T0 C) Flow

Figure 4. Schematic of glass-rods bed geometry [49] The cylindrical rods were positioned in a rectangular box of 5.2 by 6.5 cm inside cross section. The box contained 40 rows of cylinders, each row containing 9 or 10 cylinders in width. The dimension of the smallest opening, d, varied randomly between about 0.045 and 0.079 cm. Therefore, the bed is considered homogeneous on a macroscopic scale, but on small or microscopic scale there are considerable variations in pore diameter. A similar model with cylinders of shorter length and larger diameter was used by Kyle and Perrine [50]. 3.1.3 Porous Media Made out of Glass Beads Visualization of three-dimensional flow in porous media can be achieved by physical models made out of glass beads. Vossoughi [47] used a packed glass beads model in his study on viscous fingering in immiscible displacement in porous media. The model was made up of two transparent plates of 18 by 24 inches. The plates were spaced V2 inch apart, and two sizes of beads were studied. The larger bead size was 0.47 cm and the smaller was 0.15 cm in diameter. Use of glass-bead packs to study fluid flow through porous media is a common practice among investigators. Arman [51] used glass-bead pack in his study of relative permeability curves. Naar, et al. [52] used a laboratory five-spot model, made up of glass beads, to study the areal sweep efficiency of waterflooding. Rapoport, et al. [53] performed waterflooding in oil-wet glass-bead packs to

1192

establish that a laboratory-developed relationship between linear and five-spot flooding behavior holds regardless of the wettability or of the porous medium. In general, flow of Newtonian and non-Newtonian fluids through granular and non-granular packed beds is of considerable interest in many disciplines such as filtration, chemical processes involving flow through catalyst bed, and oil recovery operations. Packed beds are mainly used to study flow parameters in terms of pressure drop and flow rate. Visualization of flow distribution in displacement processes using non-glass beads pack can be achieved by X-ray shadowgraph technique developed in early 1950's [54]. In this technique, an X-ray absorber is added to either displaced or displacing phase. The phase containing the X-ray absorbent casts a shadow on a piece of film upon exposure to an X-ray beam. The density of the shadow is proportional to the saturation of that phase. 3.2 Complexity of True Porous Media The porous media presented above are over-simplified compared to the pore geometry of the actual rocks. Even the non-granular packed beds do not reflect the many complex features associated with naturally occurring rocks. It is realized that the naturally occurring rocks are composed of a variety of particle sizes with various angularity, particle size distribution, and particle arrangement. The pore space configuration is obviously different from that obtained by packing uniform spheres in a bed. Furthermore, part of the pore space is naturally filled by clay and cementing material. Cementing materials may block part of the pores and reduces, or, in some cases, eliminates interconnectivity. Figure 5 is composed of photographs of impregnated rocks revealing the complex pore geometry for the two cases of fine and coarse intergranular sandstone [55]. The complex pore geometry is arising from many factors in the geological environment of the deposit. The complexity in pore configuration becomes even more pronounced for the case of carbonate rocks. Fractures and vugs are frequently encountered in carbonate rocks as shown in Figure 6. In this figure, four typical carbonate reservoir rocks are depicted with a) vugular porosity, b) vugular with pinpoint porosity, c) fractures, and d) with conglomerate [56]. 4. FLUID/POROUS MEDIA INTERACTION

In addition to the complexity of the pore geometry, rock/fluid interaction becomes more pronounced in the case of naturally occurring rocks. Surface wettability could make a significant difference in flow behavior of wetting phase versus non-wetting phase. Rock/fluid interaction plays a more important role in injecting of polymer solution into a naturally occumng rock. In the case of naturally occurring rocks, the role of adsorption and entanglement of polymer molecules, pore blockage by polymer molecules, and inaccessible pore volumes to the polymer molecules

1193

become significantly more important and will contribute much larger share to the non-Newtonian flow behavior through porous media.

4.1 Adsorption and Mechanical Entrapment Adsorption of polymer molecules onto the rock surface plays a significant role in the fluid flow behavior of polymer solution through porous media. Adsorption is not always desirable and the loss of polymer molecules causes dilution and the apparent viscosity of the polymer solution decreases with time. This could eventually lead to the breakdown of its effectiveness as a mobility control additive. Adsorption of the macromolecules will also reduce the pore size available for the fluid to flow. The level of adsorption differs from one type of polymer to the other. Polyacrylamide-type polymers, commonly used in oil displacement processes, adsorbs strongly on mineral surfaces. The level of adsorption can be reduced by partially hydrolyzing the polyacrylamide with a base such as sodium hydroxide, potassium hydroxide, or sodium carbonate. The amount of polymer adsorbed per unit surface area increases with polymer concentration and with its molecular weight [57]. Rowland and Eirich [58] found that the thickness of the adsorbed polymer (for uncharged polymer) is in the order of the average diameter of the free polymer molecule coils in solution, i. e. approximately proportional to M ~/2, where M is the average polymer molecular weight. Michael and Morelos [59] studied adsorption of partially hydrolyzed polyacrylamide and sodium polymethyl methacrylate on kaolinite and observed higher adsorption with lower pH of the polymer solution. They concluded that the mechanism of adsorption was hydrogen bonding. Similar observation is also reported by Schmidt and Eirich [60]. There is a great deal of evidence in the literature that shows adsorption of the polymer molecules is affected by the presence of electrolytes. Polymer adsorption increases with increasing concentration of electrolytes [61-64]. Smith [62] reported an adsorption of 200 gg of a partially hydrolyzed polyacrylamide per square meter of surface area of silica powder when the polymer solvent was a 2% NaC1 solution. Mungan [63] measured an adsorption of 400 gg/rn ~of silica powder for Pusher 700 in 2% NaC1 solution. Mechanical entrapment of the polymer molecules is equally important in fluid flow behavior of macromolecule solutions through porous media. Entrapment mainly occurs at the pore throat, where the diameter becomes same or smaller than the diameter of the approaching molecule. Adsorption of the polymer molecules will definitely enhance entrapment by further reducing the pore diameter, especially near the pore throat. Dominguez [65] performed an experimental investigation of polymer retention in the absence of polymer adsorption by performing his experiments in porous media made of TEFLON | powder. Cores were prepared by compressing the powder. The

1194

Figure 5. Cast of pore space of typical reservoir rock. (a) Fine intergranular sandstone; (b) coarse intergranular sandstone. [55] Teflon powder cores have also been used by others. For example, Mungan [66] and Lefebre du Prey [67] used Teflon cores for their study of wettability effects and Sarem [68] used them to study polymer retention. Domingues [65] measured polymer retention for his Teflon core within the range of 10 to 21 gg/g of Teflon powder. Other reported values in the literature [64, 69-72] for polymer retention in

1195

Figure 6. Typical carbonate reservoir rocks (From Core Laboratories, Inc.) [56] porous media are from 6 gg/g to as high as 160 gg/g of sand grain. Pye [73] and Sandiford [74], in their experimental investigation of the flow of polyacrylamide solution through porous media, observed a reduction in water mobility of 5 to 20 times more than would be expected from the solution viscosity alone. The additional resistance to fluid flow is mainly attributed to the adsorption

1196

and mechanical entrapment of the macromolecules. This effect is characterized by introducing a resistance factor, F, defined as,

Fr=z

(-.

]

Power Law {


O or < 0, where fl/, f2/ are the derivatives with respect to the shear rate off1 and f2 respectively. Although the De Kee - Chan Man Fong viscoelastic model appears to be quite flexible, it cannot describe the pre-shearing effects reported in Figure 3.10. 4. L O W S T R A I N H A R D E N I N G

PHENOMENON

In this section, we highlight non linear effects encountered at relatively low strain with concentrated model suspensions and suspensions of industrial interest (coating colors used in the paper industry). The model suspensions discussed here consist of non colloidal particles suspended in a viscous matrix, a polybutene (PB, Indopol H100 of Stanchem), of low molecular weight, density equal to 890 kg / m 3 and viscosity of 24.5 Pa.s at 25 ~ This material is viscous enough so that the rheological properties are easily measurable, but not too much in order to highlight mechanical interactions between the particles. The industrial suspensions consist of coating colors used in the paper industry. Kaolin particles used for these suspensions are smaller, in the form of hexagonal platelets and show strong ionic interactions. Because of these strong particle-particle interactions, elastic effects are much more important, but some of the characteristics observed with the model suspensions are still detectable.

1329

4.1 Suspensions PVC particles in polybutene The non interactive PVC (polyvinyl chloride) particles were suspended in a nonpolar viscous matrix PB. The powder was an industrial non formulated PVC, with an average diameter of 12 ~tm and 50 % of the particles being smaller than 10 ~tm. A second maximum could be observed at 3 ~tm on the particle size distribution. Thus, the particle size distribution could be defined as just above the upper limit of the colloidal domain. Their shape was slightly ellipsoidal with a small to large axis ratio of 0.6. Suspensions at six different volumetric fractions (~ = 0.09, 0.18, 0.28, 0.37 and 0.47) were investigated, using a Weissenberg rheogoniometer equipped with parallel plates of 1.2 mm gap, and a Bohlin VOR with a Couette geometry. The results presented here are extracted and adapted from a previous publication on theological properties of filled polymers by Carreau et al. [20]. The specific viscosity as a function of the shear rate for the PVC suspensions is reported in Figure 4.1. For suspensions containing less than 30 vol % solids, the data fall approximately on a unique curve. The horizontal solid line corresponds to the theoretical value of 2.5 obtained from the Einstein analysis (Equation 1.1). This value represents fairly well the data for the most dilute suspensions, mainly at high shear rates. Using a maximum packing factor of 0.68, the Maron-Pierce equation (3.1) is shown to describe well the data for the 0.38 and 0.47 vol % suspension respectively at high shear rates. The shear-thinning effect obviously cannot be described by this simple relation, nor can be related to settling effects as reported by Acrivos et al. [23] and shown in Figure 2.6. These particles are non interactive, large enough so that the viscosity decrease with shear rate could not be associated with floc size variation nor to changes in the maximum packing fraction. The shape factor of 1.6 could explain the slight shear thinning observed for the lower concentrations but not the drastic effect observed for the 0.47 vol % suspension. For this suspension, the viscosity becomes unbounded as the shear rate goes to zero, indicating the presence of a yield stress and gel-type behavior, normally observed for particles having strong physical or chemical interactions. However, it is not the case here, but the yield stress is believed to be due to particle-particle steric-type interactions, as discussed below. As for time dependent models presented in Section 3.3, a structural parameter can be defined and included in a kinetic equation to follow the particles organization. With this approach, shear-thinning behavior (under steady state) is essentially based on effective volume fraction. In other words, when structural equilibrium is reached at a given shear rate, the structure is stable and can be modeled using the maximum packing fraction as a single parameter. As shown in Figure 4.2, the relative viscosity of each suspension is plotted against the volume fraction for four steady shear rates" 0.156, 1.56, 15.6 and 39.3 s~. The solid lines

1330

100 90 il 9 9

.

