Interfaces for the 21st Century: New Research Directions in Fluid Mechanics and Materials Science : A Collection of Research Papers Dedicated to Steven H. Davis in Commemoration of hi

Interfaces for st the 21 Century New Research Directions in

Fluid Mechanics and Materials Science

Marc K. Smith, Michael J. Miksis, Geoffrey B. McFadden, G. Paul Neitzel & David R. Canright editors

Imperial College Press

Interfaces for st the 21 Century New Research Directions in Fluid Mechanics and Materials Science

This page is intentionally left blank

Interfaces for st the 21 Century New Research Directions in Fluid Mechanics and Materials Science

A collection of research papers dedicated to Steven H. Davis in commemoration of his 60f birthday

editors

Marc K. Smith Georgia Institute of Technology, USA

Michael J. Miksis Northwestern University, USA

Geoffrey B. McFadden National Institute of Standards and Technology, USA

G. Paul Neitzel Georgia Institute of Technology, USA

David R. Canright Naval Postgraduate School, USA

ICP

Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

INTERFACES FOR THE TWENTY-FIRST CENTURY: NEW RESEARCH DIRECTIONS IN FLUID MECHANICS AND MATERIALS SCIENCE Copyright © 2002 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 1-86094-319-5

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PREFACE In the science of mechanics, the term interface is generally used to refer to a material boundary between two phases, i.e., liquid-gas, solid-liquid or solid-gas. Specific examples in fluid mechanics are the spreading of a liquid droplet and the dynamics of a thin liquid film; specific examples in materials science are the solidification of a solid in a melt and an elastically stressed solid interface which evolves by surface diffusion. The primary difficulty in the modeling of interfaces is the fact that there locations are, in general unknown a priori. This difficulty has resulted in the development of many analytical and numerical solution techniques. Here a number of new and exciting solution methods as well as the physical problems which have initiated them will be discussed. Our hope is that the collection of work presented here will encourage others to work in and initiate new research into this important field. The material contained in this volume is the bulk of that presented in oral form at a conference, Interfaces for the Twenty-First Century, held August 16-18, 1999, in Monterey California. These oral presentations were invited from experts renowned for their research in the field of interfacial mechanics. In addition, abstracts of contributed posters which were also presented are included. The focus of the conference was on interfacial topics in fluid mechanics and materials science. The presentations covered experimental as well as theoretical approaches with an overlying philosophy of the conference being to investigate new techniques for modeling, both theoretically and numerically, the dynamics of interfaces. The conference was supported by the National Science Foundation and the National Aeronautics and Space Administration, for which the organizing committee and conference participants are extremely grateful. The genesis of the conference which produced the present volume was to hold a meeting to pay tribute to one of the world's foremost researchers in the field of interfacial mechanics, Professor Stephen H. Davis, on the occasion of his sixtieth birthday (September 7, 1999). Steve, through his outstanding research and mentoring of a generation of graduate and post-doctoral students, has contributed profoundly to our present knowledge in the field. It is with gratitude and pleasure that we present the contributions of some of the major investigators in this field as a tribute to the work of Steve Davis. Marc K. Smith Michael J. Miksis Geoffrey B. McFadden G. Paul Neitzel David R. Canright January 2002

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CONTENTS

Preface

v

PART 1: INVITED PAPERS

1

The effect of a stabilising gradient on interface morphology T. Maxworthy

3

Spreading of a liquid drop with mass loss L. M. Hocking

21

Viscous gravity currents with solidification M. Bunk, P. Ehrhard, and U. Mutter

35

Coarsening dynamics of roll waves H.-C. Chang and E. A. Demekhin

51

Thermo capillary control with feedback of large wavelength interfacial instabilities R. E. Kelly

61

Pattern formation in thin liquid films D. Gallez and E. R. de Souza

73

Molecular aspects of contact-line dynamics J. Koplik and J. R. Banavar

89

Computational methods for advancing interfaces J. A. Sethian

105

Direct numerical simulations of multiphase flows G. Tryggvason and B. Bunner

121

A phase-field model with convection: numerical simulations D. M. Anderson, G. B. McFadden, and A. A. Wheeler

131

Phase field model of multicomponent alloy solidification with hydrodynamics R. F. Sekerka and Z. Bi

147

The effects of a stress-dependent mobility on interfacial stability P. W. Voorhees and M. J. Aziz

167

Non-constant growth characteristics of pivalic acid dendrites in microgravity J. C. LaCombe, M. B. Koss, A. O. Lupulescu, J. E. Frei, and M. E. Glicksman

177

viii

Contents

Interfaces on all scales during solidification and melting M. G. Worster

187

Phase and microstructure selection in peritectics W. Kurz and S. Dobler

203

Model phase diagrams for an FCC alloy R. J. Braun, J. Zhang, J. W. Cahn, G. B. McFadden, and A. A. Wheeler

