RIGA TECHNICAL UNIVERSITY  

 

 

JELENA LIGERE

ANALYTICAL SOLUTIONS OF

MAGNETOHYDRODYNAMICAL PROBLEMS ON A FLOW

OF CONDUCTING FLUID IN THE ENTRANCE REGION OF

CHANNELS IN A STRONG MAGNETIC FIELD

SUMMARY OF DOCTORAL THESIS

Submitted for the degree of Doctor of mathematics

Subfield of Mathematical modeling

Rīga, 2014

RIGA TECHNICAL UNIVERSITY

Faculty of Computer Science and Information Technology

Department of Engineering Mathematics

Elena Ligere

ANALYTICAL SOLUTIONS OF MAGNETOHYDRODYNAMICAL

PROBLEMS ON A FLOW OF CONDUCTING FLUID IN THE ENTRANCE

REGION OF CHANNELS IN A STRONG MAGNETIC FIELD

Summary of Doctoral Thesis

Submitted for the degree of Doctor of mathematics

Subfield of Mathematical modeling

Riga, 2014

Liģere J. Analytical solutions of magnetohydrodynamical problems on a flow of conducting fluid in the entrance region of channels in a strong magnetic field. Summary of Doctoral Thesis.- R.:RTU, 2014.- 50 lpp.

Published in accordance with the decision of the Department of Engineering Mathematics on October 3, 2014, protocol Nr. 12501-1/5

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The doctoral thesis was carried out:

at the Chair of Engineering Mathematics,

Faculty of Computer Science and Information Technology, Riga Technical University.

The thesis contains the introduction, 4 chapters, reference list, 2 appendices.

Form of the thesis: dissertation in mathematics, the subfield of mathematical

modeling.

Supervisor : Dr. habil. Phys., Professor M. Antimirov (till 20.03.2005),

Dr. math., Professor I.Volodko .

Reviewers:

1) Harijs Kalis, Dr.habil.mat., dr.habil.fiz., em. prof., Latvijas Universitāte;

2) Sergey Molokov, Dr.eng., prof., Koventri universitāte, Lielbritānija;

3) Aivars Aboltiņš, Dr.eng., vad.pētn., LLU.

The thesis will be defended at the public session of the Doctoral Committee of

mathematics, University of Latvia, at 15.00 on 24. october , 2014 in room 233 of the

Departament of Mathematics, Faculty of Physics and Mathematics, University of

Latvia in Riga, Zeļļu street 8.

The thesis is available at the Library of the University of Latvia, Kalpaka blvd. 4.

This thesis is accepted for the commencement of the degree of Doctor of mathematics

on ……………………, 2014 by the Doctoral Committee of Mathematics, University

of Latvia.

Chairman of the Doctoral Committee _______________/ Andris Buiķis/

Secretary of the Doctoral Committee _______________/ Jānis Cepītis/

© University of Latvia, 2014

© Jeļena Liģere, 2014

This work has been partly supported by the European Social Fund within the National Programme “Support for the carrying out doctoral study program’s and post-doctoral researches” project “Support for the development of doctoral studies at Riga Technical University”.

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ABSTRACT

The PhD thesis is devoted to the theoretical study of new magnetohydrodynamic (MHD) problems on a flow of a conducting fluid in an initial part of a plane or circular channel at the condition that fluid flows into the channel through a split of finite width or through a hole of finite radius on the channel’s lateral side. On the basis of the obtained solutions the velocity field of the flow is analysed numerically and asymptotic solutions at large Hartmann numbers are obtained. Moreover, in the thesis the dependence of the full pressure force in the entrance region on the profile of the inlet velocity in this region is studied analytically by means of analytical solution of two hydrodynamic problems and two MHD problems on an inflow of a viscous fluid into a half-space through a plane split of finite width or through a round hole of finite radius. Both the case of a uniform inlet velocity profile and the case of a parabolic velocity profile are considered. New asymptotic solutions for MHD problems are obtained at large Hartmann numbers for the case of parabolic inlet velocity profile. All problems in the thesis are solved in Stokes or Oseen approximation by using integral transforms.

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CONTENTS INTRODUCTION.....................................................................................................7

Actuality of research...............................................................................................7 Objectives of the thesis...........................................................................................9 Research methodology ...........................................................................................9 Scientific novelty and main results.........................................................................9 Practical importance of the work............................................................................12 List of publications .................................................................................................12 Presentations at conferences...................................................................................14

THE STRUCTURE OF THE WORK AND BRIEF CONTENT ........................14

Introduction...............................................................................................................14

Chapter 1. MHD problem on an inflow of a conducting fluid into a plane channel through a split on the channel’s lateral side ............................................15

1.1. General formulation of the problem. The case of a sloping external magnetic field....................................................................................................15

1.2. Solution of the problem for the longitudinal magnetic field ............................20

1.2.1. Solution of the odd problem with respect to y ...........................................20 1.2.2. Solution of the even problem with respect to y..........................................21 1.2.3. Numerical results and discussion ...............................................................21

1.3. Solution of the problem for the transverse magnetic field ...............................23

1.3.1. Solution of the odd problem with respect to y ..........................................23 1.3.2. Solution of the even problem with respect to y..........................................24 1.3.3. Numerical results and discussion ...............................................................24

Chapter 2. MHD problem on an inflow of conducting fluid into a channel through the channel’s lateral side in the presence of rotational symmetry ........26

2.1. MHD problem on an inflow of a conducting fluid into a plane channel through a round hole of finite radius in channel’s lateral side .........................26