.

.

.

.

80 70 ~ - 60 ,

.

.

.

,

,i.

i..m

.i..,....,

0681

Einstein's equation -Maron-Pierce(O., = . ) [] 9% PVC / PB 1 A 18%PVC/PB "~ Q 28% PVC / PB --I

_

50

....

"9

9V 437 % P V C / P B

~o 1d

mmmmmmmmImm~m_m m

40 ~=47 30 20 ~ ~_ V _ _" _ ~ ...~]E _ l"Itl?_'~..ll'..~ llt .ay_,~. , _ ~ _ 3-7~176~ 10 o ...... , . . . . . . . . , 9 . 7 . . . . . , , ...7...# 10 -2

10-t

10 o

10 2

101

'~[S "1]

Figure 4.1

Specific viscosity for five different volume fractions of PVC/PB suspensions vs. shear rate. The horizontal solid line at the value of 2.5 represents the Einstein result. The horizontal dashed lines are the results calculated via the Maron-Pierce equation using a maximum packing factor of 0.68 (Adapted from Carreau et al. [20]).

35 30 25 ~. 20

O

0 . 1 5 6 s -1

121 A

1.56 s -! 15.6 s-I

V

39.3 s "j

15 10 5 0 0

10

20

30

40

50

60

r Figure 4.2

Relative viscosity of PVC/PB suspensions for four different shear rates as a function of the volume fraction. The solid lines represent the Maron-Pierce model predictions using a maximum packing fraction of 0.575, 0.629, 0.705 and 0.756 for the lowest to the highest shear rate (Adapted from Carreau et al. [20]).

1331

represent the Maron-Pierce model predictions using the maximum packing factor as an adjustable parameter. For each shear rate, the optimal values of ~)m equal to 0.575, 0.629, 0.705, 0.756 for each shear rate respectively. Hence, the steady state viscosity of this PVC/PB system can be simply modeled by this Newtonian empirical model using an adjustable maximum packing fraction or effective volume fractions.

"~n'

......

'

'

! 9%

" 0

(Figure 4). Hence, 3 func-

tions (a1111, a2222 and a3333 ) m u s t be evaluated in this region. In order to obtain a fitted orthotropic closure, the 1st step consists in selecting an appropriate finite dimensional space for the functions a~l~l, a2222 and a3333.

1365

Quadratic polynomials in ~ and 7/ were selected in [45]. The 2 nd step consists in choosing a set of flows that generate a wide variety of orientation states. For that purpose, the simple shear, uniaxial and biaxial elongational flows, together with two sheafing/stretching flows were selected in [45] (Figure 4). In the 3 rd step, the polynomial coefficients of a l l l l , a2222 and a3333 are determined by least square best fit, with the exact values obtained by integrating the evolution equation (5). Impressive results were obtained by this technique (Figure 5), except for very low values of the interaction coefficient C I . In conclusion, the best fitted orthotropic closure approximations can easily be tuned to general or particular classes of flows as long as the fitting objectives are clear. Indeed, there is no unique solution to the closure problem and, if a general approximation is desired, there is a limit accuracy that cannot be improved on. Nevertheless, the formula proposed by Cintra and Tucker [45] today certainly represents the best choice in solving a very broad class of problems in view of its accuracy and ease of implementation. Further investigations could bear on selecting an optimal approximation space for allll, a2222 and a3333. A related question is to know whether constraints should be introduced in order to avoid discontinuities and derivative discontinuities when the solution is extended to the whole triangular domain of orientation states (Figure 4). 3.3 Natural closures The natural closure approximation of Verleye et al. [4,40-42,56,57] is defined by considering the set of canonical distribution functions introduced in Section 2.2. As this class has 5 degrees of freedom in the 3D case and 2 degrees of freedom in the 2D case, while the positive semi-definite 2nO-order orientation tensor itself has 5 or 2 degrees of freedom, respectively, since it is synunetric and its trace is l, there is a one-to-one correspondence between the set of 2nd-order orientation tensors and the subset of canonical distribution functions. The natural closure approximation is generated by non-linear projection onto this subset. More precisely, the relation (12.1), which links aij to ~ , can be inverted and

thus aij~t, which is a tensor functional of q~, can be expressed in terms of aij, and this in principle provides the natural closure. The existence of this theoretical closure was demonstrated by Lipsc0mb et al. [27]. However, no attempt was made by these authors to directly investigate and exploit the resulting relation between aijk! and aij. In theory, the natural closure approximation is completely specified by the above definition. However, direct explicit calculation of aij~:t as a function of

1366

aij is only possible (to date) in the 2D case [40]. Tuned numerical methods are therefore required in the 3D case to obtain an accurate approximation of this relation. Hence, two levels of approximation must be considered, viz., the theoretical natural closure, which provides an approximation of the 4th-order orientation tensor aijkl, and its numerical approximation (which is thus an approximation of an approximation). To avoid any misunderstanding in the sequel, the term "natural closure" will concem the theoretical closure, while the term "natural closure approximation" will be used for the numerical approximation of this theoretical closure only. It should be emphasized that the natural closure concept is based on establishing a relationship between aij and aijkl by neglecting the effect of fiber-fiber interaction and starting from isotropic orientation in a hypothetical flow. These assumptions are only necessary to define the closure, which is further used in situations where they are no longer valid. There is some physical evidence, which is corroborated by the investigations of Verleye et al. [4,40-42,56,57] and Cintra and Tucker [45] (see Figure 5), that the natural closure behaves well in many circumstances, especially for molding problems. However, this cannot be always true, and in particular the natural closure should not be used when two (or more) preferential orientations of the fibers are present. These states are highly unlikely in practical situations.

3.3.1 The 2D natural closure In 2D problems [40], the full symmetry and normalization conditions impose to

,5

1.4

i

i

i

i

a,,

1.3 1.2 1.1 1.0 -

,I

i

i

i

Linear closure ........ Quadratic closure . . . . . . . Hybrid closure Natural closure .'" Distributionfunction

. "

.

.....:.,'-- --

0.9

---"_-z_-':"-". . . . . . . .

"

. " " / . 9' " " - ~ ~ ~ ~ ' ~

08

0.7

~

ot"

0.6 ,,-'~E , , , ,Reducedtime ~: = ~/~ t 0.5 , , , 0 1 2 3 4 5 6 7 8 9 Figure 6. Prediction of 2D fiber orientation in elongational flow. (From [40]).

1367 write the natural closure in the form

aijkl - ~8 ((l + 4D)~3 -1)S(6ij61d) - (~3 -1)S(aij6kl ) + ~3 S(aijalcl ) ,

(35)

which results from (29) and (31). On the other hand, the deformation gradient F~A from fictive initial to actual time can be written in principal axes. Hence,

[FiA]_

[ J F

0

0

F -1

'

(36)

where F is a non-vanishing positive constant. Finally, the current orientation vector Pi can conveniently be expressed in polar form. Therefore, combining (36), (10) and (9.2) with the definition (12), the components aij ( - ~, 71 in principal axes) and aiju are easily calculated as a function ofF: -

,

F2+l

(37)

1

[a1111 a~122]_ 1 a1122 a2222 2(F 2 + 1)2

and thus

[F2(2F2 + l) F2

F2 1 2+ F2

[ allll a1122]- 1 I~(l+~) a1122 a2222 2 ~r/

'

~F] 1 77(1 + 7/) "

(38) (39)

Expressing (35) in principal axes and comparing the result with (39) shows that /33 - 1, from which the 2D natural closure is obtained" aij t

-

S(a j

t)

9

(40)

The 2D natural closure performs very well when compared with other 2D closures [4,40,51]. Figure 6 illustrates this behavior in simple elongational flow (v 1 = ~ x 1, v2 = 0 , v3 = - ~ x 3 ) , starting from isotropic 2D orientation (all =a22 = 0 5 , a12 = 0 ) and using the value C I =0.01. Impressive experimental validations have been obtained in the prediction of Bulk Molding Compound (BMC) injection molding with the 2D natural closure (see Section 5).

3.3.2 The 3D natural closure When the fibers can rotate in all directions, a 3D closure is needed. Unfortunately, the problem of determining accurate approximations for the coefficients

1368 fll to t6 in (28) in terms of P and D turns out to be extremely difficult in view of the presence of complex singularities in several expressions. Therefore, many different formulations were investigated until the present theory was elaborated. To summarize this work, let us first establish the following non-linear partial differential equation-

aijkq (~pm + a p#a (~im + a ipgq 3jm -- a pjmq tSik - a ipmq r +2

-- aijmp t~qk +

a ijkp 3 a ijmq 3 a ijmq Oa ijkp -- 0 Oars arsmq - 2 Oars arskp + 2 Oaks aps - 20ams aqs ,

( 41 )

which is a direct (but not obvious) consequence of equations (20.1) and (22) and the definition of the natural closure. Indeed, from the identity D F i A / D t - ~ V i / ~ X j FjA, and taking the conditions ~ - 1 and C t - 0 into account, (20.1) and (22) can easily be combined as follows in differential form :

daij - dFiA FAT a kj - aik FAT dFjA

- -aijkl (dFkA FAll + FAT dFia ) ,

(42)

where aijkt is a function of aij, which is itself a function of Fia. The key point is to observe that the differential daij in (42) must be exact when aijkl depends on

aij

through the natural closure.

o32aij/CgFkloqFmn and o32aij/cgFmn3Fkt

More precisely, the derivatives

provided by (42) must be equal for any

value of FiA when aijkt is defined by this closure. After quite long calculations (which must be performed with care for aij is constrained by (15.1) and (16.1)), this condition provides equation (41), which is the comerstone of the numerical theory of the natural closure approximation. Also, (41) can be extended to higher-order closures and, in particular, it could help in defining a closure approximation for the 6th-order orientation tensor in terms of aijkl (by extending the space of canonical distribution functions, since 14 degrees of freedom are required). This theory is under investigation. At this stage, since the invariance properties of indicial tensor notation have been used to provide (41), the calculations can be pursued with the tensors aij

and Fim in diagonal form. The following notations are used (Figure 7) : (a11,a22,a33) = (~,rl,()

,

(a2233,a1133,a1122) = ( X , Y , Z )

,

(43)

while, from (24), the invariants P and D read as (44)

1369

r/=l

~-1/8 m

r

r/=O

~=1

Figure 7. Left'triangular domain T of eigenvalues of aij. Right" isovalues of

Z-a1122 in T (the maximum Zmax is 1/8). (From [4]). Moreover, the normalization condition (16.2) becomes a1111 + Y + Z - ~ ,

a2222 + X + Z - 77 ,

a3333 + X + Y - ~" ,

and a l l l l , a2222 and a3333 can be eliminated in terms of X , Y, Z, and ~" (the natural closure is a particular orthotropic closure [45]). Furthermore, from (28) and (34), the expression of Z writes as

(45) ~, 77

Z - 112/31 + (~ + r/)/32 + 2~r//33 + (~2 + r/2)f14 + (~ + r/)~r//35 + 2~2r/Zf16], (46) 6 with similar formulae for X and Y. Combining these relations with (44) provides (X + Y+ Z), (X~ + Yr/+ Z~') and (X~ 2 -k-Y/72+Z~ 2) in terms of P , D and 131 to 136. After some calculations, and with use of (30), these relations can be cast in the form : 1A0 2

-

7

1 4

0

0

--10

/~6

-4P

6P

2D

-2

10

0

6P

1-4D

-1

-2D

1

7

o

2D

-1

-3

(D-15P)

-2

21

-35

0

1 X+Y+Z +

+

X~ 2 + g r / 2 + Z ~ . 2

, (47)

1370

with

A - - 2 7 P 2 + 18PD

+ D2

-4D 3 -4P

.

