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EARTH SCIENCES IN THE 21ST CENTURY

PERSPECTIVES IN MAGNETOHYDRODYNAMICS RESEARCH No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

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EARTH SCIENCES IN THE 21ST CENTURY

PERSPECTIVES IN MAGNETOHYDRODYNAMICS RESEARCH

VICTOR G. REYES

EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. Library of Congress Cataloging-in-Publication Data Perspectives in magnetohydrodynamics research / editor, Victor G. Reyes. p. cm. Includes index. ISBN 978-1-62100-128-7 (eBook) 1. Magnetohydrodynamics. I. Reyes, Victor G. QC718.5.M36P465 2011 538'.6--dc22 2010047794

New York

CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Index

vii MHD Free Convection in a Porous Medium Bounded by a Long Vertical Wavy Wall and a Parallel Flat Wall A.K. Tiwari Immersed Boundary Method: The Existence of Approximate Solution in Two-Dimensional Case Ling Rao and Hongquan Chen Transient Hydromagnetic Natural Convection between Two Vertical Walls Heated/Cooled Asymmetrically R.K. Singh and A.K. Singh Effect of Suction/ Injection on MHD Flat Plate Thermometer Anand Kumar and A.K. Singh Flute and Ballooning Modes in the Inner Magnetosphere of the Earth: Stability and Influence of the Ionospheric Conductivity O.K. Cheremnykh and A.S. Parnowski

1

31

57

75

87 139

PREFACE This book presents and discusses current research in the study of magnetohydrodynamics. Topics discussed include MHD free convection in a porous medium bounded by a vertical wavy wall and a parallel flat wall; the immersed boundary method; transient hydromagnetic natural convection between two vertical walls heated/cooled asymmetrically; the effect of suction/injection on MHD flat plate thermometer and flute and ballooning modes in the inner magnetosphere of the earth. Chapter 1 - This paper presents MHD free convective flow of an incompressible and electrically conducting fluid through a porous medium bounded by a long vertical wavy wall and a parallel flat wall. The shape of wavy wall is assumed to follow a profile of cosine curve and kept at constant heat flux while parallel flat wall at constant temperature. Governing systems of nonlinear partial differential equations in non-dimensional form are linearised by using perturbation method in terms of amplitude and analytical solution for velocity and temperature fields has been obtained for mean as well as perturbed part in terms of various parameters. A numerical study of the analytical solution is performed with respect to the realistic fluid air in order to illustrate the interactive influences of governing parameters on the temperature and velocity fields as well as skin friction and Nusselt number. In the case of maximum waviness (positive and negative), reverse phenomena occurs near flat wall in the parallel flow. It is observed that the parallel flow through channel at zero waviness is more than maximum waviness (positive and negative) while same trend occurs for perpendicular flow in reverse direction. Examination of Nusselt number shows that in the presence and absence of heat source, the heat flows from porous region towards the walls but in the presence of sink, the heat flows from the walls into the porous region.

viii

Victor G. Reyes

Chapter 2 - This paper deals with the two-dimensional Navier-Stokes equations in which the source term involves a Dirac delta function and describes the elastic reaction of the immersed boundary. The authors analyze the existence of the approximate solution with Dirac delta function approximated by differentiable function. The authors obtain the result via the Banach Fixed Point Theorem and the properties of the solutions to the NavierStokes equations of viscous incompressible fluids with periodic boundary conditions. Chapter 3 - The attentive work analyses a closed form solution for the transient free convective flow of a viscous incompressible and electrically conducting fluid between two vertical walls as a result of asymmetric heating or cooling of the walls in the presence of a magnetic field applied perpendicular to the walls. The convection process between the walls occurs due to a change in the temperature of the walls to that of the temperature of fluid. The solution for velocity and temperature fields are derived by the Laplace transform technique. The numerical values procure from the analytical solution show that the flow is initially in downward directions near the cooled wall for negative values of the buoyancy force distribution parameter. The effects of Hartman number, buoyancy force distribution parameter and Prandtl number on velocity profiles and skin friction are shown graphically and tabular form. Chapter 4 - The effect of suction/injection on a steady two-dimensional electrically conducting and viscous incompressible fluid owing to the flat plate thermometer is numerically analyzed. The flow is considered at small magnetic Reynolds number so that induced magnetic field is taken to be negligible. The non-linear coupled boundary layer equations are transferred to non-linear ordinary differential equations using the similarity transformation and resulting equations are solved by shooting method with fourth order Runge-Kutta algorithm. Numerical results for the dimensionless velocity and temperature profiles and skin friction coefficient are presented by graphs and table for various values of magnetic and suction/injection parameters. It is found that the effect of injection is to increase the temperature of the flat plate thermometer while suction has opposite influence. Chapter 5 - In this article the authors represent a survey of the present state and analytical methods of the theory of transversally small-scale standing MHD perturbations in the inner magnetosphere of the Earth, as well as the authors’ own views on this matter. The authors restrict their consideration by two important types of such perturbations: flute and ballooning modes.

Preface

ix

In the most general case arbitrary three-dimensional transversally smallscale standing MHD perturbations in ideal plasmas with nested magnetic surfaces are described by a pair of Dewar-Glasser equations. In some papers such equations are derived from the MHD equations by application of differential operators. This casts certain doubts upon the correctness of the resulting spectra of perturbations. The author choose a different path, using just the condition of transversally small-scaleness and longitudinal elongation of perturbations, which, nevertheless, appeared to be sufficient to derive mentioned equations. In addition the author used ideal MHD approximation, neglected the convection and considered the equilibrium to be static. Obtained equations were applied to a dipolar magnetic configuration, approximately representing the inner magnetosphere of the Earth. This made the equations much simpler and after introducing dimensionless variables they reduced to linear homogeneous ordinary differential equations of second order. When hydrodynamic pressure is considerable flute and ballooning perturbations are generated. The most unstable of them are the perturbations with a transversal to the magnetic surfaces polarization of the magnetic field. Obtained equations were supplemented with ionospheric boundary conditions derived in the same approximation. Considered perturbations appeared to be affected only by integral Pedersen conductivity of the ionosphere, which was approximated by a thin spherical layer. Moreover, when Pedersen conductivity of the ionosphere is finite flute modes can appear in the magnetosphere, determining its stability in this case. Using a modified energetic principle the author derived the corresponding stability criterion, which sets stronger restrictions on the stability than well known Gold criterion. Versions of these chapters were also published in Journal of Magnetohydrodynamics, Plasma and Space Research, Volume 14, Numbers 1-4, published by Nova Science Publishers, Inc. They were submitted for appropriate modifications in an effort to encourage wider dissemination of research.

In: Perspectives in Magnetohydrodynamics … ISBN: 978-1-61209-087-0 © 2011 Nova Science Publishers, Inc. Editor: Victor G. Reyes

Chapter 1

MHD FREE CONVECTION IN A POROUS MEDIUM BOUNDED BY A LONG VERTICAL WAVY WALL AND A PARALLEL FLAT WALL A.K. Tiwari* Department of Mathematics, Doon Institute of Engineering & Technology, Rishikesh, India

Abstract This paper presents MHD free convective flow of an incompressible and electrically conducting fluid through a porous medium bounded by a long vertical wavy wall and a parallel flat wall. The shape of wavy wall is assumed to follow a profile of cosine curve and kept at constant heat flux while parallel flat wall at constant temperature. Governing systems of nonlinear partial differential equations in non-dimensional form are linearised by using perturbation method in terms of amplitude and analytical solution for velocity and temperature fields has been obtained for mean as well as perturbed part in terms of various parameters. A numerical study of the analytical solution is performed with respect to the realistic fluid air in order to illustrate the interactive influences of governing parameters on the temperature and velocity fields as well as skin friction and Nusselt number. In the case of maximum waviness (positive and negative), reverse *

E-mail address: [email protected]

2

A.K. Tiwari phenomena occurs near flat wall in the parallel flow. It is observed that the parallel flow through channel at zero waviness is more than maximum waviness (positive and negative) while same trend occurs for perpendicular flow in reverse direction. Examination of Nusselt number shows that in the presence and absence of heat source, the heat flows from porous region towards the walls but in the presence of sink, the heat flows from the walls into the porous region.

Keywords: MHD free convection, porous media, wavy wall, perturbation method.

1. Introduction In recent years, the studies of MHD free convective flow have attracted many research workers in view of not only its own interest but also due to the applications in astrophysics, geographic and technology. Laminar free convection at vertical wall is of interest in many applications such cooling of nuclear reactors, heat exchangers, solar energy collectors, crystal growth and thermal engineering, among others. If the flow field involves an electrically conducting fluid under the influence of an external magnetic field, we then have the combined effects of viscous, buoyancy and magnetic forces on the flow. As the magnetic field and buoyancy forces can be controlled externally by changing the applied magnetic field and ramped temperature, the investigations of the effects of these forces on the flow and heat transfer characteristics of real fluids have been a subject of great research interest. Fluid flow control through external magnetic forces also find applications in the design of magnetohydrodynamic (MHD) generators and MHD devices used in various industries. It is therefore important to study the features of transport phenomena in MHD flows under practically important physical conditions for both unsteady and steady state cases. For instance, Hady et. al. [1] has studied the effect of heat generation or absorption on a free convection boundary layer flow along a vertical wavy plates embedded in electrically conducting fluid saturated porous media. The solutions of hydromagnetic free convective flows wavy channels have been discussed by Rao et. al. [2] and Tashtoush and Al-Odat[3] under different physical conditions. Transport processes as a result of free convection inside wavy–walled channel have not been investigated widely due to geometric complexity. Literatures related to this topic are not as rich as channels with flat walls. Free convection heat transfer phenomenon in a porous medium bounded by

MHD Free Convection in a Porous Medium Bounded…

3

geometries of irregular shape has attracted the attention of engineers and scientists from many varying disciplines such as chemical, civil, environmental, mechanical, aerospace, nuclear engineering, applied mathematics, geothermal physics and food science. Phenomena concerned with it include the spreading of pollutants, water movement in reservoirs, thermal insulation engineering, building science and convection in the earth’s crust etc. Geometrical complexity of such type of system affects largely the flow pattern and depends on many parameters like amplitude, wave length, phase angle, inter wall spacing etc. Each of the parameter significantly affects the hydrodynamic and thermal behavior of fluid inside it. These configurations are not idealities and its effects on flow phenomenon have motivated many researchers to perform experimental and analytical works. Ostrach [4] analyzed laminar natural convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperature. Vajravelu and Shastri [5] solved free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall. Shankar and Sinha [6] presented the flow generated in a viscous fluid by the impulsive motion of a wavy wall using perturbation method about the known solution for a straight wall. Lekoudis et al. [7] analyzed compressible viscous flows past wavy walls without restricting the mean flow to be linear in the disturbance layer. Their results agree more closely with experimental data than the results obtained by using Lighthill’s theory, which restricts the mean flow to be linear in the disturbance layer. The effect of small amplitude wall roughness on the minimum critical Reynolds number of a laminar boundary layer is studied by Lessen and Gangwani [8] under the assumptions normally employed in parallel flow stability problems. By using either analytical or numerical approaches, Singh and Gholami [9], Rees and Pop [10], and Kumar [11] have solved the natural convection problem in a fluid-saturated porous media with uniform heat flux condition. The fundamental importance of convective flow in porous media has been ascertained in the recent books by Ingham and Pop [12] and Neild and Bejan [13]. Recently, several studies by Rathish Kumar et al. [14, 15], Murthy et al. [16], Kumar and Shalini [17], Misirlioglou [18] and Sultana and Hyder [19] have been reported that were concerned with the natural convection heat transfer in wavy vertical porous enclosures. The main purpose of the present paper is to examine the MHD free convective heat transfer and fluid flow in a porous medium bounded by a long vertical wavy wall and a parallel flat wall. The wavy wall is kept at constant heat flux while parallel flat wall maintained at constant temperature. The

4

A.K. Tiwari

solution of governing equations has been obtained using perturbation technique described by Nayfeh [20] in terms of the physical parameters appearing in the governing equations. Results are presented corresponding to the velocity and temperature fields as well as skin friction and Nusselt number for different values of the governing parameters.

2. Mathematical Analysis Let us consider the two-dimensional laminar MHD free convective heat transfer flow of an incompressible and electrically conducting fluid through a porous medium bounded by a vertical wavy wall and a parallel flat wall which are maintained at constant heat flux and constant wall temperature respectively. The properties of the fluid are assumed to be constant and isotropic except the density variation in the buoyancy term in the momentum equation. The fluid and porous medium are in the local thermodynamic equilibrium. The wavy surface of the wall is described in the function form as . where, the origin of the co-ordinate system is placed at the leading edge of the vertical surface, while the flat wall which is parallel to wavy wall is taken at the distance . The fluid oncoming to the channel is still quiescent and both the fluid and flat wall have constant temperature

. A uniform magnetic

field of magnetic field strength is applied perpendicular to the channel length. In this problem, the viscous and Darcy dissipation effects are neglected and the volumetric heat source / sink is constant in the energy equation. If we define the dimensionless quantities as ,

,

,

,

,

,

5

MHD Free Convection in a Porous Medium Bounded…

,

,

,

,

, (1)

the dimensionless equations, governing the conservation of mass, momentum and energy in the channel are obtained as follows (Ingham and Pop [12]): (2)

,

(3)

,

(4)

,

(5) where, . In the dimensionless form, the boundary conditions can be written as , (6)

.

All the symbols used in the above equations are defined in the nomenclature. Under the perturbation technique, let us consider the velocity and temperature fields as , ,

(7)

6

A.K. Tiwari

where, first order quantities or perturbed parts are very small compared with the zeroth order quantities or mean parts. Using equation (7), the equations (2) - (5) reduce to the following form for zeroth order quantities (8)

. where,

is the constant pressure gradient term and is taken

equal to zero by Ostrach [4]. and for the first order quantities (9)

,

,

(10)

,

(11)

.

(12)

With the help of (7), the boundary conditions (6) can be converted into the following two parts: , (13)

, , .

(14)

7

MHD Free Convection in a Porous Medium Bounded…

2.a. Solution of Mean Part The zeroth order solutions are obtained from (8) with the help of boundary conditions (13) in the following form: (15) (16)

. The symbols expression for

used as a constant are given in appendix. The are called the zero-order solutions or mean parts.

2.b. Solution Procedure for Perturbed Part To find the solution of first order quantities from (9) - (12), let us introduce the stream function defined by (17)

.

It is obviously clear that continuity equation (9) is satisfied identically with the help of (17). Using (17) into (10) - (14) and eliminating the non-dimensional pressure , we have

(18) (19)

. Assuming , .

(20)

8

A.K. Tiwari

and using in (18) and (19), we get

,

(21) (22)

, where, a prime denotes differentiation with respect to y. Boundary conditions (14) become ,

(23) For small values of

(

), we can take (24)

Using (24) into (21) - (22), we have obtained a set of ordinary differential equations of fourth order in term of and second order in and they are not reported here for the sake of brevity. The solutions of these ordinary differential equations with their appropriate boundary conditions obtained from (23) are obtained as follows: ,

,

(25)

(26)

9

MHD Free Convection in a Porous Medium Bounded…

(27)

,

(28)

,

With the help of above obtained solutions, the first order quantities given by (17) along with (20) can be put in the following form: (29a)

,

(29b)

,

(29c)

, where, ,

(30)

. The expressions for the first-order velocity temperature

and the first-order

have been obtained with the help of eqs. (25) - (28).

3. Skin Friction and Nusselt Number at the Walls The shear stress by

at any point of the fluid in non-dimensional form is given

Using above equation, the skin friction at the flat wall (y=1) are obtained as

at the wavy wall (y=0) and

10

A.K. Tiwari

(31)

,

(32)

. The Nusselt number are obtained as

at the flat wall (y=1) in the dimensionless form

(33)

.

4. Results and Discussion The expressions for mean part

and perturbed part

have

been obtained in terms of physical parameters . The perturbed part of the solution is the contribution from the waviness of the wall. Involvement of many parameters in a study not only makes the computational works a formidable task but it also makes it difficult to incorporate a systematic parametric presentation. Thus we set Pr=0.71 corresponding to realistic fluid air, , and focus our attention on numerical computations for different values of the following table: Curves Parameter G M Da

as given in

1

2

3

4

5

6

7

8

9

10

11

12

50 5.0

50 5.0

50 5.0

50 5.0

50 5.0

50 5.0

50 2.0

50 2.0

50 2.0

100 5.0

100 5.0

100 5.0

-5

0

5

-5

0

5

-5

0

5

-5

0

5

0.01

0.01

0.01

0.1

0.1

0.1

0.01

0.01

0.01

0.01

0.01

0.01

Graphical representations of mean part as well as perturbed part of the velocity and temperature profiles of air have shown in the figs.1-5 for above data. Figure1 describes the behavior of the mean part of the velocity between vertical walls. Close examination of it reveals that in the presence of source, the

MHD Free Convection in a Porous Medium Bounded… 8

11

6

6

9 5

4

3

u0

2

0

1

-2

8

4

2 7

-4 0.0

0.2

0.4

0.6

0.8

1.0

y

Fig. 1. Zeroth-order velocity profiles Figure 1. Zeroth-order velocity profiles.

velocity profiles take parabolic shape but reverse shape in the presence of sink. The point of maxima on the curves get shifted away from parallel flat wall (y=1) as magnetic field parameter decreases and Darcy number increases. Although in the absence of source/sink, the velocity profiles are almost flat while assuming parabolic shapes due to overshooting in velocity near the flat wall (y=0) as value of magnetic field parameter decreases and Darcy number increases. It is clear from curves (3, 9) that the velocity decreases with the magnetic field parameter M for but reverse flow occurs for (see curves 1, 7). In the absence of source/sink , the velocity near the flat wall (y=0) decreases with an increase in magnetic field parameter M , a result physically equivalent to saying that fluid velocity depends fluid density inversely but near the flat wall (y=1) the velocity is approximately same as magnetic field parameter M increases. On the examination of curves (3, 6)and (2, 5), one can reveal that the velocity is increasing function of Darcy number Da in the presence and absence of source/sink while in the presence of sink, it is increasing function of Darcy number Da in the opposite direction shown by the curves (1, 4).