2.2. Analytical solution for magnetohydrodynamical problem on a flow of a conducting fluid in the initial part of a circular channel................................31

Chapter 3. The dependence between the boundary velocity profile of the fluid and the full pressure force at the entrance of region.............................................34

3.1. The dependence between the boundary velocity of the fluid and the full pressure force at the entrance of the region in hydrodynamic problems..........35

3.1.1. The problem on a plane jet flowing into a half-space through a split of finite width .................................................................................................35

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3.1.2. The problem on a round jet flowing into a half-space through a round hole of finite radius ...................................................................................37

3.2. The dependence between the the boundary velocity profile of the fluid and the full pressure force at the entrance of the region for MHD problems..........38

3.2.1. The MHD problem on a plane jet flowing fluid into half-space through a plane split of finite width ...........................................................38 3.2.2. The MHD problem on a round jet flowing into a half-space through a round hole of finite radius ......................................................................39

Chapter 4. Analytical solution of the MHD problem on the influence of a cross flow on a main flow in the initial part of a plane channel.....................................39

4.1. Formulation of the problem ..............................................................................39 4.2. The solution of the problem for the transverse magnetic field.........................41 4.3. The solution of the problem for the longitudinal magnetic field......................44

CONCLUSIONS .......................................................................................................46

REFERENCES..........................................................................................................48

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INTRODUCTION

The main theme of the thesis is the construction of analytical solutions of some new magnetohydrodynamic (MHD) problems on a flow of a conducting fluid in the entrance region of channels and derivation of asymptotic solutions for these problems for the case of strong external magnetic field.

Flows of a conducting fluid in an external magnetic field produce a variety of new effects, which are not realizable in usual hydrodynamics. Magnetohydrodynamics (MHD) analyzes these phenomena. MHD describes the frontier area combining classical fluid mechanics and electrodynamics. It is relatively young discipline in natural science and engineering starting with the pioneering work of Hartmann [11] on a liquid metal duct flow under the influence of a strong external magnetic field (1937). Nowadays MHD has developed into a vast field of applied and fundamental research in engineering and physical science. Electromagnetic methods of action on electrically conducting medium are widely used both in technical devices and industrial processes in metallurgy and material processing, in chemical industry, industrial power engineering and nuclear engineering.

The present work is devoted to a theoretical study of new magnetohydrodynamic (MHD) problems on a flow of a conducting fluid in channels in a presence of a strong magnetic field for the case where fluid is flowing into channels through a split of finite width or through a hole of finite radius in the lateral side of the channel. Plane and circular channels of infinite length are considered. Particular attention is paid to the study of the flow in the entrance region of a channel. On the basis of the obtained exact analytical solutions the velocity field of the flow is numerically analyzed and the asymptotic solution at large Hartmann numbers is obtained.

Solving problems on an inflow of conducting fluid to a half-space through a split of finite width or studying developing flows in the initial part of channels the uniform velocity profile or parabolic velocity profile usually is given at the channel’s entrance region (see for example [3], [13], [14], [23], [44]). In the thesis the dependence of the full pressure force in the entrance region on the profile of the inlet velocity in this region is studied analytically. This study is performed by solving analytically two hydrodynamic and two MHD problems on an inflow of a viscous fluid (conducting fluid for MHD problems) into a half-space through a split of finite width or through a hole of finite radius. Both the case of a uniform inlet velocity profile and the case of a parabolic velocity profile are considered. New asymptotic solutions for MHD problems are obtained at large Hartmann numbers for the case of parabolic inlet velocity profile.

Actuality of research

As it is mentioned above, nowadays magnetohydrodynamic (MHD) phenomena, caused by the interaction of electrically conducting fluids with a magnetic field, are widely exploited both in technical devices such as pumps, flow meters, generators and industrial processes in metallurgy, material processing, chemical industry, industrial power engineering in order to control and manipulate various conducting materials.

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Channels, in particular narrow channels, are common parts of many MHD devices. Therefore, investigation of MHD phenomena in plane layers and channels with conducting fluids is quite important both for understanding of the basic mechanisms and for improving of the existing industrial processes and for developing new MHD devices.

The detailed knowledge of MHD effects in ducts and channels at presence of strong external magnetic field is also a key issue for the development of the fusion reactors where plasma is confined by a strong magnetic field. Liquid metals (LM) are used in different blocks of the reactor to produce tritium that fuels the reactor, for removing plasma’s impurities and protecting solid surfaces from overheating and corrosion. One of such blocks where LM is used is the blanket of the reactor. Since the LM motion in the systems of reactor including the blanket occurs in a strong magnetic field (10Т) and leads to a formation of large electromagnetic body forces, the MHD effects become significant and it is important to take them into account already on the stage of design (see e.q. [26], [41], [16], [15], [8], [34],[39], [42]).

One of the concept of a blanket is the slotted channels concept where heat exchange is organized using a system of slotted channels (channels with high aspect ratio) considered as a method for MHD pressure drop reduction in liquid metal cooled blanket design (see e.q. [40], [43], [42]). Hence, the analysis of MHD flows in such channels in strong magnetic field is also very important. Special attention should be paid to the MHD phenomena occurring in liquid metal in transit sections of channels where a substantial change of the flow direction relative to the magnetic field direction takes place.