(48)

Equation (47) proves extremely useful in evaluating the natural closure. Finally, equation (41) can be transformed after very long calculations (involving trace operations) into the following set of equations"

+

[ (4Z + X - r/)(2Z- ~) 1

+ (rt- r162 r L_(4 z + Y - , ~ ) ( 2 Z - r l ) _ ] X - Z)(2Y - ~)(7/- ~') 1 - 0 (Y - Z)(Z X - rl)(r - ~) J

(49)

with t5 - (~ - 7"/)(7/- ( ) ( ( - ~), and A - ~ 2 . Similar pairs of equations govem the derivatives of X and Y. It can be proved that no other independent relationship than these 6 equations can be derived from (41). It should also be observed from (34) that ~, 7/ and ( are are not independent variables and only the constrained partial derivation operators (tg/0~- 0/tg(), ( 0 / 0 r / - 0/0~) and ( 0 / 0 ~ - 0/0r/) can provide meaningful results. Indeed, when a given function of and r/ (or ~ and ( , or r/ and ( ) is regularly extended to a function of ~, 7/ and ( (considered as independent variables), each of these operators provides a unique derivative along the subset of admissible values of (~, r/, ~'), as defined by (34), whatever extension is performed. 3.3.3 Numerical approximations o f the 3D natural closure In the 1 st strategy selected by Verleye et al. [41,56] to obtain a closed approxi-

mation of the natural closure, rational expressions were determined to accurately approximate the coefficients fll to f16- For that purpose, the canonical distribution function q'c was integrated numerically for many values of the cumulative deformation gradient FiA, thereby providing a set of pairs (aij, aijkt) from which a rational approximation of fll to f16 respecting (30) was further obtained by best fit. The drawback of this method (which nonetheless produced good results) is that the error decreases only slowly when the degrees of the numerator

1371

0.14

9

.

.

.

.

.

.

.

Analytical Z ~ Approximate Z - -

0.12

/

0.1

~,,

/

Anal.ytical Y - -

0.08 0.06 0.04 0.02 0.0 -0.02 0.0 0.5 -

i

i

i

i

i

i

i

0.1

0.2

0.3

0.4

0.5

0.6

0.7

i

0.8

0.9

1.0

Analytical dY/d~ Approximate dY/dr

0.4 0.3 0.2

Analytical dZ/dr Approximate dZ/d~ - -

0.1 0.0 -0.1 -0.2 ,, -0.3 Figure 8.

Comparison between the theoretical and approximated values of and Y - a l l 3 3 (top), and dZ/d~ and dY/d~ (bottom), along

Z -- a1122 - 0 . The 2 "d strategy was used. (From [4]).

and the denominator of the approximants are increased (in [41,56], these degrees were 8 and 1). This results from the singularity of the natural closure along the boundary 3/" of the triangular domain T of admissible (~,r/,~') (Figure 7). Altemative approaches, taking into account the theory of the natural closure, were thus investigated. In the 2 "a strategy [4], the following decomposition was introduced" Z-/~2~r/+PZ where

Z

,

is analytical and

(50) PZ

tends towards 0 when

This form is justified by the fact that

~r//2

P

tends towards 0.

reduces to the 2D natural closure

1372 A

(40) when P - 0 . An approximation of Z was obtained from 3 basic properties of the natural closure. First, numerical experiments showed (and this should be confirmed with the help of (49)) that 2-

3/10 ,

(51)

when ~ r / - ~,2 or, equivalently, when p _ D3

_

~3

D - ~ .

,

(52)

Secondly, the following derivatives of X , Y and Z (49) along the boundary aT when ~" - 0 :

ae, g-r z - - ~ n ,

ar ac X -

a a)z_l~r an ar

a~

(aa~ ara] x - a(an

~)

can be calculated from

y =

1 2 1

(53)

2

with similar equations when ~ = 0 or 7/= 0. Thirdly, the axisymmetric natural closure [4] (Figure 8) is easy to calculate : 9 either by solving the differential equation provided by (49) along the bisectors of the triangular domain T :

(1-3~)(8Z-(1-~))dZ/d~+40Z 2 + Z ( 8 ~ - 7 ) +

1/4 (1-~')2 - 0 ;

(54)

9 or in parametric form from (42) : _

_ (4 cosh 2 S - - 1) cosh s coshs

,6s s

S ~

sinh3s

~

s+

cosh 2s (2 cosh 2s + 1)

(55)

16sinh4s

if

"

sinh2s

o 2.5 I r Q,)

xI

2

i qlm

1.5 A

I"

0.5 0 0

0.2

0.4

0.6

0.8

1

rlR F i g u r e 6. Radial Profile of solids concentration.

[29]" 9 - dp = 0.054 m m , pp =

930 k g / m 3, U = 3.58 m/s; II - dp = 0.075 mm, pp = 608 k g / m 3, U = 3.58 m/s; ~ dp = 0.043 mm, pp = 2003 k g / m 3, U = 3.58 m/s; A - dp= 0.054 m m , pp = 930 k g / m 3, U = 1.40 m/s; "ff - dp = 0.054 mm, pp = 930 k g / m 3, U = 3.15 m/s; 9 - dp = 0.054 m m , pp = 930 k g / m 3, U = 2.8 m/s; O - dp = 0.054 mm, pp = 930 k g / m 3, U = 2.16 m/s; X - dp = 0.075 mm, pp = 608 k g / m 3, U = 2.16 m/s; [24]" dp = 0.32 mm, pp = 2593 k g / m 3, A - U = 1.25 m/s, ~ - U = 1.7 m/s, .I, - U = 2.25 m/s; theories" m m [23], ~ [24]. 1000

It.

100

10

i

100

i

*

*

Reynolds Number

.

*

9

.

1000

F i g u r e 7. Variation of friction factor with R e y n o l d s n u m b e r at spouted flow r e g i m e [30].

1409 was significantly less viscous as the gas flowrate was increased and particle size was decreased. The addition of particles of large size had little effect on the viscosity, but the addition of negligible amounts of fine particles significantly lessened the viscosity. Trawinski [42] suggested an interesting model to explain this phenomena. According to his model, fine particles act as "ball bearings" between moving surfaces. Later, Kramers [43], Furukawa and Ohmae [44], Shuster and Haas [45] and Grace [46] reported viscosity data also using Stormer viscometers obtained with fluidized beds. The viscosity data actually reported by a Stormer viscometer are the weights required to spin the immersed paddle at a certain angular velocity. A different approach using a falling sphere was applied by Peters and Schmidt [47] and Trawinski [42]. Applying the hole of the liquid state, Trawinski [42] also analytically described the fluidized bed. The following formula has been proposed to predict the effective viscosity of the fluidized system

'1]a=[TIg+64(1-Emf)Pp(pp-pg)gdp 3] [U/(Umf E m f ) - K ( U [ U m f ) ~

0"5 -

1 ].

Viscometers based on Hagen-Poiseuille laminar flow have also been used, among the first by Siemes [48] and Siemes and Hellmer [49]. The main advantages of this method are that (a) the flow process has no effect on the state of the fluidized material and (b) the fluidized material flows in inclined channels like a Newtonian liquid. However, Wu et al. [50] observed an interesting effect in sand flow through orifices. At small particle diameters (0.01 to 0.1 mm) the flow of solids is steady in the narrow range of the orifice diameter to particle diameter ratio (2 to 12). At larger values of this ratio, the oscillatory flow of sand through the orifice has been observed. Using a twophase fluid flow model based on kinetic theory for granular flow, Manger et al. [51 ] developed a numerical model of the oscillatory flow of solid particles. The simulations show qualitative agreement with the experimental data of Wu et al.

[50]. Based on the behavior of gas bubbles (shape, pressure drop, size, etc.) some researchers attempted to predict the apparent viscosity of fluidized materials. The method based on shape of bubbles in fluidized beds to predict the viscosity was proposed by Grace [46], Davies and Taylor [52] and Mendelson [53]. It was found that the bubble shape factor is a function of the Reynolds, Weber and EtStvos numbers. Using the "included angle" (0) (angle of supporting surface from horizontal) Grace [46] proposed the following empirical formula to compute the apparent viscosity Tla = 0.0435 dB UB pp (1 - a m f ) exp (0.004 0).

1410

Estimated values of the apparent viscosity were found to coincide with those obtained in [46, 54]. Murray [55] predicted the apparent viscosity of a fluidized materials by estimating the friction factor for rising bubbles. A method based on pressure measurements as a bubble rose through a fluidized bed was reported by Stewart [56]. Row and Partridge [57] estimated the apparent viscosity of a fluidized bed through the bubble size measurements. According to Sch~gerl, Merz and Fetting [54] only the viscosity data measured by a torsional pendulum oscillating at low amplitudes or by Couette viscometer at low angular velocities of the inner cylinder can be considered as quantitative. Capillary tube viscometers are preferred when the data are to be used for pipe flow problems, and rotational viscometers, which subject the material under the test to a precise and uniform rate of shear, have definite advantages in the analysis of complex systems [58]. Simple mathematical models cannot at present describe the general flow behavior of fluidized materials. The apparent viscosity of the fluidized material is a multiparametric function and is dependent on the physical and chemical properties of both the solids and the aerating fluid. Fluidized materials are particularly complex and, if numerical simulations of the behavior are to be reliable, it is critical that the measured values be consistent regardless of the measurement techniques applied. Bakhtiyarov, Overfelt and Reddy [58] presented the results of the experimental study of the "apparent viscosity" of fluidized silica sand utilizing both Poiseuille flows (capillary viscometer) and Couette flows (rotational viscometer). Viscometric measurements by capillary tube viscometers were run in the system shown schematically in Figure 8. The system consists of a clear acrylic cylindrical chamber 457.2 mm long and 69.85 mm inside diameter which is sealed at the top and bottom. A funnel with an angle of approach 57030 , was attached at the bottom of the cylinder. Precision-bore copper capillary tubes of 4.7625 mm and 7.9375 mm inside diameters and five different lengths (L/D = 15.328, 30.656, 61.333, 122.667 and 245.334) were screwed into the funnel exit. Air at a carefully controlled constant pressure was admitted through the top plug of the chamber. The rate of flow of solid particles is typically measured by collecting a sample over a measured time and determining its mass. The experiments were run using both non-coated capillary robes and capillary tubes coated (to prevent wall slippage) by sand of the same particle diameter of sample to be tested. Water and 50/50 glycerol/water mixture were used as calibrating liquids.

