Perspectives in Magnetohydrodynamics Research (Earth Sciences in the 21st Century: Space Science, Exploration and Policies)

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 <

= Lm .

Lm −kX0 kc c3

holds, for all (s,t) ∈ [0, L] × [0, T ],

X(s,t) ∈ Ω.  Follow the above argument, we can prove the main conclusion: Theorem 2. Assume that u0 ∈ H3per (Ω) ∩ H, X0 ∈ F and Condition A holds. Then there exists T 00 ∈ (0, Te] dependent on µ, Te, δh , f, u0, Lm , kX0kc , such that if

Immersed Boundary Method

51

e And X e may be T ∈ (0, T 00 ), Problem 1 has a unique pair of solutions e u and X. solved by successive approximations Yn+1 = TYn

∀Y0 ∈ B n = 0, 1, 2, . . . ,

where B = {X ∈ X | X(t) ∈ F

∀t ∈ [0, T ]}.

0 kc , Te}. By TheoProof: Below we take T satisfying 0 < T < min{ Lm −kX c3 rem 1 (b), we can observe that T maps B into itself. The set B is a bounded closed convex subset of Banach space X. We will prove that T : B → B is a contractive map, and then by the Banach Fixed-Point Theorem (see [1]), T has a unique fixed point. Arbitrarily given two elements X, Y ∈ B, we denote the corresponding solutions of (41) by uX and uY respectively. Then uX − uY solves the following equation  d ˆ ˆ    dt ((uX − uY )(t), v) + µ(∇(uX − uY )(t), ∇v) = hFˆ X (t) − Fˆ Y (t), vi, ∀v ∈ V, t ∈ (0, T ),    (uX − uY )(0) = 0.

By the definitions of Fˆˆ X , Fˆˆ Y and using Condition A, we can deduce that there exists a constant k dependent on f, δh , such that Z T 0

kFˆˆ X (t)k20dt ≤ k,

Z T 0

kFˆˆ Y (t)k20dt ≤ k.

Then according to Lemma 2, we have Z T 0

kuX (t) − uY (t)k22dt ≤ c0

Z T 0

kFˆˆ X (t) − Fˆˆ Y (t)k20dt

(48)

where constant c0 is dependent on µ, ku0 k, Te, f, δh . By the definitions of Fˆˆ X , Fˆˆ Y and using Condition A, we can deduce that there exists a constant l1 dependent on f, δh , Te, such that, for each t ∈ (0, T ), kFˆˆ X (t) − Fˆˆ Y (t)k20 ≤ l1 sup |Y(s,t) − X(s,t)|2 = l1kY(t) − X(t)k2c . 0≤s≤L

(49)

52

Ling Rao and Hongquan Chen

By (48) and (49), Z T 0

kuX (t) − uY (t)k22dt ≤ c0 l1

Z T 0

kX(t) − Y(t)k2cdt.

Let Xe = TX and Ye = TY. According to (42), we have ( e − IX (t),t) + MuX (t) ∀t ∈ (0, T ), e 0 (t) = uX (X(t) X e X(0) = X0 , (

e − IY (t),t) + MuY (t) ∀t ∈ (0, T ), e 0 (t) = uY (Y(t) Y e Y(0) = X0 ,

(50)

(51)

(52)

Subtracting (51) from (52) we obtain e 0 (t) e 0 (t) − Y X e − IX (t),t) − uY(Y(t) e − IY (t),t) + MuY (t) − MuX (t) = uX (X(t) e e − IX (t),t) = uX (X(t) − IX (t),t) − uY(X(t) e − IX (t),t) − uY(Y(t) e − IY (t),t) +uY (X(t) +MuY (t) − MuX (t) Integrating from 0 to t in the two sides of above equation, we have e − Y(t) e = X(t) +

Z t 0

Z t 0

e − IX (τ), τ) − uY (X(τ) e − IX (τ), τ)]dτ [uX (X(τ)

e − IX (τ), τ) − uY (Y(τ) e − IY (τ), τ)]dτ + [uY (X(τ)

Z t 0

[MuY (τ) − MuX (τ)]dτ.

Hence e − Y(t)k e kX(t) c≤ +

Z t

+

Z0 t 0

Z t 0

e − IX (τ), τ) − uY (X(τ) e − IX (τ), τ)kcdτ kuX (X(τ)

e − IX (τ), τ) − uY (Y(τ) e − IY (τ), τ)kcdτ kuY (X(τ) |MuY (τ) − MuX (τ)|dτ.

(53)

Immersed Boundary Method

53

By Sobolev embedding theorem, there exists a constant l2 such that e − IX (τ), τ) − uY (X(τ) e − IX (τ), τ)kc kuX (X(τ) e τ) − IX (τ), τ) − uY (X(s, e τ) − IX (τ), τ)| = sup |uX (X(s, 0≤s≤L

≤ sup |uX (z, τ) − uY (z, τ)| z∈Ω

≤ l2kuX (τ) − uY (τ)k2.

(54)

By (50) and (54), there exists a constant l3 , dependent on µ, f, δh , Te, ku0 k, such that for each t ∈ [0, T ], Rt

e

e

0 kuX (X(τ) − IX (τ), τ) − uY (X(τ) − IX (τ), τ)kcdτ

≤ l2

Rt

0 kuX (τ) − uY (τ)k2dτ

≤ l2t 1/2

R

≤ l3t 1/2

R

T 0 T 0

kuX (τ) − uY (τ)k22dτ kX(t) − Y(t)k2cdt



1/2

(55)

1 2

≤ l3T sup kX(t) − Y(t)kc. 0≤t≤T

By (46), there exists c2 dependent on µ, Te, δh , f, u0 , such that sup |DuY (z,t)| ≤ c2 . z∈Ω 0≤t≤T

Hence for τ ∈ [0, T ], e − IX (τ), τ) − uY(Y(τ) e − IY (τ), τ)kc kuY (X(τ) e e τ) − IY (τ), τ)| = sup |uY(X(s, τ) − IX (τ), τ) − uY (Y(s, 0≤s≤L

e − Y(τ)k e ≤ sup |DuY ((z, τ)|(kX(τ) c + |IY (τ) − IX (τ)|) z∈Ω 0≤τ≤T

e − Y(τ)k e ≤ c2 kX(τ) c + c2 |IY (τ) − IX (τ)|.

(56)

54

Ling Rao and Hongquan Chen

By (53), (55) and (56), for t ∈ [0, T ], we have e − Y(t)k e kX(t) ≤ l3T sup kX(t) − Y(t)kc + c2 +c2

Z 0≤t≤T t

|IY (τ) − IX (τ)|dτ +

0

Z t 0

Z t 0

e − Y(τ)kdτ e kX(τ)

|MuY (τ) − MuX (τ)|dτ.

By (39, (40),(43) and using Condition A, we can deduce that there exists constant l4 dependent on f, δh , Te , such that, for τ ∈ [0, T ], c2 |IY (τ) − IX (τ)| + |MuY (τ) − MuX (τ)| ≤ l4 sup kX(t) − Y(t)kc. 0≤t≤T

Then for t ∈ [0, T ], c2

Z t 0

|IY (τ) − IX (τ)|dτ +

Z t 0

|MuY (τ) − MuX (τ)|dτ ≤ l4T kX(t) − Y(t)kc.

Hence there exists constant l5 dependent on µ, f, δh , Te, u0 , such that e − Y(t)k e kX(t) ≤ l5T sup kX(t) − Y(t)kc + c2

Z t

0≤t≤T

0

e − Y(τ)kdτ. e kX(τ)

By Gronwall’s Lemma (A), e − Y(t)k e kX(t) ≤ l5Tec2 T sup kX(t) − Y(t)kc. 0≤t≤T

(57)

The inequality (57) implies for each constant l6 ∈ (0, 1), there exists T 0 dependent on µ, Te, δh , f, u0, such that when 0 < T ≤ T 0 , e

l5Tec2T ≤ l5Tec2 T ≤ l6, e − Y(t)k e kX(t) ≤ l6 sup kX(t) − Y(t)kc

0 ≤ t ≤ T.

0≤t≤T

Let kxkC denote the norm in X, kxkC = sup max |x(s, t)|. 0≤t≤T 0≤s≤L

Then e − Y(t)k e sup kX(t) ≤ l6 kX − YkC ,

0≤t≤T

Immersed Boundary Method

55

kTX − TYkC ≤ l6kX − YkC .

(58)

that is L −kX k

Let T 00 = min{T 0 , m c3 0 c , Te}. Notice that T 00 is dependent on µ, Te, δh, f, u0 , Lm , kX0 kc. The inequality (58) implies that if T ∈ (0, T 00 ), T : B → B is a contractive map. By the Banach Fixed-Point Theorem (see [1]), T e ∈ B, and X e may be solved by successive approximahas a unique fixed point X tions Yn+1 = TYn , ∀Y0 ∈ B, n = 0, 1, 2, . . . . e Then e e is the e be the unique solution to (41) corresponding X. Let u u and X unique pair of solutions we need. 

Remark In our conclusion we only obtain the existence of the approximate solution with Dirac function approximated by differentiable function. How the solution of the original problem is approximated by this approximate solution is still an open problem.

Acknowledgment This work is partially supported by National Natural Science Foundation of China(No: 10671092).

References [1] Ebhard Zeidler. Nonlinear functional analysis and its applications, I: Fixed-point theorems. Berlin: Springer-Verlag, 1986. [2] Peskin C.S. The immersed boundary method, Acta Numerica . Cambridge University Press, 2002. [3] Daniele Boffi, Lucia Gastaldi. A finite element approach of the immersed boundary method. Computers and Structures , 2003, Vol.81, 491-501. [4] Roger Temam. Navier-Stokes equations and nonlinear functional analysis (CBMSNSF Regional Conf. Ser. in Aplied Mathematics 66), SIAM, Philadelphia, 1983.

56

Ling Rao and Hongquan Chen

[5] Roger Temam. Navier-Stokes equations, theory and numerical analysis, Studies in Mathematics and its applications . volume 2, North-Holland Pubishing Company 1977. [6] C.Foias, O.Manley, R.Rosa, Roger Temam. Navier-Stokes equations and turbulence. Cambridge University Press 2001. [7] Ruth F.Curtain, A.J.Pritchard. Functional Analysis in Modern Applied Mathematics. Academic Press INC. (London) LTD. 1977. [8] C.S. Peskin. Numerical analysis of blood flow in the heart, J. Comp. Phys. 1977, Vol.25, 220-252. [9] Santos Alberto Enriquez-Remigio and Alexandre Megiorin Roma. Incompressible flows in elastic domains: an immersed boundary method approach. Applied Mathematical Modeling , January 2005, Volume 29, Issue 1, 35-54. [10] Lima, A.L.F., Silva, E., Silveira-Neto A.,& Damasceno J.J.R. Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method. Journal of Computational Physics , 2003, Vol.189, 351-370. [11] Arthurs, K.M., Moore, L.C., Peskin, C.S., Pitman, E.B., &Layton, H.E. Modeling arteriolar flow and mass transport using the immersed boundary method. J. Comp. Physiol. 1998, Vol.147, 402-440. [12] Roma, A.M., Peskin, C.S., & Berger, M.J. An adaptive version of the immersed boundary method. J. Comp. Physiol., 1999, Vol.153, 509-534 [13] Ling Rao and Hongquan Chen. Immersed boundary method: The existence of approximate solution of the two-dimensional heat equation. Nonlinear Analysis: Real World Applications , April 2008, Volume 9, Issue 2, 384-393.

Reviewed by Professor Yuanguo Zhu, Department of Applied Mathematics, Nanjing University of Science and Technology.

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

Chapter 3

TRANSIENT HYDROMAGNETIC NATURAL CONVECTION BETWEEN TWO VERTICAL WALLS HEATED/COOLED ASYMMETRICALLY R.K. Singh and A.K. Singh* Department of Mathematics, Banaras Hindu University, Varanasi, India

Abstract 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.

*

E-mail address: [email protected]

58

R.K. Singh and A.K. Singh

Keywords - Transient, natural convection, asymmetric heating, buoyancy force distribution, Hartman number.

1. Introduction In the laboratory many new devices have been made which utilize the MHD interaction directly, such as population units and power generators or which involve fluid-electromagnetic field interactions, such as electron beam dynamics, traveling wave tubes, electrical discharges, and many others. When the heat transfer process is initiated some time must elapse before the steadystate is reached. During this transient period, the temperature changes and analysis must take into account changes in the associated internal energy. Transient heat flow is of great practical importance in industrial heating and cooling. In addition to unsteady heat flow when the system undergoes a transition from one steady state to another , there are also many engineering problems involving periodic variation in heat flow in a building between day and night and in regenerators, where the matrix is alternative heating and cooling. Ostrach [1] has extensively presented steady laminar free-convective flow of a viscous incompressible fluid between two vertical walls. Ostrach [2] and Sparrow et al [3] have studied the combined effects of a steady free and forced convective laminar flow and heat transfer between the vertical walls. Latter on, Poots [4] and Osterle et al [5] have investigated the free convective flow between two heated vertical plates in the presence of an uniform magnetic field applied perpendicular to walls for the open and short circuit cases respectively. Sparrow and Cess [6] studied the effect of magnetic field on the natural convection heat transfer processes of the electrically conducting fluids. Anug et al [7] and Anug [8] have presented their results for a steady free convective flow between vertical walls by considering different physical situations of the transport processes. The result from a numerical study of transient natural convection flow between two vertical parallel walls has been present by Joshi [9] by considering uniform temperature and uniform heat flux on the walls. Nelson and Wood [10, 11] have presented in detail the combined effect of heat and mass transfer processes on a free convective flow between vertical plates with symmetric and asymmetric boundary conditions on the plates. Singh [12] and Singh et al [13] have studied the flow behavior of transient free convective flow of a viscous incompressible fluid in a vertical channel when one of the channel walls is moving impulsively and the walls

59

Transient Hydromagnetic Natural Convection …

are heated asymmetrically. Paul et al [14] have studied the transient free convective flow in a vertical channel having a constant temperature and constant heat flux on the channel walls. Jha et al [15] have presented an analytical solution for the transient free convection flow in a vertical channel as a result of asymmetric heating. Singh et al [16] have recently studied the transient natural convection between vertical walls heating asymmetrically. Several hydromagnetic free-convection solutions were obtained by Chandran et al [17, 18, 19, and 20], Singh et al [21] and Takhar et al [22]. In the present paper, the unsteady magnetohydrodynamic free convective flow of a viscous, incompressible and electrically conducting fluid is studied due to a change in the temperature of the walls as compared to the fluid temperature in the presence of an external uniform magnetic field directed perpendicular to the walls. The analytical solution obtained by the Laplace transform method in the present work enabled to investigate the effects of parameters such as buoyancy force distribution parameter and Hartman number

on kinematics and skin friction of the fluid.