This work is also important from a theoretical point of view. The number of exact solutions in MHD, obtained analytically, is limited due to the nonlinearity of the Navier-Stokes equations. Therefore, numerical methods are widely used for solving these problems. However, analytical solutions do not loose their importance (even if these solutions are obtained for simplified flows). First, analytical solutions give the qualitative picture of the flow and the main characteristic of the flow can be estimated. It is important for a general understanding of MHD flows. Second, analytical solutions are useful because they can be used as benchmark problems for testing numerical algorithms. Hence, it is always of interest to extend the class of problems for which an analytical solution can be found.

The geometry of the flows considered in this work for the case where a conducting fluid flows in a channel through the split on the lateral side of the channel gives an opportunity to solve these problems analytically in Stokes and Oseen approximations by means of the method of integral transforms that extends the range of problems that can be solved analytically. The analysis of dependence of the full pressure force in the entrance region on the profile of the initial velocity in this region presented in the thesis is also important for further analytical and numerical study of MHD flows in channels and ducts.

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Objectives of the thesis The aim of this work is

- an analytical study of MHD effects arising in a plane and a circular channels at the condition that conducting fluid flows into the channel though a finite width split or through a hole of finite radius on the channel’ s lateral side at the presence of strong magnetic field;

- a theoretical investigation of the dependence of the full pressure force at the entrance region on the profile of the inlet velocity;

- an analytical investigation of the influence of a cross flow on a main flow in the inital part of a plane channel in a longitudinal and transverse magnetic field.

Research methodology

Mathematical models used in this thesis are based on equations of magnetohydrodynamics containing the Navier-Stokes equation for the motion of incompressible viscous fluid with the inclusion of electromagnetic body force, the equation of mass continuity, the generalized Ohm’s law for a slowly moving medium in magnetic field and the Maxwell’s equations.

The following methods are used in the thesis: 1) All problems are solved in inductionless approximation, for which the induced

currents are taken into account, but the magnetic field created by these currents is neglected;

2) Two linearization schemes, known as Oseen and Stokes approximations, are used for the Navier-Stokes equations. In Stokes approximation the nonlinear term that corresponds to inertia force is neglected in the Navier-Stokes equations. In the Oseen approximation the inertia force is partially taken into account by means of linearization of the nonlinear term;

3) Problems are solved using Fourier and Hankel transforms; 4) Laplace transform is used for evaluation of some improper integrals; 5) On the basis of the results obtained in the form of improper integrals, the velocity

field is studied numerically using the package “Mathematica”.

Scientific novelty and main results

1) New MHD problems on an inflow of a conducting fluid into a plane channel through the channel’s lateral side are solved analytically in Stokes and inductionless approximation. Two cases are considered in detail: fluid flows into a plane channel through a split of finite width on the channel’s lateral side and through a round hole of finite radius. The cases of longitudinal and transverse madnetic fields are studied in details. The problem on an inflow of a conducting fluid into the channel through the round hole is solved only for case of transverse magnetic field.

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- Exact analytical solutions for velocity and pressure gradient are obtained in the form of convergent improper integrals;

- The velocity field is studied numerically on the basis of obtained analytical solutions. For the case where fluid flows into the channel through the split, it is shown that the effect of the longitudinal magnetic field is more pronounced than the effect of transverse magnetic field, i.e.:

• In case of longitudinal magnetic field numerical calculations give the M-shaped velocity’s profiles in the entrance region of the channel at large Hartmann numbers. The physical explanation of these velocity profiles is given in the thesis;

• It is also shown that in strong longitudinal magnetic field flows mostly occur along the wall with the split. As the Hartmann number increases, the layer of the flow is getting narrower and the velocity increases in this layer. The Poiseuille flow takes place at a large distance from the entrance split. In addition, with the increase of the intensity of the magnetic field the flow in the channel slowly approaches the Poiseuille flow;

• For the transverse magnetic field, the flow in channel differs from the Hartmann flow only near the entrance region of the channel and the Hartman flow is approached very quickly also for small Hartmann numbers.

- Asymptotic solutions of the problems at large Hartmann numbers are also obtained. An important practical result has been obtained for the case of transverse magnetic field, namely, a pressure gradient which is proportional to the square of the Hartmann number is needed for the turning the flow on the angle 90 degrees at large Hartmann numbers, while for pumping the fluid we need a pressure gradient which is proportional to only the first power of the Hartmann number.

2) Analytical solution is obtained for MHD problem on an inflow of conducting fluid into a circular channel through a split of finite width on the lateral side of the channel: - The problem is solved for the longitudinal magnetic field; - The exact analytical solution for velocity and pressure gradient is obtained in the

form of convergent improper integrals; - The velocity field is studied numerically on the basis of obtained analytical

solutions. As a result: • M-shaped velocity profiles are obtained in the entrance region of the channel at

large Hartman numbers and the physical explanation of these velocity profiles is given in the thesis;

• It is also shown that the flow in the channel slowly approaches Poiseuille flow as the Hartmann number grows.

- Asymptotic solution of the problem at large Hartmann numbers is also obtained.