1411

One of the main difficulties in capillary tube viscometry is in accurately determining the appropriate pressure drops. The corrections for the head of sample over the tube, for kinetic energy effects, and for entrance losses are required. The first correction is straightforward. The other two corrections can be respectively estimated by (1) repeating experiments in capillary tubes of different lengths and extrapolating the overall pressure drop to zero length and (2) experimental calibration with Newtonian fluids of known viscosity and density.

Figure 8. Capillary tube viscometer used in [58]. A computer controlled Brookfield HADV-II+ rotational viscometer was also used in the experimental program [58]. This device measures the torque required to rotate a spindle immersed in a fluid. For a given viscosity, the viscous drag, or resistance to flow, is proportional to the spindle's speed of rotation and is related to the spindle's geometry. Measurements made using the same spindle at different speeds are used to detect and evaluate the rheological properties of the test material. Viscosity measurements were made at spindle angular velocities in the range of 1 to 100 rpm. The viscometer was calibrated by using 50/50 and 99/1 glycerol/water mixtures and 3.5% polyacrylamide solution. A round acrylic tube with internal diameter of 58 mm and 227 mm length was used as a fluidization chamber (Figure 9). Pressure taps were provided along the column. The fluidization chamber was arranged for use with air as the operating fluid. Compressed, dried and pre-filtered air under the pressure up to 840 kPa was supplied to the bottom of the fluidization chamber.

1412

Air pressure and flowrate were measured by computer controlled pressure transducer and rotameter, respectively. A polypropylene porous plate with 0.250 mm pore size and 3.175 mm thick was used as an air distributor. All measurements were made with silica sand of average particle density 2.593 g/cm 3. The distributions of particle diameters used were as follows: less than 0.212 mm, from 0.212 mm to 0.425 mm, and from 0.425 mm to 0.710 mm. To prevent wall slippage effects, the surface of the spindle was coated by sand of the same particle diameter of sample to be tested.

Figure 9. Apparatus for fluidized sand viscosity measurements by rotational viscometer [58]. The apparent shear strain rate and the apparent shear stress at the capillary wall are defined by Ta=4 Q / p r ~ R 3 = 4 t ~ / R ,

(7)

Xa= AP R / 2 L .

(8)

The apparent viscosity is defined by the ratio of Xa over ~/a" Tla - 'l;a / ~/a 9

(9)

In the case of sand flow, the capillary viscometer technique determines the apparent viscosity as the averaged value of all inner local viscosity. The value of the apparent viscosity is obviously a function of the shear strain rate q(a-

1413

Figure 10 shows the results obtained using the Brookfield viscometer for silica sand (0.212 mm < dp < 0.425 mm) at different bed void factors. As either the shear rate or the bed void factor (air flow rate) increases, the apparent viscosity decreases. The apparent viscosity is most sensitive to shear rate and voidage effects near incipient fluidization. Thereafter, the rate of decrease lessens. However, the measured viscosity was only slightly affected by changes in particle diameter for the diameters investigated. The apparent viscosity can be correlated with the apparent shear strain rate in the usual way as [58] 'rla = K~~an-1 .

(10)

where ~ is a measure of consistency of the fluidized bed. Lower void factor of the fluidized material yields larger values of consistency, n is a measurement of the fluidization behavior of the fluidized bed and is a function of the design and operating characteristics of the fluidized bed. Fluidized beds can exhibit behavior that is almost Newtonian (n = 1) or can display several anomalous behaviors. The greater the divergence of n from unity, the more anomalous is the fluidized bed. Figure 11 relates the apparent viscosity of fluidized silica sand measured by SchOgerl, Merz and Fetting [54] and Bakhtiyarov, Overfelt and Reddy [58]. From these data one can recognize that at low air flow rates (or low void factor) the apparent viscosity increases. Data obtained with the capillary viscometer [58] show that the mass flow rate of solids does not depend on the overall pressure drop (column height) at certain drained angles of repose. Gregory [59] also observed that in stick-and slip flow of solid particles no external application of pressure is required to push the solids through the pipe, the force exerted by gravity being all that is demanded. For this type of flow he proposed the following formula Qp = kl d z'5 ,

(11)

where kl is a coefficient which depends on a tube length, particle diameter, shape factor, etc. Rausch investigated gravity flow of a wide variety of solids in 76.2 mm and 203.2 mm diameter tubes [6]. He correlated all data by the following equation Qp = f c Cw (d]dp) 2"7 g0.5 Pb dp 2"5 [ (tan 13)0.5 ,

(12)

1414

Figure 10. Apparent viscosity of fluidized sand bed as a function of the shear rate and void factor [58].

0

10000 W

o

a.

50 rpm [58]

[] 100 rpm [58]

E

[]

0

~, [ 5 4 ] o

0 o

(/)

oo

1000

[] []

/1 []

/1

eG)

12, < 100

'

~o

'

' ' ''"~

'

~oo

'

. . . . . . . . . . . . . . .

~ooo

~oooo

Air Flowrate, cm31s Figure 11. Apparent viscosity of fluidized sand as a function of air flowrate.

1415

where c is a correction factor pertaining to the angle of approach in the hopper bottom; Cw is a wall-effect correction factor; dp = (d~ d2)~ , d~ and d2 are adjacent sieve openings; f is a correction factor for tube length to diameter ratio, introduced in [58]. Variation of the Gregory coefficient (k~) and the factor (f) versus the capillary tube length to diameter ratio (L/d) are shown in Figure 12 [58]. Both correction factors appear to be approaching a limiting values as L/d increases. 1.2

~

0

.= 0.8 ll_ .=_o 0.6

...__.o...-.-=-.---

0.4 0

0.2

0

50

100

150

200

250

Lid

Figure 12. Correction factors as functions of the capillary tube length to diameter ratio [58]. The apparent viscosity of sand particles (0.425 mm < dp < 0.710 mm) obtained on both the rotational and the capillary viscometers as function of the apparent shear rate is shown for comparison in Figure 13. Regardless of the method of measurement, there is a general consistency in the data obtained with the Brookfield viscometer (low shear rates) and capillary viscometer (high shear rates). As seen from this figure, the apparent viscosity can be satisfactorily correlated with the apparent shear rate by the following empirical power law equation: 'qa = K~ ~r

.

(13)

A non-continuum approach to describe particle-gas suspension flow was presented by Cundall and Strack [60]. The method was based on the use of a numerical scheme in which the interactions of the particles are monitored contact-by-contact and the motion of the particles simulated particle-by-particle. Aizawa, Tamura and Kihara [61] calculated the particle-to-particle interaction

1416

forces by assuming that the interaction force between two solid particles is equal to the overlapped volume force from the elastic contact between the particles. To stabilize the numerical solution, they introduced a friction-like energy-dissipation mechanism in the contact algorithm. 1.00E+07 w m

a,, E

:r w o o w

",o,,

1.00E+06

',o,,

1.00E+05

~,,,,,,,

,,,,=

O

Brookfield [58]

[]

L/D=245.33 [58]

o

L/D=122.67 [58]

[]

L/D=61.33 [58]

9

L/D=30.67 [58]

9

L/D=15.33 [58]

-'-- - - E q n . (13)[58]

r., -9t~ 1.00E+04 o,, D, J

1.00E+03 0.1

a

1

,

,

|||,=!

9

10

|

J

**1.,|

|

100

9

9 , ~ , 1 m

1000

Apparent Shear Rate, 1Is

Figure 13. Apparent viscosity of sand bed as a function of the apparent shear rate [58]. Although these rigorous many-body algorithms have been demonstrated for simple geometries, excessive computational times preclude their widespread application to complex industrial problems. Continuum algorithms must be modified to enable design calculations. Central to the successful exploitation of such an approach is the measurement of appropriate transport properties (i.e., viscosity) that represent the complex rheological behavior of these multi-phase fluids. Therefore, continuum theories have been developed to describe the dynamics of particle-gas suspension flow. These theories develop hydrodynamic models based on the Navier-Stokes equations for multiphase flow. Using fluid mechanics principles, Anderson and Jackson [62] proposed a two-phase flow model. Pritchett, Black and Grag [63] applied a two-fluid model to solve the problem numerically. Gidaspow [64] presented the hydrodynamic model describing gas and solid phases in terms of separated conservation equations. Tsuo and Gidaspow [65] generalized the Navier-Stokes equations for two fluids to predict flow regimes in circulating fluidized beds. Using a two-phase 2-D computational fluid dynamics model, Samuelsberg and

1417

Hjertager [66] computed axial solid velocity, void factor and solid shear viscosity in the riser of a circulating fluidized bed. The radial profile of solid shear viscosity computed by the turbulent kinetic energy model was lower in the core than that found experimentally, but with a linear function of solid volume fraction in the measurement, the computed profile agrees well with experiments. Delassade [67] considered the fluidized bed as a mixture of a micro-stretch fluid representing the particulate phase, and a Newtonian fluid. The effect of particle inertia and other consequences of non-negligible spatial gradients of velocities, concentration and void factor on vertically propagating concentration waves of small amplitude has been analyzed. Scht~gefl [68] carried out the flow equation of fluidized systems from the shear diagrams using the method of Pawlowski [69]. According to his results, the apparent viscosity of the particle-gas suspension is independent of the flow rate at all particle diameters. However, although the viscosity is independent of the particle diameter at particle diameters dp > 0.100 mm, the viscosity decreases with decreasing particle diameter at particle diameters dp < 0.100 mm. Reviewing a number of expressions proposed for correlating fluidized bed viscosity with overall bed void factor, Johnson [70] proposed that the fluidized bed viscosity is related to bed voidage by Tla "-'Fig [1 + 0.5 (1 - e)][(1 - ~) / ~]9 / ~4.