213

PART 2: CONTRIBUTED ABSTRACTS

231

Influence of contact-angle conditions on evolution of solidification fronts V. S. Ajaev and S. H. Davis

233

Creeping steady thin film on an inclined plane with an edge N. Aksel

234

Phase-field simulation of convective effects on dendritic growth G. Amberg and R. Tonhardt

235

A model for a spreading and melting droplet on a heated substrate D. Manderson, M. G. Forest, and R. Superfine

236

Instabilities of a three-dimensional liquid droplet on a heated solid surface S. W. Benintendi and M. K. Smith

237

A laboratory model for the solidification of the Earth's inner core and the inner core's seismic anisotropy M. I. Bergman

239

Suppression of channel convection in solidifying Pb-Sn alloys via an applied magnetic field M. I. Bergman, D. R. Fearn, and J. Bloxham

240

Dynamics and stability of Van-der-Waals-driven thin film rupture A. J. Bernoff and T. P. Witelski

241

New approaches to front-tracking and front-capturing methods J. Brackbill, D. Jamet, 0. Lebaigue, and D. Torres

242

Manipulation of intravascular gas embolism dynamics with exogenous surfactants A. B. Branger and D. M. Eckmann

244

Adiabatic hypercooling of binary melts K. Brattkus

245

An insoluble surfactant model for a draining vertical liquid film R. J. Braun

246

Contents

ix

The effect of time-periodic airway wall stretch on surfactant and liquid transport in the lung J. L. Bull, D. Halpern, and J. B. Grotberg

247

The dynamic effects of surfactants on stationary gas bubbles in liquid flows D. P. Cavanagh and D. M. Eckmann

248

Buckling instabilities in thin viscous sheets S. Chaieb, R. da Siliveira, L. Mahadevan, and G. H. McKinley

249

Fluid-fluid interface experiments at the University of Chicago /. Cohen, S. R. Nagel, M. P. Brenner, J. Eggers, R. O. Grigoriev, and T. F. Dupont

250

Asymptotic estimates for 2-D sloshing modes: theory and experiment A. M. J. Davis and P. D. Weidman

251

Buoyancy-driven interactions of viscous drops with deforming interfaces R. H. Davis, J. Kushner, and M. A. Rother

252

Anomaly and uncertainty when liquid films flow over solid surfaces W. Debler

253

Flow behavior of Langmuir monolayers M. Dennin and R. S. Ghaskadvi

254

Soluble surfactants and contact-angle dynamics D. M. Eckmann

255

Experimental studies of the hydrodynamics near moving contact lines S. Garoff and E. Rame

256

Axisymmetry-breaking instabilities in axisymmetric freezing of ice A. Yu. Gelfgat, P. Z. Bar-Yoseph, A. Solan, and T. A. Kowalewski

257

Separation mechanics of thin interfacial liquid layers: the role of viscous fingering A. Gopinath

258

Large finite-element modeling of axially symmetric free-surface flows R. Grigoriev and T. Dupont

259

Molecular simulations of interface phenomena: an alternative approach N. Hadjiconstantinou

260

Experimental investigation of environment-oxygen content in solder-jet technology E. Howell, S.-Y. Lee, C. M. Megaridis, M. McNallan, and D. Wallace ...261 Droplet spreading with surfactant: modeling and simulation J. Hunter, Z. Li, and H. Zhao

263

Air entrainment at low viscosities A. Indeikina, I. Veretennikov, and H.-C. Chang

264

x

Contents

Numerical simulations of vibration-induced droplet ejection A. James, M. K. Smith, and A. Glezer

265

Interfacial dynamics associated with evaporation of LNG in a storage tank S. W. Joo, C. Park, and S. Hong

266

Oscillatory thermocapillary convection generated by a bubble M. Kassemi and N. Rashidnia

267

About computations of thin-film flows L. Kondic, J. Diez, and A. Bertozzi

268

Visualization of convection in liquid metals J. N. Koster

269

Time-evolving interfaces in viscous flows M. C. A. Kropinski

270

Influence of a nonlinear equation of state on contamination fronts at air/water interfaces J. M. Lopez and A. H. Hirsa

271

Stabilization of an electrically conducting capillary bridge far beyond the Rayleigh-Plateau limit using feedback control of electrostatic stresses M. J. Marr-Lyon, D. B. Thiessen, F. J. Blonigen, and P. L. Marston

272

Stabilization of capillary bridges in air far beyond the Rayleigh-Plateau limit in low gravity using acoustic radiation pressure M. J. Marr-Lyon, D. B. Thiessen, and P. L. Marston