2. Mathematical Formulation Here, an unsteady free convective flow of an incompressible, viscous and electrically conducting fluid between two vertical walls is taken when the temperature of the fluid is different than that of the walls. The co-ordinate system is chosen such that -axis measures the distance along the walls and -axis measures the distance normal to it and an uniform magnetic field is applied perpendicular to the walls. Initially the temperature of the fluid as well as the walls are same as at

and

. At the time

, the temperature of the walls

is instantaneously raised or lowered to

and

respectively such that which is thereafter maintained constant. The electrodes are assumed to perfect conductors and the fluid has conductivity . The applied magnetic field is steady and uniform. The magnetic Reynolds number is taken small so that the induced magnetic field can be neglected. According to Boussinesq approximation, it is assumed that all the fluid properties are constant except that the influence of the density variation with the temperature is considered in the body force term. Using the non-dimensional quantities

60

R.K. Singh and A.K. Singh

,

, (1) the momentum and the thermal energy equations in non-dimensional form for the consider model are derived as follows:

(2)

(3) The initial and boundary conditions for the velocity and temperature fields in non-dimensional form relevant to fluid flow are obtained as follows: for

;

for

,

,

,

at

,

at

(4)

Additional dimensionless parameters appearing in the above equations are the Prandtl number, distribution parameter,

, Hartman number,

and buoyancy force

defined by ,

.

(5)

61

Transient Hydromagnetic Natural Convection …

Solution of equation (3) with its appropriate initial and boundary conditions has been obtained by Singh and Paul (2006). Application of the Laplace transform, in equation (2) under initial condition (4) has given the following equation:

(6) where and are the Laplace transformation of corresponding boundary conditions are

and

respectively. The

.

at

(7)

Solution of equation (6) with boundary conditions (7) is obtained as follows:

(8) In order to determine

, we find out the inverse Laplace transform of

equation (8). By doing so, we have obtained

]} where ,

, ,

(9)

62

R.K. Singh and A.K. Singh

,

, ,

while

by

and

are functional defined as

In above equations,

is the complementary error function defined

Using equation (9), the skin-friction on both the walls in non-dimensional form are given by

(10)

Transient Hydromagnetic Natural Convection …

63

(11) where

and

are functional defined as

,

while

,

and

are constant given by ,

,

Solution for Since the expression (9) contains the term which it tends to infinite as separately derive the solution for

and as a result of

approaches to 1.0. Hence we have to to resolve this type of situation.

We put in equation (2) and find solution of velocity. Proceeding as above, we have obtained expression for the velocity in this case as follows:

(12)

64

R.K. Singh and A.K. Singh The skin-friction on both the walls for this case are given by , (13)

.

(14)

Steady State Solution To compare the steady state solution obtained from equations (9) and (12) with the exact solution for the steady case, we put in equations (2) and (3) and find the solution of the resulting equations by using only boundary conditions of (4). By doing so, the solution for the velocity field is obtained as follows:

, (15) It has been found that the numerical values of the velocity field calculated from the equation (15) have matched very well with the numerical values obtained from the equations (9) and (12) at the steady state time. Also, the steady state values of the skin-friction obtained from the equations (10, 11) and (13, 14) on the both walls are almost same as the values obtained from the following expression:

,

(16)

65

Transient Hydromagnetic Natural Convection …

. (17)

Results and Discussion It can be noted that the non-dimensional equations that govern the physical behavior of the system contain three physical parameters namely

,

and . Some numerical calculations are carried out for non-dimensional velocity profiles and skin friction of fluid for different values of buoyancy force parameter and Hartman number. The velocity profile is shown in figures 1, 2 and 3 for and in figures 4, 5 and 6 for corresponding to realistic fluid water in case of asymmetric heating or cooling of the walls. Fig. 1 shows the velocity profiles of fluid

for values of

and . It can be observed that upward flow occurs near the heated wall for all considered values of R and t. Further, we can observed that for and , the reverse flow does not occur near the cooled wall whereas reverse flow occurs for (curves 2, 3, 8 and 9). From this figure, we conclude that the influence of Hartman number is to decrease the velocity profiles (curves 1, 7; 2, 8 and 3, 9). Velocity profiles are increasing with time and then finally becoming steady state. Interestingly, reverse type flow occurs near the cooling walls (curves 1, 4, 5). Fig. 2 Illustrates the velocity profiles of fluid

for the case

. It is clear from the figure that the velocity of fluid ( and 4.0) for

is maximum near the heated wall and then it gradually

deceases towards the cooled wall because the temperature same as the temperature

of the fluid is

of cooled wall and is less then the temperature

of the heated wall (curves 1, 7). When increases, the maximum value of velocity profiles shifts in the middle region (curves 1, 4; 2, 5; 3, 6). In other

66

R.K. Singh and A.K. Singh

words, it shows tendency to achieve symmetric form about middle region of walls. In general, maximum point of velocity profiles decreases with an increase in Hartman number while it increases with increase in buoyancy force distribution parameter

.

Fig. 3 demonstrates the velocity profiles of fluid

for case

. The maximum point of velocity profiles occurs in middle region because the temperature of the both walls

and

is the

same but greater than the fluid temperature and as a result of which a symmetric flow occurs between the walls. The nature of flow is changes platykurtic (curves 1, 4, 7) to mesokurtic type (curves 2, 5, 8) as t increases. A close study of above figures indicates that the buoyancy force distribution parameter

plays a major role in controlling the velocity profiles of fluid and

that appreciates with increase in the values of

. 2

Pr=1.0 3

0.050

2 9

0.045

8

0.040

6 5

0.035

12 11

Velocity(u)

0.030 0.025 0.020 0.015

Cuvers M T R 1 0.1 0.1 -0.3 2 0.1 0.3 -0.3 3 Steady state 4 0.1 0.1 -0.7 5 0.1 0.3 -0.7 6 Steady state 7 4.0 0.1 -0.3 8 4.0 0.3 -0.3 9 Steady state 10 4.0 0.1 -0.7 11 4.0 0.3 -0.7 12 Steady state

0.010 0.005 0.000

14

7 10

-0.005 -0.010 0.0

0.2

0.4

0.6

Distance(y) Fig. 1 Figure 1.

0.8

1.0

67

Transient Hydromagnetic Natural Convection … Cuvers

0.12

Pr=1.0 6 5

0.10

R

0.1 0.1 0.0

2

0.1 0.3 0.0

8

0.06

4

0.1 0.1 0.7

5

0.1 0.3 0.7

6 Steady state 9

Velocity(u)

T

1

3 Steady state 12 11 3 2

0.08

2

M

7

4.0 0.1 0.0

8

4.0 0.3 0.0

9 Steady state

0.04

10

4.0 0.1 0.7

11

4.0 0.3 0.7

12 Steady state

0.02 1 7 10 4

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Distance(y) Fig. 2

Figure 2. 2

0.14

Cuvers M T R 1 0.1 0.1 1.0 2 0.1 0.4 1.0 9 3 Steady state 8 6 4 2.0 0.1 1.0 5 5 2.0 0.4 1.0 3 6 Steady state 2 7 4.0 0.1 1.0 8 4.0 0.4 1.0 9 Steady state

Pr=1.0

0.12

Velocity(u)

0.10 0.08 0.06 0.04 0.02

7 41 0.00 0.0

0.2

0.4

0.6

Distance(y) Fig. 3

Figure 3.

0.8

1.0

68

R.K. Singh and A.K. Singh Cuvers Pr=7.0

0.040 0.035

T

R

1

0.1 0.1 -0.3

2

0.1 0.3 -0.3

3 Steady state 12

0.030

11

0.025

6

5 2 8

0.015

4

0.1 0.1 -0.7

5

0.1 0.3 -0.7

6 Steady state

3

0.020 Velocity(u)

2

M

9

0.010

7

4.0 0.1 -0.3

8

4.0 0.3 -0.3

9 Steady state 10

4.0 0.1 -0.7

0.005

11

4.0 0.3 -0.7

0.000

12 Steady state 4 10

-0.005

1

7

-0.010 0.0 Figure 4.

0.2

0.4

0.6

0.8

1.0

Distance(y) Fig. 4

From Fig. 4 it is observed that for

and

velocity profiles of water are in upward-direction near the heated wall (curves 1, 4, 7, 10) where as velocity profiles near the cooled wall are having a downward trend (curves 1, 4, 7, 10). The maximum point of velocity profiles sifts in middle of walls as increases. Increasing the Hartman number does not alter over all shape of velocity profiles of the fluid, instead it only affects magnitude of maximum point of velocity profiles of the fluid (curves 1, 7; 2, 8 and 3, 9). It is clear from fig. 5 that for and as increases, velocity profiles (curves 1, 7; 4, 10) remain unchanged where as for as Hartman number increases the peck point of velocity profiles of fluid decreases, in other word optimum velocity of velocity profiles is shifting downwards (curves 2, 8; 5, 11). Velocity profiles of fluid for and decreases slightly as the increases (curves 4 ,10). It is also observed that velocity profiles become approximately

69

Transient Hydromagnetic Natural Convection … platykurtic for when

and

(curves 5, 11). In general we see that

increases maximum point of velocity profiles of fluid are showing the

tendency to shift towards the middle of walls. Also when (steady state) is increased, the peakedness of velocity profiles of fluid is also increased. 2

Cuvers M T R 1 0.1 0.1 0.0 2 0.1 0.4 0.0 3 Steady state 12 9 4 1.0 0.1 0.7 6 5 1.0 0.4 0.7 5 6 Steady state 3 7 4.0 0.1 0.0 2 8 8 4.0 0.4 0.0 11 9 Steady state 10 4.0 0.1 0.7 11 4.0 0.4 0.7 12 Steady state

Pr=7.0

0.10

Velocity(u)

0.08

0.06

0.04

0.02

0.00 0.0

0.2

0.4

0.6 1 7

0.8

1.0

4 10

Distance(y) Fig. 5 Figure 5.

As shown in fig. 6 for , the fluid elements near walls have velocities higher than the centerline velocity (curves 1, 4, 7). A study of figure suggests that an increase in the Hartman number, fluid elements near the middle of walls are more intensely retarded. It is clear from the figure that velocity profiles of fluid are symmetric about the line

. We also

observed that for the velocity profiles are platykurtic (curves 2, 5, 9) where as velocity profiles are parabolic (curves 3, 6, 9) for steady state.

70

R.K. Singh and A.K. Singh

Velocity(u)

2

Pr=7.0

0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00

9 6 3

1

0.0

0.2

Cuvers M T R 1 0.1 0.1 1.0 2 0.1 0.4 1.0 3 Steady state 4 2.0 0.1 1.0 5 2.0 0.4 1.0 6 Steady state 7 4.0 0.1 1.0 4.0 0.4 1.0 2 8 9 Steady state 5 8

4 7

0.4 0.6 Distance(y)

0.8

1.0

Fig. 6

Figure 6.

Table 1. Numerical values of the skin friction of

Tables 1 and 2 illustrates the influence of the buoyancy force parameter , time

and Hartman number

on dimensionless skin-friction

Transient Hydromagnetic Natural Convection …

71

of fluid on both walls for (water) and respectively. Important observations from the tables are that the numerical values of skinfriction of fluid at the both walls increases as value of the buoyancy force distribution parameter

increases. The numerical values of the skin-friction

and are same for because symmetric flow of fluid occurs for this case. In general, we also observed that numerical values of skin-friction on both walls have decreasing tendency with increasing

while it increases

with time . The skin-friction for small values of at the walls are considerable small, where as for large value of t, it is quite significant. Table 2. Numerical values of the skin friction of

Conclusion A transient magnetohydrodynamics free convective flow of a viscous incompressible and electrically conducting fluid between two vertical walls occurring as a result of asymmetric heating/cooling of the walls has been studied by defining a buoyancy force distribution parameter. It has been found

72

R.K. Singh and A.K. Singh

that the symmetric /asymmetric nature of the flow formation can be obtained by giving a suitable value to the buoyancy force distribution parameter. Formation of the upward flow occur near the heated wall while the downward flow forms near the cooled wall for negative values of the buoyancy force distribution parameter. The fluid velocity seems to decrease with increase in the Hartman number. The steady state flow corresponds to fully developed flow.