(14)

Simulations made by equation (14) show a dramatic decrease in bed viscosity as the void factor increases. A kinetic theory based on a moment expansion was developed by Tsao and Koch [71], who numerically predicted the behavior of a dilute fluidized beds subjected to a simple shear flow. The results of simulations show that as the Stokes number is increased, the velocity distribution approaches the Maxwellian and the rheology becomes more Newtonian. At small Reynolds numbers the non-linearity of the drag increases the non-Newtonian behavior of the bed. Hydrodynamics and erosion modeling of fluidized beds have been introduced by Li and Zakkay [72]. Particle velocity fields and fluidized bed dynamics (bubble formation and motion, bed expansion and collapse) were analyzed. Utilizing the Finnie erosion model, these authors show that the distance from the air distributor to the tubes, tube size, tube orientation and operating pressure have significant effect on bed dynamics and tube erosion. However, in order to accurately analyze the kinetic and transport processes in these complex gas-particle systems, it is critical to determine a rheological

1418

model to describe their flow behavior. Since the dynamics of gas-solid suspension flow is a multiparametric process as noted above, it is convenient to develop the theological model on the basis of a semi-empirical correlation for fluidized beds. Bakhtiyarov and Overfelt [73] developed a rheological model of gas-solid suspension flow based on Ergun's semi-empirical correlation [74]. Introducing two components of the apparent viscosity, the model predicts that the flow curves of gas-solid suspensions depend on the gas properties, particle diameter and sphericity, void factor and tube diameter. The following form of the theological model has been developed (15)

q;a = [Tla.sh. + Tla.k.(~a) ] ~ a ,

where two components of the apparent viscosity as apparent shear viscosity ~a.sh. and apparent kinetic viscosity ~a.k.(~a), defined as 1]a.sh. = 75 (1

- I~) 2

~l.gd 2 / 16 e 3 (0 dp)2 ,

Tla.k"---- 7 (1 - e) Og d 3 "Ya[ 1024

1~3 ~ dp,

(16) (17)

where a sphericity of particles r is defined as a ratio of the surface areas of a sphere and the particle having identical volumes: t~ = S~ / Sp. According to formula (15), the net apparent viscosity is the sum of the two components: the first is due to shear viscous pressure drop and the second due to the kinetic pressure losses. Therefore, it is of interest to establish the contribution of each viscosity component to the total apparent viscosity. The viscosity ratio ~ = 1"la.k./Tla.sh.as a function of the apparent shear rate and particle diameter is shown in Figure 14. As seen from this figure, the viscosity ratio increases as both the shear rate and the particle diameter increase. Figures 15 and 16 show the variation of the apparent shear stress and the apparent viscosity, respectively, with the apparent shear rate obtained experimentally for silica sand of different particle diameters and over-all voidage fraction measured by procedures described in [73]. The predictions of the rheological model using equation (15) [73] are also shown in Figures 15 and 16. As seen from these figures, there is a good agreement between the model and the experimental data. Figures 15 and 16 show the variation of the apparent shear stress and the apparent viscosity, respectively, with the apparent shear rate obtained experimentally for silica sand of different particle diameters and over-all

1419

voidage fraction measured by procedures described in [73]. The predictions of the rheological model using equation (15) [73] are also shown in Figures 15 and 16. As seen from these figures, there is a good agreement between the model and the experimental data.

Figure 14. Viscosity ratio ~ = Tla.k.]Tla.sh.as a function of the apparent shear rate and particle diameter [73]. An apparent viscosity model based on the integral characteristics of the radial segregated fluidized beds has been developed by Bakhtiyarov and Overfelt [75]. Assuming that (i) the flow is laminar and steady on either side of the interface between core and annular space, (ii) the interface is stable and smooth, (iii) the pressure gradient is constant, (iv) end effects and a mass transfer between core and annular flows are negligible, it has been determined that the apparent viscosity of the segregated fluidized bed is a function of the viscosity and density ratios of the core and annular flows. This embodies the effects of the grain size and shape factor of particles, rheological characteristics and relative velocity of the fluidizing fluid giving l q s e g ~-

[Xo3 q- p / 11 -

Xo 3

(3 - p) / 211 - 3 Xo (p - 1) / 2111 / 11,

(18)

1420

where 1"1 and p are viscosity and density ratios of the core and annular flows, respectively; and Xo is a dimensionless radius of the core. 50

m

Ix. 40 W G) L_

30 L_ m 0

I

... 20

I

I

G} t_ m Q.

a. lO

0

500

1000

1500

2000

2500

3000

Apparent Shear Rate, 1/s

Figure 15. Apparent shear stress of air-sand suspension as a function of the apparent shear rate [73]" r-! _ dp = 0.653 mm, ~ = 0.55; + - dp = 0.505 mm, ~ = 0.64; O - dp = 0.357 mm, ~ = 0.74; A - dp = 0.252 mm, e = 0.80; 9 - dp = 0.212 mm, e = 0.86. Figure 17 shows the variation of the dimensionless apparent viscosity (1/Tlseg) with the parameter rl, a ratio of the viscosities in core and annulus regions, and with the dimensionless radius of the core (Xo). The radius of the core plays a prevailing part in determining the apparent viscosity of the system. That is especially conspicuous at high values of the parameter 1"1. The apparent viscosity of segregated bed (1/rl~g) also shows strong subordination on the parameter 1"1 which has an increasingly larger effect on 1/l"lseg with increasing values of Xo. Theoretical predictions for the dimensionless apparent viscosity (1/Tlseg) are shown in Figure 18 together with experimental results obtained in [24] for silica sand with grain size 0.212 mm to 0.425 mm and the average particle density 2.593 g/cm 3 fluidized by air. The apparent viscosities and average densities of the segregated fluidized beds in the core and annulus regions have been determined by using the procedure described in [24]. As seen from the figure,

1421

the apparent viscosity of the segregated fluidized bed is reasonably well predicted at all values of the viscosity ratio rl. 16

w == 12 o. E

u

8

,,~ -_

13... " ~

F.




..;"

0.44

0.4

.

9 9 9

I'''l

0.01

I

l

9 ..,,.l

.

0.1

.

.

9 ,.,.

1 Air Velocity,

10

m/s

Figure 21 Variation of the void factor with imposed air velocity [78] (dp< 0.212 mm: 9 - 9.81 m/s 2, i - 11.772 m/s 2, A - 17.658 m/s2; 0.600 m m < dp< 0.710 mm: + - 9.81 rn/s 2, r-I _ 11.772 m/s 2, A - 17.658 m/s2).

o

0.8 o

E

0.6

0 ~

D4

O

_c

0.4

-~ m

8

!

i

12

16

Gravitational

Acceleration,

20 m/s 2

Figure 22. Variation of the incipient fluidization air velocity with imposed gravitational acceleration ( i - dp; ul 400 o u .w_ 300 > r

Vi/x i

200

i

D. o. 100
0, inespeclive of the heat transfer mechanism at the wall, the local Nusselt number attains an asymptotic value, about three times the value predicted for forced convection. In horizontal flow, the natural convection increases both heat transfer as well as pressure loss above those expected from the pure forced convection analyses. While from a heat transfer standpoint, the circular geometry is the most efficient one but when pressure loss is also of concern, short pipe lengths are to be preferred. Besides, under certain conditions, square pipes offer a good compromise between high heat transfer rates and low pressure drops, particularly for shear thickening fluids. Experimental work in this area of duct flow is trailing behind the great strides made in theoretical developments. Heat and mass transfer in laminar boundary layer flows have also witnessed remarkable years of progress. Consequently, sound theoretical frameworks are now available for the prediction of heat and mass transfer from simple geometries such as spheres, cylinders and plates, at least to purely viscous type nonNewtonian materials. Available experimental results, while essentially in agreelnent with the predictions, do not seem to cover anywhere near as wide ranges of kinematic conditions as encompassed by the corresponding expressions for Newtonian media. Considerable scope exists fi~rther developments in this rugged terrain of nonNewtonian fluid mechanics. For instance, the role of fluid elasticity is not at all clear both in internal flows in ducts as well as in boundary layer flows. For instance, secondary flows are ta~own to exist in non-circular ducts even in creeping regime but it is not yet fully known how these impact on heat/mass transfer. The question of stability has hardly evoked any interest. Furthermore, most heat transfer studies employ either the constant wall temperature or constant heat flux condition, whereas in practice the actual conditions may be somewhere in between these two idealizations. Also, the flow patterns and heat/mass transfer phenomena in systems of practical importance including packed and fluidized beds, heat exchangers, boiling, etc. merit much more attention than it has received in the past.

1478

11. NOMENCLATURE geometric shape factor, Table 7 B width of plate, m Br Brinkman number, = (D/2) x,,,("+l~"~/q,,,j.tl/",. b geometric shape factor, Table 7, - or coefficient in temperature dependent viscosity equation, I/K Cp heat capacity, J/kg K d particle diameter, m D tube diameter, m molecular diffusivity of A in B, m2/s DAB hydraulic diameter, m DH E activation energy for viscous flow, J/kg K, equation (5) f fanning friction factor, D (- Ap)/2L 9V 2 a

Gr Grh Gro Grl Grim Gr' Gz

g h

J k kc L 1"11 In lIl o

Nu Nul = 2Nu n q Pe Pem

Grashof number, = I3AT D 3p2g/~teff2 Grashof number, = (p2d3gl3AT/m2)(d2/002"2 modified Grashof number for sphere/cylinder, = p2Rn+2(13gAT)2-n/m2 modified Grashof number, p2d3g[ST/m2(d2/ot)2'~-2 Grashof number for mass transfer, = p2d3gAp/ln2(d2/ot)2n2 modified Grashof number, = p213gqw(Dh/2)2"+2V2"2n/k~t2 or p213g AT (Dh/2) 2n+l V22n/la2, ) =

Graetz number, = m Cp/kL acceleration due to gravity, m/s 2 heat transfer coefficient, Whn2K j-factor, = (kdV)Sc "2/3 thermal conductivity of fluid, W/InK mass transfer coefficient, m/s length in flow direction, m mass flowrate of fluid, kg/s power law consistency index, Pa.s" value of m at temperature To, Pa.s" Nusselt number, = lff)/k for pipes and hR/k for spheres and cylinders Nusselt number for spheres and cylinders power law index heat flux, W/in 2 Peclet number, p Cp VD/2k Peclet number for mass transfer (dV/DAB)

1479 Pr Pr'

Prandtl number, Cp Prandtl number = Cp m (V/Dh)"-l/k

Pra

Prandtlnumber'-Cpm l k[ 3n4,,+

Prh Pro Pr~

Prandtl number = (pC~r (1-'')/(l+,,) Prandtl number = (pCp/k)(ln/p)2/(n+l)R r 1-,,)/(l+n)(Rg[3AT)S(n-1)/2(,+I) Prandtl number = (mot/d2)n-l/pot Universal gas constant, Pa.mS/mole.K tube/sphere/cylinder radius, m Rayleigh number, = Gro Pro

R R Ra

8V]""D

R ep

Reynolds number, oV 2-n D"/8n-'m( a n + 1.]" 4n Reynolds number, pV2-n d"/m

Rel

modified Reynolds number, Rep (3,, 4nn +1)

r

radial coordinate, m Schimdt number, = (mDAB/d2)"-J/pDAB Schimdt number, = (m/pDAB)(d/V)I-" Stanton number, h/pV Cv temperature, K bulk temperature of fluid, K outlet fluid temperature, K inlet fluid temperature, K wall temperature, K mean velocity in pipe, m/s point velocity in z-direction, m/s angular velocity in 0-direction, m/s dimensionless axial distance, x/D dimensionless axial coordinate X/Pe axial coordinate, m

Rea

SClm

Scp St T Tb T~ To Zw

V Vz Vo X X* Z

Greek Symbols ot Ot.'k

13W 5

Ap

thennal diffilsivity, m2/s constant in equation (22) coefficient of thennal expansion, K l coefficient in viscosity equation (20), K -~ ((3n + 1)/4n) pressure drop, Pa