12

A.K. Tiwari

4

=5 3

2

0

=0 1

0

=-5

-1

-2 0.0

0.2

0.4

0.6

0.8

1.0

y Fig. 2. Zeroth - order temperature profiles Figure 2. Zeroth - order temperature profiles. The behavior of the mean temperature

is shown in fig. 2. From which,

it is clear that in the absence of heat source , the mean temperature is a linearly decreasing function of y (curves 2, 5, 8) while in the presence of heat source , the mean temperature is increasing from its value at the wall y=0 to a maximum temperature at around y=0.1 and then decreasing steadily its value upto y=1 (curves 3, 6, 9). In the presence of heat sink , the behavior of the mean temperature is exact opposite of that observed in the presence of source (curves 1, 4, 7). Further, we observed that there is no significant effect of on the mean temperature for all values of

.

MHD Free Convection in a Porous Medium Bounded… 0.04

8

4 0.02

3

6

7 1

u1

0.00

-0.02

9

5

2

2

8

4

3

7 1

5 9 -0.04

6 -0.06

0.0

0.2

0.4

0.6

0.8

1.0

y

(a) 3

5

4

7

u1

3

6

8

2

1

2

1

4

5 0 0.0

0.2

0.4

y (b)

Figure 3. Continued on next page.

0.6

0.8

1.0

13

14

A.K. Tiwari

0.06

6

0.04

9 5 3

8

1

4

7

u1

0.02

2

0.00

-0.02

9

2

7

8 6

5 3

0.8

1.0

1 4 -0.04 0.0

0.2

0.4

0.6

y (c)

Figure 3. First- order velocity component for (a) , (b) and (c). 6

2

7

0.00

1 4 10

-0.02

v1

5

8

-0.01

-0.03

9 -0.04

-0.05

-0.06 0.0

3

0.2

0.4

y (a)

Figure 4. Continued on next page.

0.6

0.8

1.0

MHD Free Convection in a Porous Medium Bounded…

15

4 8.0x10

-5

4.0x10

-5

7 1 2

v1

0.0 -4.0x10

-5

-8.0x10

-5

-1.2x10

-4

-1.6x10

-4

-2.0x10

-4

-2.4x10

-4

8 3

5 9

6

0.0

0.2

0.4

0.6

0.8

1.0

y (b) 0.06

0.05

3

v1

0.04

9

0.03

0.02

10

4 8

7 0.00 0.0

1

5

0.01

0.2

0.4

6 0.6

2 0.8

1.0

y (c)

Figure 4.First- order velocity component .

for (a)

, (b)

and (c)

16

A.K. Tiwari

=5

0.008

0.004

=0 0.000

-0.004

=-5

-0.008

0.0

0.2

0.4

0.6

0.8

1.0

y (a)

2,5,8,11 0.00

1,3 4,6,10,12

-0.02

v1

-0.04

-0.06

-0.08

7,8

-0.10 0.0

0.2

0.4

y (b)

Figure 5. Continued on next page.

0.6

0.8

1.0

MHD Free Convection in a Porous Medium Bounded…

17

= -5

0.008

0.004

=0 0.000

-0.004

=5

-0.008

0.0

0.2

0.4

0.6

0.8

1.0

y (c)

Figure 5. First-order temperature profiles

for (a)

, (b)

and (c)

.

4.a. Presentation of First Order Solution Figures 3(a, b, c), 4(a, b, c) and 5(a, b, c) represent the perturbed part (firstorder solution) of the velocity components and temperature respectively in the channel for three cases of the waviness of the wavy wall (y=0) and they are as follows: (i) maximum positive at , (ii) zero at and (iii) maximum negative at

. The description of first-

order solution at different types of waviness as follows; The effect of maximum waviness on the first-order velocity component is shown in figure 3(a) and it indicates that in the presence of heat sink, the velocity component increases near the wavy wall and then by decreasing becomes zero at y=0.70 approximately and thereafter reverse flow occurs. We observed from the curves (1, 4) that as increase in the Darcy number Da increases the velocity component in the two third of the channel and this behavior is reversed in the presence or absence of heat source as shown in the

18

A.K. Tiwari

curves (3, 6) or (2, 5) respectively. This behavior is reversed in the other one third of the channel. On analysis the curves (1, 7) for

, we found that as

increase in the magnetic field parameter M, the velocity component decreases in the two third of the channel and this behavior is reversed when (see curves 3, 9 and 2, 8). However in the other one third of the channel this behavior of the velocity component

with Mis reversed.

Figure 3(b), showing perturbed velocity component for zero waviness, indicates that it increases in present of heat source (curves 2, 3 and 5, 6). The effect of Darcy number is also to increase it (curves 1, 7 and 2, 8). For maximum negative waviness (fig. 3c), we found that reversed effects are observed for source and sink parameter. The effects of the parameters which appear in it are reversed of all the results found for maximum positive waviness. It is observed from fig. 4(a) that the velocity component is enhanced by an increase in the Darcy number Da in the reverse direction by the curves (3, 6) and (2, 5) in the presence of heat source and absence of heat source/sink respectively while in the presence of heat sink, it increases positively (see curves 1 and 4). In the presence of heat source and absence of heat source/sink, the velocity component is an increasing function of Da and decreasing function of M by curves ((3, 6), (2, 5)) and ((3, 9), (2, 8)) respectively in the opposite direction while in the presence of heat sink, the velocity component is also an increasing function of Da and decreasing function of M (see curves (1, 4) and (1, 7)). It is observed from the curves (1, 4, 7, and 10) of fig. 4(b) that for the velocity component

,

is an increasing function of G, Da and decreasing

function of M while for , the reverse effect can be seen by the curves (3, 6, 9, and 12) and (2, 5, 8, and 11). Close observation of figure 4(c) shows that the behavior of velocity component in case of maximum negative waviness is almost reverse to that of positive maximum waviness The behavior of the perturbed temperature with changes in is shown in figs. 5(a, b, c) according to three cases of waviness of the wavy wall. Figure 5(a) showing the perturbed part of temperature for which indicates that in the absence of heat source/sink, the perturbed temperature is zero (curves 2, 5, 8) while in the presence of heat source, it is a linearly decreasing function of y (curves 3, 6, 9). In the presence of heat sink, the behavior of the perturbed temperature is exactly opposite of that observed in the presence of source

MHD Free Convection in a Porous Medium Bounded… (curves 1, 4, 7).The parameters

19

have also negligible effect on

the perturbed temperature for all values of . For zero waviness, we observed from fig. 5(b) that the perturbed temperature is almost same up to y= 0.8 of the channel and then increases in remaining part of the channel. The behaviors of the perturbed temperature at maximum negative waviness are shown in fig. 5(c). The effect of source and sink are just reversed corresponding to maximum positive waviness but for absence of source/sink, both cases have same effect. The curves of skin friction are shown in figs. 6(a, b) only for maximum positive waviness

and zero waviness

because the

perturbed part is much smaller than mean part and curves for maximum negative waviness almost coincide with fig. 6(a). The curves in these figures are drawn based on the following data: Curves Parameter G M Da

1

2

3

4

50 5.0 0.01

50 2.0 0.01

100 5.0 0.01

100 5.0 0.1

It is observed that the skin friction at the channel walls is a linear function of the heat source parameter and the skin friction at the wall y=0 increases with the heat source parameter while the reverse is true at the other wall y=1 in both types of channel walls. The skin friction is an increasing function of the Grashof number G and Darcy number Da by the curves (1, 3) and (3, 4) at the wall y=0 while at y=1 decreases with an increase in G and Da. It is clear from curves 1 and 2 that the skin friction is unaffected by magnetic field parameter M in both cases. On comparing the skin friction for the both type of waviness of the wall at y=0 and y=1, it is observed that it is greater in case when channel has maximum positive waviness than case when channel has zero waviness. The temperature profiles of wavy wall are shown in fig.7. It is observed that the temperature of the wavy wall is a linear function of phase for all values of M, G and it becomes oscillatory when the value of Darcy number Da increases in the presence of source

and sink

. In the

absence of source/sink , the wavy wall temperature linearly varies as function of M, G and Da as the perturbed part is much smaller than mean part.

20

A.K. Tiwari 50

40

30

,

20 10

,

0

-10

-20 -6

-4

-2

0

2

4

6

(a)

50

40

30

,

20 10

,

0 -10

-20 -6

-4

-2

0

2

4

6

(b)

Figure 6. Total skin- friction at the walls for (a)

and (b)

.

MHD Free Convection in a Porous Medium Bounded…

21

3,9

6

3

2

2,5,8

1

0

4

1,7

-1

-2

0

2

4

6

8

10

12

14

16

y Fig. 7.Temperature of the wavy wall

Figure 7. Temperature of the wavy wall.

Lastly in table 1, the values of Nusselt number at the channel wall (y=1) are given only for maximum positive waviness and zero waviness for different values of M, G and Da. The Nusselt numbers for maximum negative waviness are approximately same as for maximum positive waviness. It can see from this table that the Nusselt number at the flat wall (y=1) in the both type of waviness decreases with and this decrease being least significant for Dathan G and most significant for for maximum positive waviness

. The effects of

on decreasing the value of M, the

Nusselt number has approximately same value for while for zero waviness

on the Nusselt number that and decreased value for

, the Nusselt number is exactly same

for all values of . Also, when the heat source/sink parameter takes positive increasing values, the Nusselt number at the flat wall (y=1) in the both type of waviness becomes negative, which means physically that heat flows from porous region towards the walls. However, when the heat source/sink parameter takes negative increasing values, the Nusselt number at the flat wall (y=1) in the both type of waviness is positive, which indicates physically that heat flows from the walls into the porous region.

Table 1. Numerical values of dimensionless Nusselt number for Pr=0.71 Values of Nusselt number at the flat wall (y=1) For

For

G=50 M=5.0 Da=0.01

G=50 M=2.0 Da=0.01

G=100 M=5.0 Da=0.01

G=100 M=5.0 Da=0.02

G=50 M=5.0 Da=0.01

G=50 M=2.0 Da=0.01

G=100 M=5.0 Da=0.01

G=100 M=5.0 Da=0.02

-5

3.97996

3.96992

3.96992

3.94984

3.36137

3.36137

2.72274

1.43997

-4

2.98555

2.97915

2.97914

2.96630

2.59119

2.59119

2.18238

1.36204

-3

1.99038

1.98677

1.98677

1.97954

1.77005

1.77005

1.54010

1.07933

-2

0.99439

0.99278

0.99278

0.98957

0.89795

0.89795

0.79591

0.59184

-1

-0.00240

-0.00280

-0.00280

-0.00360

-0.02509

-0.02509

-0.05019

-0.1004

0

-0.99999

-0.99999

-0.99999

-0.99999

-0.99910

-0.99910

-0.99820

-0.9974

1

-1.99840

-1.99880

-1.99880

-1.99960

-2.02406

-2.02406

-2.04813

-2.0992

2

-2.99760

-2.99921

-2.99921

-3.00242

-3.09998

-3.09998

-3.19996

-3.4058

3

-3.99761

-4.00113

-4.00123

-4.00845

-4.22685

-4.22685

-4.45371

-4.9172

4

-4.99842

-5.00485

-5.00485

-5.01770

-5.40468

-5.40468

-5.80937

-6.6333

5

-6.00004

-6.01008

-6.01008

-6.03016

-6.63347

-6.63347

-7.26694

-8.5542

MHD Free Convection in a Porous Medium Bounded…

23

Conclusion The two-dimensional MHD free convective heat transfer flow of an incompressible and electrically conducting fluid through a porous medium of a viscous and incompressible fluid through a porous medium bounded by a vertical wavy wall and a parallel flat wall which are maintained at constant heat flux and constant wall temperature respectively has been studied. The governing equations in non-dimensional form are linearised by using perturbation technique. Analytical solution for mean part as well as perturbed part have been obtained and using them detailed analysis of velocity and temperature fields are presented in graphical form for various values of the parameters. We have also discussed about the surface skin-friction coefficient as well as the Nusselt numbers at the flat wall (y=1) and temperature of the wavy wall (y=0). The important fact of this study is a comparison among three type of waviness of wall. It is observed that the parallel flow through channel at zero waviness is more than maximum waviness (positive and negative) while same trend occurs for perpendicular flow in reverse direction.

Nomenclature cp

-specific heat at constant pressure

Nu P

–distance between both walls -Darcy number -magnetic field parameter –Grashof number or free convection parameter -acceleration due to gravity -permeability of the porous medium –Nusselt number -fluid pressure

Pr q Q T

-dimensionless fluid pressure -Prandtl number – rate of heat transfer –source/sink parameter –fluid temperature

Da M G g

K

-temperature of the flat wall

24

A.K. Tiwari

u, v -dimensionless velocity components along x- and y-axis, respectively u , -velocity components along x- and y-axis, respectively x, y -dimensionless Cartesian coordinates , y - Cartesian coordinates

Greek Symbols

-dimensionless source/sink parameter -volumetric coefficient of thermal expansion - electrical conductivity - constant magnetic field flux density -dimensionless amplitude parameter -amplitude parameter

-dimensionless fluid temperature -thermal conductivity -dimensionless frequency parameter -frequency parameter -dynamic viscosity -kinematic viscosity -fluid density -skin friction or dimensionless shear stress –dimensionless stream function

Subscript

-zero-order quantity –first-order quantity s p

–static fluid -wavy wall –flat wall

25

MHD Free Convection in a Porous Medium Bounded…

Appendix

,

,

,

,

, , , , , ,

26

A.K. Tiwari

„

,

,

, , , ,

,

27

MHD Free Convection in a Porous Medium Bounded…

, , , ,

,

,

,

,

,

,

28

A.K. Tiwari

References [1] Rao, D. R. V. P., Krishna, D. A. and Debnath L., Free convection in hydromagnetic flows in a vertical wavy channel, International Journal of Engineering Science, Vol. 21 (9), pp. 1025-1039, 1983. [2] Hady, F. M., Mohamed R. A. and Mahdy, A., MHD free convection flow along a vertical wavy surface with heat generation or absorption effect, Int. Comm. in Heat and Mass Transfer, Vol. 33 (10), pp. 1253-1263, 2006. [3] Tashtoush B. and Al-Odat, Magnetic field effect on heat and fluid flow over a wavy surface with a variable heat flux, Journal of Magnetism and Magnetic Materials, Vol. 268 ( 3), pp. 357-363, 2004. [4] Ostrach, S., Laminar natural convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperature, N.A.S.A. Tech. Note No. 2863, 1952. [5] Vajravelu, K. and Sastri K. S., Free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall, J. Fluid Mech., Vol.86 (2), pp.365-383, 1976. [6] Shankar P. N. and Sinha U. N., The Rayleigh problems for a wavy wall, J. Fluid Mech., Vol.77, pp.243-256, 1976. [7] Lekoudis, S. G., Nayfeh, A. H. and Saric, W. S., Compressible boundary layers over wavy walls, Phys. Fluids, Vol.19, pp.514-519, 1976. [8] Lessen, M. and Gangwani, S. T., Effect of small amplitude wall waviness upon the stability of the laminar boundary layer, Phys. Fluids, Vol.19, pp.510-513, 1976. [9] Singh, A. K. and Gholami, H. R., Unsteady free convective flow through a porous medium bounded by an infinite vertical porous plate with constant heat flux, Rev. Roum. Sci. Techn. Mec. Appl., Vol.35, pp.337382, 1990. [10] Rees, D. A. and Pop, I., Free convection induced by a vertical wavy surface with uniform heat flux in a porous medium, ASME J. Heat Transfer, Vol.117, pp.547-550, 1995. [11] Kumar, B. V. R., A study of free convection induced by a vertical wavy wall with heat flux in a porous enclosure, Numer. Heat Transfer, Vol.37 (part A), pp.493-510, 2000. [12] Ingham, D. B. and Pop, I., Transport Phenomena in Porous Media, Pergamon Press, Oxford, 2002. [13] Nield, D. A. and Bejan, A., Convection in Porous Media, Springer, New York, 2006.

MHD Free Convection in a Porous Medium Bounded…

29

[14] Kumar, B. V. R., Singh, P. and Murthy, P. V. S. N., Effect of surface undulations on natural convection in a porous square cavity, ASME J. Heat Transfer, Vol.119, pp.848-851, 1997. [15] Kumar, B. V. R., Murthy, P. V. S. N. and Singh, P., Free convection heat transfer from an isothermal wavy surface in a porous enclosure, Int. J. Numer. Meth. Fluids, Vol.28, pp.633-661, 1998. [16] Murthy, P. V. S. N., Kumar, B. V. R. and Singh, P., Natural convection heat transfer from a horizontal wavy surface in a porous enclosure, Numar. Heat Transfer, Vol.31 (Part A), pp.207-221, 1997. [17] Kumar, B. V. R. and Shalini, Free convection in a non-Darcian wavy porous enclosure, Int. J. Energy Sci., Vol.41, pp.1827-1848, 2003. [18] Misirlioglu, A., Baytes, A. C. and Pop, I., Free convection in a wavy cavity filled with a porous medium, Int. J. Heat and Mass Transfer, Vol.48, pp.1840-1850, 2005. [19] Sultana, Z. and Hyder, M. N., Non-Darcy free convection inside a wavy enclosure, Int. Communications in Heat and Mass Transfer, Vol.34 (2), pp.136-146, 2007. [20] Nayfeh, A. H., Perturbation Methods, Willey-Interscience, New York, 1973.

Perspectives in Magnetohydrodynamics ISBN: 978-1-61209-087-0 c 2011 Nova Science Publishers, Inc. Editor: Victor G. Reyes

Chapter 2

I MMERSED B OUNDARY M ETHOD : T HE E XISTENCE OF A PPROXIMATE S OLUTION IN T WO -D IMENSIONAL C ASE Ling Rao1,2 and Hongquan Chen 1 1 College of Aerospace Engineering Nanjing University of Aeronautics and Astronautics Nanjing 210016, China 2 Department of Applied Mathematics Nanjing University of Science and Technology Nanjing 210094, China

Abstract This paper deals with the two-dimensional Navier-Stokes equations in which the source term involves a Dirac delta function and describes the elastic reaction of the immersed boundary. We analyze the existence of the approximate solution with Dirac delta function approximated by differentiable function. We obtain the result via the Banach Fixed Point Theorem and the properties of the solutions to the Navier-Stokes equations of viscous incompressible fluids with periodic boundary conditions.