273

Interactions between Hele-Shaw flows and directional solidification: numerical simulations E. Meiburg

274

Modelling the contact region of an evaporating meniscus with a view to applications S. J. S. Morris

275

Aspects of vortex dynamics at a free surface B. Peck, L. Sigurdson, P. Koumoutsakos, and J. Walther

277

Instabilities at the "interface" between miscible fluids — emergence of an effective surface tension P. Petitjeans, P. Kurowski, and J. Fernandez

279

Secondary instabilities of falling films using models C. Ruyer-Quil and P. Manneville

280

Interfacial phenomena in suspensions U. Schaflinger and G. Machu

282

Sessile drop solidification W. W. Schultz, M. G. Worster, and D. M. Anderson

283

A homogenized Monte-Carlo model for film growth T. Schulze

284

Contents

xi

Influence of the surrounding conditions near the interface on the stability of liquid bridges V. M. Shevtsova, M. Mojahed, and J. C. Legros

285

Convective-diffusive lattice models of interfacial and wetting dynamics Y. Shnidman

286

A level-set approach to domain growth in multicomponent fluids K. A. Smith, F. J. Solis, and M. 0. de la Cruz

287

Morphological instability in strained alloy films B. J. Spencer, P. W. Voorhees, and J. Tersoff

288

Stability issues in spin-casting molten metals P. H. Steen

289

The stability of thermocapillary convection in half zones with deformed free-surface profiles L. B. S. Sumner and G. P. Neitzel

291

Direct write of passive circuitry using ink-jet technology J. Szczech, C. M. Megaridis, D. Gamota, and J. Zhang

292

Stability of the meniscoid particle band at advancing interfaces in Hele-Shaw suspension flows H. Tang, W. D. Grivas, T. J. Singler, J. F. Geer, and D. Homentcovschi

293

Instabilities in thin fluid sheets B. S. Tilley, D. T. Papageorgiou, and R. V. Samulyak

295

Thermal effects of internal interfaces: equilibrium microstructure and kinetics A. Umantsev

296

The role of long-range forces in the stability of nematic films M. P. Valignat, F. Vandenbrouck, and A. M. Cazabat

297

Vibration-induced drop atomization B. Vukasinovic, M. K. Smith, and A. Glezer

298

Capillarity-driven instabilities and the evolution of solid thin films H. Wong, M. J. Miksis, P. W. Voorhees, and S. H. Davis

300

Interfacial wave theory for dendrite growth J.-J. Xu

301

Uniformly valid asymptotic solutions for dendrite growth with convection J.-J. Xu and D.-S. Yu

302

Comparison of asymptotic solutions of phase-field models to a sharp-interface model G. W. Young and S. I. Hariharan

303

xii

Contents

A new approach to measure the contact angle and the evaporation rate with flow visualization in a sessile drop N. Zhang and D. F. Chao

PART 3: PANEL DISCUSSION SESSION New research directions in interfacial science M. K. Smith

304

305 307

PART 1

INVITED PAPERS

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THE EFFECT OF A STABILISING G R A D I E N T O N INTERFACE MORPHOLOGY

T. MAXWORTHYt Department

of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1191, U.S.A.

We have been studying the motion of sedimenting, surface gravity currents and the resultant motion of particles through the interface between a heavy, ambient fluid and a lighter surface layer (Maxworthy [1,2]). As noted by Green [3] and Chen [4] this latter motion has many features in common with double-diffusive interfaces, but further study has revealed a similarity to a wide range of problems involving the stability and morphology of interfaces in general. Some of these similarities have been discussed before (Michalland [5] and others) in another context and we discuss these and related applications.

1

Experiments to Study the Dynamics and Stability of a Sedimenting Interface

We have undertaken a series of experiments on the dynamics of sedimenting, surface gravity currents in a rectangular tank 220 cm long, 57 cm deep, and 15.7 cm wide (Maxworthy [1]). The physical situation concerns a current, with total density pc, evolving at the surface of a fluid of greater density, p^. In turn, pc is made up of interstitial fluid of density pi and heavy particles with a concentration, by weight, c and a density pp. The relevant dimensionless parameter is denoted as R = iPA—pc)/(pc—pi), which has a close relationship [R = (Rp — l)} to the densitydifference parameter, Rp, used in the study of double diffusion (Turner [6], p. 274). In summary it was found that the sedimentation of the particles, plus some of the interstitial fluid, through the interface between the two fluids has a profound effect upon the motion of the current. In order to study this trans-interfacial particle/fluid motion in detail a subsidiary experiment was set-up by running a mixture of particles and pure water on top of a deep layer of salt water in a smaller, narrower tank (10 x 7 x 0.4 cm) (Maxworthy [2]). In order to observe the resultant flow only relatively large values of R were used, since, for small values, the instability evolved so quickly that interpretation was difficult. In Fig. la, a series of photographs of the resultant particle motion is shown for R = 3.1. In the first of these photographs one can see a number of the larger particles falling out of the upper region, the instability then appears on the interface after a sufficient number of the smaller particles in the distribution had entered the lower layer and its effective density, and thickness, was large enough for the layer to undergo a Rayleigh-Taylor type of instability. Of especial interest is the observation that the instability is locked in its axial location by the underlying stable fluid distribution. In Fig. lb we compare the shape of a typical cell with the Saffman T