Nomenclature g

specific heat at constant pressure acceleration due to gravity distance between two vertical walls thermal conductivity Hartman number Nusselt number at the wall Nusselt number at the wall Prandtl number buoyancy force distribution parameter time in non-dimensional form time temperature of the wall at temperature of the wall at initial temperature of the fluid fluid velocity in non-dimensional form velocity of fluid dimensionless co-ordinate perpendicular to the walls co-ordinate perpendicular of the walls

Greek Symbols coefficient of thermal expansion

Transient Hydromagnetic Natural Convection …

73

dynamic viscosity of the fluid temperature of the fluid in non-dimensional form kinematic viscosity of the fluid skin-friction in non-dimensional form at the wall skin-friction in non-dimensional form at the wall

References [1] S. Ostrach, 1952, Laminar natural convection flow and heat transfer of the fluids with and without heat sources in channels with constant wall temperature, NASA Technical Note No. 2863. [2] S. Ostrach, 1954, Combined Natural and Forced Convection Laminar Flow and Heat Transfer of Fluids With and Without Heat Sources in Channels with Linearly Varying Wall Temperature, NASA Technical Note No. 3141. [3] E. M. Sparrow, T. Eichhorn and J. L. Gregg, 1959, Combined forced and free convection in a boundary layer flow, Physics of Fluids, No. 2, 319328. [4] G. Poots, 1961, Laminar Natural Convection Flow in MagneticHydrodynamics, International Journal of Heat and Mass Transfer, 3, 125. [5] J. F. Osterle and F. J. Young, 1961, Natural Convection between Heated Vertical Plates in Horizontal Magnetic Field, Journal of Fluid Mechanics, 11, 512-518. [6] E. M. Sparrow and R. D. Cess, 1961, Effect of Magnetic on Free Convection Heat Transfer, International Journal Heat and Mass Transfer, 3, 267-274. [7] W. Anug, L. S. Fletcher and V. Sernas, 1972, Developed laminar free convection between vertical parallel plates asymmetric heating, International Journal of Heat Mass Transfer, No. 15, 2293-2308. [8] W. Anug, (1972), Fully developed laminar free convection between vertical plates heated asymmetrically, International Journal of Heat Mass Transfer, No. 15, 1577-1580. [9] H. M. Joshi, 1988, Transient effects in natural convection cooling of vertical plates, Int. comm. Heat Mass Transfer, No. 15, 883-896.

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[10] D. J. Nelson, B. D. Wood, 1989a, Combined Heat and Mass Transfer Natural Convection between Vertical Parallel Plates, International Journal of heat and Mass Transfer, 32, 1779-1787. [11] D. J. Nelson, B. D. Wood, 1989b, Fully developed Combined Heat and Mass Transfer Natural Convection between Vertical Parallel Plates with Symmetric Boundary Conditions, International Journal of Heat and Mass Transfer, 33, 1789-1792. [12] K. Singh, 1988, Natural convection in unsteady Couette motion, Defence Science Journal, No. 34, pp. 35-41. [13] K. Singh, H. R. Gholami and V. M. Soundalgekar, 1996, Transient free convection flows between two vertical parallel plates, Heat and Mass Transfer, No. 31, pp.329-332. [14] T. Paul, B. K. Jha and A. K. Singh, 1996, Transient free convection flow in vertical channel with constant heat flux on walls, Heat and Mass Transfer, No. 32, pp. 61-63. [15] K. Jha, A. K. Singh and H. S. Takhar, 2003, Transient free convection flows in vertical channel due to symmetric heating, International Journal of Applied Mechanics and Engineering, No. 8, pp. 497-502. [16] K. Singh and T. Paul, 2006, Transient natural convection between two vertical walls heated/cooled asymmetrically, International Journal of Applied Mechanics and Engineering, N0. 11(2006), pp 67-73. [17] P. Chandran, N. C. Sacheti and A. K. Singh, 1993, Effects of Rotation on Unsteady Hydrodynamic Couette Flow, Astrophysics and Space Science, 202, 1-10. [18] P. Chandran, N. C. Sacheti and A. K. Singh, 1996, Haydromagnetic Flow and Heat Transfer past a Continuously moving Porous Boundary, International Communication in Heat & Mass Transfer,23, 889-898. [19] P. Chandran, N. C. Sacheti and A. K. Singh, 1998, Unsteady Hydromagnetic Free Convection Flow with Heat Flux and Accelerated Boundary Motion, Journal of Physical Society of Japan, 67, 124-129. [20] P. Chandran, N. C. Sacheti and A. K. Singh, 2001, An Undefined approach to Analytical Solution of a Hydromagnetic Free Convection Flow, Scientiae Mathematicae Japonicae, 53, 467-476. [21] K. Singh, P. Chandran and N. C. Sacheti, 2000, Effects of Transverse Magnetic Field on a Flat Plate Thermometer Problem, International Journal of Heat and Mass Transfer, 43, 3253-3258. [22] H. S. Takhar, A. K. Singh and G. Nath, 2002, Unsteady MHD Flow and Heat Transfer on a Rotating Disk in an ambient Fluid, International Journal of Thermal Sciences, 41, 147-155.

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

Chapter 4

EFFECT OF SUCTION/ INJECTION ON MHD FLAT PLATE THERMOMETER Anand Kumara and A.K. Singhb Department of Mathematics, Banaras Hindu University, Varanasi, India

Abstract 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.

Keywords: Magnetohydrodynamics, flat plate, suction, injection, shooting method, Runge-Kutta method.

a b

E-mail address: [email protected] E-mail address: [email protected]

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Anand Kumar and A.K. Singh

Introduction The studies of magnetohydrodynamics flows have many important applications in engineering devices such as power generator, the cooling of reactors the design of heat exchangers, MHD accelerators and boundary layer control in the field of aeronautics and aerodynamics. The processes of suction and injection have also its importance in many engineering activities such as in the design of thrust and radial diffusers, and thermal oil recovery. Suction is applied to chemical processes to remove reactants while blowing is used to add reactants, cool the surface, prevent corrosion and reduce the drag [1] Callegari and Friedman [2] have obtained an exact analytical solution for laminar boundary-layer flow over a semi-infinite flat plate having similarity preserving suction. A boundary layer solution corresponding to the flow of an electrically conducting fluid over a semi-infinite flat plate in the presence of a transverse magnetic field has been studied by Ingham [3]. The numerical solution of similarity equations describing MHD flow and heat transfer of an incompressible viscous and electrically conducting fluid past a semi-infinite flat plate have been presented by Soundalgeker and Vighnesam [4]. Singh and Dikshit [5] have studied hydromagnetic flow generated by a continuously moving semi-infinite porous plate in the presence of a magnetic field. AlSanea [6] has shown that the suction tends to increase the skin friction and heat transfer coefficients, where as injection acts in the opposite manner. Singh et al [7, 8] have shown the effects of magnetic field on the velocity and temperature profiles in the boundary layer formed over a flat plate thermometer when plate is at rest and moving with a constant velocity. We present in this paper the flow and heat transfer phenomena of a viscous incompressible and electrically conducting fluid over a flat plate thermometer with an imposed transverse, uniformly magnetic field when there is suction/injection at the plat. The problem is formulated in such a way that the non-linear partial differential equations describing the flow and temperature fields are reduced to non-linear ordinary differential equations using the similarity transformation. The similarity equations are solved by numerically, using the shooting method employing the Runge-Kutta fourth order method. Finally results are presented through graphs and table. The present analysis may be useful as a simple model in understanding more complicated applications to practical problems.

Effect of Suction/ Injection on MHD Flat Plate Thermometer

77

Problem Formulation Let us consider the flow of a viscous incompressible and electrically conducting fluid past a semi-infinite flat plate when a uniform magnetic field is applied transversely. The x-axis is taken along the plate and the y-axis is normal to the plate. The magnetic Reynolds number of the flow is assumed to be small enough so that the induced magnetic field can be neglected [9]. If and are the velocity components along x- and y-axes, respectively, the twodimensional flow is governed by the equations

,

(1)

,

(2)

,(3) where is the fluid density, is the electrical conductivity of the fluid, the kinematic viscosity, B the magnetic field strength, U the free stream velocity, T the fluid temperature, the fluid thermal conductivity, the specific heat at constant pressure. The second and third terms on the righthand side of equation (3) are due to viscous and Ohmic dissipations, respectively. The appropriate boundary conditions for the velocity and temperature fields are: at ,

as

Here is the normal velocity at the plate and corresponding to the fluid suction at the plate surface while

 (4) is

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Anand Kumar and A.K. Singh

corresponding to the fluid blowing or injection at the wall. is the temperature of the fluid away from the boundary. Equations (1) to (3) can be reduced to a system of ordinary differential equations by using the following similarity transformation:

; .

(5)

When we use equation (5), the continuity equation (1) will be automatically satisfied, while equations (2) and (3) will be reduced respectively, to (6)

, ,

(7)

having boundary conditions at at

; (8)

In above equations primes denote derivative with respect to similarity variable

In equation (5),

which can also be expressed as

is the non-dimensional magnetic parameter where

is the Hartmann number

Effect of Suction/ Injection on MHD Flat Plate Thermometer

79

and is the Reynolds number, is the Prandtl number of the fluid, is the dimensionless temperature, is the dimensionless suction/injection parameter and it corresponds to suction when and injection when . The third-order non-linear ordinary differential equation (6) is first converted in the form of three first-order simultaneous equations. But the numerical integration of eq. (6) can’t be started at because is not known there so the eq. (6) is solved by shooting method in which Runge-Kutta method of fourth order has been employed. The half-interval method is used in the shooting process. Eq. (7) with the boundary conditions (8) is a second order ordinary differential equation coupled with the eq. (6). Using the implicit finite difference method, it reduces to a set of equations in tridiagonal form. These equations have been solved by the Gaussian elimination method. As , must be finite for the numerical computation, hence instead of extending to infinity, we chose a reasonably large value of as and detailed description of which is given by Chow [10]. In numerical solution, we have considered the values of

as 0.1 and 0.6, while the values of Prandtl

number Pr have been taken as ,0 , , and the mercury, air, sulphur dioxide and water respectively. The skin friction coefficient , is given by

corresponding to

(9) where stress.

is the Reynolds number,

is the shear

Results and Discussion The numerical results are presented though graphs and table in order to carry out a parametric study showing influences of non-dimensional parameters namely Prandtl number , magnetic number , and suction/injection parameter . Fig. 1 depicts the velocity profiles corresponding to

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Anand Kumar and A.K. Singh

different values of suction/injection parameter (

= 0.7, 0.3, -0.1, -0.5) when

takes values 0.1 and 0.6. We observe that the influence of the magnetic field strength on the velocity profile is to increase it. The effect of suction is to decrease velocity profiles while injection has opposite effects. Thus the boundary layer thickness decreases in the case of suction while increases in the case of injection. Figs. 2 to 5 show the effects of magnetic and suction/injection parameters on the temperature profiles of mercury (

=0.044), air (

=0.71), sulphur

dioxide ( =2.0), and water ( =7.0) respectively. These figures reveal that the magnetic parameter increases the temperature profiles of the above mentioned fluids. As the value of the suction parameter increases the temperature profile decreases, where as the effect of injection is to increase it. For large values of suction parameter and small value of magnetic parameter, the temperature profiles are nearly flat, but for large value of injection and magnetic parameters, the temperature profiles are going to be of curved shape i.e. the temperature gradient at the wall increases. Thus the thermal boundary layer thickness increases as the magnetic and injection parameter increases. For low Pradtl number ( ), the thermal boundary layer formed is large but for large Pradtl number it is small. The thermal boundary layer thickness is large in case of mercury but smallest for water. It can be see that the temperature profiles are very sensitive with the Pradtl number. 1.0

0.8

5 6 7 8 1 2 3 4

f ()

0.6

/

N0. 1 2 3 4 5 6 7 8

0.4

0.2

M 0.1 0.1 0.1 0.1 0.6 0.6 0.6 0.6

fw -0.5 -0.1 0.3 0.7 -0.5 -0.1 0.3 0.7

0.0 0

1

2

3



Figure 1. Velocity Profile.

4

5

6

Effect of Suction/ Injection on MHD Flat Plate Thermometer

6

7 8

12

No. 1. 2. 3. 4. 5. 6. 7. 8.

10

8



81

6

4

M 0.1 0.1 0.1 0.1 0.6 0.6 0.6 0.6

Fw 0.7 0.3 -0.1 -0.5 0.7 0.3 -0.1 -0.5

1 2 3 4 5

2

0 0

1

2

3

4

5

6



Figure 2. Temperature profile for mercury, Pr=0.044.

25

No. 1. 2. 3. 4. 5. 6. 7. 8.

20 8

7



15

M 0.1 0.1 0.1 0.1 0.6 0.6 0.6 0.6

10 1 2 3

6 5

4

5

0 0

1

2

3



Figure 3. Temperature profile for Pr=0.71.

4

5

6

Fw 0.7 0.3 -0.1 -0.5 0.7 0.3 -0.1 -0.5

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Anand Kumar and A.K. Singh

30

No. 1. 2. 3. 4. 5. 6. 7. 8.

25 7 8



20

15 1 2

10

5 34

M 0.1 0.1 0.1 0.1 0.6 0.6 0.6 0.6

Fw 0.7 0.3 -0.1 -0.5 0.7 0.3 -0.1 -0.5

6

5

0 0

1

2

3

4

5

6



Figure 4. Temperature profile for Sulphur dioxide, Pr=2.0. 50 45 40



35









No. 1. 2. 3. 4. 5. 6. 7. 8.





30 25 20

M 0.1 0.1 0.1 0.1 0.6 0.6 0.6 0.6

Fw 0.7 0.3 -0.1 -0.5 0.7 0.3 -0.1 -0.5

 

15 10 5 0 0

1

2

3



Figure 5. Temperature profile for water, Pr =7.0.

4

5

6

83

Effect of Suction/ Injection on MHD Flat Plate Thermometer Table 1. Skin friction

and plate temperature

M

0.1 0.1 0.1 0.1 0.6 0.6 0.6 0.6

-0.5 -0.1 0.3 0.7 -0.5 -0.1 0.3 0.7

0.698352 0.552338 0.420389 0.306891 1.041650 0.917959 0.805375 0.704797

Pr=0.044

Pr=0.71

Pr=2.0

Pr=7.0

1.56218 0.69429 0.23225 0.07239 11.6653 7.23954 3.96775 1.83276

5.79073 2.75092 1.03121 0.36925 22.4088 13.2454 6.06798 5.61730

9.73988 4.16069 1.41589 0.46300 29.2606 16.7275 7.63661 6.71962

25.6191 7.18194 1.88955 0.53771 50.3900 22.6537 9.46767 7.68027

Finally, the skin friction coefficient and the plate temperature are presented in the table 1. The effect of magnetic field on the skin friction is to increase it. In the case of suction, the skin friction decreases but in case of injection it is reverse. Next with increase in the Pradtl number and magnetic parameter, the temperature of plate increases. The effect of suction is to decrease the temperature of plate while injection has opposite effect.