Keywords: Navier-Stokes Equations; Immersed boundary method; Nonlinear ordinary differential equations. AMS subject classifications: 35K10, 46N20, 34A12.

32

1.

Ling Rao and Hongquan Chen

Introduction

Problems involving the interaction between a fluid flow and elastic interfaces may appear in several branches of science such as engineering, physics, biology, and medicine. Regardless the field, as a rule, they share in common a high degree of complexity, often displaying intricate geometry or time-dependent elastic properties, turning the problem into a real challenge for applied scientists, from both the mathematical modeling and the numerical simulation points of view. In the early 70s, Peskin [2, 8] introduced a mathematical model and a computational method to study the flow patterns of the blood around the heart valves. Through years, Peskin’s immersed boundary (IB) method was developed for the computer simulation of general problems [9, 10, 11, 12] involving a transient incompressible viscous fluid containing an immersed elastic interface, which may have time-dependent geometry or elastic properties, or both. The IB method is at the same time a mathematical formulation and a numerical scheme. The mathematical formulation is based on the use of Eulerian variables to describe the dynamic of fluid and of Lagrangean variables along the moving structure. The force exerted by the structure on the fluid is taken into account by means of a Dirac delta function constructed according to certain principles. The main idea is to use a regular Eulerian mesh for the fluid dynamics simulation, coupled with a Lagrangian representation of the immersed boundary. The advantage of this method is that the fluid domain can have a simple shape, so that structured grids can be used. The Lagrangian mesh is independent of the Eulerian mesh. The interaction between the fluid and the immersed boundary is modeled using a well-chosen discrete approximation to the Dirac delta function. Although the immersed boundary method is a particularly effective approach in scientific computation, but very little theoretical analysis has been performed on either the underlying model equations or numerical methods. Daniele Boffi [3] gave a variational formulation of the problem and provided a suitable modification of the IB method which made use of finite elements method. Because the source term in the Navier-Stokes equations involves a Dirac delta function, the problem is highly nonlinear and presents several difficulties related with the lacking of regularity of the solution of the Navier-Stokes equations due to such source term. Daniele Boffi analyzed the existence of the solution in a very simple one-dimensional heat equation. In [13] we dealt with

Immersed Boundary Method

33

the two-dimensional heat equation. We analyzed the existence of the approximate solution with Dirac delta function approximated by differentiable function. In this paper will deal with two-dimensional Navier-Stokes equations. We analyze the existence of the approximate solution still with Dirac delta function approximated by differentiable function. We obtain the results via the Banach Fixed Point Theorem and a theorem in nonlinear ordinary differential equations in abstract space and the properties of the solutions to the Navier-Stokes equations of viscous incompressible fluids with periodic boundary conditions. In section 2 we will present the mathematical model of the problem. In order to prove the existence of the approximate solution, some properties of the Navier-Stokes equations with the space-periodic boundary conditions are reviewed in section 3. In section 4 we will introduce the corresponding variational formulation of the problem. In section 5 we will prove the existence of the approximate solution of of the problem via a fixed point argument.

2.

Problem

The authors (see [3]) considered the model problem of a viscous incompressible fluid in a two-dimensional square domain Ω containing an immersed massless elastic boundary in the form of a curve. To be more precise, for all t ∈ [0, T ], let Γt be a simple curve, the configuration of which is given in a parametric form, X(s,t), 0 ≤ s ≤ L, X(0,t) = X(L,t). The equations of motion of the system are ∂u − µ∆u + u · ∆u + ∆p = F ∂t

in Ω × (0, T ),

(1)

∆·u = 0

in Ω × (0, T ),

(2)

∀x ∈ Ω, t ∈ (0, T ),

(3)

F(x,t) =

Z L

f(s,t)δ(x − X(s,t))ds

0

∂X (s,t) = u(X(s,t),t) ∂t

∀s ∈ [0, L], t ∈ (0, T ).

(4)

Here u is the fluid velocity and p is the fluid pressure. The coefficient µ is the fluid viscosity constant. Eqs. (1) and (2) are the usual incompressible NavierStokes equations. F is the force density generated by the boundary on the fluid. The force exerted by the element of boundary on the fluid is f. The function δ

34

Ling Rao and Hongquan Chen

in the integrals is the two-dimensional Dirac delta function concentrate at the origin. Eq. (4) is equivalent to the no-slip condition that the fluid sticks to the boundary. In [3] and [13] the problem was supplemented with the following boundary and initial conditions: u(x, 0) = u0 (x)

∀x ∈ Ω,

(5)

X(s, 0) = X0(s)

∀s ∈ [0, L],

(6)

u(x,t) = 0

∀(x,t) ∈ ∂Ω × (0, T ).

(7)

When we perform the numerical simulation with the IB method the fluid domain Ω often can be chosen to have a simple big enough shape such as square or rectangle, then we can make space-periodic extension of the problem and suppose that the fluid satisfies space-periodic boundary condition. Such assumption is also useful for idealizations and needed in the following proofs. In this paper, we use the space-periodic boundary condition instead of (7). We use Ω to denote the space-period: L1 L1 L2 L2 Ω = (− , ) × (− , ). 2 2 2 2 We assume that the fluid fills the entire space R2 but with condition that u, F and p are periodic in each direction oxi , i = 1, 2, with corresponding periods Li > 0. It is sometimes useful and simpler to assume that average flow is zero, that is

1 |Ω|

Z Ω

u(x)dx = 0.

Moreover, the general case-where the volume forces and the initial condition do not average to zero-can be reduced to this case. We may refer to [4, 6] for the method. We show it as follows. First, averaging each term in (1). Using integration by parts and periodicity condition, several terms vanish, we get d dt Therefore if we set

Z Ω

u(x,t)dx =

1 Mu(t) = |Ω|

Z

Z Ω

Ω

F(x,t)dx.

u(x,t)dx

(8)

Immersed Boundary Method and

1 MF (t) = |Ω|

we obtain

thus

Z Ω

35

F(x,t)dx,

dMu(t) = MF (t), dt 1 Mu (t) = |Ω|

Z Ω

u0 (x)dx +

Z t 0

MF (t)dt.

Let uˆ = u − Mu , Fˆ = F − MF , R Iu(t) = 0t Mu(s)ds, ˆˆ u(x,t) = u(x ˆ + Iu (t),t), ˆp(x, ˆ t) = p(x + Iu (t),t), ˆF(x,t) ˆ ˆ + Iu (t),t). = F(x

(9)

ˆˆ and Fˆˆ are also periodic with period Ω and u, ˆˆ p, ˆˆ Fˆˆ have zero space Notice that u, ˆˆ we need to solve the ˆˆ pˆˆ and F, average. With u, p and F in (1)-(6) replaced by u, ˆ ˆˆ pˆ and X: following equations instead of (1)-(6) for u, ∂u − µ∆u + u · ∆u + ∆p = Fˆˆ ∂t

in Ω × (0, T ),

(10)

∆·u = 0

in Ω × (0, T ),

(11)

∀s ∈ [0, L],

(12)

∂X (s,t) = u(X(s,t) − Iu(t),t) + Mu (t) ∂t u(x, 0) = uˆˆ 0 (x), X(s, 0) = X0 (s)

∀x ∈ Ω,

(13)

∀s ∈ [0, L],

(14)

X(0,t) = X(L,t)

∀t ∈ (0, T ),

(15)

t ∈ (0, T ),

36

Ling Rao and Hongquan Chen

where for given f and u0 , uˆˆ 0 and Fˆˆ may be calculated by (3) and (9), 1 uˆˆ 0 (x) = u0 (x) − Mu (0) = u0 (x) − |Ω| ˆF(x,t) ˆ = F(x + Iu(t),t) − MF (t) =

Z L 0

−

Z

Ω

u0 (x)dx, (16)

f(s,t)δ(x + Iu(t) − X(s,t))ds

1 |Ω|

Z L

Z

f(s,t)(

0

Ω

δ(x − X(s,t))dx)ds.

ˆˆ p(= p) ˆˆ of (10)-(16) we can recover the solutions And with the solutions u(= u), u, p of (1)-(6) by (9). In order to obtain the existence of the approximate solution to equations (10)-(16), we review some properties of the Navier-Stokes equations with the space-periodic boundary conditions as follows.

3.

Review of the Mathematical Theory of the NavierStokes Equations with the Space-Periodic Boundary Conditions

We introduce some basic mathematical properties about Navier-Stoks equations with the space-periodic boundary conditions. For more details of them, the readers may refer to [4, 6]. We shall be concerned with the spaces of twodimensional vector functions. We use the notations Hm (Ω) = {H m (Ω)}2,

L2 (Ω) = {L2 (Ω)}2 ,

and we suppose that these product spaces are equipped with the usual product norm. The norm on L2 (Ω) is denoted by | · | (also denoted by k · k0 ). The norm on Hm (Ω) is denoted by k · km. (·, ·) stands for the scalar product on L2 (Ω). n We denote by Hmper (Ω), m ∈ N, the space of functions which are in Hm Loc (R ) (i.e. u|O ∈ Hm (O) for every open bound set O) and which are periodic with period Ω. For m = 0, H0per (Ω) is also denoted by L2per (Ω) and coincides simply with L2 (Ω) (the restrictions of the functions in H0per (Ω) to Ω are the whole space L2 (Ω)). For an arbitrary m ∈ N, Hmper (Ω) is a Hilbert space for the scalar product and the norm (u, v)m =

∑

Z

[α]≤m Ω

Dα u(x)Dαv(x)dx,

1

kukm = {(u, u)m } 2 ,

Immersed Boundary Method

37

∂[α] . α1 . . .∂αn ∂x√ We work with complex representation, for which we take i = −1. Then a square integrable vector field u = u(x) on Ω can be represented by the Fourier series expansion k u(x) = ∑ ck e2πi L ·x , where α = (α1, . . ., αn), αi ∈ N, [α] = α1 + . . . + αn and Dα =

k∈Zn

where

k1 k2 k = ( , ). L L1 L2

The functions in Hmper (Ω) are easily characterized by their Fourier series expansion Hmper (Ω) = {u, u =

∑

k

ck e2πi L ·x

ck = c−k ,

k∈Zn

∑

|k|2m|ck |2 < ∞},

(17)

k∈Zn 1

and the norm kukm is equivalent to the norm {∑k∈Zn (1 + |k|2m )|ck |2} 2 . We denote by V the space of smooth( C ∞) divergence-free vector fields on R2 that are periodic with period Ω. Let H be the closure of V in L2 (Ω) and let V be the closure of V in H1 (Ω). The space H is equipped with the scalar product (·, ·) induced by L2 (Ω); the space V with the scalar product Z

2

∂u ∂v

∑ ∂xi · ∂xi dx Ω

((u, v)) =

i=1

and the associated norms, denoted by 1

|u| = (u, u) 2 for u ∈ H,

1

kuk = ((v, v)) 2 for v ∈ V.

For the sake of simplicity, we restrict ourselves to the space-periodic case with vanishing space average. For the vanishing space average case, we have the additional condition Z c0 = u(x)dx = 0. Ω

Then H = {u ∈ L2per (Ω); ∇ · u = 0, V = {u ∈

H1per (Ω);

∇ · u = 0,

Z Ω

Z

Ω

˙ u(x) = 0}(, H); u(x) = 0}(, V˙ );

38

Ling Rao and Hongquan Chen

and ˙ mper (Ω)). Hmper (Ω) = {u ∈ Hmper (Ω) of type (17), c0 = 0}(, H The functions in H˙ and V˙ also can be characterized by there Fourier series expansion

∑

H˙ = {u, u =

k

ck e2πi L ·x ,

ck = c−k ,

k∈Z2 \{0}

V˙ = {u, u =

∑

k

ck e2πi L ·x ,

k · ck = 0, L

∑

|ck |2 < ∞};

k∈Z2 \{0}

k · ck = 0, L

k | |2 |ck |2 < ∞}. L k∈Z2 \{0}

k · ck = 0, L

k | |2s |ck |2 < ∞}. L k∈Z2 \{0}

ck = c−k ,

k∈Z2 \{0}

∑

For all s ∈ R we may consider the space ˙ sper (Ω), divu = 0} = Vs = {u ∈ H {u, u =

∑

k

ck e2πi L ·x ,

ck = c−k ,

k∈Z2 \{0}

∑

˙ sper (Ω). Vs1 ⊂ Vs2 for s1 ≥ s2 , V1 = V˙ and Note that, Vs is a closed subspace of H ˙ Hence, V˙ ⊂ Vs ⊂ H˙ for 0 ≤ s ≤ 1,Vs ⊂ V˙ for s ≥ 1, and Vs ⊃ H˙ for V0 = H. s < 0. It can be shown that Vs is a Hilbert space for the norm 1 k | |2s |ck |2) 2 . L k∈Z2 \{0}

kukVs = (

∑

Of particular interest are the spaces V2 and V−1. The space V−1 is the dual space of V˙ , usually denoted V 0 ; this is the space of linear continuous forms on V˙ . More generally, for all s ≥ 0,V−s is the dual of Vs. We have H ⊥ = {u ∈ L2 (Ω); u = ∇p, p ∈ H1per (Ω)}. We definite the so-called Leray projector by PL : L2 (Ω) → H, which is the orthogonal projector onto H in L2 (Ω). We definite the Stokes operator A by Au = −PL 4u

˙ 2per (Ω), ∀u ∈ D(A) = V˙ ∩ H

Immersed Boundary Method

39

where 4 is the Laplacian. And we can see that ˙ 2per (Ω). ∀u ∈ D(A) = V˙ ∩ H

−PL 4u = −4u

The Stokes operator is a positive self-adjoint operator, so we can work with fractional powers of A. A is just the mapping u=

∑

k

ck e2πi L ·x → Au =

k∈Z2 \{0}

k k | |2ck e2πi L ·x. L k∈Z2 \{0}

∑

Ar u is the operator u=

∑

k

ck e2πi L ·x → Ar u =

k∈Z2 \{0}

k k | |2r ck e2πi L ·x . L k∈Z2 \{0}

∑

It is straightforward to see that Ar maps V2s continuously onto V2s−2r (s, r ∈ R). In particular, for s ≥ 0, we have AsV2s = H˙ and so V2s = D(As ) is the domain of ˙ The norm |As u|(= kukV2 s ) is equivalent to the (unbounded) operator As in H. 2s ˙ the norm induced by H per (Ω), ckuk2s ≤ |Asu| ≤ c0 kuk2s

∀u ∈ D(As),

(18)

with positive constants c, c0 depending on L1 , L2 and s. We use some notations as follows. Let a, b be two extended real numbers, −∞ ≤ a < b ≤ +∞, and let X be a Banach space. For given α, 1 ≤ α < +∞, Lα (a, b; X) denotes the space of Lα -integrable functions from [a,b] into X, which is equipped with the Banach norm |

Z b a

k f (t)kαX dt | α . 1

The space C([a, b]; X) is the space of continuous functions from [a, b] (−∞ < a < b < ∞) into X and is equipped with the Banach norm sup k f (t)kX . t∈[a,b]

For u, v, w ∈ L1 (Ω), we set 2

b(u, v, w) =

∑

Z

i, j=1 Ω

ui (Di v j )w j dx,

40

Ling Rao and Hongquan Chen

whenever the integrals make sense. The trilinear operator b = b(u, v, w) can be extended to a continuous trilinear operator defined on V . Moreover, b(u, v, v) = 0 b(u, v, w) = −b(u, w, v)

u, v ∈ V. u, v, w ∈ V.

(19)

In the periodic case, ˙ 2per (Ω) ∩ V˙ . b(v, v, Av) = 0 v ∈ D(A) = H Now we consider the Navier-Stokes equations ∂u − µ∆u + u · ∆u + ∆p = f, ∂t ∆ · u = 0,

(20)

(21)

in

L1 L1 L2 L2 , ) × (− , ), 2 2 2 2 with µ > 0 and require that Ω = (−

L1 , L2 > 0,

u = u(x,t), p = p(x, t), and f = f(x,t) with x = (x1 , x2 ), are Li -periodic in each variable xi and

Z Ω

f(x,t)dx =

Z Ω

u(x,t)dx = 0.

Let T > 0 be given. Assume that u ∈ C2 (R2 × [0, T ]), p ∈ C1(R2 × [0, T ]) are the classical solutions of (21). For each v ∈ V, by multiplying momentum equation in (21) by v, integrating over Ω and using the Stokes formula we find that (cf. [5] Chap.III for the details) d (u, v) + µ((u, v)) + b(u, u, v) = hf, vi dt

∀ v ∈ V.

(22)

By continuity, (22) holds also for each v ∈ V . This suggests the following variational formulation of (21) ( strong solutions). For f and u0 given, ˙ f ∈ L2 (0, T ; H),

Immersed Boundary Method

41

u0 ∈ V˙ , find u satisfying

u ∈ L2 (0, T ; D(A) ∩ L∞(0, T ; V˙ )

such that ( d

(u, v) + µ((u, v)) + b(u, u, v) = hf, vi dt u(0) = u0 .