(12 (1 - e) / e")""

1480

porosity of packed or fluidised beds rl ~tcff ~-to

P 17rz B "Co

Subscripts b H1 H2 iso L S SIIS

T vp w X Z

"roB/Tw

viscosity, Pa.s effective viscosity, = re(V/d) nl, (Pa. s) viscosity at reference temperature, Pa.s dimensionless radial coordinate, r/R density, kg/m 3 shear stress, Pa Bingham yield stress, Pa wall shear stress, Pa volume fraction bulk pertains to thermal condition H l pertains to thermal condition H~_ pertains to isothennal case liquid solid suspension constant temperature variable property case wall condition local value local value average value

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1485

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149. Pak,B.,Cho,Y.I.,Choi,S.U.S., lnt.J.Heat Mass Transf.,34(1991 ) 1195. 150. Wronski,S.,Jastrzebski,M., Int.J.Heat MassTransf.,33(1990) 1. 151. Kozicki,W.,Tiu,C.,Can.J.Chem.Engng.,45(1967) 127. 152. Dunwoody,N.T.,Hamill,T.A.,ZAMP,30(1979)587. 153. Dunwoody,N.T.,Hamill,T.A., lnt.J.Heat Mass Transf.,23(1980)943. 154. Oliver,D.R.,Trans.Inst.Chem.Engns.,47(1969)T18. 155. Oliver,D.R.,Karim,R.B., Can.J.Chem.Engng.,49(1971 )236. 156. Oliver,D.R., Asghar, S.M., Trans.Inst.Chem.Engrs., 54(1976)218. 157. Mena,B.,Best,G.,Bautista,P.,Sanchez,T.,Rheol.Acta, 17(1978)454. 158. Lawal,A., J.Heat Transf.(ASME) 111(1989)399. 159. Naccache,M.,Souza Mendes,P.R.,lnt.J. Heat Fluid Flow, l 7(1996)613. 160. Naccache,M.,Souza Mendes,P.R., J.Thermophysics Heat Transf., 11(1997)98. 161. Shenoy,A.V.,Mashelkar,R.A.,Adv.Heat Transf.,15(1982)143. 162. Chhabra,R.P.,Bubbles,Drops and Particles in non-Newtonian Fluids,CRC Press, Boca Raton,FL(1993). 163. Chhabra, R.P.,Adv.Heat Transf.,23(1993) 187. 164. Chhabra,R.P.,Adv.Transport Processes,9(1993)501. 165. Ghosh,U.K., Chhabra,R.P.,Upadhyay,S.N.,Adv.HeatTrans.,25(1994)251. 166. Schowalter,W.R., AIChE J.,6(1960)24. 167. Bizzell,G.D.,Sl attery,J. C.,Chem.Eng. Sci., 17( 1962) 177. 168. Na,T.Y.,Hansen,A.G., lnt.J.Heat Mass Transf.,9(1966)261. 169. Lee,S.Y.,Ames,W.F.,AIChE J.,12(1966)700. 170. Chen,J.L.S.,Radulovic,P.T.,J.Heat Transf(ASME).,95( 1973)498. 171 Denn,M.M.,Chem.Eng. Sci.,22(1967)395. 172. Serth,R.W., AIChE J.,19(1973)1275. 173. Verlna,R .L.,Rheol.Acta, 16( 1977) 510. 174. Acrivos,A.,Shah,M.J.,Petersen,E.E.,AIChE J.,6(1960)312. 175. Acrivos,A., Shah,M. J. ,Pet ersen,E .E.,Chem .Eng. Sci. ,20( 1965) 101. 176. Nakayama,A.,Koyama,H., Warme-und-Stoffubertragung,20(1988)29. 177. Nakayama,A., Shenoy,A.V., Int.J.Heat Fluid Flow, 12(1991)336. 178. Nakayama,A., Shenoy,A. V., J.Heat Transf.(ASME), 114(1992) 127 179. Acrivos,A.,AIChE J.,6(1960)584. 180. Stewart,W.E., Int.J.HeatMassTransf., 14(1971 ) 1013. 181. Liew,K. S.,Adelman,M .,Can. J. Ch era. En gqlg.,53( 1975)494. 182. Amato,W.S.,Tien,C.,Int.J.Heat Mass Transl.,19(1976)1257. 183. Alhamdan,A.,Sastry,S.K.,J.Food Process Engng., 13(1990) 113. Process Engng., 184. Awuah'G'B"Ramaswamy'H'S"Simps~176176 16(1 993)39.

1486

185. Lee,T.-L., Donatelli,A.A.,lnd.Eng.Chem. Res.,28(1989) 1056.

186. Churchill,S.W.,Chem.Engng.Commun.,24(1983)339. 187. Westerberg,K.W.,Finlayson,B.A.,Num.Heat Transfer, l 7A(1990)329. 188. Shanna,O.P.,Bhatnagar,R.K.,ZAMM,55(1975)235. 189. Kawase,Y., Mashelkar,R.A., Uibrecht,J.J.,Int.J.Multiphase Flow, 8(1982)433. 190. Ogawa,K.,Kuroda,C.,lnoue,l.,J.Chem.Engng.Jpn., 17(1984)654. 191. Morris,S.,J.Fluid.Mech., 124(1982) 1. 192. Andersson,H.l.,lnt.J.Heat Fluid Flow.,9(1988)343. 193. Wang,T.-Y.,Kleinstreuer,C.,Int.J.Heat Fluid Flow,9(1988) 182. 194. Meissner,D.L.,Jeng,D.R.,DeWitt,K.J., Int.J.Heat Mass Transf., 37(1994)1475. 195. Kawase,Y., Ulbrecht,J.J., Chem.En~nlg.Commun.,8(1981 )213. 196. Lochiel,A.C.,Calderbank,P.H.,Chem.Eng.Sci., 19(1964)471. 197. Moo-Young,M., Hirose,T.,Ali,S.,Proc.5th hat.Con g.Rheol.,Kyoto, 1( 1968)233. 198. Hyde,M.A.,Donatelli,A.A.,Ind.Eng.Chem.Fundam.,22( 1983)502. 199. Carberry,J.J.,AIChE J.,6(1960)460. 200. Mixon,F.O.,Carberry,J.J., Chem.Eng.Sci., 13(1960)30. 201. Kawase,Y., Ulbrecht,J.J., Electrochimica Acta,28(1983)643. 202. Kumar, S.,Mall,B.K.,Upadhyay,S.N.,Lett.Heat Mass Transf.,7(1980)55. 203. Chandarana,D.l.,Gavin,A.,Wheaton,F.W.,J.Food Process En~nag., 13(1990)191. 204. Zuritz,C.A.,McCoy,S.C.,Sastry,S.K.,J.Food Eng.,11(1990)159. 205. Bhamidipati'S"Singh'R'K"Trans ASAE,38(1995)857. 206. Yamanaka,A.,Mitsuishi,N.,Heat Transf.-Jpn.Res.,6(1977)85. 207. Zitoun,K.B.,Sastry,S.K.,J.Food Process Engug.,l 7(1994)209 & 229. 208. Balasubramaniam,V.M.,Sastry,S.K.,J.Food Engng.,23(1994) 169. 209. Gadonna,J.P.,Pain,J.P.,Barigou,M.,Trans.l.Chem.E.,C74(1996)27.

210. Baptista,P.N.,Oliveira,F.A.R.,Oliveira,J.C.,Sastry,S.K.,J.Food Engng.,31 (1997)199. Also see ibid 125. 211 .Ghosh,U.K.,Kumar, S.,Upadhyay,S.N.,Polym.Plast.Tech.Eng.,31 (1992)271. 21 2. Keey,R.B.,Mandeno,P.,Tuoc,T.K.,l.Chem.E. Sym.Ser.,33(1971 )53. 21 3. Lal,P.,Upadhyay,S.N.,Chem.Eng. Sci.,36(1981) 1865. 21 4. Ng,M.L., Hartnett,J.P., lnt.Comm.Heat Mass Transf.,l 3(1986)115. 21 5. Gorla,R.S.R.,Polym.-Plast.Teclmol.Eng.,30(1991 )37. 21 6. Gentry'CC"W~ Transf.(ASME),96(1974)3. 21 7. Ng,M.L., Hartnett,J.P., lnt.J.Heat Mass Transf.,31 (1988)441. 21 8. Ng,M.L., Hartnett,J.P., J.Heat Transf.(ASME), 108(I 986)790. 21 9. Ng,M.L., Hartnett,J.P., Int.Comm.Heat Mass Transf.,l 5(1988)293.

1487

220. Lyons,D.W.,White,J.W.,Hatcher,J.D.,Ind.Eng.Chem.Fund.,l 1(1972)586. 221. Shenoy,A.V.,Mashelkar,R.A.,Chem.Engng.Sci.,33(1978)769. 222. Chhabra,R.P.,Proc. 14th Nat.Heat & Mass Transf. Conf.,Kanpur (1997)in press. 223. Shah,M.J.,Petersen,E.E.,Acrivos,A., AIChE J.,8(1962)542. 224. Kim,B.K.,Lee,H.S.,Korean J.Chem.Eng.,6(1989)227. 225. Wolf,C.J.,Szewcyzk,A.A.,Proc.3rd Int.Heat Tranf.Conf. ,Chicago,1( 1966)388. 226. Mizushina,T., Usui,H.,Heat Transf.-Jap.Res.,7(1978a)83. 227. Mizushina,T., Usui,H., Kagaku Kogaku Ronbunshu,4(1978b)166. 228. Schlichting,H.,BoundaryLayer Theory,McGraw Hill,New York(1968). 229. Dienemann,W.,ZAMM,33(1953)89. 230. Shenoy,A.V., AIChE J.,26(1980)505. 231. Wang,T.-Y.,Kleinstreuer,C., J.Heat Transf(ASME).,l 12(1990)939. 232. Mizushina,T., Usui,H.,Kato,T., Kagaku Kogaku Ron.,4(1978)173. 233. Mizushina,T., Usui,H.,Kato,T., Kagaku Kogaku Ron.,4(1978)214. 234. Luikov,A.V.,Shuhnan,Z.P.,Puris,B.l.,Heat Transf.Sov.Res., 1(1969) 121. 235. Luikov,A.V.,Shuhnan,Z.P.,Puris,B.I., Zhdanovich,N.V.,Prog.Heat Mass Transf.,2(1969)262. 236. Takahashi'K"Maeda'M"lkai'S"Preprint 14th Heat Transf.Sym.Jpn. (1977)A-306. 237. Kumar,S., Tripathi,P.K., Upadhyay,S.N.,Lett.Heat MassTransf., 7(1980)43. 238. Ghosh,U.K.,Gupta,S.N.,Kumar,S.,Upadhyay,S.N., Int.J.Heat Mass Transf.,29(1986b)955. 239. James,D.F.,Acosta,A.J.,J.Fluid Mech.,42(1970)269. 240. James,D.F.,Gupta,O.P.,Chem.Eng.Prog.Sym.Ser.,67,No.111 (1971 )62. 241. Hoyt,J.W.,Sellin,R.H.J.,Exp.Heat Transf.,2(1989)113. 242. Ruckenstein,E., Ramgopal,A., J.Non-Newt.Fluid Mech.,l 7(1985) 145. 243. A1Taweel,A.M.,Sedahmed,H.G.,Abdel-Khalik,A.,Farag,H.A., Chem.Eng.J., 15(1978)81. 244. Kawase,Y., Ulbrecht,J.J., J.Appl.Electrochem.,13(1983)289. 245. Garg,N. S.,Tripathi,G.,Ind.J.Technol., 19(1981) 131. 246. Fuj ii,T.,Bull.JSME.,2(1959)365. 247. Reilly,I.G.,Tien,C.,Adehnan,M.,Can. J. Chenl.Engqlg.,43(1965)157. 248. Huang,M._j.,Chen,C.K.,int.j.Heat Mass Transf.,33(1990)119. 249. Shenoy,A.V.,Ulbrecht,J.J.,Chem. Engng.Commun.,3( 1979)303. 250. Pittman'J'F'T"Richards~ Int.J.Heat Mass Tran sf.,37(suppl. 1)( 1994)333. 251. Mishra,l.M., Singh,B.,Mishra,P.,lndian J.Technol.,14(1976)322.