3) As it is mentioned above, on solving problems on an inflow of conducting fluid into a half-space through a split of finite width or studying a developing flow in the initial

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part of channels, a uniform velocity profile or parabolic velocity profile is usually given at the entrance of region (for example [3], [13], [14], [23], [35], [44]). In some works (for example [9], [44]) the case of an arbitrary velocity profile at the channel’s inlet region has been also considered. In all these works the velocity distribution, the distribution of pressure or the temperature field have been studied, but to the author’s knowledge, the dependence of full pressure force at the entrance on the inlet velocity profile has been not studied in literature. In the PhD thesis this dependence is studied analytically by solving two hydrodynamic and two MHD problems on an inflow of viscous fluid (or conducting fluid for MHD problems) into a half-space through a split of finite width or through a hole of finite radius. Both the case of a uniform inlet velocity profile and the case of a parabolic inlet velocity profile are considered in the thesis: - The problems are solved in the Stokes approximation (and inductionless

approximation for MHD problems). In the case of plane split, the solution of the hydrodynamic problem is obtained in terms of elementary functions, but in the case of round hole the solution is obtained in the form of convergent integrals involving Bessel functions. For the MHD problems the solution is obtained in the form of improper convergent integrals;

- It is shown that the full pressure force at the entrance region is equal to infinity if the boundary velocity has a uniform profile and, consequently, such a problem is physically unrealistic. But if the boundary velocity has a parabolic profile, the full pressure force at the entrance region has a finite value and the problem is physically realistic. To the author knowledge, this fact isn’t reflected in the literature;

- New asymptotic solutions for MHD problems are obtained at large Hartmann numbers for the case of parabolic inlet velocity profile.

4) Analytical solution is obtained for MHD problem on a flow of conducting fluid in the initial part of a plane channel at the presence of cross flow in an external magnetic field. - The problem is solved for transverse and longitudinal magnetic field in Oseen and

inductionless approximation. - The solution of the problem for velocity and pressure gradient is obtained in the

form of convergent improper integrals. - The vector field of perturbation velocity is studied for different Hartmann and

Reynolds numbers. Numerical calculations give the M-shaped velocity profile at the initial part of channel at large Hartmann numbers.

- The length of the initial part of the channel ( initL ), at which the x-component of the velocity differs from velocity ∞V by less than 1%, is studied ( ∞V is the velocity far from the entrance region, i.e. ∞V is the velocity of the Hartmann flow for the transverse magnetic field and ∞V is the velocity of the Poiseuille flow for the longitudinal magnetic field).

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- It is shown that in transverse magnetic field the increase in the Reynolds number leads to the increase in the length of the initial part of the channel and with the increase in the intensity of magnetic field, the flow of conducting fluid in channel faster approaches Hartmann flow. The cross-flow in the transverse magnetic field generates vortices in the channel.

- In longitudinal magnetic field the increase in the Reynolds number leads to the increase in the length of the initial part initL . In addition, the length of the initial part of the channel initL increases when the Hartmann number grows and the flow in the channel slower approaches the Poiseuille flow. The dependence of the value

initL on the Reynolds number becomes linear in a strong magnetic field. Practical importance of the work

The results obtained in the thesis can be used for analyzing the behaviour of the MHD flow of a conducting fluid in channels with more complicated configuration than the one considered in this work.

Channels, in particular narrow channels, are common parts of many MHD devices used in metallurgy and material processing where MHD phenomena are exploited to control and manipulate metals. Results obtained in the thesis can be also used for engineering design of new MHD devices as well as for the improvement of different parts of already existing devices.

Since the results are obtained also for the case of strong magnetic field, they can be used in projecting of blanket of reactor Tokamak.

In MHD the number of exact solutions obtained analytically is limited due to the nonlinearity of the Navier-Stokes equations. Therefore, numerical methods are widely used for solving these problems, especially 3D problems (see [28], [12]). Nowadays numerical codes are developed that are able to simulate MHD flows in arbitrary geometries and for any orientation of the applied magnetic field. During the last decade few attempts are made to develop 3D software also for analysis of MHD flows in strong magnetic field (see for example [1], [24], [25], [29]). Analytical solutions of the problems solved in this work can be used as benchmark problems for complete numerical algorithms.

The analysis of the dependence of the full pressure force in the entrance region on the inlet velocity profile in this region presented in the thesis is important for further analytical and numerical study of MHD flows in channels and ducts. List of publications.

1) M.Ya.Antimirov, E.S Ligere. Analytical solutions for the problems of the flowing into of the conducting fluid through the lateral side of the plane channel in a strong magnetic field. Magnetohydrodynamics. Vol. 36, No. 1, pp. 47-60, 2000.

2) M.Ya. Antimirov, E.S Ligere. Analytical solutions for the problems of the flowing into of the conducting fluid through the lateral side of the round channel in a strong

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magnetic field. Proceedings for the 4th International Conference MHD at dawn of 3rd Millenium. “Pamir 2000”- Giens, France.

3) M.Ya. Antimirov, E.S Ligere. Analytical solution for magnetohydrodynamical problems at flows of conducting fluid in the initial part of round and plane channels. Magnetohydrodynamics. Vol. 36, No. 3, pp. 241-250, 2000.

4) Ligere J. On a dependence between the smoothness of the boundary velocity of the fluid and the full pressure force at the entrance of the region. Scientific Proceeding of Riga Technical University. Series-Computer Science. 43rd thematic issue.2001.-pp.48-57.

5) Antimirov M.Ya., Ligere E.S. On dependence between smoothness of the boundary velocity of the fluid and the full pressure force at the entrance of the region in MHD problems. Proceedings of the “5th International PAMIR Conference on Fundamental and Applied MHD”, Ramatuelle, France, September 16-20, 2002.-Vol.1, pp. I-217- I-222.

6) Antimirov M. Ya., Ligere E.S. Analytical solution of the problem on the magnetohydrodynamic flow in the initial part of the plane channel in the Oseen approximation. Scientific Proceedings of Riga Technical University, 5th series: Computer Science, 46th thematic issue: Boundary Field Problems and Computer Simulation, vol.16, Riga Technical University, ISSN 1407 - 7493, Riga, pp.113-122 pp. 106-112, 2004.