∀ v ∈ V, t ∈ (0, T ),

(23)

If f is square integrable but not in H then we can replace it by its Leray projection on H, so that f is always assumed to be in H. We may refer to [5], p.307 for the relation of equations (21) and its variational formulation (23). One can show that if u is a solution of (23) then there exists p such that (21) is satisfied in a weak sense. Lemma 1. (Existence and Uniqueness of Strong Solutions in Two Dimensions) [4, 6] (i) Assume that u0 , f, T > 0 are given and satisfy ˙ f ∈ L2 (0, T ; H).

u0 ∈ V˙ ,

Then there exists a unique solution u of (23), satisfying ui ,

∂ui ∂ui ∂2 ui , , ∈ L2 (Ω × (0, T )), ∂t ∂x j ∂x j ∂xk

i, j, k = 1, 2,

and u is a continuously function from [0,T] into V˙ . We also can see that ˙ u ∈ L2 (0, T ; D(A)) ∩ C([0, T], V). And u satisfies the following priori estimates 1 |u(s)| ≤ |u(0)| + µ 2

2

2

Z s

sup ku(t)k ≤ K1 , t∈[0,T ]

0

kf(t)kV2 0 dt Z T 0

∀s ∈ [0, T ],

|Au(t)|2dt ≤ K2, R

where constants K 1 , K2 are dependent on ku0 k, 0T |f(t)|2dt, µ, L1 , L2 . (ii) For r ≥ 1, if the initial data u0 ∈ Vr and f ∈ L∞ (0, T ;Vr−1), then the solution

42

Ling Rao and Hongquan Chen

u = u(t) belongs to C([0, T];Vr ). And there exists constant K dependent on µ, ku0 kr , |f|L∞ (0,T ;Vr−1 ) , such that sup ku(t)kr ≤ K.

(24)

t∈[0,T ]

Gronwall’s Lemma: (A)[7] Let a ∈ L1 (τ, T ), b be absolutely continuous on [τ, T ]. If x ∈ L∞ (τ, T ) satisfies Z t

x(t) ≤ b(t) +

a(s)x(s)ds,

τ

then for t ∈ (0, T ) x(t) ≤ b(τ)exp

Z t τ

a(s)ds +

Z t τ

b0 (s)exp

Z

t

a(ρ)dρ ds.

s

(B)[4] Let a, θ ∈ L1 (0, T ). If y satisfies y0 ≤ a + θy, then for t ∈ (0, T ) y(t) ≤ y(0)exp(

Z t 0

θ(τ)dτ) +

Z t

Z t

a(s)exp(

0

s

θ(τ)dτ)ds.

Lemma 2. Assume that 0 < T ≤ Te, c is a positive constant and u0 , f1 , f2 satisfy 2

u0 ∈ V˙ ,

˙ fi , ∈ L (0, T ; H),

Z T 0

|fi (t)|2dt ≤ c

f or i = 1, 2.

Suppose that u1 , u2 are the two solutions to (23) corresponding forcing terms f1 , f2 respectively. Then we have the following estimate Z T 0

ku1 (t) − u2 (t)k22dt

≤ c1

Z T 0

|f1(t) − f2 (t)|2dt

where constant c 1 is dependent on µ, ku0k, Te, c.

Immersed Boundary Method Proof

43

According to the assumptions and Lemma 1, we have u1 , u2 ∈ L2 (0, T ; D(A))

and 2

sup ku1 (t)k ≤ K1 ,

Z T

t∈[0,T ]

0

|Au1(t)|2dt ≤ K2 ,

where constants K1 , K2 are dependent on µ, ku0 k, c. By assumption, we have d dt ((u1 − u2 )(t), v) + µ((u1 − u2 , v)) + b(u1, u1, v) ∀ v ∈ V, t ∈ (0, T ), −b(u2, u2 , v) = hf1 − f2 , vi (u1 − u2 )(0) = 0.

(25)

(26)

Let u = u1 − u2 , f = f1 − f2 . Using (19), we have b(u1 , u1, u1 − u2 ) − b(u2 , u2, u1 − u2 ) = −b(u1 , u1, u2 ) + b(u2 , u1, u2 ) = −b(u, u1, u2 ) = b(u, u1 , u1 ) − b(u, u1 , u2) = −b(u, u, u1 ) Replacing v by u in (26), we obtain 1d |u(t)|2 + µkuk2 − b(u, u, u1 ) = hf, ui ∀t ∈ (0, T ). 2 dt

(27)

Using Poicare inequality (see [6]), we obtain hf, ui ≤ |f||u| ≤

1 1 2

|f|kuk ≤

λ1

1 µ |f|2 + kuk2. µλ 1 4

(28)

And by the following inequality (see[4], page 13) 1

1

|b(u, u, u1)| ≤ k|u|kukku1k 2 |Au1 | 2 , we obtain

µ k2 |b(u, u, u1)| ≤ kuk2 + ku1 k|Au1 ||u|2. 4 µ

(29)

By (27),(28),(29) we obtain, for t ∈ [0, T ], 2 2k2 d |u(t)|2 + µkuk2 ≤ ku1 k|Au1||u|2. |f|2 + dt µλ 1 µ

(30)

44

Ling Rao and Hongquan Chen

By Gronwall’s Lemma (B) and (25), we deduce that, for t ∈ [0, T ], |u(t)|2 ≤

Z t 2

(

0

µλ 1

|f(s)|2)(exp

Z t 2 2k

µ

s

then sup |u(t)|2 ≤ K3 0∈[0,T ]

Z T

ku1 (τ)k|Au1(τ)|dτ)ds,

|f(s)|2ds,

(31)

0

where constant K3 is dependent on µ, ku0k, c. Replacing v by Au in (26), we have, for t ∈ (0, T ), 1d ku(t)k2 + µ|Au|2 + b(u1 , u1 , Au) − b(u2, u2 , Au) = hf(t), Aui. 2 dt Since b(u1, u1 , Au) − b(u2, u2 , Au) = b(u1 , u1, Au) − b(u2 , u1, Au) + b(u2 , u1 , Au) − b(u2, u2 , Au) = b(u, u1 , Au) + b(u2, u, Au), then 1d ku(t)k2 + µ|Au|2 + b(u, u1 , Au) + b(u2 , u, Au) = hf(t), Aui. 2 dt

(32)

And using (20), we obtain the following equality (see [6], page 135) 1d ku(t)k2 + µ|Au|2 = b(u, u, Au1) + hf(t), Aui. 2 dt

(33)

Notice that

1 µ hf, Aui ≤ |f||Au| ≤ |f|2 + |Au|2 . µ 4 And by the following inequality (see[4], page 13) 1

(34)

1

|b(u, u, Au1)| ≤ k|u| 2 |Au| 2 kuk|Au1|, we obtain

µ 1 k4 |b(u, u, Au1)| ≤ |Au|2 + |Au1 |2 kuk2 + |u|2. 4 2 4µ By (33),(34),(35), we deduce that d k4 2 ku(t)k2 + µ|Au|2 ≤ |Au1|2 kuk2 + |u|2 + |f|2. dt 2µ µ

(35)

(36)

Immersed Boundary Method

45

By Gronwall’s Lemma (B), we obtain, for t ∈ [0, T ], 2

ku(t)k ≤ (

Z t 4 k 0

2 ( |u(s)| + |f(s)|2)exp( 2µ µ

Z t s

|Au1(τ)|2dτ)ds.

Then Z T

sup ku(t)k2 ≤ exp(

0

t∈[0,T ]

Z T 4 k

|Au1(τ)|2dτ)(

(

0

2 |u(s)| + |f(s)|2)ds). 2µ µ

By (25), (31), there exists constant K4 dependent on µ, ku0 k, c, Te, such that sup ku(t)k2 ≤ K4

Z T

t∈[0,T ]

|f(s)|2ds.

(37)

0

Then by integration in t of (36) from 0 to T, after dropping unnecessary terms we obtain Z Z T

0

that is

Z T 0

|Au(t)|2dt ≤ K5

|Au1(t) − Au2 (t)|2dt ≤ K5

T

|f(t)|2 dt,

0

Z T 0

|f1 (t) − f2 (t)|2dt,

(38)

where constant K5 is dependent on µ, ku0 k, c, Te. Then using (18) with s=1, we obtain Z Z T

0

ku1 (t) − u2 (t)k22dt ≤ c1

T

0

|f1(t) − f2 (t)|2dt

where constant c1 is dependent on µ, ku0 k, c, Te.

4.

Variational Formulation of the Problem

C[0, L] denotes the space of continuous functions from [0, L] into R2 and is equipped with the norm kxkc = max |x(s)|. 0≤s≤L

In what follows, we always suppose that Te > 0 is a given constant, and constant T ∈ (0, Te]. Let G = L([0, T ]; C[0, L]);

E = {x ∈ C[0, L] : x(0) = x(L)};

46

Ling Rao and Hongquan Chen F = {x ∈ E|x(s) ∈ Ω, 0 ≤ s ≤ L},

where Ω = (−

L2 L2 L1 L1 , ) × (− , ). 2 2 2 2

Notice that F = {x ∈ E|x(s) ∈ Ω, 0 ≤ s ≤ L}. For x ∈ G, x(·,t) ∈ C[0, L], ∀ t ∈ [0, T ], x(·,t) is simply denoted by x(t). Given X0 ∈ F, let X = {X ∈ L([0, T ]; E) : X(0) = X0 }. X is a Banach subspace of G. Given δh : R2 → R2 , f : [0, L] × [0, Te] → R2, we assume that they satisfy Condition A: e δh ∈ H20 (R2 ), f ∈ C([0, L] × [0, T]).

In this paper, we let Dirac delta function δ be approximated by differentiable function δh . In fact, some authors do so when performing the numerical simulation in order to get the approximate solution of the problem (1)-(7). For example, δh in [9] is chosen as follows δh (x) = dh (x)dh(y), where

∀x ∈ R2

h πz i 0.25 1 + cos( ) |z| ≤ 2h, dh (z) = h 2h 0, |z| > 2h.

It is clear that δh ∈ H20 (R2). According to the theories and notations in Section 2, Section3, we introduce the corresponding variational formulation of the equations (10)-(16): Problem 1. Given u0 ∈ V, X0 ∈ F and δh : R2 → R2 , f : [0, L] × [0, Te] → R2 satisfying Condition A, find u ∈ L2 (0, T ; D(A)) ∩ L∞(0, T ; V˙ ), and X : [0, L] × [0, T ] → Ω, such that d ˆˆ (u(t), v) + µ((u, v)) + b(u, u, v) = hF(t), vi dt ∂X (s,t) = u(X(s,t) − Iu(t),t) + Mu(t) ∂t u(x, 0) = uˆˆ 0 (x) X(s, 0) = X0(s) X(0,t) = X(L,t)

∀ v ∈ V, t ∈ (0, T ), ∀ s ∈ [0, L],t ∈ (0, T ), ∀ x ∈ Ω, ∀s ∈ [0, L], ∀t ∈ (0, T ),

Immersed Boundary Method

47

where 1 uˆˆ 0(x) = u0 (x) − |Ω| ˆF(x,t) ˆ =

Z L 0

5.

Z Ω

u0 (x)dx,

1 f(s,t)δh(x + Iu (t) − X(s,t))ds − |Ω|

Z L

Z

f(s,t)(

0

Ω

δh (x − X(s,t))dx)ds.

Conclusion

In this section we shall prove the existence of the solution of Problem 1 via a fixed point argument. We define an operator T on X as follows. Given u0 ∈ V, X ∈ X. ∀x ∈ Ω,t ∈ (0, T ), let FX (x,t) =

Z L 0

f(s,t)δh(x − X(s,t))ds.

Hence Fˆˆ X (x,t) =

Z L 0

f(s,t)δh(x+IX (t)−X(s,t))ds −

1 |Ω|

Z L 0

Z

f(s,t)(

Ω

δh (x−X(s,t))dx)ds

where, according to (9), t IX (t) = |Ω| 1 MFX (t) = |Ω|

Z

Z Ω

u0 (x)dx +

1 FX (x,t)dx = |Ω| Ω

Z L 0

Z tZ s 0

0

MFX (τ)dτds,

Z

f(s,t)(

Ω

δh (x − X(s,t))dx)ds.

(39) (40)

˙ 2per (Ω)). If Condition A is satisfied, we can verify that Fˆˆ X ∈ L∞ (0, T ; H Then let u (also denoted by uX ) be the solution to the following problem: Find u ∈ L∞ (0, T ; V˙ ) such that d (u(t), v) + µ(∇u, ∇v) = hFˆˆ X (t), vi ∀v ∈ V, t ∈ (0, T ), (41) dt u(0) = uˆˆ . 0 Finally, Let X be the solution X ∈ X of 0 X (t) = u(X(s,t) − IX (t),t) + MuX (t)) ∀t ∈ (0, T ), X(0) = X0,

(42)

48

Ling Rao and Hongquan Chen

where

1 MuX (t)) = |Ω|

Z Ω

u0 (x)dx +

Z t 0

MFX (τ)dτ.

(43)

Then let T(X) = X. We recall that the definition of X is X = {X ∈ L([0, T ]; E) : X(0) = X0 }. For t ∈ [0, T ], X(t) = X(·,t) ∈ E. (42) is an ordinary equation in the Banach space E and equivalent to ∂X (s,t) = u(X(s,t) − IX (t),t) + MuX (t)) ∀s ∈ [0, L], ∀t ∈ (0, T ), ∂t ∀s ∈ [0, L], X(s, 0) = X0 (s) X(0,t) = X(L,t) ∀t ∈ (0, T ). e in X, then this fixed point and the We observe that, if T has a fixed point X e give the solutions to Problem 1. solution e u to (41) corresponding to X = X, The following lemma guarantees the existence of the solution to (42): Lemma 3. (see [1]) Let x : [t0,t0 + a] → Y be a mapping into the Banachspace Y. Consider the initial value problem dx = f (x,t), dt

x(t0) = x0 .

(44)

Let Q = [t0,t0 + a] ×Y . Suppose f : Q → Y is continuous and satisfies k f (t, x) − f (t, y)k ≤ Lkx − yk,

f or all

(t, x), (t, y) ∈ Q, and f ixed L ≥ 0.

Then initial problem (44) has exactly one continuously differential solution on [t0,t0 + a] for each initial value x 0 ∈ Y. Theorem 1. Assume that T ∈ (0, Te], u0 ∈ H3per (Ω) ∩ H, X0 ∈ F and Condition A holds. (a) For given X ∈ X, (41) has a unique solution u ∈ L([0, T ], C1(Ω)). Furthermore there exist constants c 1 , c2, c3 dependent on e µ, T , δh , |fkC([0,L]×[0,Te]) , ku0kV3 , such that sup |u(z,t)| ≤ c1 < ∞, z∈Ω 0≤t≤T

(45)

Immersed Boundary Method sup |Du(z,t)| ≤ c2 < ∞,

49 (46)

z∈Ω 0≤t≤T

sup |u(z,t) + MuX (t)| ≤ c3 < ∞. z∈Ω 0≤t≤T

(b) Suppose that u is the solution of (41). Then (42) has a unique solution X ∈ X. 0 kc e , T } holds, where L m = min{L1 , L2}, then Furthermore, if 0 < T < min{ Lm −kX c3

X(s,t) ∈ Ω, for all (s,t) ∈ [0, L] × [0, T ].

Proof: (a) Since u0 ∈ H3per (Ω) ∩ H, then uˆˆ 0 ∈ V3. Since Condition A is satisfied, ˙ 2per (Ω)) and PL Fˆˆ X ∈ L∞ (0, T ;V2). We can verify that there then Fˆˆ X ∈ L∞ (0, T ; H exists constant C dependent on δh , such that kPL Fˆˆ X kL∞ (0,T ;V2 ) ≤ kFˆˆ X kL∞ (0,T ;H˙ 2per (Ω)) ≤ CkfkC([0,L]×[0,Te]) .

(47)

By Lemma 1 (ii) with r = 3, there exists u belonging to C([0, T];V3) satisfying d (u(t), v) + µ(∇u,∇v) = hPL Fˆˆ X (t), vi ∀v ∈ V, t ∈ (0, T ), dt u(0) = uˆˆ . 0 Since PL is the orthogonal projector onto H in L2 (Ω), we have hPL Fˆˆ X (t), vi = hFˆˆ X (t), vi ∀v ∈ V. Then u is the solution of (41). Since u belonging to C([0, T];V3 ) and V3 is a ˙ 3per (Ω), we obtain that u ∈ L([0, T ], C1(Ω)) due to (18) closed subspace of H and Sobolev embedding theorem (see [7]). Furthermore by (24) and (47) there exist constants c1 , c2, c3 dependent on µ, Te, δh , |fkC([0,L]×[0,Te]) , ku0 k3 , such that sup |u(z,t)| ≤ c1 < ∞, z∈Ω 0≤t≤T

sup |Du(z,t)| ≤ c2 < ∞, z∈Ω 0≤t≤T

50

Ling Rao and Hongquan Chen sup |u(z,t) + MuX (t)| ≤ c3 < ∞. z∈Ω 0≤t≤T

(b) We shall prove the conclusion by Lemma 3 with Y = E, f = u, x0 = X0,t0 = 0. Below we verify that the conditions in Lemma 3 hold. First we prove that u : E × [0, T ] → E is continuous. Given (x0 ,t0) ∈ E × [0, T ], (x,t) ∈ E × [0, T ]. It is easy to see u(x,t) ∈ E, u(x0,t0) ∈ E. We have ku(x,t) − u(x0,t0)kc = sup |u(x(s),t) − u(x0 (s),t0)| 0≤s≤L

≤ sup |u(z,t) − u(z,t0)| z∈Ω

+ sup |u(z,t0) − u(z0,t0)|. z∈Ω

Then u(x,t) is continuous at (x0 ,t0) due to u ∈ L([0, T ], C1(Ω)). For t ∈ [0, T ], x, y ∈ E, by (46) we have ku(x,t) − u(y,t)kc = sup |u(x(s),t) − u(y(s),t)| 0≤s≤L

≤ sup |Du(z,t)|kx − ykc ≤ c2 kx − ykc . z∈Ω 0≤t≤T

Hence the conditions in Lemma 3 hold. (42) has a unique solution X ∈ X. By (42), we have X(t) = X0 +

Z t 0

(u(X(s, τ) − IX (τ), τ) + MuX (τ))dτ.