This contribution is dedicated to my good friend Professor Stephen H. Davis on the occasion of his sixtieth birthday. He has been and continues to be a source of inspiration to us all.

3

4

T.

Maxworthy

Figure 1. a) Sequence of photographs of an unstable, sedimenting interface between heavier-lower and lighter-upper fluid layers. Instability first appears between times ii) and iii). The majority of the heavy particles drain from the upper layer through thin fingers that are about 0.5-0.7 mm wide. During this motion they drag a substantial amount of upper-layer fluid downwards with them, b) Comparison between the experimental interface shape and that due to Saffman and Taylor [7] for an immiscible interface in a Hele-Shaw cell. For a value of A = 0.95. c) Idealised representation of the interface before sedimentation starts, d) After a time At the monodispersed particles have formed a heavier layer in the lower layer that is v$ At thick. This is then susceptible to a gravitational instability of the Rayleigh-Taylor type.

and Taylor [7] solution for an air finger penetrating into a viscous fluid with the same width ratio (A), i.e., the width of the particle-free region divided by the total

The Effect of a Stabilising

Gradient on Interface Morphology

5

width of the cell. Here, we have chosen a value for A — 0.95 that appears to be about the correct value for the sedimenting finger. The solution given is for a theory that does not take surface tension into account and gives the flattened shape shown in Fig. lb. The finger shape including this latter effect is more rounded (see e.g., Pitts [8], McLean and Saffman [9]) and such finger shapes are similar to the shapes found in flows with free surfaces at which surface tension is an important dynamical effect, as discussed below. In summary, the sedimenting interface is then one example of a class of problems in which the axial gradient of a stabilising quantity sets the morphology of the resultant instability and often this morphology is similar to that found in Saffman-Taylor fingers. In the photographs, note also the lateral interaction of the cells as the finite-amplitude instability establishes itself. This behaviour is also characteristic of systems stabilised by an axial gradient, as discussed in what follows. Based on these observations, we have explored possible theoretical explanations for the instability of the particle-laden layer (Maxworthy [2]). For our present purposes, we consider only an idealized situation in which the initial density distribution is two-layered, with an interface of zero thickness and with a monodispersed particle distribution, as shown in Fig. lc. After a short time, At, the particle-front enters the lower layer to produce an unstable, heavy layer of thickness, vsAt, and a density close to [PA + (pc — Pi)], sandwiched within the stable fluid density distribution (Fig. Id). Here, vs is the settling velocity of the particles. After some time this heavy layer becomes unstable and produces a sequence of fingers that drain particles and fluid from the upper layer. For this simple model, the use of the theoretical results of Chandler and Redekopp [10] gives a time to instability that is somewhat smaller than that measured experimentally. In a more complete discussion of this problem (Maxworthy [2]), we have obtained better agreement between theory and experiment by considering a number of extra effects: e.g., the unsteady growth of the intermediate layer, the finite thickness of the density interface, the initial particle distribution within that interface, the polydispersion of that particle distribution, and the fact that the collective effect of the particles is to drag some of the light, upper-layer fluid downwards with them, reducing their effective buoyancy and hence velocity, as they traversed the stable layer. We wish to extract two points from this introductory section: firstly, to show the basic physical mechanisms that lead to instability in this previously unexplored example and, secondly, to show the morphology of the finite-amplitude interface that is formed under the conditions of these experiments. It is this second point that we wish to explore further in what follows. That is, we present diverse physical situations that produce similar morphologies, at finite amplitude, to the ones shown in Fig. la. The requirement to do so seems to be that there is some physical parameter representing a gradient of force that acts normally to the interface to prevent axial competition between the growing instability waves and, secondarily, reduces the growth rate. In the case discussed above this gradient is created by the difference in density between the upper and lower fluid layers. In what follows, we look at different mechanisms that create similar patterns. In order to inter-compare mechanisms that inhibit axial competition, it is probably appropriate to first show examples of this competition between growing insta-

6

T.