Conclusion The effect of suction/ injection on the steady two-dimensional laminar hydromagnetic flow on a flat plate thermometer has been studied. The similarity equations, obtained by using suitable similarity variable, are solved numerically using shooting method. Numerical results are given for the dimensionless velocity profiles, the dimensionless temperature profiles as well as the skin friction coefficient for various values of the Pradtl number, suction/ injection and magnetic parameter. We can conclude from the results that the suction parameter decreases the temperature of the flat plate thermometer while injection and magnetic parameters have increasing effects on it. The injection and magnetic parameters increase the velocity profiles while suction decreases it.

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Nomenclature −magnetic induction −specific heat at constant pressure −skin friction coefficient −dimensionless suction/ injection coefficient −Hartmann number −fluid thermal conductivity −magnetic parameter −Prandtl number −temperature of the fluid −temperature of the free steam −free stream velocity −horizontal velocity −vertical velocity −co-ordinate along the plate −co-ordinate normal to the plate

Greek Symbols −dimensionless similarity variable −dimensionless temperature −coefficient of viscosity −kinematic viscosity of the fluid −fluid density −fluid electrical conductivity −shear stress

Subscripts −condition at the free stream

Effect of Suction/ Injection on MHD Flat Plate Thermometer

85

Superscript −denotes derivative with respect to

References [1] Labropulu, F., Dorrepaal, J. M., and Chandna, O. P., Oblique flow impinging on a wall with suction or blowing, Acta Mechanica, Vol. 115, pp. 15-25, 1996. [2] Callegari, A. J., and Friedman, M. B., An analytical solution for the laminar flow over a flat plate with similarity preserving suction, International Journal of Non-Linear-Mechanics, Vol. 11, pp 147-154, 1976. [3] Ingham, D. B., MHD flow in the presence of a transverse magnetic field, NuclearEngineering and Design, Vol. 52, pp. 325-329, 1979. [4] Soundalgekar, V. M., and Vighnesam, N. V., MHD flow past a semiinfinite flat plate with heat transfer, Nuclear Engineering and Design, Vol. 58, pp. 109-112, 1980. [5] Singh, A. K., and Dikshit, C. K., Hydromagnetic flow past a continuously moving semi-infinite plate for large suction, Astrophysics and Space Science, Vol. 148, pp. 249-256, 1988. [6] Al-Sanea, S. A., Mixed convection heat transfer along a continuously moving heated vertical plate with suction or injection, International Journal of Heat and Mass transfer, Vol. 47, pp. 1445-1465, 2004. [7] Singh, A. K., Chandran, P., and Sacheti, N. C., Effect of transverse magnetic field on a flat plate thermometer, International Journal of Heat and Mass transfer, Vol. 43, pp. 3253-3257, 2000. [8] Singh, A. K., Chandran, P. and Sacheti, N. C., On the thermal boundary layer on a moving flat plate thermometer in the presence of Ohmic heating, Journal of AppliedMechanics and Engineering, Vol. 5 (2000), pp. 303-315. [9] Shercliff, J. A., A text book of Magnetohydrodynamics, Pergamon Press, London, 1965. [10] Chow, C. Y., An Introduction to Computational Fluid Mechanics, Wiley, New York, 1979.

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

Chapter 5

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 Space Plasma Department, Space Research Institute of NASU and NSAU, 40, prosp. Akad. Glushkova, Kyiv-187, MSP, 03680, Ukraine

Abstract In this article we 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 our own views on this matter. We restrict our consideration by two important types of such perturbations: flute and ballooning modes. In the most general case arbitrary three-dimensional transversally small-scale standing MHD perturbations in ideal plasmas with nested magnetic surfaces are described by a pair of Dewar-Glasser equations [34]. 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. We 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.

88

O.K. Cheremnykh and A.S. Parnowski In addition we 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 we derived the corresponding stability criterion, which sets stronger restrictions on the stability than well known Gold criterion.

Introduction Plasma confined by the geomagnetic field is thermodynamically nonequilibrium like any other plasma with pressure. Therefore, according to general concepts of theory of plasma instabilities, it can spontaneously acquire collective degrees of freedom, becoming unstable. Magnetospheric plasma never occupies a low-energy state due to constant flow of energy from outside. Trying to reach the minimum energy state magnetosphere constantly generates perturbations, which change its configuration and transport properties. Instabilities in plasma can be divided by two main classes — magnetohydrodynamic (MHD) and kinetic ones. MHD instabilities are usually associated with motions of macroscopic volumes of plasma. They can be described with MHD equations [1]. The other instabilities, which essentially depend on difference in motions of different groups of particles in the same volume, are called kinetic instabilities. These are microscopic in comparison to large-scale slow MHD motions. The greatest threat to plasma stability is known to be posed by MHD instabilities with large growth rates, which lead to a rapid reconstruction of the initial equilibrium. As magnetospheric plasma pressure grows, the main role goes to MHD instabilities, which are powered by plasma’s thermal energy. These are flute and ballooning modes, which are driven by pressure and magnetic field line curvature. These instabilities were first investigated in papers [2, 3]. Kruskal and Schwarzschild [2] were first to demonstrate the development of

Flute and Ballooning Modes in the Inner Magnetosphere …

89

instabilities shaped in a form of flutes elongated along the field lines (hence the name ―flute modes‖) in plasma with sharp boundaries. The criterion of stability of magnetospheric plasma against flute perturbations was derived for the first time by Gold [3]. This criterion is best understood in terms of virtual flux tube interchange calculations made by Chandrasekhar [4], hence the second name of these modes, ―interchange modes‖. Later on this criterion acquired an apt physical interpretation [5]: among all motions of flux tubes across the magnetic field lines, only those, which do not perturb the magnetic field, are allowed. Note that this criterion is analogous to the convective stability condition of gas, compressed by a gravity field. Gold’s assumption of large-scale interchange motions of magnetic flux tubes being responsible for realignment of mass of planetary magnetosphere was further developed in papers [6 — 11]. In 1966 Furth et al [12] considered plasma stability in a flat geometry replacing field line curvature with variable gravity and discovered ballooning instability. Flute modes in this case appeared to be not constant, increasing in unfavourable curvature area and decreasing in favourable curvature area. Ballooning instability occurs when plasma pressure exceeds a certain critical pressure, which depends on ratio of lengths of favourable and unfavourable curvature sections of the field line. Ballooning modes are well known in the magnetic fusion literature as pressure driven instabilities [13, 14], analogous to the instability developing at a weak spot of a pressurized elastic container. Ballooning perturbations are also sometimes called transversally small-scale modes. The assumption of smallness of transversal scale of perturbations has a profound physical meaning: (fast) magnetosonic waves, exhibiting themselves in compression and depression of field lines and being suppressed by stationary magnetic field of the Earth at low plasma pressure, are thrown away and only Alfvén and slow magnetosonic (ionosonic) waves, which are the most dangerous to the stability, are considered. So, perturbations of plasma, submersed in a stationary magnetic field, are a hybrid of Alfvén and slow magnetosonic waves. It’s just these waves, which form perturbations of ballooning type. In the case of the magnetosphere such stationary field is the geomagnetic one, and in the case of fusion research — the stationary field of a device. This is one of the reasons for ballooning modes to be so important in magnetospheric and fusion research. Many theoreticians were attracted by the problem of ballooning modes in the magnetosphere [15 — 29]. These investigations were motivated by the responsibility of ballooning modes for some observed geomagnetic pulsations

90

O.K. Cheremnykh and A.S. Parnowski

[30]. Besides, interchange instability can explain strong ejections of plasma at the time of magnetospheric substorms [31]. An interesting result was obtained in paper [17], where flute perturbations were shown to be a special class of ballooning perturbations being described by the same equations. Note that the equation for ballooning perturbations, as it was shown in [17], generalizes differential operators for Alfvén waves derived by several authors [32, 33]. It was specified ibid that the stability is mostly determined by large-scale perturbations, whose structure essentially depends on the distribution of plasma parameters and ionospheric boundary conditions. At the same time intensive research of ballooning stability of toroidal fusion devices were carried out [13, 14, 34 — 36]. In particular, paper [34] introduced equations for ballooning modes in arbitrary magnetic geometry with nested magnetic surfaces. For this reason, the approach of paper [34] was later successfully applied to magnetospheric plasma. All attempts of deriving an analytical stability criterion against such perturbations had no success until recently, so large-scale ballooning modes were usually studied numerically. In last years an interest for generation of flute and ballooning perturbations in the Earth’s magnetosphere was also prompted by an investigation of mechanisms of self-organisation of non-linear structures in the magnetosphere [37] and their influence on the redistribution of mass of magnetospheric plasma. Hitherto the matter concerned the stability of magnetospheric plasma without detailed elaboration of ionospheric boundary conditions. Real ionospheric plasmas, which bound magnetospheric plasma, are always of finite conductivity and thus are not completely tied to the magnetic field lines in the ionosphere. This causes dissipative instabilities, which develop much easier than ideal ones. The first articles dealing with derivation of ionospheric boundary conditions were also the first ones investigating the influence of dissipation caused by finite ionospheric conductivity on ballooning (and interchange) plasma stability. The boundary problem for magnetospheric perturbations is discussed for half a century, starting from pioneer articles by Dungey and Nishida [38, 39]. Earlier investigators considered either isolating boundary conditions, or no boundary conditions at all (local perturbations, see e.g. [5]). More complicated boundary conditions, alternative to [38, 39], were later derived by Maltsev [40]. It was only at the beginning of nineties, when boundary conditions for magnetospheric perturbations were thoroughly analyzed in paper [41] as part of an investigation of magnetospheric pulsations excitement mechanisms [42]. Different attempts to describe the influence of

Flute and Ballooning Modes in the Inner Magnetosphere …

91

the ionosphere on magnetospheric waves using the boundary conditions are given out in details in paper [43] and references therein. Boundary conditions for ballooning perturbations were derived in paper [25]. The aim of this work is to derive a stability criterion of magnetospheric plasma against flute and ballooning instabilities. This research was motivated by three reasons. The first one is the result of paper [25], which states that ballooning stability threshold essentially depends on ionospheric conductivity. The second one is our earlier research of magnetospheric plasma’s ballooning stability with perfect ionospheric conductivity, which showed that stability is determined by ballooning modes, and interchange modes are absent [27]. This reminded us of paper [44] by Furth, Killen and Rosenbluth about interchange plasma stability with dissipation caused by finite conductivity, which demonstrated for the first time that finite conductivity leads to flute instability of plasma, considered to be interchange stable in the framework of ideal MHD. We think that this situation applies to magnetospheric plasma stability research as well. The third reason is that scientific publications on ballooning modes do not always present consistent and thorough descriptions of mathematical techniques, involved in ballooning perturbations research. We decided to fill this gap and try to give out in fine detail the mathematical part of the problem. For the same reason we give a detailed description of a plasma equilibrium, dipole model of the geomagnetic field and ionospheric boundary conditions, as well as a brief overview of the energetic principle ( W ) and its generalisations for the case of a non-Hermitian operator.

1. Main Equations The Earth’s magnetosphere represents an apt illustration of a system, which is describable in terms of ideal MHD. The simplest estimates [45] show that MHD approximation is applicable for describing large-scale structures with spatial scales above 100 km and temporal scales above 1 sec. These include the long-period magnetospheric perturbations of interest, which are observed both from the ground and in the space. We shall investigate their stability in the framework of linear perturbation theory. The description of wave processes in inhomogeneous medium with magnetic fields, such as magnetospheric plasma, even in the framework of linear theory of MHD perturbations is a complicated mathematical problem. Further we shall derive equations of small perturbations for a special class of perturbations called the ballooning ones in scientific literature. When deriving

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O.K. Cheremnykh and A.S. Parnowski

these equations we shall use the approaches developed in papers [34, 35]. For all this, we shall not use variational methods, as in paper [34], or apply differential operators to the initial equations, as in paper [35], because it can distort the perturbations spectrum. Instead, we derive these equations using only the approximation of transversal localization of perturbations and their elongation along the magnetic field lines. We shall use the derived equations to analyze plasma stability in the inner magnetosphere of the Earth.

1.1. Initial Equations In the framework of MHD plasma is usually considered as an ideally conductive isotropic quasineutral fluid described by equations

  v 1   H     p  J  H  ,  rot v  H , t c t      p  div v  0 ,   v      0 , t  t 

(1.1)

 4    1   rot H  J , div H  0 ,    v  H , c c



where  is plasma density, v is a velocity of an elementary plasma







volume,  is a ratio of specific heats. Symbols p , J ,  and H denote plasma pressure, current density, and electric and magnetic intensities. For the sake of brevity we introduce new unit-independent variables

   j , B and E as

follows

   c   4  H  4 B , J  E. j,   c 4

(1.2)

One can easily express the same variables in MKS units. As a result, we obtain from (1.1) a set of equations, which we shall use for further analysis

Flute and Ballooning Modes in the Inner Magnetosphere …

93

    B v     p   j  B ,  rot v  B , t t      p  div v  0 ,   v      0 , t  t 

(1.3)

      rot B  j , div B  0 , E  v  B. Taking into account linearized versions of Faraday induction law, combined with the condition of perfect conductivity

  ~ B  rot   B, adiabatic equation

  ~ p      p  p div  , and Ampere’s law

~ ~ j  rot B ,     and introducing a plasma displacement vector  , v  , we obtain that t

all perturbed quantities in a linearized Euler equation (the first of (1.3)) can be



expressed through  and the whole set (1.3) reduces to a single vector equation



  2 t 2



  

  1 ~ 1  ~     p    p div    j  B  j  B . c c

(1.4)

94

O.K. Cheremnykh and A.S. Parnowski

Fig. 1. A label of magnetic surface.