Notice that X0 ∈ F, then Lm − kX0 kc > 0. If 0 < T < t ∈ [0, T ], we have

Lm −kX0 kc c3

holds, for

kX(t)kc ≤ kX0 kc + t sup |u(z, τ) + MuX (τ)| < kX0 kc + c3

z∈Ω 0≤τ≤T Lm −kX0 kc

c3

Then X(t) ∈ F. That is, if 0 < T

PERSPECTIVES IN MAGNETOHYDRODYNAMICS RESEARCH No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

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EARTH SCIENCES IN THE 21ST CENTURY

PERSPECTIVES IN MAGNETOHYDRODYNAMICS RESEARCH

VICTOR G. REYES

EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. Library of Congress Cataloging-in-Publication Data Perspectives in magnetohydrodynamics research / editor, Victor G. Reyes. p. cm. Includes index. ISBN 978-1-62100-128-7 (eBook) 1. Magnetohydrodynamics. I. Reyes, Victor G. QC718.5.M36P465 2011 538'.6--dc22 2010047794

New York

CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Index

vii MHD Free Convection in a Porous Medium Bounded by a Long Vertical Wavy Wall and a Parallel Flat Wall A.K. Tiwari Immersed Boundary Method: The Existence of Approximate Solution in Two-Dimensional Case Ling Rao and Hongquan Chen Transient Hydromagnetic Natural Convection between Two Vertical Walls Heated/Cooled Asymmetrically R.K. Singh and A.K. Singh Effect of Suction/ Injection on MHD Flat Plate Thermometer Anand Kumar and A.K. Singh Flute and Ballooning Modes in the Inner Magnetosphere of the Earth: Stability and Influence of the Ionospheric Conductivity O.K. Cheremnykh and A.S. Parnowski

1

31

57

75

87 139

PREFACE This book presents and discusses current research in the study of magnetohydrodynamics. Topics discussed include MHD free convection in a porous medium bounded by a vertical wavy wall and a parallel flat wall; the immersed boundary method; transient hydromagnetic natural convection between two vertical walls heated/cooled asymmetrically; the effect of suction/injection on MHD flat plate thermometer and flute and ballooning modes in the inner magnetosphere of the earth. Chapter 1 - This paper presents MHD free convective flow of an incompressible and electrically conducting fluid through a porous medium bounded by a long vertical wavy wall and a parallel flat wall. The shape of wavy wall is assumed to follow a profile of cosine curve and kept at constant heat flux while parallel flat wall at constant temperature. Governing systems of nonlinear partial differential equations in non-dimensional form are linearised by using perturbation method in terms of amplitude and analytical solution for velocity and temperature fields has been obtained for mean as well as perturbed part in terms of various parameters. A numerical study of the analytical solution is performed with respect to the realistic fluid air in order to illustrate the interactive influences of governing parameters on the temperature and velocity fields as well as skin friction and Nusselt number. In the case of maximum waviness (positive and negative), reverse phenomena occurs near flat wall in the parallel flow. It is observed that the parallel flow through channel at zero waviness is more than maximum waviness (positive and negative) while same trend occurs for perpendicular flow in reverse direction. Examination of Nusselt number shows that in the presence and absence of heat source, the heat flows from porous region towards the walls but in the presence of sink, the heat flows from the walls into the porous region.

viii

Victor G. Reyes

Chapter 2 - This paper deals with the two-dimensional Navier-Stokes equations in which the source term involves a Dirac delta function and describes the elastic reaction of the immersed boundary. The authors analyze the existence of the approximate solution with Dirac delta function approximated by differentiable function. The authors obtain the result via the Banach Fixed Point Theorem and the properties of the solutions to the NavierStokes equations of viscous incompressible fluids with periodic boundary conditions. Chapter 3 - The attentive work analyses a closed form solution for the transient free convective flow of a viscous incompressible and electrically conducting fluid between two vertical walls as a result of asymmetric heating or cooling of the walls in the presence of a magnetic field applied perpendicular to the walls. The convection process between the walls occurs due to a change in the temperature of the walls to that of the temperature of fluid. The solution for velocity and temperature fields are derived by the Laplace transform technique. The numerical values procure from the analytical solution show that the flow is initially in downward directions near the cooled wall for negative values of the buoyancy force distribution parameter. The effects of Hartman number, buoyancy force distribution parameter and Prandtl number on velocity profiles and skin friction are shown graphically and tabular form. Chapter 4 - The effect of suction/injection on a steady two-dimensional electrically conducting and viscous incompressible fluid owing to the flat plate thermometer is numerically analyzed. The flow is considered at small magnetic Reynolds number so that induced magnetic field is taken to be negligible. The non-linear coupled boundary layer equations are transferred to non-linear ordinary differential equations using the similarity transformation and resulting equations are solved by shooting method with fourth order Runge-Kutta algorithm. Numerical results for the dimensionless velocity and temperature profiles and skin friction coefficient are presented by graphs and table for various values of magnetic and suction/injection parameters. It is found that the effect of injection is to increase the temperature of the flat plate thermometer while suction has opposite influence. Chapter 5 - In this article the authors represent a survey of the present state and analytical methods of the theory of transversally small-scale standing MHD perturbations in the inner magnetosphere of the Earth, as well as the authors’ own views on this matter. The authors restrict their consideration by two important types of such perturbations: flute and ballooning modes.

Preface

ix

In the most general case arbitrary three-dimensional transversally smallscale standing MHD perturbations in ideal plasmas with nested magnetic surfaces are described by a pair of Dewar-Glasser equations. In some papers such equations are derived from the MHD equations by application of differential operators. This casts certain doubts upon the correctness of the resulting spectra of perturbations. The author choose a different path, using just the condition of transversally small-scaleness and longitudinal elongation of perturbations, which, nevertheless, appeared to be sufficient to derive mentioned equations. In addition the author used ideal MHD approximation, neglected the convection and considered the equilibrium to be static. Obtained equations were applied to a dipolar magnetic configuration, approximately representing the inner magnetosphere of the Earth. This made the equations much simpler and after introducing dimensionless variables they reduced to linear homogeneous ordinary differential equations of second order. When hydrodynamic pressure is considerable flute and ballooning perturbations are generated. The most unstable of them are the perturbations with a transversal to the magnetic surfaces polarization of the magnetic field. Obtained equations were supplemented with ionospheric boundary conditions derived in the same approximation. Considered perturbations appeared to be affected only by integral Pedersen conductivity of the ionosphere, which was approximated by a thin spherical layer. Moreover, when Pedersen conductivity of the ionosphere is finite flute modes can appear in the magnetosphere, determining its stability in this case. Using a modified energetic principle the author derived the corresponding stability criterion, which sets stronger restrictions on the stability than well known Gold criterion. Versions of these chapters were also published in Journal of Magnetohydrodynamics, Plasma and Space Research, Volume 14, Numbers 1-4, published by Nova Science Publishers, Inc. They were submitted for appropriate modifications in an effort to encourage wider dissemination of research.

In: Perspectives in Magnetohydrodynamics … ISBN: 978-1-61209-087-0 © 2011 Nova Science Publishers, Inc. Editor: Victor G. Reyes

Chapter 1

MHD FREE CONVECTION IN A POROUS MEDIUM BOUNDED BY A LONG VERTICAL WAVY WALL AND A PARALLEL FLAT WALL A.K. Tiwari* Department of Mathematics, Doon Institute of Engineering & Technology, Rishikesh, India

Abstract This paper presents MHD free convective flow of an incompressible and electrically conducting fluid through a porous medium bounded by a long vertical wavy wall and a parallel flat wall. The shape of wavy wall is assumed to follow a profile of cosine curve and kept at constant heat flux while parallel flat wall at constant temperature. Governing systems of nonlinear partial differential equations in non-dimensional form are linearised by using perturbation method in terms of amplitude and analytical solution for velocity and temperature fields has been obtained for mean as well as perturbed part in terms of various parameters. A numerical study of the analytical solution is performed with respect to the realistic fluid air in order to illustrate the interactive influences of governing parameters on the temperature and velocity fields as well as skin friction and Nusselt number. In the case of maximum waviness (positive and negative), reverse *

E-mail address: [email protected]

2

A.K. Tiwari phenomena occurs near flat wall in the parallel flow. It is observed that the parallel flow through channel at zero waviness is more than maximum waviness (positive and negative) while same trend occurs for perpendicular flow in reverse direction. Examination of Nusselt number shows that in the presence and absence of heat source, the heat flows from porous region towards the walls but in the presence of sink, the heat flows from the walls into the porous region.

Keywords: MHD free convection, porous media, wavy wall, perturbation method.

1. Introduction In recent years, the studies of MHD free convective flow have attracted many research workers in view of not only its own interest but also due to the applications in astrophysics, geographic and technology. Laminar free convection at vertical wall is of interest in many applications such cooling of nuclear reactors, heat exchangers, solar energy collectors, crystal growth and thermal engineering, among others. If the flow field involves an electrically conducting fluid under the influence of an external magnetic field, we then have the combined effects of viscous, buoyancy and magnetic forces on the flow. As the magnetic field and buoyancy forces can be controlled externally by changing the applied magnetic field and ramped temperature, the investigations of the effects of these forces on the flow and heat transfer characteristics of real fluids have been a subject of great research interest. Fluid flow control through external magnetic forces also find applications in the design of magnetohydrodynamic (MHD) generators and MHD devices used in various industries. It is therefore important to study the features of transport phenomena in MHD flows under practically important physical conditions for both unsteady and steady state cases. For instance, Hady et. al. [1] has studied the effect of heat generation or absorption on a free convection boundary layer flow along a vertical wavy plates embedded in electrically conducting fluid saturated porous media. The solutions of hydromagnetic free convective flows wavy channels have been discussed by Rao et. al. [2] and Tashtoush and Al-Odat[3] under different physical conditions. Transport processes as a result of free convection inside wavy–walled channel have not been investigated widely due to geometric complexity. Literatures related to this topic are not as rich as channels with flat walls. Free convection heat transfer phenomenon in a porous medium bounded by

MHD Free Convection in a Porous Medium Bounded…

3

geometries of irregular shape has attracted the attention of engineers and scientists from many varying disciplines such as chemical, civil, environmental, mechanical, aerospace, nuclear engineering, applied mathematics, geothermal physics and food science. Phenomena concerned with it include the spreading of pollutants, water movement in reservoirs, thermal insulation engineering, building science and convection in the earth’s crust etc. Geometrical complexity of such type of system affects largely the flow pattern and depends on many parameters like amplitude, wave length, phase angle, inter wall spacing etc. Each of the parameter significantly affects the hydrodynamic and thermal behavior of fluid inside it. These configurations are not idealities and its effects on flow phenomenon have motivated many researchers to perform experimental and analytical works. Ostrach [4] analyzed laminar natural convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperature. Vajravelu and Shastri [5] solved free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall. Shankar and Sinha [6] presented the flow generated in a viscous fluid by the impulsive motion of a wavy wall using perturbation method about the known solution for a straight wall. Lekoudis et al. [7] analyzed compressible viscous flows past wavy walls without restricting the mean flow to be linear in the disturbance layer. Their results agree more closely with experimental data than the results obtained by using Lighthill’s theory, which restricts the mean flow to be linear in the disturbance layer. The effect of small amplitude wall roughness on the minimum critical Reynolds number of a laminar boundary layer is studied by Lessen and Gangwani [8] under the assumptions normally employed in parallel flow stability problems. By using either analytical or numerical approaches, Singh and Gholami [9], Rees and Pop [10], and Kumar [11] have solved the natural convection problem in a fluid-saturated porous media with uniform heat flux condition. The fundamental importance of convective flow in porous media has been ascertained in the recent books by Ingham and Pop [12] and Neild and Bejan [13]. Recently, several studies by Rathish Kumar et al. [14, 15], Murthy et al. [16], Kumar and Shalini [17], Misirlioglou [18] and Sultana and Hyder [19] have been reported that were concerned with the natural convection heat transfer in wavy vertical porous enclosures. The main purpose of the present paper is to examine the MHD free convective heat transfer and fluid flow in a porous medium bounded by a long vertical wavy wall and a parallel flat wall. The wavy wall is kept at constant heat flux while parallel flat wall maintained at constant temperature. The

4

A.K. Tiwari

solution of governing equations has been obtained using perturbation technique described by Nayfeh [20] in terms of the physical parameters appearing in the governing equations. Results are presented corresponding to the velocity and temperature fields as well as skin friction and Nusselt number for different values of the governing parameters.

2. Mathematical Analysis Let us consider the two-dimensional laminar MHD free convective heat transfer flow of an incompressible and electrically conducting fluid through a porous medium bounded by a vertical wavy wall and a parallel flat wall which are maintained at constant heat flux and constant wall temperature respectively. The properties of the fluid are assumed to be constant and isotropic except the density variation in the buoyancy term in the momentum equation. The fluid and porous medium are in the local thermodynamic equilibrium. The wavy surface of the wall is described in the function form as . where, the origin of the co-ordinate system is placed at the leading edge of the vertical surface, while the flat wall which is parallel to wavy wall is taken at the distance . The fluid oncoming to the channel is still quiescent and both the fluid and flat wall have constant temperature

. A uniform magnetic

field of magnetic field strength is applied perpendicular to the channel length. In this problem, the viscous and Darcy dissipation effects are neglected and the volumetric heat source / sink is constant in the energy equation. If we define the dimensionless quantities as ,

,

,

,

,

,

5

MHD Free Convection in a Porous Medium Bounded…

,

,

,

,

, (1)

the dimensionless equations, governing the conservation of mass, momentum and energy in the channel are obtained as follows (Ingham and Pop [12]): (2)

,

(3)

,

(4)

,

(5) where, . In the dimensionless form, the boundary conditions can be written as , (6)

.

All the symbols used in the above equations are defined in the nomenclature. Under the perturbation technique, let us consider the velocity and temperature fields as , ,

(7)

6

A.K. Tiwari

where, first order quantities or perturbed parts are very small compared with the zeroth order quantities or mean parts. Using equation (7), the equations (2) - (5) reduce to the following form for zeroth order quantities (8)

. where,

is the constant pressure gradient term and is taken

equal to zero by Ostrach [4]. and for the first order quantities (9)

,

,

(10)

,

(11)

.

(12)

With the help of (7), the boundary conditions (6) can be converted into the following two parts: , (13)

, , .

(14)

7

MHD Free Convection in a Porous Medium Bounded…

2.a. Solution of Mean Part The zeroth order solutions are obtained from (8) with the help of boundary conditions (13) in the following form: (15) (16)

. The symbols expression for

used as a constant are given in appendix. The are called the zero-order solutions or mean parts.

2.b. Solution Procedure for Perturbed Part To find the solution of first order quantities from (9) - (12), let us introduce the stream function defined by (17)

.

It is obviously clear that continuity equation (9) is satisfied identically with the help of (17). Using (17) into (10) - (14) and eliminating the non-dimensional pressure , we have

(18) (19)

. Assuming , .

(20)

8

A.K. Tiwari

and using in (18) and (19), we get

,

(21) (22)

, where, a prime denotes differentiation with respect to y. Boundary conditions (14) become ,

(23) For small values of

(

), we can take (24)

Using (24) into (21) - (22), we have obtained a set of ordinary differential equations of fourth order in term of and second order in and they are not reported here for the sake of brevity. The solutions of these ordinary differential equations with their appropriate boundary conditions obtained from (23) are obtained as follows: ,

,

(25)

(26)

9

MHD Free Convection in a Porous Medium Bounded…

(27)

,

(28)

,

With the help of above obtained solutions, the first order quantities given by (17) along with (20) can be put in the following form: (29a)

,

(29b)

,

(29c)

, where, ,

(30)

. The expressions for the first-order velocity temperature

and the first-order

have been obtained with the help of eqs. (25) - (28).

3. Skin Friction and Nusselt Number at the Walls The shear stress by

at any point of the fluid in non-dimensional form is given

Using above equation, the skin friction at the flat wall (y=1) are obtained as

at the wavy wall (y=0) and

10

A.K. Tiwari

(31)

,

(32)

. The Nusselt number are obtained as

at the flat wall (y=1) in the dimensionless form

(33)

.

4. Results and Discussion The expressions for mean part

and perturbed part

have

been obtained in terms of physical parameters . The perturbed part of the solution is the contribution from the waviness of the wall. Involvement of many parameters in a study not only makes the computational works a formidable task but it also makes it difficult to incorporate a systematic parametric presentation. Thus we set Pr=0.71 corresponding to realistic fluid air, , and focus our attention on numerical computations for different values of the following table: Curves Parameter G M Da

as given in

1

2

3

4

5

6

7

8

9

10

11

12

50 5.0

50 5.0

50 5.0

50 5.0

50 5.0

50 5.0

50 2.0

50 2.0

50 2.0

100 5.0

100 5.0

100 5.0

-5

0

5

-5

0

5

-5

0

5

-5

0

5

0.01

0.01

0.01

0.1

0.1

0.1

0.01

0.01

0.01

0.01

0.01

0.01

Graphical representations of mean part as well as perturbed part of the velocity and temperature profiles of air have shown in the figs.1-5 for above data. Figure1 describes the behavior of the mean part of the velocity between vertical walls. Close examination of it reveals that in the presence of source, the

MHD Free Convection in a Porous Medium Bounded… 8

11

6

6

9 5

4

3

u0

2

0

1

-2

8

4

2 7

-4 0.0

0.2

0.4

0.6

0.8

1.0

y

Fig. 1. Zeroth-order velocity profiles Figure 1. Zeroth-order velocity profiles.

velocity profiles take parabolic shape but reverse shape in the presence of sink. The point of maxima on the curves get shifted away from parallel flat wall (y=1) as magnetic field parameter decreases and Darcy number increases. Although in the absence of source/sink, the velocity profiles are almost flat while assuming parabolic shapes due to overshooting in velocity near the flat wall (y=0) as value of magnetic field parameter decreases and Darcy number increases. It is clear from curves (3, 9) that the velocity decreases with the magnetic field parameter M for but reverse flow occurs for (see curves 1, 7). In the absence of source/sink , the velocity near the flat wall (y=0) decreases with an increase in magnetic field parameter M , a result physically equivalent to saying that fluid velocity depends fluid density inversely but near the flat wall (y=1) the velocity is approximately same as magnetic field parameter M increases. On the examination of curves (3, 6)and (2, 5), one can reveal that the velocity is increasing function of Darcy number Da in the presence and absence of source/sink while in the presence of sink, it is increasing function of Darcy number Da in the opposite direction shown by the curves (1, 4).