1489

A ONE-DIMENSIONAL M O D E L FOR VISCOELASTIC DIFFUSION IN POLYMERS C.J. Durning 1, P.H. Tang I and R.A. Cairncross z 1Department of Chemical Engineering and Applied Chemistry, Columbia University, New YorkNY 10027 2Department of Chemical Engineering, Drexel University, Philadelphia PA 19104

1. I N T R O D U C T I O N Very complex motions can result when a viscoelastic fluid is put under external forces. In general, for a given set of forces one has to solve continuity, momentum and constitutive equations simultaneously to determine the details. The majority of effort along these lines has focused on incompressible media, which execute isochoric motions; viscoelastic flows with a strong dilational component have not received as much attention, primarily because there are relatively few applications demanding their study. One situation involving viscoelastic media where the resulting flow is dominated by the dilational component is inter-diffusion. Here, flow occurs in a mixture with relative motion between components, driven by a combination of thermodynamic and mechanical forces. The simplest case involves just two components with no applied mechanical tractions; the relative motion is mutual diffusion, generally resulting in flow of each component dominated by dilation. We discuss this situation for one-dimensional motions in the case of a binary mixture with one viscoelatic component and the other an ordinary viscous fluid. A quantitative understanding of mutual diffusion in concentrated polymer/viscousfluid mixtures is of great practical importance, in order to understand and control a number of polymer processing operations and end-use applications. For example, it

1490

is a controlling physical process in devolatilization, fiber spinning, coating, drying, microlithography and several control release schemes; accurately modeling the process is essential to the design and control of these operations. Experimental studies indicate that unsteady, one-dimensional diffusion in concentrated polymer/viscous-fluid mixtures ranges from ordinary Fickian behavior to strongly non-Fickian depending on the conditions (see, for example, references [ 1] -[4] ). At temperatures far above the glass transition (greater than 50~ above Ta), diffusion follows Fick's law on all accessible length scales, reminiscent of the behavior in mixtures of simple viscous fluids. However, at lower temperatures and high polymer densities, the dynamics begin to deviate from Fickian on small length scales. Dramatic non-Fickian effects appear on relatively large scales throughout the concentrated regime, near and below the glass transition. It is known that the polymer's viscoelasticity, causes the deviations from Fickian response [5, 6], hence the non-Fickian diffusion in such cases is often called viscoelastic diffusion. In this work we demonstrate the capabilities of a relatively simple one-dimensional model to account for the most obvious viscoelastic, non-Fickian effects observed experimentally. This model allows a straitforward calculation of the diffusioninduced flow of the polymer component during mutual diffusion. Mutual diffusion in concentrated polymer/viscous-fluid mixtures is often studied by a vapor sorption experiment. Since this technique is the main focus of our modeling effort we describe it briefly, along with key experimental findings. The mixture is confined in a thin film, initially in equilibrium with a large reservoir of the vapor of the viscous fluid at activity a - . At time t = 0 the activity is suddenly increased to a +, driving the system to a new equilibrium, with kinetics controlled by diffusion into the mixture. This causes "swelling" or distention of the film in the lateral direction, corresponding to longitudinal flow of polymer in the mixture. The kinetics are usually tracked by measuring the evolution of concemration profiles or by following the mass of the mixture as a function of time. For a "differential" or "interval" sorption, a + - a - > 1.

(23)

MlnM

Duming et al. [28] also show si ~ 0

(24)

according to the TW model. In the present model, three parameters a, 0, and M appear in the equation governing surface concentration, equation (19). One sees that the effect of 0 is only to rescale s, which implies that si ~ 0 as in the TW model (equation (24)). The parameter M describes the dependence of the relaxation time on concentration. Figure (3) shows its effect on surface swelling with a fixed at 0.99, and 0 set to 1.0; lnM ranges from 2 to 3.25. When in M < 1.5 ( M < 4.48), there is no discemable induction time; the Ub vs. s curves in this case do not show upward curvature. For In M >_ 1.5 ( M >_ 4.48), autoacceleration of Ub occurs, and an induction time can be calculated using a process analogous to that illustrated in figure (2b). Figure (4) plots si vs. ( M in M ) -1 for a - 0.99 and shows a linear relationship. Taking differences in 0 into account, the slope agrees in order of magnitude with the value obtained by Fu and Duming [23] from the TW model. In the current model the parameter a also affects the induction time. a shows up as a measure of the instantaneous elasticity of the polymer in the mixture; there is no analog for a in the TW model, since it presumes the polymer is purely viscous. a -- 0 means a perfectly compliant polymer and corresponds to ordinary diffusion; for finite a instantaneous elasticity and relaxation effects are switched on. Figure (5) shows the effect of a on surface concentration with M fixed at 10.0. For finite a one sees an initial jump, permitted by the instantaneous elasticity, which decreases with

1502

increasing c~. In the linearized version of the model, the surface concentration jumps initially to 1 - a (Tang et al. [ 12] ). In the non-linear version, the value given by the dimensionless version of equation (7) is near, but not exactly equal to 1 - c~. From figure (5) one sees that when a is less than 0.1, the relaxation of UD is minimal and there is no discernible induction time. As c~ increases from 0.1 to 0.9, the amplitude of the relaxation in Ub increases, induction times si appear and increase with a. The si increase monotonically with a, but the numerical data do not suggest an unambiguous analytical representation for si(a).

1.0

!

I

'

0.9 0.8 0.7 .Q

0.6 0.5 0.4 0.3 0.2

A

M = 12.18

v

M= 15.64

0.1 0.0

,

0

~

~

I

2

,

~

,

I

4

J

,

J

I

6

,

J

'

8

Figure 3. Effect of M on surface concentration (Ub vs. s) for a -- 0.99 and 0 -- 1.0.

1503

9

,

,

!

,

,

,

!

3

0

i

0.00

0.02

0.04

i

I

I

=

0.06

|

0.08

1/MIn(M)

Figure 4. Linear relation between induction time from surface concentration kinetics, si, and ( M In M) -1. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0"3 I

a

a=0.3

0.2 ~ [ 0.1 [

"

a=?.5

o

a=0.7 a=0.9

/

0.0 0.000

0.005

0.010

0.015

Figure 5. Effect of a on surface concentration kinetics O - 1.0.

0.020

(Ub VS. 8) for M - 10.0 and

1504

4.3 Profiles and Weight Uptake in the Case II Limit In what follows we systematically investigate the effects of a, 0, M and K on the concentration profiles (u(s, x)), nonequilibrium chemical potential profiles (OI/Os(x, s) ), weight uptake kinetics (W(s)) and several other characteristic features, including the concentration front's speed and the induction time based on weight gain kinetics. All of the calculations are on parameter ranges where Case II transport is predicted, u0 and ~ are fixed at 0 and 1, respectively, which corresponds to a dry polymer film suddenly exposed to unit fluid activity.

4. 3.1

Effect of a

Trial calculation showed that concentration profiles only begin to resemble Case-II when a is near 1. To determine its effect on the process, we varied it over the range 0.9 to 0.999. Provided 0, M and K also have appropriate values (see later sections), the weight uptake plots in this range are near linear with time, and show an induction as in figure (2b). From this, we conclude that in order to observe Case-II phenomena D~ >> Do (see equation (16)). Figure (6) shows the effect of a on weight uptake in detail for 0 = 0.01, K - M -- 5.0. From the plot one sees that, when a is small, there is practically no induction. The induction time shows up more clearly when a increases; at the same time the weight uptake curves become more nearly linear with time. Induction times for the process were calculated from figure (6) as illustrated in figure (2b); The results are qualitatively, consistent with those for induction times determined from Ub, i.e. the induction times from weight gain kinetics and surface concentration both increase monotonically with a, but they do not agree quantitatively. The quantitative discrepancy warns that experimental induction times determined from weight gain measurements not be interpreted in terms of surface concentration kinetics. Figures (7) show u and 0~ profiles for a - 0.9 with 0 - 0.01 and K and M fixed at 5. Recall ~oI is the (dimensionless) nonequilibrium contribution to the local chemical potential, and is part of the driving force for diffusion. Initially a large peak in 0I develops at the surface. As time increases, the peak decays, broadens, and then propagates into the film with fixed shape, at nearly constant speed. Eventually, the two peaks from both sides of the film combine into one at the film center and slowly relax while concentration reaches equilibrium. The peak in 0I is responsible for the dominant feature of Case II transport, i.e.

1505

1.0

0.8

0.6

0.4

/

/

/

/

=

o~= 0.92

.

-o.oo

i. i 0.0

,

0.0

I

,

,

0.2

Figure 6. Effect of a on W vs. s for 0 1.0.

,

,

,

0.4

0.01, K -

0.6

5.0, M -

5.0, Uo -

0.0,

the sharp concentration front which establishes near the surface after the external activity is switched on and propagates into the polymer film at a constant speed. We conducted systematic calculations to investigate the effect of a on the key features of OI the from: The front speed v and the values of u and 8-; at the front (uf and (o~):), oI Figure with the front's position being defined by the position of the maximum in N. (8) displays a typical trace of u: and (oi) i with time, for M - / f i - 5.0, a - 0.9 and 0 = 0.01. The plot shows that, after an initial induction at the surface both u / and ( ~ ) I achieve nearly steady values and move at nearly constant speed into the film. When the front reaches the center of the film, (oi N ) : stops and sinks to zero gradually. At the same time, uf relaxes to the equilibrium concentration. Two measures of the moving front speed were calculated. One, v:, is the slope of the linear part of front position versus time, and the other, Vw, is the slope of the linear portion of W(s) vs. s. Since the moving front invades the film from both sides, vw corresponds to about twice Vv (the values ofvf and vw/2 agree within • 8%).