1.2. Magnetic Label Further we shall assume that equilibrium magnetic field has magnetic surfaces, i.e. surfaces, containing magnetic field lines and current lines. In a magnetic field with flux surfaces the following useful geometrical relations that do not depend on plasma pressure hold

    B  a  0 , div B  0 , j  a  div B  a  0 ,





(1.5)

where a is a value, traditionally called as magnetic label, which is constant on any given magnetic surface, see Fig. 1. It was introduced by Kruskal and Kulsrud in paper [46]. A magnetic label can be arbitrary chosen. In particular, it can be associated with toroidal magnetic flux or with a volume of plasma, confined inside this surface. We shall return to this matter in section 2. The knowledge of the precise value of a is not necessary for deriving the equations of small perturbations. In this sense (1.5) is independent from the equilibrium equation





  2 p a  rot B  B  a,

(1.6)

Flute and Ballooning Modes in the Inner Magnetosphere …

95



where p   dp da and which is needed to find a r  . Note that a ,

B  a and B directions are orthogonal and can be chosen as a local flux coordinate system.

1.3. Equations of Small Oscillations In this subsection we shall follow a procedure set out in [35]. It is

~

convenient to express the values of perturbed magnetic field B and current



~

density j in (1.4) through a vector T defined as

 



    j  a  T  rot   B    a . 2 a



(1.7)

Introducing (1.7) into (1.4) transforms the latter into



  2 t 2

       a      rot T  B    j   T  T  a     p div   a 2     2     j j  a        p   a  rot  B  .  a 2 a 2     (1.8)

When obtaining (1.8) we used the equation of hydrostatic equilibrium (1.6) and vector equality

   T  a  B    a  , which follows from (1.7).







(1.9)

Let us decompose vectors T і  into three orthogonal vectors a ,

B  a and B : 

96

O.K. Cheremnykh and A.S. Parnowski

    B  a  T  T1a  T2  T B , 3 a 2     a B  a  B     2   2 . a 2 B B

(1.10)

The components of (1.10) have the form

T1 

 B   a 2

, T2 

1  B     s  S  ,

s

 a   B  a   T3    2     2 div  s a   2 p, 2 a B B

(1.11)

where

    B  a  B  a  S  rot , a

2

a

2

s



2 B a

2

, s 

  j B a

2

.

(1.12)

Substituting (1.10) — (1.12) into (1.8) we get   2

   a  T0  a  s a  T3   2 pT3  T2  S   s   B  T1  K   2 t a      B    B  a   B  a  T0    B  T2  B  a T3    2 B   T0 . 2  B a 2  B  



2

(1.13) Here the following denotations are used:

97

Flute and Ballooning Modes in the Inner Magnetosphere …

2 a   2 p  B   p  . K  s  s  S    2 2 s a B

 T0  p div  ,

(1.14) Equation (1.13) does not depend on the coordinate system, because it is based only on the general properties of differential operators in an arbitrary flux coordinate system. For this reason it is exact in such a way that it describes arbitrary low-frequency plasma perturbations. In review [35] it was noted that equation (1.13) is too difficult for further analysis, because it contains high-order derivatives of T3 . Really, only this quantity contains transversal derivatives of  and  , which are large for transversally small-scale perturbations. This complicates the analysis of (1.13). For this reason, authors of [34] used different equations. They decomposed





(1.13) into a and B  a directions, and then subtracted one component from another. As a result, they got an equation, which did not contain transversal derivatives of T3 . So, their set of equations consisted from this new equation and two equations, following from (1.13) and proportional to

B  a and B . In our opinion, such procedure may significantly distort the

spectrum of perturbations. Thus we shall use the original equation (1.13) and shall not apply any differential operators to it.

1.4. Ballooning Approximation Equation (1.13) is still cumbersome because it describes arbitrary MHD perturbations. Here and later we restrict to considering pressure driven transversal small-scale perturbations. Further for the description of the



1





perturbations we turn to the approximation k   b , k|| , where k|| and

 k  are the longitudinal and the transversal wave vectors, and b is a

characteristic spatial scale of variation of equilibrium quantities. This approximation, which is usually called ballooning (see [35] and references therein), will significantly simplify the analysis of equation (1.13). In our

98

O.K. Cheremnykh and A.S. Parnowski

problem’s geometry ballooning approximation has a simple physical meaning: any component X of the displacement vector satisfies an inequality

a  X a

,

B  a X  B a

 1 B  X  ,  , b B

(1.15)

where b is a characteristic spatial scale of the system. For description of perturbations (1.15) in an inhomogeneous medium it seems natural to use eikonal approximation. However, the wavelengths of perturbations of interest are of the same order as the spatial scale of the magnetosphere. Thus, usual eikonal approximation is inapplicable. However, it is possible to apply it to transversal components [34], leaving the spatial part of the longitudinal components unchanged. In this case magnetospheric plasma perturbations are described in terms of transversal rays and longitudinal MHD waves. Following Dewar and Glasser [34], we search for solution of (1.13) in the transversal eikonal form

 

ˆ 

 (r , t )   (r ) exp it  i /   ,

ˆ   Ti r ,t   Ti r exp it  i /   ,

(1.16)

where   1 is a characteristic transversal spatial scale, and  is a transversal eikonal, which satisfies an equation

   k   B  0 , k    .

ˆ 

ˆ 

(1.17)

We assume the amplitudes  (r ) and T r  to be smooth coordinate functions. One can easily see that the relation (1.15) holds due to smallness of . Perturbed components Ti from (1.16) are equal to (see (1.11))

 ˆ i  ˆ  B  ˆ  , Tˆ0  p div   k     , Tˆ1  2    a

Flute and Ballooning Modes in the Inner Magnetosphere …



  Tˆ0   ˆ   ˆ ˆ T3   B    2   2   , p B   



1  Tˆ2  B  ˆ   s  S ˆ ,

s

99

(1.18)

  where     

  B B       is a magnetic field line curvature. Taking (1.18) B B 

and (1.13) into account and omitting terms of order

 , we obtain

2 Tˆ0  B Tˆ3  0 .

(1.19)

Taking into account the relation between Tˆ3 and Tˆ0 in (1.18) we find from (1.19)

2     ˆ   ˆ   ˆ T0   2  B    2   2    . B  p  B     

p B

(1.20)

Comparing the obtained expression for Tˆ0 to (1.18) we see that the following relation holds to within



 ˆ k    0 .

(1.21)

ˆ

Substituting  for its expression given by (1.16) we obtain

 k  a , ˆ  ˆ  2 k  a where

(1.22)

100

O.K. Cheremnykh and A.S. Parnowski

k 





  k   B  a .  2 B

(1.23)

From (1.10), (1.16), and (1.22) we get





  k  B ˆ    2  ˆ k  B ˆ

 B 2 . B

(1.24)

Then





    ˆ ˆ k   B       2 . k  B

(1.25)

As a result, the expression (1.20) for Tˆ0 can be rewritten as

2    B   Tˆ0  2    p  B   

p B



   k ˆ   B   2   2 2 B  k  B  

ˆ 

  Tˆ B  

3

2

.

(1.26)

Using two easily verifiable equalities

   k   a     Sk B  k   0 , B     a 2   

(1.27)

and equation (1.22) we obtain

  k  a  B  ˆ . B  ˆ  Sˆ   2 k   a Taking into account (1.28) we find from (1.18)

(1.28)

Flute and Ballooning Modes in the Inner Magnetosphere …



101



 k  a B  ˆ . Tˆ2  s ˆ    2 s s k B

(1.29)



Casting (1.16) and (1.24) in (1.13) we get    k  B 2 ˆ    B  2ˆ  B         B  Tˆ0    k B 2 t 2 B 2 t 2  B 2    (1.30)      ˆ     k   a 2    a B       a  Tˆ0  p ˆ    2 B  ˆ  S   s   B    a 2    a 2 k B          B  a   B  ˆ   S   2 B  a  Tˆ  pˆ  k   a B   B  ˆ .  s 0 2    a 2   s k   a 2  s  B  





















Now we shall decompose this equation. First, we obtain the longitudinal



component (along B )

 2   2  B  Tˆ0 . t 

Then we obtain the transversal component (along



 k k 2 

2

 2ˆ

2 2 B t

(1.31)

k   B  ) 

2 k B

       k   a 2  B  ˆ  .(1.32)   k  B ˆ ˆ   2T0  p     B   1  2  2  k   B  a   a 2  k B     





Using an obvious relation





 2 2  2 2 k 2  B  k   a  k  a we rewrite (1.32) in the form

(1.33)

102

O.K. Cheremnykh and A.S. Parnowski



 k

2

 2ˆ

2 2 k 2  B t





      k  B    k  ˆ .  2Tˆ0  pˆ     B   B    2  2 

k   B  a 

2

 k B 

 

(1.34)

Equations (1.26) and (1.34) coincide with those obtained by Dewar and Glasser [34]. They are principal for further analysis. Equation (1.8), supplied with corresponding boundary conditions, can be used to obtain a spectrum of ballooning perturbations.

2. Dipole Model of the Geomagnetic Field The obtained equations of small oscillations (1.26) and (1.34) should be adjusted with the equilibrium plasma configuration. Therefore, prior to analyzing them we shall briefly discuss the magnetospheric equilibrium.

Fig. 2. Plasma structure of the Earth’s magnetosphere.

At present time, the configuration of large-scale magnetic field in the Earth’s magnetosphere (see Fig. 2) is already studied well enough to create its global empirical model [47, 48]. Nevertheless, this model is very unwieldy for analytical description of magnetospheric perturbations. To illustrate the

Flute and Ballooning Modes in the Inner Magnetosphere …

103

influence of electromagnetic forces and pressure gradient on the stability of MHD perturbations in the inner magnetosphere of the Earth and not to complicate the consideration we shall use the simplest three-dimensional model of the geomagnetic field — axially symmetric magnetic dipolar field [49] with nested magnetic surfaces (Fig. 3). This model is very convenient for the analysis of geophysical phenomena and approximates the real geomagnetic field well enough up to 6 RE. At greater distances this model becomes less and less accurate due to the influence of magnetospheric currents, becoming inapplicable at 10 RE.

2.1. Magnetic Field The Earth’s magnetic field is mostly determined by internal currents and can be approximated by a multipole expansion with a dominant dipole term.



For this reason this field can be assumed to be provided by a point dipole M located at origin and pointed southwards, i.e.

   M  r   B  rot A , A  , r3 

(2.1)



where A is a vector potential, and r is radius vector. In spherical coordinates r ,  ,  with   0 being an equatorial plane equation (2.1) has the form

 B     , where



(2.2)

 M cos 2  is a poloidal magnetic flux, and M  M is the r

norm of the magnetic dipole momentum of the Earth. Note that in spherical



coordinates   r cos   e , and the components of the magnetic field 1

have the form

 2M   M cos   Br   3 sin  er , B  e , r3 r

(2.3)

104

O.K. Cheremnykh and A.S. Parnowski



and its norm B equals to

 M b  B  , r3 where b   1 3 sin shown on Fig. 3.

2

.

(2.4)

The field lines of such a magnetic field are

Further we will also need a field line equation. The arc element ds of a field line satisfies a condition

ds  B   0 . For the considered axially symmetric field this condition reduces to an equation

dr rd ,  Br B whence

r  L cos 2  ,

(2.5)

where L is a radial distance to the field line in the equatorial plane, measured in RE. This value can be expressed through the magnetic latitude

0

of the field line at the Earth’s surface as

L  RE cos 2  0 , where RE is the Earth’s radius. The length element of the field line is given by

Hence

dl 2  dr 2  r 2 d 2  L2 cos 2 b d 2 .

105

Flute and Ballooning Modes in the Inner Magnetosphere …

dl  L b  cos d .

(2.6)

Integrating this equation gives us the field line length at any given value of McIlwain parameter.

Fig. 3. Dipolar magnetic field lines.

2.2. Equilibrium



We assume magnetospheric plasma convection to be small ( v  0 ), so that magnetospheric equilibrium can be approximated by a static MHD equilibrium





  p  j  B .

(2.7)

It follows from (2.7) that there is a transversal current with density

   B  p  j   2 . B

(2.8)

Besides the transversal current (2.8) there can be a longitudinal current

106

O.K. Cheremnykh and A.S. Parnowski

  j||  B . It should be of such value that the total current is divergenceless





  div j||  j   0 .

(2.9)

As before, we assume that equilibrium magnetic field has no curls and can



be approximated by vacuum magnetic dipolar field BD

 rot BD  0 .

(2.10)

   B  p  B   2 B   3 . B

(2.11)

Then we obtain from (2.9)

Following [50], we shall assume that equilibrium field-aligned current is absent (   0 ), and the magnetospheric equilibrium is provided mostly by the toroidal current

  j  j r , e .

(2.12)

Then from (2.1), (2.10) and (2.11) one can derive the following useful relations





    j  B  0 , B     B  0 ,

(2.13)

which will be used later. When deriving (2.12) we used that

p  p  ,

(2.14)

which follows from (2.1) і (2.7). One can see from (2.7), (2.8) and (2.13) that equilibrium toroidal ring current is equal to

Flute and Ballooning Modes in the Inner Magnetosphere …

j 

dp r cos  d

107 (2.15)

and depends on particle distribution in the magnetosphere.



 

In particular, since B    B  e  0 , magnetic field lines and the lines of constant  coincide. Thus it follows from (2.15) that the quantity

dp / d is constant along the field line, and, therefore, current (2.15) reaches its maximum value at the equator, and, besides, equation (2.15) correctly describes eastward and westward ring currents, flowing in the magnetosphere [50]. All stated witnesses that the model of magnetospheric equilibrium, which we based upon, correctly describes main observed phenomena. Note that in the considered model of magnetospheric plasma with isotropic pressure the density of plasma is constant along the magnetic field lines. Really, from the equation of equilibrium (2.7) it follows that pressure is



constant along the magnetic field line ( B  p  0 , see (2.7)) and then from the relation

d p  dt   

   0 follows the mentioned statement. 