12

A.K. Tiwari

4

=5 3

2

0

=0 1

0

=-5

-1

-2 0.0

0.2

0.4

0.6

0.8

1.0

y Fig. 2. Zeroth - order temperature profiles Figure 2. Zeroth - order temperature profiles. The behavior of the mean temperature

is shown in fig. 2. From which,

it is clear that in the absence of heat source , the mean temperature is a linearly decreasing function of y (curves 2, 5, 8) while in the presence of heat source , the mean temperature is increasing from its value at the wall y=0 to a maximum temperature at around y=0.1 and then decreasing steadily its value upto y=1 (curves 3, 6, 9). In the presence of heat sink , the behavior of the mean temperature is exact opposite of that observed in the presence of source (curves 1, 4, 7). Further, we observed that there is no significant effect of on the mean temperature for all values of

.

MHD Free Convection in a Porous Medium Bounded… 0.04

8

4 0.02

3

6

7 1

u1

0.00

-0.02

9

5

2

2

8

4

3

7 1

5 9 -0.04

6 -0.06

0.0

0.2

0.4

0.6

0.8

1.0

y

(a) 3

5

4

7

u1

3

6

8

2

1

2

1

4

5 0 0.0

0.2

0.4

y (b)

Figure 3. Continued on next page.

0.6

0.8

1.0

13

14

A.K. Tiwari

0.06

6

0.04

9 5 3

8

1

4

7

u1

0.02

2

0.00

-0.02

9

2

7

8 6

5 3

0.8

1.0

1 4 -0.04 0.0

0.2

0.4

0.6

y (c)

Figure 3. First- order velocity component for (a) , (b) and (c). 6

2

7

0.00

1 4 10

-0.02

v1

5

8

-0.01

-0.03

9 -0.04

-0.05

-0.06 0.0

3

0.2

0.4

y (a)

Figure 4. Continued on next page.

0.6

0.8

1.0

MHD Free Convection in a Porous Medium Bounded…

15

4 8.0x10

-5

4.0x10

-5

7 1 2

v1

0.0 -4.0x10

-5

-8.0x10

-5

-1.2x10

-4

-1.6x10

-4

-2.0x10

-4

-2.4x10

-4

8 3

5 9

6

0.0

0.2

0.4

0.6

0.8

1.0

y (b) 0.06

0.05

3

v1

0.04

9

0.03

0.02

10

4 8

7 0.00 0.0

1

5

0.01

0.2

0.4

6 0.6

2 0.8

1.0

y (c)

Figure 4.First- order velocity component .

for (a)

, (b)

and (c)

16

A.K. Tiwari

=5

0.008

0.004

=0 0.000

-0.004

=-5

-0.008

0.0

0.2

0.4

0.6

0.8

1.0

y (a)

2,5,8,11 0.00

1,3 4,6,10,12

-0.02

v1

-0.04

-0.06

-0.08

7,8

-0.10 0.0

0.2

0.4

y (b)

Figure 5. Continued on next page.

0.6

0.8

1.0

MHD Free Convection in a Porous Medium Bounded…

17

= -5

0.008

0.004

=0 0.000

-0.004

=5

-0.008

0.0

0.2

0.4

0.6

0.8

1.0

y (c)

Figure 5. First-order temperature profiles

for (a)

, (b)

and (c)

.

4.a. Presentation of First Order Solution Figures 3(a, b, c), 4(a, b, c) and 5(a, b, c) represent the perturbed part (firstorder solution) of the velocity components and temperature respectively in the channel for three cases of the waviness of the wavy wall (y=0) and they are as follows: (i) maximum positive at , (ii) zero at and (iii) maximum negative at

. The description of first-

order solution at different types of waviness as follows; The effect of maximum waviness on the first-order velocity component is shown in figure 3(a) and it indicates that in the presence of heat sink, the velocity component increases near the wavy wall and then by decreasing becomes zero at y=0.70 approximately and thereafter reverse flow occurs. We observed from the curves (1, 4) that as increase in the Darcy number Da increases the velocity component in the two third of the channel and this behavior is reversed in the presence or absence of heat source as shown in the

18

A.K. Tiwari

curves (3, 6) or (2, 5) respectively. This behavior is reversed in the other one third of the channel. On analysis the curves (1, 7) for

, we found that as

increase in the magnetic field parameter M, the velocity component decreases in the two third of the channel and this behavior is reversed when (see curves 3, 9 and 2, 8). However in the other one third of the channel this behavior of the velocity component

with Mis reversed.

Figure 3(b), showing perturbed velocity component for zero waviness, indicates that it increases in present of heat source (curves 2, 3 and 5, 6). The effect of Darcy number is also to increase it (curves 1, 7 and 2, 8). For maximum negative waviness (fig. 3c), we found that reversed effects are observed for source and sink parameter. The effects of the parameters which appear in it are reversed of all the results found for maximum positive waviness. It is observed from fig. 4(a) that the velocity component is enhanced by an increase in the Darcy number Da in the reverse direction by the curves (3, 6) and (2, 5) in the presence of heat source and absence of heat source/sink respectively while in the presence of heat sink, it increases positively (see curves 1 and 4). In the presence of heat source and absence of heat source/sink, the velocity component is an increasing function of Da and decreasing function of M by curves ((3, 6), (2, 5)) and ((3, 9), (2, 8)) respectively in the opposite direction while in the presence of heat sink, the velocity component is also an increasing function of Da and decreasing function of M (see curves (1, 4) and (1, 7)). It is observed from the curves (1, 4, 7, and 10) of fig. 4(b) that for the velocity component

,

is an increasing function of G, Da and decreasing

function of M while for , the reverse effect can be seen by the curves (3, 6, 9, and 12) and (2, 5, 8, and 11). Close observation of figure 4(c) shows that the behavior of velocity component in case of maximum negative waviness is almost reverse to that of positive maximum waviness The behavior of the perturbed temperature with changes in is shown in figs. 5(a, b, c) according to three cases of waviness of the wavy wall. Figure 5(a) showing the perturbed part of temperature for which indicates that in the absence of heat source/sink, the perturbed temperature is zero (curves 2, 5, 8) while in the presence of heat source, it is a linearly decreasing function of y (curves 3, 6, 9). In the presence of heat sink, the behavior of the perturbed temperature is exactly opposite of that observed in the presence of source

MHD Free Convection in a Porous Medium Bounded… (curves 1, 4, 7).The parameters

19

have also negligible effect on

the perturbed temperature for all values of . For zero waviness, we observed from fig. 5(b) that the perturbed temperature is almost same up to y= 0.8 of the channel and then increases in remaining part of the channel. The behaviors of the perturbed temperature at maximum negative waviness are shown in fig. 5(c). The effect of source and sink are just reversed corresponding to maximum positive waviness but for absence of source/sink, both cases have same effect. The curves of skin friction are shown in figs. 6(a, b) only for maximum positive waviness

and zero waviness

because the

perturbed part is much smaller than mean part and curves for maximum negative waviness almost coincide with fig. 6(a). The curves in these figures are drawn based on the following data: Curves Parameter G M Da

1

2

3

4

50 5.0 0.01

50 2.0 0.01

100 5.0 0.01

100 5.0 0.1

It is observed that the skin friction at the channel walls is a linear function of the heat source parameter and the skin friction at the wall y=0 increases with the heat source parameter while the reverse is true at the other wall y=1 in both types of channel walls. The skin friction is an increasing function of the Grashof number G and Darcy number Da by the curves (1, 3) and (3, 4) at the wall y=0 while at y=1 decreases with an increase in G and Da. It is clear from curves 1 and 2 that the skin friction is unaffected by magnetic field parameter M in both cases. On comparing the skin friction for the both type of waviness of the wall at y=0 and y=1, it is observed that it is greater in case when channel has maximum positive waviness than case when channel has zero waviness. The temperature profiles of wavy wall are shown in fig.7. It is observed that the temperature of the wavy wall is a linear function of phase for all values of M, G and it becomes oscillatory when the value of Darcy number Da increases in the presence of source

and sink

. In the

absence of source/sink , the wavy wall temperature linearly varies as function of M, G and Da as the perturbed part is much smaller than mean part.

20

A.K. Tiwari 50

40

30

,

20 10

,

0

-10

-20 -6

-4

-2

0

2

4

6

(a)

50

40

30

,

20 10

,

0 -10

-20 -6

-4

-2

0

2

4

6

(b)

Figure 6. Total skin- friction at the walls for (a)

and (b)

.

MHD Free Convection in a Porous Medium Bounded…

21

3,9

6

3

2

2,5,8

1

0

4

1,7

-1

-2

0

2

4

6

8

10

12

14

16

y Fig. 7.Temperature of the wavy wall

Figure 7. Temperature of the wavy wall.

Lastly in table 1, the values of Nusselt number at the channel wall (y=1) are given only for maximum positive waviness and zero waviness for different values of M, G and Da. The Nusselt numbers for maximum negative waviness are approximately same as for maximum positive waviness. It can see from this table that the Nusselt number at the flat wall (y=1) in the both type of waviness decreases with and this decrease being least significant for Dathan G and most significant for for maximum positive waviness

. The effects of

on decreasing the value of M, the

Nusselt number has approximately same value for while for zero waviness

on the Nusselt number that and decreased value for

, the Nusselt number is exactly same

for all values of . Also, when the heat source/sink parameter takes positive increasing values, the Nusselt number at the flat wall (y=1) in the both type of waviness becomes negative, which means physically that heat flows from porous region towards the walls. However, when the heat source/sink parameter takes negative increasing values, the Nusselt number at the flat wall (y=1) in the both type of waviness is positive, which indicates physically that heat flows from the walls into the porous region.

Table 1. Numerical values of dimensionless Nusselt number for Pr=0.71 Values of Nusselt number at the flat wall (y=1) For

For

G=50 M=5.0 Da=0.01

G=50 M=2.0 Da=0.01

G=100 M=5.0 Da=0.01

G=100 M=5.0 Da=0.02

G=50 M=5.0 Da=0.01

G=50 M=2.0 Da=0.01

G=100 M=5.0 Da=0.01

G=100 M=5.0 Da=0.02

-5

3.97996

3.96992

3.96992

3.94984

3.36137

3.36137

2.72274

1.43997

-4

2.98555

2.97915

2.97914

2.96630

2.59119

2.59119

2.18238

1.36204

-3

1.99038

1.98677

1.98677

1.97954

1.77005

1.77005

1.54010

1.07933

-2

0.99439

0.99278

0.99278

0.98957

0.89795

0.89795

0.79591

0.59184

-1

-0.00240

-0.00280

-0.00280

-0.00360

-0.02509

-0.02509

-0.05019

-0.1004

0

-0.99999

-0.99999

-0.99999

-0.99999

-0.99910

-0.99910

-0.99820

-0.9974

1

-1.99840

-1.99880

-1.99880

-1.99960

-2.02406

-2.02406

-2.04813

-2.0992

2

-2.99760

-2.99921

-2.99921

-3.00242

-3.09998

-3.09998

-3.19996

-3.4058

3

-3.99761

-4.00113

-4.00123

-4.00845

-4.22685

-4.22685

-4.45371

-4.9172

4

-4.99842

-5.00485

-5.00485

-5.01770

-5.40468

-5.40468

-5.80937

-6.6333

5

-6.00004

-6.01008

-6.01008

-6.03016

-6.63347

-6.63347

-7.26694

-8.5542

MHD Free Convection in a Porous Medium Bounded…

23

Conclusion The two-dimensional MHD free convective heat transfer flow of an incompressible and electrically conducting fluid through a porous medium of a viscous and incompressible fluid through a porous medium bounded by a vertical wavy wall and a parallel flat wall which are maintained at constant heat flux and constant wall temperature respectively has been studied. The governing equations in non-dimensional form are linearised by using perturbation technique. Analytical solution for mean part as well as perturbed part have been obtained and using them detailed analysis of velocity and temperature fields are presented in graphical form for various values of the parameters. We have also discussed about the surface skin-friction coefficient as well as the Nusselt numbers at the flat wall (y=1) and temperature of the wavy wall (y=0). The important fact of this study is a comparison among three type of waviness of wall. It is observed that the parallel flow through channel at zero waviness is more than maximum waviness (positive and negative) while same trend occurs for perpendicular flow in reverse direction.

Nomenclature cp

-specific heat at constant pressure

Nu P

–distance between both walls -Darcy number -magnetic field parameter –Grashof number or free convection parameter -acceleration due to gravity -permeability of the porous medium –Nusselt number -fluid pressure

Pr q Q T

-dimensionless fluid pressure -Prandtl number – rate of heat transfer –source/sink parameter –fluid temperature

Da M G g

K

-temperature of the flat wall

24

A.K. Tiwari

u, v -dimensionless velocity components along x- and y-axis, respectively u , -velocity components along x- and y-axis, respectively x, y -dimensionless Cartesian coordinates , y - Cartesian coordinates

Greek Symbols

-dimensionless source/sink parameter -volumetric coefficient of thermal expansion - electrical conductivity - constant magnetic field flux density -dimensionless amplitude parameter -amplitude parameter

-dimensionless fluid temperature -thermal conductivity -dimensionless frequency parameter -frequency parameter -dynamic viscosity -kinematic viscosity -fluid density -skin friction or dimensionless shear stress –dimensionless stream function

Subscript

-zero-order quantity –first-order quantity s p

–static fluid -wavy wall –flat wall

25

MHD Free Convection in a Porous Medium Bounded…

Appendix

,

,

,

,

, , , , , ,

26

A.K. Tiwari

„

,

,

, , , ,

,

27

MHD Free Convection in a Porous Medium Bounded…

, , , ,

,

,

,

,

,

,

28

A.K. Tiwari

References [1] Rao, D. R. V. P., Krishna, D. A. and Debnath L., Free convection in hydromagnetic flows in a vertical wavy channel, International Journal of Engineering Science, Vol. 21 (9), pp. 1025-1039, 1983. [2] Hady, F. M., Mohamed R. A. and Mahdy, A., MHD free convection flow along a vertical wavy surface with heat generation or absorption effect, Int. Comm. in Heat and Mass Transfer, Vol. 33 (10), pp. 1253-1263, 2006. [3] Tashtoush B. and Al-Odat, Magnetic field effect on heat and fluid flow over a wavy surface with a variable heat flux, Journal of Magnetism and Magnetic Materials, Vol. 268 ( 3), pp. 357-363, 2004. [4] Ostrach, S., Laminar natural convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperature, N.A.S.A. Tech. Note No. 2863, 1952. [5] Vajravelu, K. and Sastri K. S., Free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall, J. Fluid Mech., Vol.86 (2), pp.365-383, 1976. [6] Shankar P. N. and Sinha U. N., The Rayleigh problems for a wavy wall, J. Fluid Mech., Vol.77, pp.243-256, 1976. [7] Lekoudis, S. G., Nayfeh, A. H. and Saric, W. S., Compressible boundary layers over wavy walls, Phys. Fluids, Vol.19, pp.514-519, 1976. [8] Lessen, M. and Gangwani, S. T., Effect of small amplitude wall waviness upon the stability of the laminar boundary layer, Phys. Fluids, Vol.19, pp.510-513, 1976. [9] Singh, A. K. and Gholami, H. R., Unsteady free convective flow through a porous medium bounded by an infinite vertical porous plate with constant heat flux, Rev. Roum. Sci. Techn. Mec. Appl., Vol.35, pp.337382, 1990. [10] Rees, D. A. and Pop, I., Free convection induced by a vertical wavy surface with uniform heat flux in a porous medium, ASME J. Heat Transfer, Vol.117, pp.547-550, 1995. [11] Kumar, B. V. R., A study of free convection induced by a vertical wavy wall with heat flux in a porous enclosure, Numer. Heat Transfer, Vol.37 (part A), pp.493-510, 2000. [12] Ingham, D. B. and Pop, I., Transport Phenomena in Porous Media, Pergamon Press, Oxford, 2002. [13] Nield, D. A. and Bejan, A., Convection in Porous Media, Springer, New York, 2006.

MHD Free Convection in a Porous Medium Bounded…

29

[14] Kumar, B. V. R., Singh, P. and Murthy, P. V. S. N., Effect of surface undulations on natural convection in a porous square cavity, ASME J. Heat Transfer, Vol.119, pp.848-851, 1997. [15] Kumar, B. V. R., Murthy, P. V. S. N. and Singh, P., Free convection heat transfer from an isothermal wavy surface in a porous enclosure, Int. J. Numer. Meth. Fluids, Vol.28, pp.633-661, 1998. [16] Murthy, P. V. S. N., Kumar, B. V. R. and Singh, P., Natural convection heat transfer from a horizontal wavy surface in a porous enclosure, Numar. Heat Transfer, Vol.31 (Part A), pp.207-221, 1997. [17] Kumar, B. V. R. and Shalini, Free convection in a non-Darcian wavy porous enclosure, Int. J. Energy Sci., Vol.41, pp.1827-1848, 2003. [18] Misirlioglu, A., Baytes, A. C. and Pop, I., Free convection in a wavy cavity filled with a porous medium, Int. J. Heat and Mass Transfer, Vol.48, pp.1840-1850, 2005. [19] Sultana, Z. and Hyder, M. N., Non-Darcy free convection inside a wavy enclosure, Int. Communications in Heat and Mass Transfer, Vol.34 (2), pp.136-146, 2007. [20] Nayfeh, A. H., Perturbation Methods, Willey-Interscience, New York, 1973.

Perspectives in Magnetohydrodynamics ISBN: 978-1-61209-087-0 c 2011 Nova Science Publishers, Inc. Editor: Victor G. Reyes

Chapter 2

I MMERSED B OUNDARY M ETHOD : T HE E XISTENCE OF A PPROXIMATE S OLUTION IN T WO -D IMENSIONAL C ASE Ling Rao1,2 and Hongquan Chen 1 1 College of Aerospace Engineering Nanjing University of Aeronautics and Astronautics Nanjing 210016, China 2 Department of Applied Mathematics Nanjing University of Science and Technology Nanjing 210094, China

Abstract This paper deals with the two-dimensional Navier-Stokes equations in which the source term involves a Dirac delta function and describes the elastic reaction of the immersed boundary. We analyze the existence of the approximate solution with Dirac delta function approximated by differentiable function. We obtain the result via the Banach Fixed Point Theorem and the properties of the solutions to the Navier-Stokes equations of viscous incompressible fluids with periodic boundary conditions.