Maxworthy

Figure 2. Growth of a gravitationally unstable, immiscible interface in a Hele-Shaw cell, showing the axial competition between fingers that results in one finger dominating the others. Reprinted with permission from The Physics of Fluids.

bility waves so that their characteristics are evident and it is clear what we mean by the term. There are numerous examples, so we show just one that is the result of non-linear growth of an immiscible interface in a Hele-Shaw cell of constant gap width (Fig. 2). In this case, when one wave grows a little faster than its neighbours it produces a flow field that prevents the growth of these neighbours so that it can outrun them more readily the larger it becomes. In the literature on the subject this is sometimes called a "shielding" effect. If the growth discussed above takes place in a gap with a width that is not constant, the instability is regularised if the front is moving perpendicular to, and in the direction of, the negative gradient in gap width. That is, the attempt by any one wave to outrun neighbouring waves, that are slightly retarded, is inhibited by the weaker backfiow it generates at their location where the gap width is larger. The result is a growth that is uniform along the length of the interface. Contrariwise, we assume that competition in the direction of the positive gradient in gap width is enhanced since the flow field produced at the retarded perturbation is increased in magnitude by the decrease in width at that location. The parameter that seems to organise this behaviour, in the case of immiscible Hele-Shaw flows, is the ratio T of the wavelength of the instability to a length scale characteristic of the gap-width variation, ho/(dh/dx), where x is the distance perpendicular to the interface, h is the local gap height, and the O-subscript indicates the gap width at the x location of the interface. Since, from linear theory at low values of the capillary number Ca, the most unstable wavelength scales approximately

The Effect of a Stabilising Gradient on Interface Morphology

7

as ho/Ca1/2 (Maxworthy [11]), then Y = {dh/dx)/Ca1/2, where Ca = Ufi/a, U is a characteristic interface velocity, /i the absolute viscosity of the fluid, and a the surface tension at the interface. On the other hand, at values of Ca of order one and greater the most unstable wavelength scales as the width of the gap ho (Maxworthy [11]) so that r = dh/dx. In the region around Y = 0, classical SafFman-Taylor growth takes place and there is weak axial competition. For Y relatively large and negative, i.e., the unstable interface movement is in the direction of the negative gradient of height, axial competition is inhibited and the cells can only interact laterally. It is this case that interests us most in what follows. For Y relatively large and positive one would expect that axial competition would be enhanced, although, as far as we know, no experiments have been undertaken to look at this case. 2

Interface Stability in a Tapered Gap, as in a Journal Bearing, a.k.a.: Printer's Instability, Coating Instability, and Directional Viscous Fingering

In what follows, we discuss a number of examples of the effect of a stabilising gradient in layer thickness starting with the one that appears to have chronological priority: "Journal-Bearing Instability," sometimes called "Printer's Instability" (Michalland [5]), "Coating Instability," or "Directional Viscous Fingering," the latter name to emphasis the analogy with the problem of directional solidification, to be discussed later. This type of instability occurs under many different circumstances, with the section title revealing several examples. The generation of streakiness in fluid films in many industrial processes has been observed for many years, but it was with Pearson [12], who first analysed the streaks generated by a simplified version of a paint brush, that attempts were made to place these on a scientific footing. Recent work on this type of instability in journal bearings appears to have started with the papers of Banks and Mill [13] and Cole and Hughes [14], followed by Pitts and Greiller [15], Floberg [16], and Taylor [17]. Of special interest in the latter case are his Plates 1 and 2, with the latter being reproduced in Fig. 3. These are photographs of the fingers found in a tapered gap of divergence angle 2.8°. Comparison with the finger shape found in the study of SafFman and Taylor [7] is not satisfactory. However, when this theory is modified to take the effect of surface tension into account (Pitts [8] and McLean and Saffman [9]) the agreement is much better, as shown in Fig. 3. This is confirmed by the experiments of Michalland [5], in which the shapes were compared with a theory by Ben Amar (unpublished, Fig. 4) that is apparently similar to that of McLean and Saffman [9]. Michalland's experiments are among the most comprehensive known to us and will form the basis for much of the discussion that follows. Parts of the thesis have been extracted as published papers and these are the most readily available [18-23]. In Fig. 5a, we show a simple schematic of the double-roller apparatus used by Michalland, and in Fig. 5b a sequence of the shapes taken by the trailing interface (location xm\ of Fig. 5a) as the speed of the inner cylinder is increased, with a non-rotating outer cylinder. The progression from a sinusoidal shape through

8

T.