1506

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.20 0.18 0.16

s = 0.0004

0.14

o.12 o.lo

s=oos

s=o.o3

s= 0.011 /~/I

0.08 0.06 0.04 0.02 0.00 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 7. (a) Concentration profiles (u(z, s)) determined by two different numerical schemes. Hollow symbols correspond to collocation method while filled symbols correspond to Galerkin method. The model parameters are a = 0.9, 0 -- 0.01, K = 5.0, M = 5.0, Uo = 0.0, ~b = 1.0.; (b). c9I/c3s profiles from collocation for the same set of parameters.

1507 Results for effects of a on both v~ and "of with K = M = 5.0 and 0 = 0.01 are summarized in figure (9), which shows that the v decrease linearly with increasing a. That plot suggests v f ~ (1 - a) since the speed gets vanishly small near a - 1.

0.3 ~

Uf

0.2

? 0.1

0.0

0.0

0.1

Fig 8. u f , ( O I / O s ) s vs. x ; a

4.3.2

0.2

0.3

0.4

0.5

- 0.9, 0 -- 0.01, K - M - 5.0, Uo - 0.0, r -

1.0.

Effect of O

0 is a Deborah number for this model. It corresponds to the ratio of a characteristic relaxation time of the polymer to the characteristic diffusion time, evaluated for the dry film. For nonlinear diffusion, the behavior cannot be anticipated from 0 along, as in the linear limit, since the actual relaxation and diffusion times vary with concentration making the instantaneous, local value of the Deborah number vary significantly. However, for this model in the special case M = K the concentration dependence of the relaxation and diffusion times cancel, and 0 does correspond to the dimensionless relaxation time at all compositions. It happens that M ___ K is required for Case II (see later), so we may view 0 as a key parameter for Case II.

1508

14[:

....

I ....

t ....

I ....

= ....

d -i

12O ~ 10 8 [-

4_

~.

Vw

vf

-2 0.90

0.92

0.94

0.96

0.98

Gt

Figure 9. v f and v~ vs. a with 0 - 0.01, K - 5.0, M - 5.0, Uo - 0.0, r - 1.0.

We found that 0 must be _< O(0.01) in order to predict Case II, consistent with Fu and Durning's [23] analysis which derived the TW model of Case II as a small Deborah number limit of the model considered here. We picked the range 0.008 _< 0 _< 0.022 to study the effect of 0 on the process. From calculations of concentration profiles and weight uptake with a -- 0.9, / ( = M = 5.0, one finds that an increase of 0 causes a proportional increase in the induction time calculated from weight ~ain kinetics, i.e. si ~ 0, exactly as anticipated from analysis of the surface concentration. 0 also decreases front speed, and therefore the slope of the linear portion of W ( s ) vs s plots. The results for v f and vw are plotted in figure (10); we find v f ~ 0 -0.52 under these conditions. Fu and Durning [23] studied the TW model numerically and found v ~ 0 -~ identical with the dependence of in the current model in the Case II limit. One can anticipate this result from dimensional analysis.

1509

1.2

9

9

l--

w

1

w

i

i

w

1.1 1.0 0.9 c

0.8 b.O 9

z

0.7 0.6

Iogvf

0.5 L 0.4

-

3

1

I

I

-5.0

I

I

I

-4.5

-4.0

-3.5

logo

Figure 10. vf and v~ vs. 0 with a - 0.9, .K - 5.0, M - 5.0, Uo - 0.0, ~b - 1.0.

4.3.3

Effects of M and K

The parameter M controls how fast the local relaxation time decreases with concentration; it characterizes one of the nonlinearities in the model. The effect of M on weight uptake kinetics was determined for a -- 0.9, 0 = 0.01 and K -- 5.0 for 2 < M < 10. The data showed that increasing M decreases the induction time, in qualitative, but not quantitative agreement with the result from the analysis of surface concentration (figure (4)). When M is large, the average relaxation time in the surface layers is low, and it takes less time to establish equilibrium at the surface. Increasing M also increases the slope of the linear portion of weight uptake plots, corresponding to an increase in front speed. The effect of M on the front speed is summarized figure (11) for a = 0.9, 0 = 0.01 and K = 5.0; the data for, M = 6, 8, 10, were generated by Galerkin's method. The front speed increases monotonically with M, with the data for vf being nearly linear with M. Duming et al.'s [28] asymptotic prediction

1510

from the TW model, that v ~ ( K + M ) a/2 is consistent with figure (11).

22

. . . .

1

'

'

'"

I

. . . .

I

. . . .

I

. . . .

I

'

'

'

r

20

II

18

16 ~

vw

14 12

r-I

lO ~

_

o

-

8. 6~

El

-

9

[3

ooOO

. ,

0

-

0

0

4 2

9

n

=

j

,

I

2

,

,

,

I

0 O0

1

4

. . . .

9

-

v,

:

I

6

,

,

,

,

I

8

,

,

=

i

I

10

. . . .

12

M

Figure 11. Dependence of v / a n d vw on M with a - 0.9, 0 - 0.01, K - 5.0, Uo 0.0, ~b - 1.0. Filled symbols represent values calculated by Galerkin's method.

The numerical solutions give evidence that M has to be sufficiently large in order to predict Case II 9 We calculated u and 0I profiles with 3/1 -- 2 and other parameters as above. By comparing the results with figure (7), one finds that the decrease of M from 5 to 2 suppresses the Case II features. The decrease results in considerable broadening of the concentration front; at the same time, the peak in 0I continuously decays at M = 2 and spreads progressively more along the spatial coordinate during sorption. The parameter K govems how fast the diffusion coefficient increases with concentration and characterizes the other key nonlinearity in the model. Figure (12) shows the dependence of induction time from weight gain on K with ce = 0.9, 0 = 0.01 and M = 5.0. The four data points in the high K range were generated

1511

by the Galerkin code. One expects that the induction time from weight gain be independent of K, if it truly characterizes the surface relaxation process, i.e. if it corresponds closely to the induction calculated from the surface relaxation kinetics (e.g. figure (4)). However, we find that the induction time from weight gain increases linearly with K until near the value of M (5.0 in figure (12)) after which it becomes independent of K as expected. This indicates that diffusion in the surface layers can limit the induction time determined from weight gain kinetics. We note, however, that compared with the effect of M, which reduces the induction time even if the value exceeds K, the influence of K on the induction from weight gain is much weaker, as one expects intuitively. Calculations show a roughly linear relationship between front speed and K, as shown in figure (13) for M = 5, a - 0.9, 0 = 0.01. This finding is again consistent with that by a perturbation analysis ofthe TW model, which predicts v ~ ( K + M ) 1/2. Calculations showed that in order to have Case II, M needs to be at least 2 - 3 and K has to be at least equal to M. For example, it was found that if K is kept at 5.0, when M rises above 8, the predicted behavior deviates from Case II in that the linear weight uptake kinetics were not predicted. This is because when M exceeds K, the front speed becomes so fast that diffusion behind the front quickly becomes rate limiting. Consequently, the front begins to show diffusive dynamics (v ~ sl/2).

5. C O N C L U S I O N A one-dimensional nonlinear model for viscoelastic diffusion in concentrated polymer-fluid mixtures was constructed by an ad-hoc generalization of the linear response model by Durning and Tabor [18] . The nonlinearities were introduced by retaining concentration dependencies of physical properties. In order to get a numerical solution for sorption in films, the diffuson equation was cast as coupled partial differential equations by introducing a new dependent variable. Finite element methods were used to discretize the spatial domain, converting the PDEs into an ODE system, which was solved by a time integrator package. In one scheme, orthogonal collocation on Hermite cubic basis functions was used to discretize and LSODI was adopted to do the time integration. An alternative integration technique employed Galerkin's method on quadratic basis functions together with the DASSL integrator. The two schemes were shown to agree. There are six dimensionless parameters in the model, a, 0, M, K, u0 and ~b. The

1512

0.0092 0.0088

O

9

9

@

0.0084 0.0080 0.0076

0.0072

3

4

5

6

7

8

Figure 12. Dependence of the induction time from weight gain kinetics, si, on K with a - 0.9, 0 - 0.01, K - 5.0, Uo - 0.0, ~b - 1.0. Filled symbols represent values calculated by Galerkin's method.

first four characterize the mixture and the last two define the initial and final states for sorption. We first investigated the predictions for two well-studied situations, where all the parameters could be calculated apriori: Differential sorption in polystyreneethylbenzene (PS-EB) solutions and integral sorption in poly(methylmethacrylate)liquid methanol (PMMA-MeOH). For PS-EB, the two-stage sorption process was correctly predicted for differential sorptions in thin films at concentrations just below Tg. For PMMA-MeOH, the Case II diffusion was correctly predicted for immersion conditions in thick, dry plates at room temperature. The calculation shows that the model can predict the most well documented and striking non-Fickian effects observed in sorption w i t h o u t empiricism. A systematic investigation was conducted of the effects of the materials parameters

1513

22

'""

'

'

I

'

i

,

,

I

'

"'

'

I

'

;

;

'

I

'

"'

'

I ''w

'

'

"

i

20 18 16 Vw

14 12 10 8

0

9

D

9 0

6 0

0

0

9

9

Vf

4 4

5

6

7

8

9

Figure 13. Dependence of vf and vw on K with a - 0.9, 0 - 0.01, M - 5.0, 1.0. Filled symbols represent values calculated by Galerkin's method.

Uo - 0.0, ~b -

on Case II diffusion. The study should facilitate analytical asymptotic work, and provide guidelines for design and control of systems relying on Case II. Case II diffusion appears only when a, a measure of the instantaneous elasticity of the system, is close to 1. This implies D~ > > Do and physically means that the osmotic modulus is weak compared to the mixture's shear modulus. The situation occurs when the fluid is a poor solvent or swelling agent. It was found that the front's speed decreases nearly linearly with increasing a. 0, the diffusion Deborah number, has to be ~ O(0.01) for Case II to appear, indicating that Case II is a slow-motion limit of the model. 0 affects the front speed according to v ~ 0 -1/2. The numerical study shows that strong nonlinearities in the relaxation time and diffusion coefficient, represented by large M and /4, are both essential for the prediction of Case II transport. Importantly, the values of M and K should be about the same, and at least 2 - 3. I f M exceeds K by too much, the process rapidly becomes

1514

controlled by diffusion behind the moving front. Regarding the numerical methods used, collocation can handle parameters K an M only up to about 5, beyond which the calculation becomes unstable. For modelin more severe non-linearities, one should employ Galerkin's method, which seems t have a wider range of stability in parameter space. It was demonstrated that the model predictions qualitatively match the behavk observed experimentally. A forthcoming publication [29] reports in detail o practical procedures for evaluating the model parameters, and on the quantitatN capabilities of the model.

6. Acknowledgment 1LAC acknowledges support from Sandia National Laboratory.

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