7) Antimirov M.Ya., Dzenite I.A. and Ligere E.S. Application of integral transforms to some problems of nondestructive testing and magnetohydrodynamics. Research report CM05/I-27, Department of Mathematics, University of Aveiro, Aveiro (Portugal), June 2005, 71 page, online version is available at http://www.pisharp.org/dspace/handle/2052/76 (under support of CTS fellowship).

8) Elena Ligere and Maximilian Antimirov. Analytical solution of the problem on a magnetohydrodynamic flow in the initial part of a plane channel in a transverse magnetic field in Oseen approximation. Taming Heterogeneity and Complexity of Embedded Control. CTS-HYCON Workshop on Nonlinear and Hybrid Control. Publisher: International Scientific & Technical Encyclopedia (ISTE), London, 2006, pp. 409-418.

9) Ligere E., Dzenite I. Analytical solution to the MHD problem on the influence of cross flow on the main flow in the plane channel at the Hartman large numbers. Proceedings (book & CD) of the 7th Pamir International Conference on Fundamental and Applied MHD, Presqu’ile de Giens (France), September 8-12, 2008.-Vol.2, pp.541-551

10) Ligere E., Dzenite I. On a round jet flowing into a plane channel through the hole of finite radius on the channel’s lateral side. Proceedings of the “8th International PAMIR Conference on Fundamental and Applied MHD”, Borgo, Corsica, France, September 5-9, 2011.- Vol.1, pp.151-155

11) Ligere E. Remarks to the Solution of MHD Problem on an Inflow of Conducting Fluid into a Plane Channel through the Channel’s Lateral Side. Scientific Journal of Riga Technical University. Computer Science. Boundary Field Problems and Computer Simulation. -2011.-Vol. 50, pp. 30-39.

12) Ligere E., Dzenite I. Application of Integral Transforms for Solving Some MHD Problems. Advances in Mathematical and Computational Methods: Proceedings of the 14th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering (MACMESE’12).- Sliema (Malta): WSEAS Press, September 7-9, 2012, - pp. 286.-291.

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Presentations at conferences.

1) 3th Latvian Mathematical Conference, April 14-15, 2000, Jelgava, Latvia; 2) 5th International Conference Mathematical Modeling and Applications, June 8-9, 2000,

Jurmala, Latvia; 3) 3rd European Congress of Mathematics, July 10-15, 2000, Barcelona, Spain; 4) 6th International Conference Mathematical Modeling and Analysis, May 31-June 2,

2001, Vilnius, Lithuania; 5) 42nd International scientific conference of Riga Technical University, October 11-13,

2001, Riga, Latvia; 6) 5th International Conference on Fundamental and applied MHD, September 2002,

France; 7) 4th European Congress of Mathematics, June 27- July 2, 2004, Stockholm, Sweden; 8) 45th International Scientific Conference of Riga Technical University, October 14 - 16,

2004, Riga, Latvia; 9) 6th Latvian Mathematical Conference, April 7 - 8, 2006, Liepaja, Latvia; 10) Join CTS-HYCON Workshop on Nonlinear and Hybrid Control, 10-12 July 2006,

Paris, France; 11) 47th International scientific conference of Riga Technical University, October 12-14,

2006, Riga, Latvia; 12) 7th Pamir International Conference on Fundamental and Applied MHD, September 8-

12, 2008, Presqu’ile de Giens (France); 13) 8th International PAMIR Conference on Fundamental and Applied MHD”, September

5-9, 2011, Borgo, Corsica, France; 14) 52th International scientific conference of Riga Technical University, Subsection

«Boundary Field Problems and Computer Simulation». October 12-14, 2011, Riga, Latvia;

15) 14th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering (MACMESE’12), September 7-9, 2012, Sliema, Malta;

16) 53th International scientific conference of Riga Technical University, Subsection «Boundary Field Problems and Computer Simulation». October 14-16, 2013, Riga, Latvia.

THE STRUCTURE OF THE WORK AND BRIEF CONTENT The thesis consists of 4 chapters, introduction and conclusions. It is written on 138 pages. Thesis contains 34 figures and 145 references. The thesis is written in English. Introduction. In this part of the thesis the actuality, theoretical and practical importance of this work and the structure of the theses is described. In addition, the governing equations of MHD used in the work are presented and important dimensionless parameters and assumptions used in this work are discussed. Literature review reflecting the topic of the thesis is also given in this part of the work.

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Chapter 1. MHD PROBLEM ON AN INFLOW OF A CONDUCTING FLUID INTO A PLANE CHANNEL THROUGH A SPLIT ON THE CHANNEL’S LATERAL SIDE

This chapter is devoted to an analytical study of the MHD problem on an inflow of conducting fluid in a plane channel through a split of finite width located on the channel’s lateral side. The cases of longitudinal and transverse magnetic fields are studied in details. Exact analytical solution for velocity and pressure gradient is obtained in Stokes and inductionless approximation in the form of convergent improper integrals. The Fourier transform is used to solve the problem. On the basis of the obtained solutions the velocity field of the flow is analyzed numerically. The asymptotic solution at large Hartmann numbers is also obtained for transverse magnetic field.

Section 1.1. General formulation of the problem. The case of a sloping external magnetic field.