Maxworthy

Figure 3. The form of the fingers found in a journal bearing with the shape compared with the solutions of McLean and Saffman [9] and Pitts [8].

shapes that are more and more non-linear is apparent, with the final ones being characteristic of a wide range of problems with a stabilising axial gradient (in this case of gap thickness), as will be shown in what follows. This very basic system displays a wide range of interesting behaviour. In particular, various types of lateral interaction is possible between cells. In Fig. 6, we show an example of space-time traces that exhibit cell splitting, wave propagation, chaos, etc. Michalland also presents the formal similarity between these types of instability and those found in directional solidification, to be discussed in more detail in Sec. 3. 2.1

Peeling Instability

The Peeling Instability is intimately related to the cases discussed above and was explored in detail by McEwan and Taylor [24]. The theory, the apparatus devised for the experiments, and the experiments themselves, show the imagination and skill inherent in the body of work from both authors. A horizontal roller supports the free end of a sheet of plastic that is, at the same time, loaded with a weight of known magnitude. The other end of the sheet is "glued" to a flat surface using a Newtonian fluid of known properties that was rolled to a constant, known thickness before an experiment began. An experiment was started by placing weights in the

The Effect of a Stabilising

Gradient on Interface Morphology

9

Figure 4. Comparison between the interface shapes found by Michalland [5] and the theory of Ben Amar (unpublished) for the double-roller apparatus of Fig. 5a.

pan and then raising the roller at a known rate. The speed of peeling and the angle of the sheet (5 were then set by these external parameters and the properties of the meniscus at the line of detachment. For our present purposes, we are interested in the shape of the instability cells formed at the line of detachment. Since the photographs taken from the original paper are hard to reproduce informatively, a photograph, taken using the apparatus described in Sec. 2.2, which is dynamically identical to the McEwan-Taylor arrangement, is shown in Fig. 7. The correspondence between these shapes and those found in the printer's instability is quite clear, especially when one inter-compares them with the Saffman-Taylor solutions with surface tension (McLean and Saffman [9] and Pitts [8]), for the shape of a finger with the same width ratio. 2.2

A Simple Home/'Lecture

Demonstration

The examples given above are easily demonstrated by an experiment that can be performed with a very simple apparatus that is easy to transport. All that is needed is two pieces of thin plastic sheet approximately 30 x 15 x 0.15 cm. To each of the two short ends of one is attached a commercial spring (paper) clip. The sheet is then bent and a piece of wire, tied to the clips, is used to maintain the curvature. A small

10

T.

Maxworthy

Figure 5. a) A schematic of the double-roller, journal-bearing apparatus used by Michalland [5] to study stability of the interface at the leading edge ( x m i ) . b) Photographs of the interface shapes found by Michalland [5] for various values of the velocity of the inner cylinder. In this case, the outer cylinder was stationary. Velocity increases from top to bottom. At the highest velocity the cell shapes are well represented by the Pitts [8], McLean and Saffman [9], and Ben Amar (unpublished ) solutions as shown in Fig. 4.

amount of a readily available, but transparent, viscous fluid (honey, syrup, or olive, canola, motor oils, etc., shampooing liquid, oil painting oil, etc.) is poured onto the flat sheet and the curved one is pressed into it (see Fig. 8). By rocking the latter back and forth or dragging it over the fluid layer one can generate the instability and cells of interest. The onset of instability, the spacing between cells, the lateral interaction between cells, and other behaviour can be seen to depend on the speed of rocking or towing, the viscosity and surface tension of the oil, its thickness far from the gap and the minimum value under the curved sheet. By working out the relative velocities between the two sheets one can determine that the system involving rocking the sheet is an exact analoque of the peeling instability as well as of some variations of the printer's instability. Photograph of typical "experiments" using this apparatus are shown in Figs. 7 and 8b. It is especially useful as a lecture demonstration since it can be placed on an overhead projector and viewed, at large scale, on a lecture-hall screen. 2.3

Experiments in a Tapered, Hele-Shaw Cell

A useful review of the effects of various perturbing mechanisms on unsteady interface dynamics in a Hele-Shaw cell has been presented by McCloud and Meyer [25]. In particular, they look at the effects of a uniform gradient in thickness on the

The Effect of a Stabilising

Gradient on Interface Morphology

11

Figure 6. A spatio-temporal representation of the regime of intermittant behaviour in the experiments of Michalland [5]. Locations A correspond to regions of regular behaviour and B to regions of chaotic behaviour. Regions of wave propagation and cell splitting and coalescence can also be seen clearly. From Michalland [5].