Keywords: Navier-Stokes Equations; Immersed boundary method; Nonlinear ordinary differential equations. AMS subject classifications: 35K10, 46N20, 34A12.

32

1.

Ling Rao and Hongquan Chen

Introduction

Problems involving the interaction between a fluid flow and elastic interfaces may appear in several branches of science such as engineering, physics, biology, and medicine. Regardless the field, as a rule, they share in common a high degree of complexity, often displaying intricate geometry or time-dependent elastic properties, turning the problem into a real challenge for applied scientists, from both the mathematical modeling and the numerical simulation points of view. In the early 70s, Peskin [2, 8] introduced a mathematical model and a computational method to study the flow patterns of the blood around the heart valves. Through years, Peskin’s immersed boundary (IB) method was developed for the computer simulation of general problems [9, 10, 11, 12] involving a transient incompressible viscous fluid containing an immersed elastic interface, which may have time-dependent geometry or elastic properties, or both. The IB method is at the same time a mathematical formulation and a numerical scheme. The mathematical formulation is based on the use of Eulerian variables to describe the dynamic of fluid and of Lagrangean variables along the moving structure. The force exerted by the structure on the fluid is taken into account by means of a Dirac delta function constructed according to certain principles. The main idea is to use a regular Eulerian mesh for the fluid dynamics simulation, coupled with a Lagrangian representation of the immersed boundary. The advantage of this method is that the fluid domain can have a simple shape, so that structured grids can be used. The Lagrangian mesh is independent of the Eulerian mesh. The interaction between the fluid and the immersed boundary is modeled using a well-chosen discrete approximation to the Dirac delta function. Although the immersed boundary method is a particularly effective approach in scientific computation, but very little theoretical analysis has been performed on either the underlying model equations or numerical methods. Daniele Boffi [3] gave a variational formulation of the problem and provided a suitable modification of the IB method which made use of finite elements method. Because the source term in the Navier-Stokes equations involves a Dirac delta function, the problem is highly nonlinear and presents several difficulties related with the lacking of regularity of the solution of the Navier-Stokes equations due to such source term. Daniele Boffi analyzed the existence of the solution in a very simple one-dimensional heat equation. In [13] we dealt with

Immersed Boundary Method

33

the two-dimensional heat equation. We analyzed the existence of the approximate solution with Dirac delta function approximated by differentiable function. In this paper will deal with two-dimensional Navier-Stokes equations. We analyze the existence of the approximate solution still with Dirac delta function approximated by differentiable function. We obtain the results via the Banach Fixed Point Theorem and a theorem in nonlinear ordinary differential equations in abstract space and the properties of the solutions to the Navier-Stokes equations of viscous incompressible fluids with periodic boundary conditions. In section 2 we will present the mathematical model of the problem. In order to prove the existence of the approximate solution, some properties of the Navier-Stokes equations with the space-periodic boundary conditions are reviewed in section 3. In section 4 we will introduce the corresponding variational formulation of the problem. In section 5 we will prove the existence of the approximate solution of of the problem via a fixed point argument.

2.

Problem

The authors (see [3]) considered the model problem of a viscous incompressible fluid in a two-dimensional square domain Ω containing an immersed massless elastic boundary in the form of a curve. To be more precise, for all t ∈ [0, T ], let Γt be a simple curve, the configuration of which is given in a parametric form, X(s,t), 0 ≤ s ≤ L, X(0,t) = X(L,t). The equations of motion of the system are ∂u − µ∆u + u · ∆u + ∆p = F ∂t

in Ω × (0, T ),

(1)

∆·u = 0

in Ω × (0, T ),

(2)

∀x ∈ Ω, t ∈ (0, T ),

(3)

F(x,t) =

Z L

f(s,t)δ(x − X(s,t))ds

0

∂X (s,t) = u(X(s,t),t) ∂t

∀s ∈ [0, L], t ∈ (0, T ).

(4)

Here u is the fluid velocity and p is the fluid pressure. The coefficient µ is the fluid viscosity constant. Eqs. (1) and (2) are the usual incompressible NavierStokes equations. F is the force density generated by the boundary on the fluid. The force exerted by the element of boundary on the fluid is f. The function δ

34

Ling Rao and Hongquan Chen

in the integrals is the two-dimensional Dirac delta function concentrate at the origin. Eq. (4) is equivalent to the no-slip condition that the fluid sticks to the boundary. In [3] and [13] the problem was supplemented with the following boundary and initial conditions: u(x, 0) = u0 (x)

∀x ∈ Ω,

(5)

X(s, 0) = X0(s)

∀s ∈ [0, L],

(6)

u(x,t) = 0

∀(x,t) ∈ ∂Ω × (0, T ).

(7)

When we perform the numerical simulation with the IB method the fluid domain Ω often can be chosen to have a simple big enough shape such as square or rectangle, then we can make space-periodic extension of the problem and suppose that the fluid satisfies space-periodic boundary condition. Such assumption is also useful for idealizations and needed in the following proofs. In this paper, we use the space-periodic boundary condition instead of (7). We use Ω to denote the space-period: L1 L1 L2 L2 Ω = (− , ) × (− , ). 2 2 2 2 We assume that the fluid fills the entire space R2 but with condition that u, F and p are periodic in each direction oxi , i = 1, 2, with corresponding periods Li > 0. It is sometimes useful and simpler to assume that average flow is zero, that is

1 |Ω|

Z Ω

u(x)dx = 0.

Moreover, the general case-where the volume forces and the initial condition do not average to zero-can be reduced to this case. We may refer to [4, 6] for the method. We show it as follows. First, averaging each term in (1). Using integration by parts and periodicity condition, several terms vanish, we get d dt Therefore if we set

Z Ω

u(x,t)dx =

1 Mu(t) = |Ω|

Z

Z Ω

Ω

F(x,t)dx.

u(x,t)dx

(8)

Immersed Boundary Method and

1 MF (t) = |Ω|

we obtain

thus

Z Ω

35

F(x,t)dx,

dMu(t) = MF (t), dt 1 Mu (t) = |Ω|

Z Ω

u0 (x)dx +

Z t 0

MF (t)dt.

Let uˆ = u − Mu , Fˆ = F − MF , R Iu(t) = 0t Mu(s)ds, ˆˆ u(x,t) = u(x ˆ + Iu (t),t), ˆp(x, ˆ t) = p(x + Iu (t),t), ˆF(x,t) ˆ ˆ + Iu (t),t). = F(x

(9)

ˆˆ and Fˆˆ are also periodic with period Ω and u, ˆˆ p, ˆˆ Fˆˆ have zero space Notice that u, ˆˆ we need to solve the ˆˆ pˆˆ and F, average. With u, p and F in (1)-(6) replaced by u, ˆ ˆˆ pˆ and X: following equations instead of (1)-(6) for u, ∂u − µ∆u + u · ∆u + ∆p = Fˆˆ ∂t

in Ω × (0, T ),

(10)

∆·u = 0

in Ω × (0, T ),

(11)

∀s ∈ [0, L],

(12)

∂X (s,t) = u(X(s,t) − Iu(t),t) + Mu (t) ∂t u(x, 0) = uˆˆ 0 (x), X(s, 0) = X0 (s)

∀x ∈ Ω,

(13)

∀s ∈ [0, L],

(14)

X(0,t) = X(L,t)

∀t ∈ (0, T ),

(15)

t ∈ (0, T ),

36

Ling Rao and Hongquan Chen

where for given f and u0 , uˆˆ 0 and Fˆˆ may be calculated by (3) and (9), 1 uˆˆ 0 (x) = u0 (x) − Mu (0) = u0 (x) − |Ω| ˆF(x,t) ˆ = F(x + Iu(t),t) − MF (t) =

Z L 0

−

Z

Ω

u0 (x)dx, (16)

f(s,t)δ(x + Iu(t) − X(s,t))ds

1 |Ω|

Z L

Z

f(s,t)(

0

Ω

δ(x − X(s,t))dx)ds.

ˆˆ p(= p) ˆˆ of (10)-(16) we can recover the solutions And with the solutions u(= u), u, p of (1)-(6) by (9). In order to obtain the existence of the approximate solution to equations (10)-(16), we review some properties of the Navier-Stokes equations with the space-periodic boundary conditions as follows.

3.

Review of the Mathematical Theory of the NavierStokes Equations with the Space-Periodic Boundary Conditions

We introduce some basic mathematical properties about Navier-Stoks equations with the space-periodic boundary conditions. For more details of them, the readers may refer to [4, 6]. We shall be concerned with the spaces of twodimensional vector functions. We use the notations Hm (Ω) = {H m (Ω)}2,

L2 (Ω) = {L2 (Ω)}2 ,

and we suppose that these product spaces are equipped with the usual product norm. The norm on L2 (Ω) is denoted by | · | (also denoted by k · k0 ). The norm on Hm (Ω) is denoted by k · km. (·, ·) stands for the scalar product on L2 (Ω). n We denote by Hmper (Ω), m ∈ N, the space of functions which are in Hm Loc (R ) (i.e. u|O ∈ Hm (O) for every open bound set O) and which are periodic with period Ω. For m = 0, H0per (Ω) is also denoted by L2per (Ω) and coincides simply with L2 (Ω) (the restrictions of the functions in H0per (Ω) to Ω are the whole space L2 (Ω)). For an arbitrary m ∈ N, Hmper (Ω) is a Hilbert space for the scalar product and the norm (u, v)m =

∑

Z

[α]≤m Ω

Dα u(x)Dαv(x)dx,

1

kukm = {(u, u)m } 2 ,

Immersed Boundary Method

37

∂[α] . α1 . . .∂αn ∂x√ We work with complex representation, for which we take i = −1. Then a square integrable vector field u = u(x) on Ω can be represented by the Fourier series expansion k u(x) = ∑ ck e2πi L ·x , where α = (α1, . . ., αn), αi ∈ N, [α] = α1 + . . . + αn and Dα =

k∈Zn

where

k1 k2 k = ( , ). L L1 L2

The functions in Hmper (Ω) are easily characterized by their Fourier series expansion Hmper (Ω) = {u, u =

∑

k

ck e2πi L ·x

ck = c−k ,

k∈Zn

∑

|k|2m|ck |2 < ∞},

(17)

k∈Zn 1

and the norm kukm is equivalent to the norm {∑k∈Zn (1 + |k|2m )|ck |2} 2 . We denote by V the space of smooth( C ∞) divergence-free vector fields on R2 that are periodic with period Ω. Let H be the closure of V in L2 (Ω) and let V be the closure of V in H1 (Ω). The space H is equipped with the scalar product (·, ·) induced by L2 (Ω); the space V with the scalar product Z

2

∂u ∂v

∑ ∂xi · ∂xi dx Ω

((u, v)) =

i=1

and the associated norms, denoted by 1

|u| = (u, u) 2 for u ∈ H,

1

kuk = ((v, v)) 2 for v ∈ V.

For the sake of simplicity, we restrict ourselves to the space-periodic case with vanishing space average. For the vanishing space average case, we have the additional condition Z c0 = u(x)dx = 0. Ω

Then H = {u ∈ L2per (Ω); ∇ · u = 0, V = {u ∈

H1per (Ω);

∇ · u = 0,

Z Ω

Z

Ω

˙ u(x) = 0}(, H); u(x) = 0}(, V˙ );

38

Ling Rao and Hongquan Chen

and ˙ mper (Ω)). Hmper (Ω) = {u ∈ Hmper (Ω) of type (17), c0 = 0}(, H The functions in H˙ and V˙ also can be characterized by there Fourier series expansion

∑

H˙ = {u, u =

k

ck e2πi L ·x ,

ck = c−k ,

k∈Z2 \{0}

V˙ = {u, u =

∑

k

ck e2πi L ·x ,

k · ck = 0, L

∑

|ck |2 < ∞};

k∈Z2 \{0}

k · ck = 0, L

k | |2 |ck |2 < ∞}. L k∈Z2 \{0}

k · ck = 0, L

k | |2s |ck |2 < ∞}. L k∈Z2 \{0}

ck = c−k ,

k∈Z2 \{0}

∑

For all s ∈ R we may consider the space ˙ sper (Ω), divu = 0} = Vs = {u ∈ H {u, u =

∑

k

ck e2πi L ·x ,

ck = c−k ,

k∈Z2 \{0}

∑

˙ sper (Ω). Vs1 ⊂ Vs2 for s1 ≥ s2 , V1 = V˙ and Note that, Vs is a closed subspace of H ˙ Hence, V˙ ⊂ Vs ⊂ H˙ for 0 ≤ s ≤ 1,Vs ⊂ V˙ for s ≥ 1, and Vs ⊃ H˙ for V0 = H. s < 0. It can be shown that Vs is a Hilbert space for the norm 1 k | |2s |ck |2) 2 . L k∈Z2 \{0}

kukVs = (

∑

Of particular interest are the spaces V2 and V−1. The space V−1 is the dual space of V˙ , usually denoted V 0 ; this is the space of linear continuous forms on V˙ . More generally, for all s ≥ 0,V−s is the dual of Vs. We have H ⊥ = {u ∈ L2 (Ω); u = ∇p, p ∈ H1per (Ω)}. We definite the so-called Leray projector by PL : L2 (Ω) → H, which is the orthogonal projector onto H in L2 (Ω). We definite the Stokes operator A by Au = −PL 4u

˙ 2per (Ω), ∀u ∈ D(A) = V˙ ∩ H

Immersed Boundary Method

39

where 4 is the Laplacian. And we can see that ˙ 2per (Ω). ∀u ∈ D(A) = V˙ ∩ H

−PL 4u = −4u

The Stokes operator is a positive self-adjoint operator, so we can work with fractional powers of A. A is just the mapping u=

∑

k

ck e2πi L ·x → Au =

k∈Z2 \{0}

k k | |2ck e2πi L ·x. L k∈Z2 \{0}

∑

Ar u is the operator u=

∑

k

ck e2πi L ·x → Ar u =

k∈Z2 \{0}

k k | |2r ck e2πi L ·x . L k∈Z2 \{0}

∑

It is straightforward to see that Ar maps V2s continuously onto V2s−2r (s, r ∈ R). In particular, for s ≥ 0, we have AsV2s = H˙ and so V2s = D(As ) is the domain of ˙ The norm |As u|(= kukV2 s ) is equivalent to the (unbounded) operator As in H. 2s ˙ the norm induced by H per (Ω), ckuk2s ≤ |Asu| ≤ c0 kuk2s

∀u ∈ D(As),

(18)

with positive constants c, c0 depending on L1 , L2 and s. We use some notations as follows. Let a, b be two extended real numbers, −∞ ≤ a < b ≤ +∞, and let X be a Banach space. For given α, 1 ≤ α < +∞, Lα (a, b; X) denotes the space of Lα -integrable functions from [a,b] into X, which is equipped with the Banach norm |

Z b a

k f (t)kαX dt | α . 1

The space C([a, b]; X) is the space of continuous functions from [a, b] (−∞ < a < b < ∞) into X and is equipped with the Banach norm sup k f (t)kX . t∈[a,b]

For u, v, w ∈ L1 (Ω), we set 2

b(u, v, w) =

∑

Z

i, j=1 Ω

ui (Di v j )w j dx,

40

Ling Rao and Hongquan Chen

whenever the integrals make sense. The trilinear operator b = b(u, v, w) can be extended to a continuous trilinear operator defined on V . Moreover, b(u, v, v) = 0 b(u, v, w) = −b(u, w, v)

u, v ∈ V. u, v, w ∈ V.

(19)

In the periodic case, ˙ 2per (Ω) ∩ V˙ . b(v, v, Av) = 0 v ∈ D(A) = H Now we consider the Navier-Stokes equations ∂u − µ∆u + u · ∆u + ∆p = f, ∂t ∆ · u = 0,

(20)

(21)

in

L1 L1 L2 L2 , ) × (− , ), 2 2 2 2 with µ > 0 and require that Ω = (−

L1 , L2 > 0,

u = u(x,t), p = p(x, t), and f = f(x,t) with x = (x1 , x2 ), are Li -periodic in each variable xi and

Z Ω

f(x,t)dx =

Z Ω

u(x,t)dx = 0.

Let T > 0 be given. Assume that u ∈ C2 (R2 × [0, T ]), p ∈ C1(R2 × [0, T ]) are the classical solutions of (21). For each v ∈ V, by multiplying momentum equation in (21) by v, integrating over Ω and using the Stokes formula we find that (cf. [5] Chap.III for the details) d (u, v) + µ((u, v)) + b(u, u, v) = hf, vi dt

∀ v ∈ V.

(22)

By continuity, (22) holds also for each v ∈ V . This suggests the following variational formulation of (21) ( strong solutions). For f and u0 given, ˙ f ∈ L2 (0, T ; H),

Immersed Boundary Method

41

u0 ∈ V˙ , find u satisfying

u ∈ L2 (0, T ; D(A) ∩ L∞(0, T ; V˙ )

such that ( d

(u, v) + µ((u, v)) + b(u, u, v) = hf, vi dt u(0) = u0 .