At this point we would like to discuss the accuracy of our calculations. We based our derivation of the static equilibrium upon (2.7) and

  j  rot B .

(2.16)

   B  BD  B1 ,

(2.17)

Setting





where B1 and j1 are respectively the magnetic field and current density regarded to pressure gradient. They satisfy the equations

    p  j1  BD  j1  B1 ,   j1  rot B1 .

(2.18)

108

O.K. Cheremnykh and A.S. Parnowski As distinct from Chan et al. [20], we consider only the low-pressure case (

  p B02  1 , where B0  M L3 is a value of the magnetic intensity at equatorial plane) and thus cancel all terms of order equilibrium is described by an equation

 2 . In that way, plasma

  p  j1  BD .

(2.19)

It follows from (2.18) and (2.19) that j1 ~ B1 ~   1. Obviously, one should carry out all the calculations with the same accuracy. That is why we should linearize all equilibrium quantities in pressure.

2.3. Equations of Small Oscillations in Dipole Geometry While discussing the distribution of pressure in the magnetosphere, we



noted that B    0 . This condition, combined with (2.2) allows us to set the poloidal magnetic flux  equal to a magnetic label

a  .

(2.20)

In this case (1.6) describes the toroidal current (2.15), which provides plasma equilibrium. Then due to (2.13), coefficients (1.11), (1.12) and (1.14) of the equation of small oscillations become much simpler, because both the local magnetic shear and the longitudinal current vanish

S  0 ,  s  0.

(2.21)

In this case eikonal  satisfies the equation (1.17)

  B  k          0 .



(2.22)

Since eikonal is constant on a given field line, k  can be represented without the loss of generality as

Flute and Ballooning Modes in the Inner Magnetosphere …

 k     ˆ ,

109 (2.23)

where ˆ is a constant quantity along a field line, whose meaning will be explained further. From (1.22) and (2.23) we obtain

ˆ  ˆˆ  0

(2.24)

 k   a ˆ   . k  

(2.25)

and

Then the equations (1.26) and (1.34) have the form 2   2ˆ 2      ˆ0  p ˆ  B   1  ˆ2    T 2  t 2  2  2        2ˆ  ˆ  2    ˆ   2  B  Tˆ0 , Tˆ0  p  B    2    . 2   B  t     

 

2  1  ˆ2    2 





  B  ˆ   ,   2   

(2.26)

 

Note that this equation contains only the B   differential operator. Thus it seems convenient to replace this operator with the longitudinal derivatives, i.e. write that

B    B dld , 



(2.27)

d is a derivative along the field line. Taking into account that one dl  should cast into the expression (2.4) for the norm of dipole field B the value where

of r on a magnetic field line (2.5), and that the arc element of the magnetic field line has the form (2.6), we finally obtain

110

O.K. Cheremnykh and A.S. Parnowski

M B    L cos 

4

7

d .  d

(2.28)

Let us also note that along the field line



2

2 B 



M 2b L cos  4

6

M 2b L6 cos12 

,

, 

2

 2   

2





1 L cos 6  2

,

4 L cos 2  Mb 2

.

(2.29)

Introducing dimensionless quantities

2 

M2 L dp pL6 2 2   , , , ,      A L8  p dL  A2 M2

  ˆ

L2 L ,   ˆ ,   ˆ , M M

(2.30)

L6 M , T0  Tˆ0 , L M2

(2.31)

  ˆ we transform (2.24) and (2.26) into



4  b cos

 2 1  2 b  

   b   1  2 b     T    0, 0 4   cos13    b cos     

    0 ,  2 

1  T0  0 , 7 cos  

 1    cos12   4 cos 2      T0     . 7 2     b  cos      b     

(2.32)

Flute and Ballooning Modes in the Inner Magnetosphere …

111

According to subsection 2.3, we dropped all terms of order less than  in (2.32). Alfvén frequency  A is assumed constant along the field line (see note at the end of subsection 2.2). One can easily see that (2.32) contains only even powers of sin  and

1   . Thus it  cos   sin   seems logical to introduce a new variable x  sin  and relieve (2.32) of

cos and the derivatives have the form

trigonometric functions:





4   1  b  2 bc 2

2

    b 1  2 b   T0      0 ,   c 6  b  

    0 ,  2 

T0  0, c3

(2.33)

   1  c 6  4c   T0   3     2  , c  b  b    2 2 where b  1  3x , c  1  x , and a prime indicates a derivative with respect to x .

2.4. Polarization of Ballooning Modes Equations (2.33) can be expressed as

Lˆ P ,    2 LˆT ,    Fˆ ,  ,  ,   0 ,

    0 , Lˆ S ,  ,  ,   0 ,

(2.34)

112

O.K. Cheremnykh and A.S. Parnowski where

 b    Lˆ P ,     2  6   , LˆT ,     2  6 , c c b

T Lˆ S ,  ,  ,    2  03 , Fˆ  ,  ,   4  T0     . c   bc 2 

(2.35)

This set contains two parameters: 0    1 and       . While the physical meaning of parameter  is obvious, the latter one requires some explanations. The last of equations (1.3) yields





   E  i    B . Hence

  i E  B    .  B 2 

(2.36)

From (1.17), (1.22) and (2.36) we obtain to within

    ˆ i B  k   E k    0. 2  B















(2.37)



Since k   B , (2.37) can hold only when k   E  0 , so

  (2.38) E || k      . ˆ  In such a manner, when   0 we have E ||  ,   ||  , and when      we have E ||  , ˆ ||  . At intermediate values of 

Flute and Ballooning Modes in the Inner Magnetosphere …

113



vector E lies in the same plane as vectors  and  . Thus, parameter  describes the polarization of perturbations. When   0 , as follows from (2.34),   0 and equations (2.33) reduce to a set of equations

 4    b     2 ˆ LP  ,      2  T0     0,   c 6  b  bc 

  0,  2 

(2.39)

T0  0, c3

obtained earlier in paper [26] and describing coupled slow magnetosonic and poloidal Alfvén wave, modified by pressure and field line curvature. The case    yields   0 ,   0 , but     0 , and equation (2.33) reduces to an equation

  LˆT     2  6  0 , c

(2.40)

obtained earlier in papers [32, 51] and describing toroidal Alfvén waves.

3. Ionospheric Boundary Conditions Due to a common magnetic field and large longitudinal (Spitzer) conductivity, the magnetosphere and the ionosphere constitute a joint electrodynamical system. A perturbed electrical field created in the magnetosphere propagates along the field lines and provides magnetosphereionosphere coupling. Magnetospheric events usually affect ionospheric processes weakly. This is due to characteristic temporal scale of magnetospheric perturbations being much larger than that of ionospheric ones. For this reason, the magnetosphere can be affected by the ionosphere, whereas reverse influence is negligible. That is why it is necessary to take into account

114

O.K. Cheremnykh and A.S. Parnowski

electrodynamical properties of ionospheric plasma, first of all, its conductivity, when considering the generation of magnetospheric MHD perturbations.

Fig. 4. A sketch of bounding layers.

Further we shall take this influence into account in the form of boundary conditions or equations, describing the propagation of ballooning perturbations in the magnetosphere. We shall derive these boundary conditions taking into account the existence of two near-Earth layers: insulating atmosphere and partially ionized ionosphere (see Fig. 4). We can speak about the ionospheric layer because for the real ionosphere the transversal conductivity essentially differs from zero only in a layer on heights from 150 km to 350 km. Below this layer the ionosphere can be regarded weakly different from neutral atmosphere.

3.1. Boundary Conditions The magnetosphere-ionosphere coupling is determined by properties of magnetospheric and ionospheric plasmas. The magnetosphere is filled with totally ionized plasma with low concentration, and the ionosphere is a layer of weakly ionized plasma with higher concentration with thickness of order 100 km. Ohm’s law in the ionosphere can be written as [1]

Flute and Ballooning Modes in the Inner Magnetosphere …

  j  E , where

115 (3.1)



 is a unit-independent tensor of conductivity given by  P     H  0  

H P

0  0 .  || 

0

(3.2)

Here  || is a longitudinal (Spitzer) conductivity along the magnetic field,

 P is Pedersen conductivity along the electric field, and  H is Hall conductivity along the vector product of electric and magnetic fields. Note that  in accordance to (1.2) conductivity  is scaled as follows

 4    . 2 c



Spitzer conductivity significantly exceeds the transversal conductivity   , which equals to

       P  H

H    P 

(3.3)

and is of essence only in the ionospheric layer. Further we will also need a tensor of integral transversal conductivity, equal to h           dz   P  H a

H  ,  P 

(3.4)

where a and h are the heights of respectively the lower and the upper boundaries of the ionosphere,  P 

h

h

a

a

  P dz and  H    H dz are integral

Pedersen and Hall conductivities of the ionosphere.

116

O.K. Cheremnykh and A.S. Parnowski

The problem of propagation of perturbations in the ionosphere in the general case is too complex for direct solution. Nevertheless, as shown in paper [41], one could significantly simplify it if he notifies that it contains two small parameters — a ratio of transversal components of the conductivity to a longitudinal one (~10–2 — 10–4) and the ratio of the height of the ionosphere to the magnetospheric wavelength (~10–1). Decomposing Maxwell equations by these two parameters leads to exclusion of longitudinal conductivity and the condition of conservation of horizontal relatively to ground surface component of the electric field in the ionosphere. This allows us to represent ionosphere as an infinitely thin layer, which is characterized only by its integral transversal conductivity. Using also the conditions of continuity of perturbed magnetic field and the horizontal component of the electric field, one can obtain that ionospheric currents, which flow horizontally in this layer, close the magnetospheric currents, which flow to the surface of the ionosphere. These estimates were obtained in papers [25, 41]. We will use them in further considerations to avoid unwieldy calculations. After choosing the appropriate boundary conditions, we will formulate them mathematically. We start from the ―non-penetration‖ boundary condition. Its most logical notation would be

 

 n  0,

(3.5)

b



where n is an external (earthward) unit normal to the ionospheric layer, subscript b indicates that value should be taken at the boundary, more precisely, at z  a . This condition controls vertical motions of plasma and holds due to large neutrals population in the ionosphere and to the presence of the insulating atmosphere. The second boundary condition requires that total current is divergenceless, or, in different way, that electric charge of the ionospheric layer is constant. To derive this boundary condition we expand a condition





j z     s   s  Es  0 , z

 div j  0

(3.6)

where subscript s indicates a horizontal component, and subscript z stands for a vertical one. Height-integrating this equation over the ionospheric

Flute and Ballooning Modes in the Inner Magnetosphere …

117

layer, we derive that upward component of the ionospheric current, assuming the longitudinal current to vanish at z  a ,

jz

z h









h         s   s  E s dz   s   s E s ,

(3.7)

a



where  S 

h



 a



S

dz is an integral horizontal conductivity of the 

ionosphere. Since n  e z , we obtain the second boundary condition in a form, given in [41]



    jM  n   S  S  ES b



b

.

(3.8)

Subscript b in (3.8) is due to thinness of the ionospheric layer, and j M is a boundary value of the magnetospheric current. Note that in addition to boundary conditions (3.1) and (3.8) there is also a

 

boundary condition B  n

b

 0 , connected with the ―freezing‖ of the

magnetic flux. In paper [17] it was shown that this condition corresponds to fast magnetosonic waves and thus can be rejected in the ballooning approximation.

 3.2. Expression for  S Despite its simple form, boundary condition (3.8) is quite unwieldy. This



unwieldiness is caused by the presence of an unknown tensor  S . However,



the tensor   is known to us and we shall try to express the former through the latter.





To find the relation between  S and   we should find a similar relation





for the corresponding currents in the ionosphere ( I S  n ) and in the plasma





layer, perpendicular to the magnetic field ( I   b )

118

O.K. Cheremnykh and A.S. Parnowski

   I S   S ES ,    I     E .

(3.9)

 One can write the relation between currents using the projector Q , which transforms the horizontal ionospheric current into magnetospheric one and likewise for the electric intensity:

    QI S  I  , QE S  E  .

a

transversal

(3.10)

Casting (3.8) and (3.9) into (3.10) yields

      Q S E S    E    QE S ,

(3.11)

   Q S    Q .

(3.12)

whence





Since projector Q has an inverse operator Q 1 , the sought-for tensor has the form

 Now all that remains is to find Q .

Fig. 5. To derivation of projector

 Q.

     S  Q 1  Q .

(3.13)

119

Flute and Ballooning Modes in the Inner Magnetosphere …

 3.3. Derivation of Q 

To find the projector Q we consider the following problem. Let an





arbitrary vector a lie in a horizontal plane, perpendicular to the unit normal n .



We seek the operator Q , which uniformly projects this vector on the



 

transversal plane, perpendicular to the unit vector b  B B (see Fig. 5.).



Obviously, this would be the operator we seek. To define the vector a  ,









satisfying the relation Qa  a , we introduce the vector c  n , where



   is an arbitrary scalar, and vector d    a  c . Let us find such a value       that d  b . For that we set their scalar product equal to zero:

 

      d  b  a  b   n  b  0 , whence

 

   a b    . n  b 

(3.14)

(3.15)



Thus, the sought-for vector a  d    has the form

     b a  a   a  n    Qa , b n

(3.16)

   n  b QI    , n b

(3.17)

therefore,

 







where I is an identity tensor, and n  b is a dyadic. Equation (3.17) coincides with the analogous expression from paper [41].

120

O.K. Cheremnykh and A.S. Parnowski

3.4. Boundary Conditions for Ballooning Modes in Dipole Geometry Now we shall combine (3.5), (3.8), (3.9), and (3.17) to find the boundary conditions for our specific problem. Equation (3.5) yields

B  ˆ  ˆ r  0 r s

(3.18)

Casting (1.3) into (3.7), we obtain

~    rot B  n   s  s  E s 

(3.19)

b

Considering the ballooning approximation (1.16) and dropping terms of order  we get from (3.19)    ˆ   Br  ˆ   B  1 rot B  n  i   r B  ˆ  B      2 2 2   s r r cos    2   B    ˆ  ˆ div     p  ,   s 2 B2 B 



(3.20)



     s   s  E s    ˆ ˆ  s    s   s    ˆ ˆ  s    s   s       ˆ   s    s   s    ˆ  s    s   s   .