stability of an immiscible interface. The basic conclusion from the theoretical work is that the effect is small unless the gap slope is very large. They report on some experimental work in a relatively narrow cell, using a very small slope in thickness, in which only a few waves could be produced. This made the accurate determination of the instability wavelength very difficult as is apparent in their large data scatter. Recently, we have extended the work reported by Maxworthy [11] to include the effect of gap-width variation. Here, we have manufactured a gap that varies radially in a cylindrical geometry at an angle of 2.50° and used the same techniques as Maxworthy [11]. A photograph of the interface evolution in such a case is shown in Fig. 9. At values of Ca below about 0.2 it was impossible to generate a clean, unstable interface. As a result, a different method was used that involved tilting a flat plate in a thin pool of viscous fluid. Here, the angle was varied as the plate was tilted, but the gap width at the location of the interface remained approximately constant at 6.0 mm. Typically, the instability took place when the angle was approximately 1.50°, while the values of the most unstable wavelength were a seamless continuation of the values from the radial experiment (see Fig. 10). The most unstable wavelength, L, divided by the gap width, 6, as a function of Ca is shown in Fig. 10 (Maxworthy [26]), where they are compared with the results

12

T. Maxworthy

Figure 7. Three examples of the interface shapes found in the peeling instability. The apparatus used here is described in Sec. 2.2 and is formally identical to that used in the original experiments of McEwan and Taylor [24].

using a gap of constant width (Maxworthy [11]). It appears that with a gap gradient of between 1.5 and 2.50 the region of L/b approximately equal to 5-6 is extended to smaller values of Ca than in the case of constant gap width, with no indication of a tendency towards the linear instability results that give values of L/b greater than approximately 20-25. This would suggest, in line with previous results, that three- dimensional, viscous effects (Paterson [27]) dominate the dynamics at these small, but finite angles and at all values of Ca that result in an instability. As found by Michalland [5], in the case of the Printer's Instability, it was not possible to create an unstable interface below values of Ca of about 0.1, probably due to the stabilising effect of gravity on the non-uniform layer in the gap (Maxworthy [26]). 3

Effect of a Stabilising Temperature Gradient on a Solidifying Interface: Directional Solidification

The stability of a solidifying interface is of great practical importance as demonstrated by the enormous literature it has generated. Here, we concentrate on a small corner of this research and look at the effects of a stabilising temperature gradient on the stability of such a front. A typical apparatus consists of a HeleShaw cell filled with a transparent, organic, liquid alloy, e.g., succinonitrile/acetone, that is traversed through the temperature gradient formed between a cold and a hot reservoir. The relevent stability parameter is the morphological number M: M =

mVC0{k~l) kDLG

(1)

where V is the towing speed, G the temperature gradient, m the liquidus slope, k the segregation coefficient, CQ the solute concentration at infinity, and DL the

The Effect of a Stabilising

Gradient on Interface Morphology

Rocking Motion \

Spring Clip

13

.Cord Spring Clip

TT^'X^s

/ / s /\s / s /1}

Fluid

Spacer

r

~=~s b). Figure 8. a) A sketch of the apparatus used to demonstrate the Peeling Instability, b) A threequarter view of an experiment performed in the apparatus of Fig. 8a.

solute's diffusivity. For a given alloy when V and G are such that M is slightly above unity the interface becomes unstable to a sinusoidal instability that evolves over time to the non-linear shapes shown in Fig. lla,b; c.f., compare to Fig. 5b for the case of the Printer's Instability. This asymptotic shape is in every way similar to those discussed before, and a direct comparison with the Saffman-Taylor finger shape, with surface tension, is favourable (Fig. lie). If M is substantially above unity then the cells eventually develope perturbations at their tips that evolve into dendrites (Fig. lid) that also suppress the growth of their neighbours. In this case, the temperature gradient is not large enough initially to prevent axial competition between cells, but as the cell spacing becomes larger this is no longer true and axial competi-

14

T.

Maxworthy

m®&

Figure 9. Images from a video recording of the evolution of an immiscible interface in a circular Hele-Shaw cell with a gap thickness that varies radially (Maxworthy [26]). In this case L/b as 5.2.

20 L/b (402&230cP;Circular Plate) L/b(137cP) L/b(63cP)

15

10

L/b (63cp;Tilting Plate)

-

0.01

0.1

Ca

10

Figure 10. The variation of L/b with Ca over a range of the latter of almost two orders of magnitude. The results for a constant gap width are shown by the curve marked Maxworthy [11]. The error bars are ± one standard deviation in both cases. Note that L/b has a value close to that found at large Ca for constant gap width at all values of Ca for which the interface is unstable.

tion is inhibited. In the fluid dynamical systems we have considered up to now the dendritic form of instability seems to only occur if some mechanical anisotropy is placed at the tip of the growing cell in the Hele-Shaw apparatus. Examples of such disturbances include, a bubble, a longitudinal wire, a groove in one of the plates of a Hele-Shaw cell or in one of the cylinders in Michalland's experiments, etc. An