∀ v ∈ V, t ∈ (0, T ),

(23)

If f is square integrable but not in H then we can replace it by its Leray projection on H, so that f is always assumed to be in H. We may refer to [5], p.307 for the relation of equations (21) and its variational formulation (23). One can show that if u is a solution of (23) then there exists p such that (21) is satisfied in a weak sense. Lemma 1. (Existence and Uniqueness of Strong Solutions in Two Dimensions) [4, 6] (i) Assume that u0 , f, T > 0 are given and satisfy ˙ f ∈ L2 (0, T ; H).

u0 ∈ V˙ ,

Then there exists a unique solution u of (23), satisfying ui ,

∂ui ∂ui ∂2 ui , , ∈ L2 (Ω × (0, T )), ∂t ∂x j ∂x j ∂xk

i, j, k = 1, 2,

and u is a continuously function from [0,T] into V˙ . We also can see that ˙ u ∈ L2 (0, T ; D(A)) ∩ C([0, T], V). And u satisfies the following priori estimates 1 |u(s)| ≤ |u(0)| + µ 2

2

2

Z s

sup ku(t)k ≤ K1 , t∈[0,T ]

0

kf(t)kV2 0 dt Z T 0

∀s ∈ [0, T ],

|Au(t)|2dt ≤ K2, R

where constants K 1 , K2 are dependent on ku0 k, 0T |f(t)|2dt, µ, L1 , L2 . (ii) For r ≥ 1, if the initial data u0 ∈ Vr and f ∈ L∞ (0, T ;Vr−1), then the solution

42

Ling Rao and Hongquan Chen

u = u(t) belongs to C([0, T];Vr ). And there exists constant K dependent on µ, ku0 kr , |f|L∞ (0,T ;Vr−1 ) , such that sup ku(t)kr ≤ K.

(24)

t∈[0,T ]

Gronwall’s Lemma: (A)[7] Let a ∈ L1 (τ, T ), b be absolutely continuous on [τ, T ]. If x ∈ L∞ (τ, T ) satisfies Z t

x(t) ≤ b(t) +

a(s)x(s)ds,

τ

then for t ∈ (0, T ) x(t) ≤ b(τ)exp

Z t τ

a(s)ds +

Z t τ

b0 (s)exp

Z

t

a(ρ)dρ ds.

s

(B)[4] Let a, θ ∈ L1 (0, T ). If y satisfies y0 ≤ a + θy, then for t ∈ (0, T ) y(t) ≤ y(0)exp(

Z t 0

θ(τ)dτ) +

Z t

Z t

a(s)exp(

0

s

θ(τ)dτ)ds.

Lemma 2. Assume that 0 < T ≤ Te, c is a positive constant and u0 , f1 , f2 satisfy 2

u0 ∈ V˙ ,

˙ fi , ∈ L (0, T ; H),

Z T 0

|fi (t)|2dt ≤ c

f or i = 1, 2.

Suppose that u1 , u2 are the two solutions to (23) corresponding forcing terms f1 , f2 respectively. Then we have the following estimate Z T 0

ku1 (t) − u2 (t)k22dt

≤ c1

Z T 0

|f1(t) − f2 (t)|2dt

where constant c 1 is dependent on µ, ku0k, Te, c.

Immersed Boundary Method Proof

43

According to the assumptions and Lemma 1, we have u1 , u2 ∈ L2 (0, T ; D(A))

and 2

sup ku1 (t)k ≤ K1 ,

Z T

t∈[0,T ]

0

|Au1(t)|2dt ≤ K2 ,

where constants K1 , K2 are dependent on µ, ku0 k, c. By assumption, we have d dt ((u1 − u2 )(t), v) + µ((u1 − u2 , v)) + b(u1, u1, v) ∀ v ∈ V, t ∈ (0, T ), −b(u2, u2 , v) = hf1 − f2 , vi (u1 − u2 )(0) = 0.

(25)

(26)

Let u = u1 − u2 , f = f1 − f2 . Using (19), we have b(u1 , u1, u1 − u2 ) − b(u2 , u2, u1 − u2 ) = −b(u1 , u1, u2 ) + b(u2 , u1, u2 ) = −b(u, u1, u2 ) = b(u, u1 , u1 ) − b(u, u1 , u2) = −b(u, u, u1 ) Replacing v by u in (26), we obtain 1d |u(t)|2 + µkuk2 − b(u, u, u1 ) = hf, ui ∀t ∈ (0, T ). 2 dt

(27)

Using Poicare inequality (see [6]), we obtain hf, ui ≤ |f||u| ≤

1 1 2

|f|kuk ≤

λ1

1 µ |f|2 + kuk2. µλ 1 4

(28)

And by the following inequality (see[4], page 13) 1

1

|b(u, u, u1)| ≤ k|u|kukku1k 2 |Au1 | 2 , we obtain

µ k2 |b(u, u, u1)| ≤ kuk2 + ku1 k|Au1 ||u|2. 4 µ

(29)

By (27),(28),(29) we obtain, for t ∈ [0, T ], 2 2k2 d |u(t)|2 + µkuk2 ≤ ku1 k|Au1||u|2. |f|2 + dt µλ 1 µ

(30)

44

Ling Rao and Hongquan Chen

By Gronwall’s Lemma (B) and (25), we deduce that, for t ∈ [0, T ], |u(t)|2 ≤

Z t 2

(

0

µλ 1

|f(s)|2)(exp

Z t 2 2k

µ

s

then sup |u(t)|2 ≤ K3 0∈[0,T ]

Z T

ku1 (τ)k|Au1(τ)|dτ)ds,

|f(s)|2ds,

(31)

0

where constant K3 is dependent on µ, ku0k, c. Replacing v by Au in (26), we have, for t ∈ (0, T ), 1d ku(t)k2 + µ|Au|2 + b(u1 , u1 , Au) − b(u2, u2 , Au) = hf(t), Aui. 2 dt Since b(u1, u1 , Au) − b(u2, u2 , Au) = b(u1 , u1, Au) − b(u2 , u1, Au) + b(u2 , u1 , Au) − b(u2, u2 , Au) = b(u, u1 , Au) + b(u2, u, Au), then 1d ku(t)k2 + µ|Au|2 + b(u, u1 , Au) + b(u2 , u, Au) = hf(t), Aui. 2 dt

(32)

And using (20), we obtain the following equality (see [6], page 135) 1d ku(t)k2 + µ|Au|2 = b(u, u, Au1) + hf(t), Aui. 2 dt

(33)

Notice that

1 µ hf, Aui ≤ |f||Au| ≤ |f|2 + |Au|2 . µ 4 And by the following inequality (see[4], page 13) 1

(34)

1

|b(u, u, Au1)| ≤ k|u| 2 |Au| 2 kuk|Au1|, we obtain

µ 1 k4 |b(u, u, Au1)| ≤ |Au|2 + |Au1 |2 kuk2 + |u|2. 4 2 4µ By (33),(34),(35), we deduce that d k4 2 ku(t)k2 + µ|Au|2 ≤ |Au1|2 kuk2 + |u|2 + |f|2. dt 2µ µ

(35)

(36)

Immersed Boundary Method

45

By Gronwall’s Lemma (B), we obtain, for t ∈ [0, T ], 2

ku(t)k ≤ (

Z t 4 k 0

2 ( |u(s)| + |f(s)|2)exp( 2µ µ

Z t s

|Au1(τ)|2dτ)ds.

Then Z T

sup ku(t)k2 ≤ exp(

0

t∈[0,T ]

Z T 4 k

|Au1(τ)|2dτ)(

(

0

2 |u(s)| + |f(s)|2)ds). 2µ µ

By (25), (31), there exists constant K4 dependent on µ, ku0 k, c, Te, such that sup ku(t)k2 ≤ K4

Z T

t∈[0,T ]

|f(s)|2ds.

(37)

0

Then by integration in t of (36) from 0 to T, after dropping unnecessary terms we obtain Z Z T

0

that is

Z T 0

|Au(t)|2dt ≤ K5

|Au1(t) − Au2 (t)|2dt ≤ K5

T

|f(t)|2 dt,

0

Z T 0

|f1 (t) − f2 (t)|2dt,

(38)

where constant K5 is dependent on µ, ku0 k, c, Te. Then using (18) with s=1, we obtain Z Z T

0

ku1 (t) − u2 (t)k22dt ≤ c1

T

0

|f1(t) − f2 (t)|2dt

where constant c1 is dependent on µ, ku0 k, c, Te.

4.

Variational Formulation of the Problem

C[0, L] denotes the space of continuous functions from [0, L] into R2 and is equipped with the norm kxkc = max |x(s)|. 0≤s≤L

In what follows, we always suppose that Te > 0 is a given constant, and constant T ∈ (0, Te]. Let G = L([0, T ]; C[0, L]);

E = {x ∈ C[0, L] : x(0) = x(L)};

46

Ling Rao and Hongquan Chen F = {x ∈ E|x(s) ∈ Ω, 0 ≤ s ≤ L},

where Ω = (−

L2 L2 L1 L1 , ) × (− , ). 2 2 2 2

Notice that F = {x ∈ E|x(s) ∈ Ω, 0 ≤ s ≤ L}. For x ∈ G, x(·,t) ∈ C[0, L], ∀ t ∈ [0, T ], x(·,t) is simply denoted by x(t). Given X0 ∈ F, let X = {X ∈ L([0, T ]; E) : X(0) = X0 }. X is a Banach subspace of G. Given δh : R2 → R2 , f : [0, L] × [0, Te] → R2, we assume that they satisfy Condition A: e δh ∈ H20 (R2 ), f ∈ C([0, L] × [0, T]).

In this paper, we let Dirac delta function δ be approximated by differentiable function δh . In fact, some authors do so when performing the numerical simulation in order to get the approximate solution of the problem (1)-(7). For example, δh in [9] is chosen as follows δh (x) = dh (x)dh(y), where

∀x ∈ R2

h πz i 0.25 1 + cos( ) |z| ≤ 2h, dh (z) = h 2h 0, |z| > 2h.

It is clear that δh ∈ H20 (R2). According to the theories and notations in Section 2, Section3, we introduce the corresponding variational formulation of the equations (10)-(16): Problem 1. Given u0 ∈ V, X0 ∈ F and δh : R2 → R2 , f : [0, L] × [0, Te] → R2 satisfying Condition A, find u ∈ L2 (0, T ; D(A)) ∩ L∞(0, T ; V˙ ), and X : [0, L] × [0, T ] → Ω, such that d ˆˆ (u(t), v) + µ((u, v)) + b(u, u, v) = hF(t), vi dt ∂X (s,t) = u(X(s,t) − Iu(t),t) + Mu(t) ∂t u(x, 0) = uˆˆ 0 (x) X(s, 0) = X0(s) X(0,t) = X(L,t)

∀ v ∈ V, t ∈ (0, T ), ∀ s ∈ [0, L],t ∈ (0, T ), ∀ x ∈ Ω, ∀s ∈ [0, L], ∀t ∈ (0, T ),

Immersed Boundary Method

47

where 1 uˆˆ 0(x) = u0 (x) − |Ω| ˆF(x,t) ˆ =

Z L 0

5.

Z Ω

u0 (x)dx,

1 f(s,t)δh(x + Iu (t) − X(s,t))ds − |Ω|

Z L

Z

f(s,t)(

0

Ω

δh (x − X(s,t))dx)ds.

Conclusion

In this section we shall prove the existence of the solution of Problem 1 via a fixed point argument. We define an operator T on X as follows. Given u0 ∈ V, X ∈ X. ∀x ∈ Ω,t ∈ (0, T ), let FX (x,t) =

Z L 0

f(s,t)δh(x − X(s,t))ds.

Hence Fˆˆ X (x,t) =

Z L 0

f(s,t)δh(x+IX (t)−X(s,t))ds −

1 |Ω|

Z L 0

Z

f(s,t)(

Ω

δh (x−X(s,t))dx)ds

where, according to (9), t IX (t) = |Ω| 1 MFX (t) = |Ω|

Z

Z Ω

u0 (x)dx +

1 FX (x,t)dx = |Ω| Ω

Z L 0

Z tZ s 0

0

MFX (τ)dτds,

Z

f(s,t)(

Ω

δh (x − X(s,t))dx)ds.

(39) (40)

˙ 2per (Ω)). If Condition A is satisfied, we can verify that Fˆˆ X ∈ L∞ (0, T ; H Then let u (also denoted by uX ) be the solution to the following problem: Find u ∈ L∞ (0, T ; V˙ ) such that d (u(t), v) + µ(∇u, ∇v) = hFˆˆ X (t), vi ∀v ∈ V, t ∈ (0, T ), (41) dt u(0) = uˆˆ . 0 Finally, Let X be the solution X ∈ X of 0 X (t) = u(X(s,t) − IX (t),t) + MuX (t)) ∀t ∈ (0, T ), X(0) = X0,

(42)

48

Ling Rao and Hongquan Chen

where

1 MuX (t)) = |Ω|

Z Ω

u0 (x)dx +

Z t 0

MFX (τ)dτ.

(43)

Then let T(X) = X. We recall that the definition of X is X = {X ∈ L([0, T ]; E) : X(0) = X0 }. For t ∈ [0, T ], X(t) = X(·,t) ∈ E. (42) is an ordinary equation in the Banach space E and equivalent to ∂X (s,t) = u(X(s,t) − IX (t),t) + MuX (t)) ∀s ∈ [0, L], ∀t ∈ (0, T ), ∂t ∀s ∈ [0, L], X(s, 0) = X0 (s) X(0,t) = X(L,t) ∀t ∈ (0, T ). e in X, then this fixed point and the We observe that, if T has a fixed point X e give the solutions to Problem 1. solution e u to (41) corresponding to X = X, The following lemma guarantees the existence of the solution to (42): Lemma 3. (see [1]) Let x : [t0,t0 + a] → Y be a mapping into the Banachspace Y. Consider the initial value problem dx = f (x,t), dt

x(t0) = x0 .

(44)

Let Q = [t0,t0 + a] ×Y . Suppose f : Q → Y is continuous and satisfies k f (t, x) − f (t, y)k ≤ Lkx − yk,

f or all

(t, x), (t, y) ∈ Q, and f ixed L ≥ 0.

Then initial problem (44) has exactly one continuously differential solution on [t0,t0 + a] for each initial value x 0 ∈ Y. Theorem 1. Assume that T ∈ (0, Te], u0 ∈ H3per (Ω) ∩ H, X0 ∈ F and Condition A holds. (a) For given X ∈ X, (41) has a unique solution u ∈ L([0, T ], C1(Ω)). Furthermore there exist constants c 1 , c2, c3 dependent on e µ, T , δh , |fkC([0,L]×[0,Te]) , ku0kV3 , such that sup |u(z,t)| ≤ c1 < ∞, z∈Ω 0≤t≤T

(45)

Immersed Boundary Method sup |Du(z,t)| ≤ c2 < ∞,

49 (46)

z∈Ω 0≤t≤T

sup |u(z,t) + MuX (t)| ≤ c3 < ∞. z∈Ω 0≤t≤T

(b) Suppose that u is the solution of (41). Then (42) has a unique solution X ∈ X. 0 kc e , T } holds, where L m = min{L1 , L2}, then Furthermore, if 0 < T < min{ Lm −kX c3

X(s,t) ∈ Ω, for all (s,t) ∈ [0, L] × [0, T ].

Proof: (a) Since u0 ∈ H3per (Ω) ∩ H, then uˆˆ 0 ∈ V3. Since Condition A is satisfied, ˙ 2per (Ω)) and PL Fˆˆ X ∈ L∞ (0, T ;V2). We can verify that there then Fˆˆ X ∈ L∞ (0, T ; H exists constant C dependent on δh , such that kPL Fˆˆ X kL∞ (0,T ;V2 ) ≤ kFˆˆ X kL∞ (0,T ;H˙ 2per (Ω)) ≤ CkfkC([0,L]×[0,Te]) .

(47)

By Lemma 1 (ii) with r = 3, there exists u belonging to C([0, T];V3) satisfying d (u(t), v) + µ(∇u,∇v) = hPL Fˆˆ X (t), vi ∀v ∈ V, t ∈ (0, T ), dt u(0) = uˆˆ . 0 Since PL is the orthogonal projector onto H in L2 (Ω), we have hPL Fˆˆ X (t), vi = hFˆˆ X (t), vi ∀v ∈ V. Then u is the solution of (41). Since u belonging to C([0, T];V3 ) and V3 is a ˙ 3per (Ω), we obtain that u ∈ L([0, T ], C1(Ω)) due to (18) closed subspace of H and Sobolev embedding theorem (see [7]). Furthermore by (24) and (47) there exist constants c1 , c2, c3 dependent on µ, Te, δh , |fkC([0,L]×[0,Te]) , ku0 k3 , such that sup |u(z,t)| ≤ c1 < ∞, z∈Ω 0≤t≤T

sup |Du(z,t)| ≤ c2 < ∞, z∈Ω 0≤t≤T

50

Ling Rao and Hongquan Chen sup |u(z,t) + MuX (t)| ≤ c3 < ∞. z∈Ω 0≤t≤T

(b) We shall prove the conclusion by Lemma 3 with Y = E, f = u, x0 = X0,t0 = 0. Below we verify that the conditions in Lemma 3 hold. First we prove that u : E × [0, T ] → E is continuous. Given (x0 ,t0) ∈ E × [0, T ], (x,t) ∈ E × [0, T ]. It is easy to see u(x,t) ∈ E, u(x0,t0) ∈ E. We have ku(x,t) − u(x0,t0)kc = sup |u(x(s),t) − u(x0 (s),t0)| 0≤s≤L

≤ sup |u(z,t) − u(z,t0)| z∈Ω

+ sup |u(z,t0) − u(z0,t0)|. z∈Ω

Then u(x,t) is continuous at (x0 ,t0) due to u ∈ L([0, T ], C1(Ω)). For t ∈ [0, T ], x, y ∈ E, by (46) we have ku(x,t) − u(y,t)kc = sup |u(x(s),t) − u(y(s),t)| 0≤s≤L

≤ sup |Du(z,t)|kx − ykc ≤ c2 kx − ykc . z∈Ω 0≤t≤T

Hence the conditions in Lemma 3 hold. (42) has a unique solution X ∈ X. By (42), we have X(t) = X0 +

Z t 0

(u(X(s, τ) − IX (τ), τ) + MuX (τ))dτ.

Notice that X0 ∈ F, then Lm − kX0 kc > 0. If 0 < T < t ∈ [0, T ], we have

Lm −kX0 kc c3

holds, for

kX(t)kc ≤ kX0 kc + t sup |u(z, τ) + MuX (τ)| < kX0 kc + c3

z∈Ω 0≤τ≤T Lm −kX0 kc

c3

Then X(t) ∈ F. That is, if 0 < T

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