(3.21)



A term in (3.21), proportional to  s    s   s  , can be transformed as











        s   s   s   s Q 1   Q    Q       Q  



(3.22)

Flute and Ballooning Modes in the Inner Magnetosphere …

121

 

When obtaining (3.22) we took into account that Q  n  0 , therefore,

  Q   s  Q   . Since

 Q     ,

(3.23)

it is finally rewritten in the form









   s   s   s        .

(3.24)

Taking into account (3.3), we have from (3.22)

    s   s    P  2 .



(3.25)



A term, proportional to  s   s   s transforms in a similar way to the form





 2  s   s   s   P  .

(3.26)

The other two terms in (3.21) vanish, because      0 . As a result, we get from (3.21) — (3.26)

 

   s  E s   ˆ P 

2





 ˆ 2  P  . 

(3.27)

Equating (3.20) and (3.27), we get ˆˆ   2



 1 2 r r cos 2 

ˆ r 2 cos 2 





i P

  ˆ  B  ˆ  Br B  ˆ  Br  2  s  



(3.28) 0

  ˆ  B    ˆ ˆ             div     p  0.   s  2 2 2 B B        0

Casting (2.24) into (3.28), we derive the second boundary condition in the form

122

O.K. Cheremnykh and A.S. Parnowski

1  ˆ  cos  ˆ

r2

2

2

2



cos 2  





 i  Br B  ˆ 2  1  ˆ2  r 2 cos 2    P   2 



0



(3.29)

  ˆ ˆ    1  p 4r  p B   ˆ   p     0. 2   2   r r 2 cos 2   B 2 Mb 2 B 2  B  B     0

The boundary conditions (3.18) and (3.29) should be made dimensionless. Following the same procedure, as in section 2, considering (2.28) — (2.30) and

Br  

 M 2M x,  2 , 3 3 r Lc Lc

(3.30)

we finally obtain the boundary conditions in the form

   2 5    0, 2 x 1   b    c  T0    x   x   0

 1  2 b 

i b

  2 xc

 0,





x   x0

where  

1  P A





(3.31)

is a unit-independent squared skin depth.

Linear homogenous ordinary differential equations (2.33) with boundary conditions (3.31) are an eigenvalue problem. Its eigenvalues are the perturbation frequencies  , and eigenfunctions are the corresponding amplitudes  and  . Thus the formulated problem describes a spectrum of MHD perturbations of plasma equilibrium in dipole geometry of the magnetic field with a resistive ionosphere.

In the case of a conducting ionosphere (  P   ,   0 ), equations (2.33) hold true, and the boundary conditions (3.31) acquire the form



x  x0

0,

Flute and Ballooning Modes in the Inner Magnetosphere …



x  x0

 0.

123 (3.32)

For an insulating ionosphere (  P  0 ,

   ), boundary conditions

can be written as

   2 x 1  2 b    c 5  T0    0,   x  x  0





  2 xc

x   x0

 0.

(3.33)

In conclusion of this section we would like to mention that equations (2.33) with boundary conditions (3.31) represent an eigenvalue problem for a non-Hermitian operator, while the same equations with boundary conditions (3.32) or (3.33) — for a Hermitian one [52].

4. Magnetospheric Plasma Stability From the mathematic point of view, the stability problem of MHD perturbations comes to investigation of small oscillations in the vicinity of the equilibrium. Only in several simplest cases the corresponding eigenvalue problem can be completely solved. These solutions contain full information about low-frequency oscillations of plasma and its stability. For more complicated cases, like ours, this problem is too complex for direct solving. For this reason we would like to investigate its stability without solving the complete problem. This can be accomplished with an energetic principle.

4.1. Energetic Analysis In the common form the energetic method [53] lies in studying the potential energy of perturbations

     2  2  F    Kˆ  , t



(4.1)

124

O.K. Cheremnykh and A.S. Parnowski where Kˆ is an operator, which can be identified as a coefficient of



 

resilience when plasma displacement is small. The quantity F  is the force, affecting the plasma elementary volume. For ―good‖ boundary conditions without dissipation (i.e., without imaginary quantities) the operator Kˆ is selfadjoint [5, 52], and equation (4.1) allows to investigate the stability of the system in any geometry. Truly, after choosing the time dependence of all quantities in the form exp   it  , equation of small oscillations (4.1)

acquires the form

  

2 

  Kˆ  dr 

.

(4.2)

2   dr

2 This means that  is purely real due to self-adjointness of Kˆ and the condition of stability reads

  

W    Kˆ  dr  0 .

(4.3)

Since we are not concerned with the value of the growth rate, we can now simply minimize W . Naturally, we can minimize W only partially, because its full minimization is equivalent to finding an exact solution of (4.1) for all three components. In our case, equations of small oscillations (2.33) have ―bad‖ boundary conditions (3.21) and we can not directly use (4.3). Fortunately, the structure of the equations and boundary conditions allows us to make certain deductions about plasma stability. For energetic analysis we multiply the first equation of (2.33) by asterisk indicates a complex conjugate), and the second one — by

*

(an

c6 *  , add b

them up and integrate with respect to x over the field line. This yields

where

K 2  iS  W  0 ,

(4.4)

Flute and Ballooning Modes in the Inner Magnetosphere …

K

 x0

c6 2  2 3 2 c 1   b     dx ,   b   x0





 1  2 b , S 2 x

125

(4.5)

2

W

 x0



 x0

1   b   2

b

2

 x0

dx   c 3 T0 dx  2

 x0

 x0

(4.6)

 x0

 4c 4 2   c 5 2  . (4.7)   dx    0 2   x b   2 xb   x 0

0

Here we used boundary conditions (3.31).

Fig. 6. A solution of the equation (4.4).

Notice that all coefficients in (4.4) are purely real and have simple physical meanings. K is a kinetic energy of the perturbation, S is a normal Poynting vector and describes energy losses in the ionospheric layer due to its finite conductivity, W is a potential energy of the perturbation.

126

O.K. Cheremnykh and A.S. Parnowski A graphical solution of (4.4) is plotted on Fig. 6. One can see that W  0

yields Im   0 and the perturbations are stable. When W 

S2 , all 4A

oscillations decay. The case W  0 leads to Im   0 and the perturbations become unstable. Thus, an equation

W  ,   0

(4.8)

describes a stability threshold regardless of the ionospheric conductivity. Usually [17, 28], considered perturbations become unstable due to the common influence of unfavourable curvature of magnetic field lines and of outward pressure gradient. Now, in addition to that source of instability, we have a new one, caused by pressure gradient in the ionospheric layer.

4.2. Boundary Problem for the Stability Threshold One can see from Fig. 6, that growing perturbations have zero frequencies, so functional W is self-adjoint in the region of instability [52]. For this reason, we can replace  with  , and assume  and 2

2

 to be purely

real. In that way, the stability threshold is determined by an equation  x0



 x0

1   b   dx  2

2

b

 x0

 c T dx    x 3

2 0

0

 c05  2    2bx0  where c0  1  x02 ,

 x0

 2

 x0

 x0

4c 4 2  x b 2  dx  0

  0 , 

(4.9)

x0 is a boundary value of x .

With the help of boundary conditions (3.32) one can easily persuade himself that the last destabilizing term is absent when the ionosphere is perfectly conductive (   0 ). When its conductivity is finite (   0 ), the two last terms are destabilizing. In addition, the most unstable situation corresponds to   0 , which conforms to the result of paper [54]. Thus we should minimize the positive summands in the functional

Flute and Ballooning Modes in the Inner Magnetosphere …

    b 3 W  dx   c  6 b  x0  x0  b  c  x0

  

2

 x0

127

   6  1  c    4c    dx   c 3  b  b 2     (4.10) 2

 x0

4c 4 2  c05  2  2  dx   2b0 x0    x0 b

 x0

 2

 x0

  0 . 

To minimize this functional we shall follow paper [53] and cancel all strictly positive terms by proper selection of  and  . Setting the minimized functional equal to zero, we get a stability threshold and amplitudes  and



of the most unstable perturbations. Here we consider  and  to be independent. Firstly, we minimize (4.10) over  . Since  is present only in the second term of (4.10), proportional to T02 , the functional W can be made minimal over  regardless of  . Varying W over  , we obtain

    6   1 c 4 c T0    3     2    0 . c  b  b   

(4.11)

 1  c 6  4c     D, c 3  b  b 2

(4.12)

Hence

where D is a constant of integration, equal to



D

c05  2b0 x0

 x0





x0

 x0

 

c 3 dx

4c 4 dx  x0 b 2 x0

.

 x0

Casting (4.12) into (4.10), we get the following expression for W

(4.13)

128

O.K. Cheremnykh and A.S. Parnowski

W

  2  4c 4 2   x  b   b 2   dx  0

 x0



 c05   x0  3 2b0 x0  c dx 





 x0

 x0



 x0

 c05  2    2b0 x0 

 x0

2

 4c 4     dx  x0 b 2  x0

 2

 x0

(4.14)

 . 

The stability threshold of MHD perturbations is now given by

W    0 .

(4.15)

In paper [25] the stability threshold of ballooning perturbations was proved to be independent from the value of ionospheric conductivity. Equation (4.15) confirms this result. Further minimization of (4.14) depends on  , which should satisfy a boundary condition

2 x  

 5 c   c 5 



c05  2b0 x0



x0

 x0 3



c dx

 x0

 x0



4c 4 dx  x0 b 2  c 5 x0 3  c dx



x0

0

.

(4.16)

 x0

x   x0

Now we should derive a differential equation for  , casting (4.14) into (4.15) with boundary condition (4.16). This is a variational problem, so we must ensure that (4.15) holds not only at given  , but also at its small fluctuations. To find  we calculate the first variational derivative, which we set equal to zero. This gives us a differential equation for  on the stability threshold 5



c0 4c 4  x   dx 4 0 2    x 2 x   4 c 4 c 4 c   0 b 0 b0         2 2 x0 2 x0 3 3  b b b b  c dx  c dx 

x0

4

4

 x0

 x0

 x0



0,

(4.17)

Flute and Ballooning Modes in the Inner Magnetosphere …

129

which together with boundary condition (4.16) is equivalent to the mentioned variational problem. Equation (4.17) with boundary condition (4.16) has solutions only at certain relation between  and  . This relation determines the stability threshold. It is easy to see from (2.33), (3.33) that equations (4.16), (4.17) hold for the insulating boundary (  P  0 ,

   ) as well. For this reason the

stability threshold in cases 0   P   and  P  0 will be the same. This result conforms to Theorem 2 from paper [25], which states that a ballooning perturbation occurs for resistive bounding ends if, and only if it occurs when bounding ends are insulators.

Fig. 7. A flute perturbation.

4.3. Stability Threshold Prior to solving the boundary problem (4.16), (4.17), let us analyze functional (4.14). It contains two stabilizing terms. The first one, proportional

to   , describes stabilization by boundary conductivity. The last summand describes stabilization by compressibility. These two terms cannot vanish simultaneously. We shall consider two cases when one of these terms 2

130

O.K. Cheremnykh and A.S. Parnowski

vanishes. Of course, these cases do not cover all possibilities, but one of them proves to be the most unstable.

4.3.1. Flute Perturbations The first stabilizing term in (4.14) will be minimal (equal to zero) when

 ~    0 . For this reason one should expect first to become unstable will l

be perturbations of the form

  constx  ,

(4.18)

which remind of flutes (see Fig. 7), stretched along the field lines, hence the name ―flute modes‖. For these modes elasticity of plasma is caused only by its compressibility. Thus, they will grow at high pressure profile, pointed towards the unfavourable field line curvature. Usually perturbations (4.18) exist in magnetic systems with closed field lines, for example, in fusion devices, but are not always suitable for magnetospheric equilibrium, because they do not satisfy the boundary conditions in several cases. At the same time, they can be used to obtain useful estimates. The first stability criterion for flute modes was obtained in a pioneering paper [3] in the form

  4. 

(4.19)

Although being the main interchange criterion for a long time, this criterion became a very controversial subject, enlisting both supporters and opponents. A very similar criterion was obtained in paper [8], which dealt with finite pressure plasma. The latter was then criticized in paper [6]; its author disputed the integrability of the energetic principle along the field lines, which would led to a completely different criterion than (4.19). This statement was later supported by Liu [11], who also criticized the confirmation of (4.19) by the authors of paper [7], stating that it was only due to an incidental property of their toroidal model. We tend to agree with Liu [11], because, as it was shown in [20], the dipole flux surface is pushed outward by the finite pressure

Flute and Ballooning Modes in the Inner Magnetosphere …

131

and taking this effect into account would lead, in our opinion, to a significant distortion of stability criterion. Functions (4.18) satisfy equations (4.17), (4.18) when

c05 4c 4 dx   x0 b 2 x0 b0 x0

  



x0

c 3 dx

.

(4.20)

 x0

Equation (4.20) essentially differs from (4.19) and describes less restricting conditions of the development of instability.

Fig. 8. Stability thresholds for different perturbations. 1 — ballooning perturbations with perfectly conductive ionosphere (a: L=2, b: L=2.5, c: L=4, d: L=7); 2 — flute perturbations (a: Gold criterion (4.19), b: criterion (4.20)); 3 — ballooning modes with resistive or insulating ionosphere (a: compressible, b: incompressible).

132

O.K. Cheremnykh and A.S. Parnowski

4.3.2. Incompressible Perturbations Now we consider incompressible ballooning perturbations which induce

 div  ~ T0 ~ C  0

(4.21)

and cancel the term, proportional to T02 in W . It means that



c05    x 0 2 x0b0  x0



4c 4 x 2 dx  0 x0b 0. 3 c dx  x0

(4.22)

 x0

For such perturbations (4.17) takes the form

 4      4c   0,    b2 b

(4.23)

and boundary conditions (4.16) become

2 x  

 5 c 

 0.