The mathematical and geometric formulation of the problem is given in this part of the thesis and a main idea of solution is described for the case of sloping magnetic field. Formulation of the problem: A plane channel with conducting fluid is located in the region D={- h < y~ < h , ∞<<∞− x~ , ∞<<∞− z~ }. On the channel’s lateral side hy −=~ there is a split in the region { LxL ~~~ <<− , hy −=~ , ∞<<∞− z~ }. Conducting fluid flows into the channel through the split with constant velocity yeV0 . A strong uniform

external magnetic field eB is applied under the angle α to the split, i.e. ( )yx

e eeBB ⋅+⋅= αα sincos0 . The geometry of the flow is shown in Fig.1.1. The case of nonconducting walls

hy ±=~ and perfectly conducting lateral sidewalls ±∞=z~ is considered in the thesis. In this case one can assume that

0== yx EE and constEz = (see [38]). For determination of the constant zE we used the fact that the Hartmann flow takes the place in a plane channel in external magnetic field y

e eBB ⋅= αsin0 at

x→∞. One more assumption is used for the problem: induced streams do not flow through the split hy −=~ ,

LxL ~~~ <<− in the region hy −<<∞− ~ .

y~

x~

h

D

α

eBα

eBy

hyeVV 0

~ =−=

L~−h−

L~

Fig.1.1. The geometry of the flow in the plane

channel with a split on the channel’s lateral side.The case of sloping magnetic field.

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The dimensionless variables are introduced using the half-width of the channel h as the scale of length, the magnitude of the velocity of fluid in the entrance region 0V as the scale of the velocity and 0B , 00 BV , hV /0ρν as the scales of magnetic field, electrical field and pressure, respectively, where σ, ρ, ν are, respectively, the conductivity, the density and the viscosity of the fluid.

We use MHD equations in Stokes and inductionless approximation (see [38]). Projections of these equations onto the x and y axes for the problem described above are:

0)cossin(sin2 =−⋅−∆+− ααα∂

∂yxx

m VVHaVx

P , (1.1)

0cossincos2 =−⋅+∆+− )V(VHaVy

Pyxy

m ααα∂

∂ , (1.2)

0=+y

Vx

V yx

∂∂

∂∂ , (1.3)

where yyxx eyxVeyxVV ),(),( += is the velocity of the fluid,

),( yxP is the pressure,

∆ is the Laplace operator , i.e. 2

2

2

2

yf

xff

∂∂

∂∂

+=∆ ,

ρνσ /0hBHa = is the Hartmann number, which characterizes the ratio of

electromagnetic force to viscous force,

)sign(2

cossin2sin 222

xyHaxHaPPm ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅⋅−⋅

⋅−=

ααα . (1.4)

The boundary conditions are:

⎩⎨⎧

−∈−∉

==−=),(,1),(,0

,0:1LLxLLx

VVy yx , (1.5)

0:1 === yx VVy , (1.6)

)sign()(: xyVVx x ⋅→±∞→ ∞ , )()( xsignAxsignx

PxP

⋅=⋅∂

∂→

∂∂ ∞ , (1.7)

where constA = , hLL /~= and xeyVyV ⋅= ∞∞ )()( is the velocity of the flow in the channel at the sufficient distance from the entrance region. The functions )(yV∞ and xP ∂∂ ∞ / at ∞→x depend on the external magnetic field and satisfy the following equation (see [36],[37]):

17

AconstyVHady

yVdx

P≡=⋅−=

∂∂

∞∞∞ )(sin)( 22

2

2

α (1.8)

with the boundary conditions: 1±=y : 0)( =∞ yV .

1) In the case of longitudinal external magnetic field xe eBB ⋅= 0 ( 0=α ) the Poiseuille

flow takes place at ±∞→x . 2) In the case of transverse magnetic field y

e eBB ⋅= 0 ( 2/πα = ) the Hartmann flow takes place at ±∞→x . Solution of the problem: To obtain the solution of the problem, the complex Fourier transform with respect to x is used:

dxeyxfyxfFyf xi∫+∞

∞−

−== λ

πλ ),(

21)],([),(ˆ . (1.9)

Since xV and xP ∂∂ / do not tend to zero at ±∞→x , we introduce the following new functions for the velocity and pressure gradient before using the Fourier transform:

)()arctan(2 yVxVV new∞⋅−=

π and Ax

xP

xP m

new

⋅−∂

∂=

∂∂ )arctan(2

π (1.10)

The problem (1.1)-(1.3) for the new functions has the form:

0)()1(

4)cossin(sin 222 =

+−−⋅−∆+− ∞ yV

xxVVHaV

xP

yxnew

x

newm

πααα

∂∂ , (1.11)

0)()arctan(2sincossincos2

2 =⋅+−⋅+∆+− ∞ yVxHa)V(VHaVy

Pyxy

newm

παααα

∂∂ , (1.12)

0)(1

122 =

+++ ∞ yV

xyV

xV y

newx

π∂∂

∂∂ . (1.13)

Boundary conditions are:

⎩⎨⎧

−∈−∉

==−=),(,1),(,0

,0:1LLxLLx

VVy ynew

x ; (1.14)

0:1 === ynew

x VVy ; (1.15)

0: →±∞→ newxVx , 0→

∂∂

xPnew

. (1.16)

Applying the complex Fourier transform (1.9) to Eqs.(1.11)-(1.16) we get the system of ordinary differential equations for the Fourier transforms )],([),(ˆ yxVFyV new

xx =λ ,

)],([),(ˆ yxVFyV yy =λ , )],([),(ˆ yxPFyP newm=λ with the corresponding boundary

conditions:

18

0)()(ˆ)cosˆsinˆ(sinˆˆ3

2 =+−⋅−+− ∞ yVfVVHaVPi yxx λαααλ L , (1.17)

0)()(ˆsincos)cosˆsinˆ(cosˆˆ1

22 =⋅+−⋅++− ∞ yVfHaVVHaVdyPd

yxy λαααααL , (1.18)

0)()(ˆˆ

ˆ2 =++ ∞ yVf

dyVd

Vi yx λλ , (1.19)

( )λλ

πLVVy yx

sin2ˆ,0ˆ:1 ==−= ; (1.20)

0ˆ,0ˆ:1 === yx VVy , (1.21)

where 222 / dyfdff +−= λL ,

λππλ

λ−

⋅−=⎥⎦⎤

⎢⎣⎡ ⋅=

eixFf 2)arctan(2)(1̂ , λ

ππλ −⋅=⎥⎦

⎤⎢⎣⎡

+⋅= e

xFf 2

112)(ˆ

22

( )λ

ππλ λ ⋅⋅⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+⋅

−= −ei

xxFf 2

14)(ˆ

223 .

The general solution of this system for yV̂ has the form:

)cosh()cosh()sinh()sinh(),(ˆ44332211 ykCykCykCykCyVy +++=λ (1.22)

where 41 CC − are arbitrary constants, 24,312,1 sin,sin DkDk ±−=±= αµαµ ,

)cos2(sin 222,1 αµλλαµ iD ±+= , µ2=Ha .

One can get the solution of the problem by determining xV̂ from Eq.(1.19) and then P̂ from Eq.(1.17) and applying the inverse Fourier transform in the form

( )dxxixyfdxeyfyfFyxf xi ∫∫+∞

∞−

+∞

∞−

− +=== λλλπ

λπ

λ λ sincos),(ˆ21),(ˆ

21)],(ˆ[),( 1 (1.23)

In the next two paragraphs two special cases are considered in detail:

1) the external magnetic field xe eBB 0= is parallel to the x-axis (so-called

longitudinal magnetic field ( 0=α )) 2) the external magnetic field y

e eBB 0= is parallel to the y-axis (so-called transverse magnetic field ( 2/πα = ))

In order to simplify the solution of these problems and reduce the number of constants

41 CC − in Eq.(1.22), the problems are divided into two sub-problems: odd and even problems with respect to y, by considering a plane channel with two splits on its lateral sides hy ±=~ in the region LxL ~~~ <<− .

19

1) The odd problem (Fig.1.2): the fluid with velocities ( ) 2/0 yeV∓ , flows into the channel through both splits on hy ±=~ . The boundary conditions are:

:1±=y

⎪⎩

⎪⎨⎧

−∈

−∉== ),(,

21

),(,0,0 LLx

LLxVV yx ∓ (1.24)

)sign()(: xyVVx x ⋅→±∞→ ∞ ,

)()( xsignAxsignx

PxP

⋅=∂

∂→

∂∂ ∞ . (1.25)

In order to solve this problem, new functions for the velocity and pressure gradient are introduced according to (1.10). Boundary conditions for these functions are:

⎩⎨⎧

−∈−∉

==±=),(,2/1

),(,0,0:1

LLxLLx

VVy ynew

x ∓ (1.26)

0: →±∞→ newxVx , 0/ →∂∂ xPnew . (1.27)

The boundary conditions in the transformed space have the form:

( )λλ

πLVy y

sin221ˆ:1 ∓=±= , 0ˆ =xV . (1.28)

In this problem yV is an odd function with respect to y, therefore, the coefficients 043 == CC in Eq.(1.22). The coefficients 1C and 2C are determined from boundary

condition (1.28) and the following additional boundary condition, derived from Eq.(1.19):

1at0/ˆ ±== ydyVd y . (1.29)

2) The even problem (Fig.1.3): the fluid with velocity ( ) 2/0 yeV flows into the channel through a split in plane hy −=~ and flows out with the same velocity through the split in plane hy =~ .

The boundary conditions are:

:1±=y

⎪⎩

⎪⎨⎧

−∈

−∉== ),(,

21

),(,0,0 LLx

LLxVV yx (1.30)

0: →±∞→ xVx , 0→∂∂

xP . (1.31)

0

yhy

y eVV 02

1~ =−=

x0e e=BB

yhy

y eVV 02

1~ −==

L~− L~

h

h−

y~

x~

Fig.1.2. Channel with the split. The geometry of the flow for the odd problem.

0

yhy

y eVV 02

1~ =−=

x0e e=BB

yhy

y eVV 02

1~ ==

L~− L~

h

h−

y~

x~

Fig.1.3. Channel with the split. The geometry of the flow for the even problem.

20

There is no need to introduce new functions in this case, because xV and xP ∂∂ / tend to zero at ±∞→x . Therefore, one has to solve problem (1.1)-(1.3) with boundary conditions (1.30),(1.31) or problem (1.11)-(1.13) with the same boundary conditions and with

0)( =∞ yV and 0/ ==∂∂ ∞ AxP (1.32)

After applying the Fourier transform to the boundary conditions (1.30), we have:

( )λλ

πLVy y

sin221ˆ:1 =±= , 0ˆ =xV . (1.33)

Solving the even problem, one must take into account that in this case yV is an even function with respect to y and the coefficients 021 == CC in Eq.(1.22). The constants

3C and 4C in Eq.(1.22) are determined from boundary conditions (1.33) and (1.29).

3) The solution of the general problem is equal to the sum of solutions for odd and even problems with respect to y.