The Effect of a Stabilising

=>•

Gradient on Interface Morphology

15

d).1§^il/U,

Figure 11. a, b) Examples of interface shapes in directional solidification, (Billia and Trevedi [28]). Note the similarity to previous examples, c) Comparison of interface shape with the McLean and Saffman [9] solution for an interface with surface tension, d) Axial competition between cells in directional solidification when the axial temperature gradient is not large enough to prevent it. Eventually, when the largest dendrites have emerged, the temperature gradient is large enough to prevent competition and the cells evolve uniformly, e) "Dendritic" growth of a gravity-driven liquid/air interface in a Hele-Shaw cell with an anisotropic tip condition, in this case a small tip-bubble.

example is shown in Fig. l i e for the case of a large bubble with a small tip-bubble in a Hele Shaw cell. The latter subject is covered extensively in Couder et al. [29]. Also, McCloud and Meyer [25] explore such matters in considerable detail and give numerous examples of these effects, while Billia and Trivedi [28] have discussed the qualitative similarity between Saffman-Taylor fingers and solidifying interfaces and Michalland [5] the mathematical analogy.

16

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Maxworthy

4

Combustion-Front Instability

The stability of pre-mixed, combustion fronts has been treated in the aerodynamic, thin-flame limit by Markstein [30]. The effect that gives rise to the appearance of quasi-steady, cellular flames is the difference in diffusivity between the fuel and oxidiser in a curved flame. In the theory, this is parameterised by allowing the local flame speed to be a function of flame curvature: U = U0(1+L/R),

(2)

where U is the local flame speed, UQ the speed of a flat flame, R the flame radius of curvature, and L a phenomenological parameter that describes the effect of the preferential diffusion on flame speed. One can, then, derive an effective surface tension a for a flame (Maxworthy [31]): o = 2PU*L(\-l),

(3)

where p is the unburned gas density and A is the density ratio across the flame. The presence of this effect acts to stabilise the flame, generates a cut-off wavelength and a wavelength at which the growth-rate is maximum. At finite amplitude, the shape saturates at a certain amplitude, a measure of which has been calculated by Markstein [?] for example. Experimentally it is possible to generate finite-amplitude, unstable flames on a slot-burner of the type shown in Fig. 12a. A photograph of a finite-amplitude flame is shown in Fig. 12b, where the similarity to the Printer's Instability front of Fig. 5b should be noted. Agreement with the calculated shapes is excellent. In some regions of the flow-rate/equivalence-ratio parameter space the flame becomes unsteady with cells appearing and disappearing as shown in the location vs. time photographs of Figs. 12c,d. The similarity to cell motion in the Printer's Instability, Fig. 6 for example, is again striking (Michalland [5]). 5

Viscous Gravity Current Instability

The instability of interest here is well known to anyone who has attempted to paint a vertical surface and found, on applying an excess of paint, that it forms rivulets that destroy the smooth appearance of the surface. The first attempt at a scientific explanation of the phenomenon was due to Huppert [32], as far as we are aware. A two-dimensional, viscous gravity current was formed on a sloping surface by releasing a known volume of viscous fluid from behind a dam. Initially, the current was two-dimensional, but it finally became unstable to a fingering instability. Huppert presented an expression that scaled the wavelength of the disturbance, L*, as follows:

gp sin a where A is the cross-sectional area of the current, a is the surface tension of the fluid, g is the acceleration of gravity, p is the density of the fluid, and a is the slope of the plate over which the current is flowing. Johnson et al. [33] have considered the case with a constant Row rate per unit width, Q (units of cm 2 /s). Their

The Effect of a Stabilising

Gradient on Interface Morphology

17

Figure 12. a) A slot burner used to study the stability of two-dimensional flames, b) T h e flamefront shape for a rich propane-air mixture (Maxworthy [31]). Note the similarity in shape to examples given in prior sections, c and d) Spatio-temporal photograph of a part of a cellular flame front (Markstein [30]). Note examples of cell coalescence, cell propagation, and oscillatory behaviour.

experiments gave L* = 13.9d/(3Ca)1^3, if Huppert's scaling arguments were used, where Ca — Qn/da, and d is the layer thickness equal to (3QfJ,/pg sin a ) 1 / 3 . The best fit to the data is actually given by L* = 9.2d/(3Ca)0A5, which suggests that a scaling on Ca~1^2 is probably closer to the true behaviour. A photograph of the instability by Johnson et al. [33], is shown in Fig. 13 using a fluorescence technique to determine the fluid depth. This method had been used previously by Goodwin [34] in the case of release of a constant volume of fluid. Troian, Herbolzheimer, Safran, and Joanny [35] have solved the linear-stability problem for this flow noting that the instability is due to the development of a raised nose at the front of the current which then becomes unstable under the ac-

18

T.

Maxworthy

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