(4.24)

x   x0

One could think that equation (4.23) has an excessive number of boundary conditions. Nonetheless, equation (4.22) directly follows from (4.23) and (4.24), and the problem is self-consistent.

4.3.3. Numerical Analysis Quite obviously, the two considered types of MHD perturbations are not the only ones possible. Their stability threshold can be apparently calculated only numerically from (4.16) and (4.17). The results of these calculations are plotted on Fig. 8. The flute modes are the easiest to exceed their stability threshold (4.20) and become unstable. They are absent when the ionosphere is perfectly conductive. In this case, stability is determined by ballooning modes

Flute and Ballooning Modes in the Inner Magnetosphere …

133

[28] (see Fig. 9). Less prone to instability are incompressible modes, and the most stable are compressible modes. Thus we have obtained that general MHD stability of the magnetospheric plasma is determined by flute modes at arbitrary finite ionospheric conductivity with threshold (4.20). The latter lies lower than Gold criterion (4.19). As it was shown in our previous works [28], the same stability criterion (4.20) applies to an insulating boundary case, which fully agrees with the analytical analysis of paper [25].

Fig. 9. A ballooning perturbation.

4.4. Stability Criterion of Ballooning Modes with Perfectly Conductive Boundary In general case stability analysis of MHD perturbations is a complex problem. However, in some special case it can be solved without involvement of the energetic principle. One of such special cases is ballooning modes with perfectly conductive ionosphere. It corresponds to boundary conditions (3.32) and all quantities in (2.33) are purely real. Therefore, the stability threshold corresponds to a condition   0 . Numerical calculations show [28] that the instability occurs first on the lowest even eigenmode at all considered values of  and  . Equations (2.33) on the stability threshold have the form 2

134

O.K. Cheremnykh and A.S. Parnowski

  1  2 b  4c 4         2  T0     0,   a   b 

T0  0 ,

(4.25)

and can be analytically solved by reducing to an integral equation. Integrating (4.25) with respect to x twice taking into account boundary conditions (3.32) we derive a final expression for the stability threshold x0



x 0

4c 4

b

x0

b



dtdx

6  4c b 4c 4 b x c   dtdx  x 30 dtdx. 2 2 2    6 x  0 b 1   b b 1  2 b   c a   c a x 0 x 0 a x c 6 dtdx x0

4

x0

2

0

x0

x0

(4.26)

0

x2 , which approximates the x02 numerical results to within a few percent with normalization  x 0  1 , into Substituting a test function

  1

equation (4.26) gives a following expression for the stability threshold:



K1



 K2 .

(4.27)

The exact form of coefficients K 1 and K 2 is too cumbersome to publish, and we will restrict by presenting their values for some magnetic shells in Table 4.1. At latitudes around 70 degrees (6 — 10 RE) they are roughly the same. The plots of stability thresholds calculated analytically after (4.27) and numerically after (2.33), (3.32) coincide. Table 4.1. Values of coefficients in (4.27).

0 , º

L

K1

K2

45

2.00

1.83

5.2

50

2.42

1.54

5.0

60

4.00

1.17

4.8

72

10.47

0.95

4.7

Flute and Ballooning Modes in the Inner Magnetosphere …

135

Conclusion In this chapter we investigated the influence of interchange and ballooning modes on plasma stability in the inner magnetosphere of the Earth. Staying in the framework of linear MHD we tried to find the stability threshold of these perturbations in dependence of ionospheric conductivity. We applied both analytical and numerical methods to solve this problem. They allowed to understand the main mechanisms of destabilization of these perturbations. Firstly, we discovered that finite ionospheric conductivity permits the interchange modes to exist, in which case they define the general MHD stability of the magnetosphere. Secondly, their threshold does not depend on the ionospheric conductivity. We derived a stability criterion (5.20) for this threshold. This threshold appeared to be lower than predicted by Gold [3] and Kadomtsev [5]. The stability threshold of ballooning modes lies higher than that of interchange modes. It is hard to tell which role interchange and ballooning modes play in the magnetospheric processes. Only by investigating a non-linear stage of development of instability and interaction of perturbations with hot particles, this question could be answered. Only these effects and, naturally, space experiments can provide some information on macroscopic effects. Certain information can be gained from papers [55 — 57]. They conclude that interaction of interchange and ballooning modes with high-energy particles could lead to a substorm onset. Multi-satellite observations in the magnetosphere [58] witness that ballooning instability in the field-reversal region of the magnetotail [21] is connected to substorm detonations [19, 59]. The picture of collective processes, caused by these perturbations, can become much more complicated in presence of convection.

References [1] [2] [3] [4]

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INDEX A absorption, 2, 28 aerospace, 3 amplitude, vii, 1, 3, 24, 28 analytical solution, vii, viii, 1, 57, 59, 76, 85 applied mathematics, 3 atmosphere, 114, 116

B behaviors, 19 blood, 32, 56 blood flow, 56 boundary conditions, viii, ix, 5, 6, 7, 8, 31, 33, 36, 58, 60, 61, 64, 77, 78, 79, 88, 90, 91, 102, 114, 116, 117, 120, 122, 123, 124, 125, 126, 130, 132, 133, 134 boundary method, vii, 31, 32, 55, 56 buoyancy force distribution parameter, viii, 57, 59, 60, 66, 71, 72

C casting, 128 chemical, 3, 76 Chicago, 135 China, 31, 55 class, 90, 91

classes, 88 closure, 37 combined effect, 2, 58 complexity, 2, 32 compressibility, 129, 130 compression, 89 computation, 32, 79 conductivity, ix, 24, 59, 72, 77, 84, 88, 90, 91, 93, 113, 114, 115, 116, 117, 125, 126, 128, 129, 133, 135 conductors, 59 configuration, ix, 3, 33, 88, 102 conservation, 5, 116 controversial, 130 cooling, viii, 2, 57, 58, 65, 71, 73, 76 corrosion, 76 crust, 3 crystal growth, 2

D decay, 126 depression, 89 depth, 122 derivatives, 97, 109, 111 Dewar-Glasser equations, ix, 87 differential equations, vii, viii, ix, 1, 8, 75, 76, 78, 88, 122 dipolar magnetic configuration, ix, 88 discharges, 58

140

Index

displacement, 93, 98, 124 distortion, 131 distribution, viii, 57, 58, 59, 60, 66, 71, 72, 90, 107, 108 divergence, 37 dynamic viscosity, 24, 73

E eigenvalues, 122 elaboration, 90 electric charge, 116 electric field, 115, 116 electrical conductivity, 24, 77, 84 electrodes, 59 electromagnetic, 58, 103 electromagnetic field, 58 electron, 58 elongation, ix, 87, 92 energy, 2, 4, 5, 58, 60, 88, 123, 125, 135 engineering, 2, 3, 32, 58, 76 equilibrium, ix, 4, 88, 91, 94, 95, 97, 102, 105, 106, 107, 108, 122, 123, 130 exclusion, 116

F flat plate thermometer, vii, viii, 75, 76, 83, 85 flat wall, vii, 1, 2, 3, 4, 9, 10, 11, 21, 22, 23, 24, 28 flow field, 2 fluctuations, 128 fluid, vii, viii, 1, 2, 3, 4, 9, 10, 11, 23, 24, 28, 32, 33, 34, 57, 58, 59, 60, 65, 66, 68, 69, 71, 72, 73, 75, 76, 77, 79, 84, 92 force, viii, 32, 33, 57, 58, 59, 60, 65, 66, 70, 71, 72, 124 formation, 72 formula, 40 free convection, vii, 2, 23, 28, 29, 59, 73, 74 freezing, 117 frequencies, 122, 126

friction, vii, viii, 1, 4, 9, 19, 20, 23, 24, 57, 59, 62, 64, 65, 70, 71, 73, 75, 76, 79, 83, 84 functional analysis, 55 fusion, 89, 90, 130

G geometry, 32, 89, 90, 98, 122, 124 gravity, 23, 72, 89 grids, 32 growth, 2, 88, 124

H heart valves, 32 heat transfer, 2, 3, 4, 23, 28, 29, 58, 73, 76, 85 height, 116 Hermitian operator, 91, 123 Hilbert space, 36, 38 hybrid, 89 hydrodynamic pressure, ix, 88

I impulsive, 3 independent variable, 92 India, 1, 57, 75 induction, 84, 93 industries, 2 inequality, 43, 44, 54, 55, 98 inner magnetosphere, vii, viii, ix, 87, 88, 92, 103, 135 insulation, 3 insulators, 129 integration, 34, 45, 79, 127 interface, 32 ionospheric boundary conditions, ix, 88, 90, 91

141

Index

J Japan, 74

L laminar, 3, 4, 28, 58, 73, 76, 83, 85 Laplace transform, viii, 57, 59, 61 lead, 88, 131, 135 linear function, 19 longitudinal elongation, ix, 87 LTD, 56

M magnet, 90 magnetic field, viii, ix, 2, 4, 11, 18, 19, 23, 24, 57, 58, 59, 75, 76, 77, 80, 83, 85, 88, 89, 90, 91, 94, 95, 99, 102, 103, 104, 105, 106, 107, 109, 113, 115, 116, 117, 122, 126 magnetic fusion, 89 magnetohydrodynamics, vii, 71, 76 magnetosphere, vii, viii, ix, 87, 88, 89, 90, 91, 92, 98, 102, 107, 108, 113, 114, 135 magnitude, 68 mapping, 39, 48 mass, 5, 56, 58, 89, 90 matrix, 58 matter, viii, 87, 90, 94 maximum waviness, vii, 1, 17, 18, 23 Maxwell equations, 116 medicine, 32 mercury, 79, 80, 81 MHD, vii, viii, ix, 1, 2, 3, 4, 23, 28, 58, 74, 75, 76, 85, 87, 88, 91, 92, 97, 98, 103, 105, 114, 122, 123, 128, 132, 133, 135 MHD free convective flow, vii, 1, 2 modifications, ix, 31 momentum, 4, 5, 40, 60, 103

N natural convection, vii, 3, 28, 29, 58, 73, 74 nonlinear partial differential equations, vii, 1 numerical analysis, 56 numerical computations, 10 Nusselt number, vii, 1, 4, 10, 21, 22, 23, 72

O oil, 76 ordinary differential equations, viii, ix, 8, 31, 33, 75, 76, 78, 88, 122 oscillations, 102, 108, 123, 124, 126

P parallel, vii, 1, 3, 4, 11, 23, 28, 58, 73, 74 partial differential equations, vii, 1, 76 Pedersen conductivity, ix, 88, 115 periodicity, 34 permeability, 23 perpendicular flow, vii, 2, 23 perturbation method, vii, 1, 2, 3 Philadelphia, 55 physics, 3, 32, 138 polarization, ix, 88, 113 pollutants, 3 population, 58, 116 porous media, 2, 3 porous medium, vii, 1, 2, 3, 4, 23, 28, 29 pressure gradient, 6, 103, 107, 126 propagation, 114, 116

R radial distance, 104 radius, 103, 104 reactants, 76 reconstruction, 88 recovery, 76 redistribution, 90

142

Index

restrictions, ix, 36, 88 roughness, 3

S scientific publications, 91 shape, vii, 1, 3, 11, 32, 34, 68, 80 shear, 9, 24, 79, 84, 108 simulation, 32, 34, 46, 56 skin, vii, viii, 1, 4, 9, 19, 20, 23, 24, 57, 59, 62, 64, 65, 70, 71, 73, 75, 76, 79, 83, 84, 122 skin friction, vii, viii, 1, 4, 9, 19, 24, 57, 59, 65, 70, 71, 75, 76, 79, 83, 84 solution, vii, viii, 1, 3, 4, 7, 10, 17, 23, 31, 32, 33, 36, 41, 42, 46, 47, 48, 49, 50, 55, 56, 57, 59, 63, 64, 76, 79, 85, 98, 116, 124, 125, 126 specific heat, 23, 72, 77, 84, 92 stability, ix, 3, 28, 88, 89, 90, 91, 92, 103, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135 stabilization, 129 state, viii, 2, 58, 64, 65, 69, 72, 87, 88 states, 91, 129 stress, 9, 24, 79, 84 structure, 32, 90, 102, 124 successive approximations, 51 suction/injection, vii, viii, 75, 76, 79, 80 sulphur, 79, 80 surface component, 116 survey, viii, 87

T techniques, 91 temperature, vii, viii, 1, 2, 3, 4, 5, 9, 10, 12, 17, 18, 19, 23, 24, 28, 57, 58, 59, 60, 65, 66, 72, 73, 75, 76, 77, 78, 79, 80, 83, 84

theory of transversally small-scale standing, viii, 87 thermal energy, 60, 88 thermal expansion, 24, 72 thermodynamic equilibrium, 4 thermometer, vii, viii, 75, 76, 83, 85 three-dimensional model, 103 transformation, viii, 61, 75, 76, 78 transport, 2, 56, 58, 88 transport processes, 58 turbulence, 56

U Ukraine, 87

V variables, ix, 32, 88, 92 vector, 36, 37, 93, 95, 98, 103, 113, 115, 119, 125 velocity, vii, viii, 1, 4, 5, 9, 10, 11, 14, 15, 17, 18, 23, 24, 33, 57, 60, 63, 64, 65, 66, 68, 69, 72, 75, 76, 77, 79, 83, 84, 92 velocity profiles, viii, 11, 57, 65, 66, 68, 69, 79, 83 viscosity, 24, 33, 73, 77, 84

W wall temperature, 3, 4, 19, 23, 28, 73 water, 3, 65, 68, 71, 79, 80, 82 wave vector, 97 wavelengths, 98 wavy wall, vii, 1, 2, 3, 4, 9, 17, 18, 19, 21, 23, 24, 28