Section 1.2. Solution of the problem for the longitudinal magnetic field

In this paragraph the case of longitudinal magnetic field is considered. The problem is described by system (1.1)-(1.3) at 0=α and boundary conditions (1.5)-(1.7). 1.2.1. Solution of the odd problem with respect to y

The odd problem with respect to y is solved, i.e. the system (1.11)-(1.13) at 0=α with boundary conditions (1.26), (1.27). The corresponding system of ordinary

differential equations for the Fourier transforms ),(ˆ yVx λ , ),(ˆ yVy λ , ),(ˆ yP λ consists of Eqs. (1.17)-(1.19) at 0=α and boundary conditions (1.28), (1.29). As a result, the solution to this problem is obtained in the form of convergent improper integrals:

λλ

λλλπ

dxLykkykkHayxVxsinsin)coshcoshcosh(cosh1),( 2112

0 1

22

⋅⋅−⋅∆+

= ∫∞

, (1.34)

λλ

λλπ

dxLykkkykkkyxVycossin)sinhcoshsinhcosh(11),( 122211

0 1

⋅⋅−⋅∆

= ∫∞

, (1.35)

λλλλπ

dxLykkykkHaiHaxP sinsin)coshcoshcosh(cosh 1221

0 1

22

⋅⋅+⋅∆+

=∂∂

∫∞

, (1.36)

λλ

λλλλπ∂

∂ dxLykkkHaiykkkHaiHayP cossin)sinhcosh)(sinhcosh)((1

2111220 1

⋅⋅++⋅−∆

−= ∫∞

(1.37) where 2111221 sinhcoshsinhcosh kkkkkk ⋅−⋅=∆ , (1.38)

λλλλ HaikHaik ⋅−=⋅+= 22

21 , . (1.39)

21

1.2.2 Solution of the even problem with respect to y

The problem (1.1)-(1.3) at 0=α with boundary conditions (1.30),(1.31) is solved in this part of the thesis. The system of ordinary differential equations for the Fourier transforms ),(ˆ yVx λ , ),(ˆ yVy λ , ),(ˆ yP λ for the problem has the form of (1.17)-(1.19) with 0=α and with the conditions (1.32). Boundary conditions for this system are given by (1.33) and (1.29). The solution of the even problem is also obtained in the form of convergent improper integrals, similarly to the solution of the odd problem.

1.2.3. Numerical results and discussion

On the basis of the obtained solutions, the velocity field is studied numerically by using the package “Mathematica”. The profiles of the x-component of the velocity are presented graphically for the odd, even and general problems for different Hartmann numbers. The odd problem: Fig.1.4 presents the profiles of the velocity component xV calculated by means formula (1.34) for Ha=10, Ha=50. The component xV is shown only for 10 ≤≤ y since it is an even function with respect to y. It can be seen from Fig.1.4 that at 10≥Ha the velocity component xV has the M-shaped profile in the channel’s entrance region. The M-shaped profiles become more pronounced as the Hartmann number increases. The qualitative explanation of this phenomenon is given in this part of the thesis. In addition, the Hartmann number is larger the farther away from the entrance region the flow approaches the Poiseille’s flow.

0.2 0.4 0.6 0.8 1y

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Vx

x=1.1

x=1.5x=2

x=3

Ha=10Vp

Fig.1.4. Profiles of the velocity component xV for the odd problem and x

e eBB 0= (Vp is the Poiseuille flow).

Vx

0.2

0.4

0.6

0.8

1.2

0.2 0.4 0.6 0.8 1y

1

Ha=50

x=1.1

x=9

x=5

x=3x=2

Vp

22

The even problem: Fig.1.5 presents the profiles of the velocity component xV for the even problem with respect to y. The component xV is an odd function with respect to y.

-1 -0.8 -0.6 -0.4 -0.2

0.1

0.2

0.3

0.4

0.5

Ha=10x=1.1

x=1.5

x=2

x=3

x=4x=5

xV

y

Fig.1.5. Profiles of the velocity component xV for the even problem and x

e eBB 0= .

The general problem: The solutions of general problem (1.1)-(1.3), (1.5)-(1.7) at

0=α are equal to the sum of solutions for the even and odd problems with respect to y. The profiles of the velocity component xV are shown in Fig.1.6. One can see that near the entrance split the flow mostly occurs along the wall with the split at 1−=y . As the Hartmann number increases, the layer of the flow is getting narrower and the velocity increases in this layer. The Poiseuille flow takes place at a distance from the entrance split. In addition, when the Hartmann number grows the flow in the channel slowly approaches Poiseuille flow.

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1

1.2 Ha=10Vx

y

x=1.1x=1.5

x=2

x=3x=5 Vp

Fig.1.6. Profiles of the velocity component xV for the general problem and x

e eBB 0= . ( is the Poiseuille flow)

-1 -0.8 -0.6 -0.4 -0.2

0.2

0.4

0.6

0.8

1

1.2

Ha=50

y

Vxx=1.1

x=1.5

x=2

x=3

x=5x=7

x=9

-1 -0.5 0.5 1

0.5

1

1.5

2

2.5 Ha=50Vx

y

x=1.1

x=1.5

x=2

x=3

x=5x=7

x=20x=10

23

Section 1.3. Solution of the problem for the transverse magnetic field

In this paragraph the problem (1.1)-(1.3) with boundary conditions (1.5)-(1.7) is solved for the transverse magnetic field ( 2/πα = ). Similarly as it is done for the longitudinal magnetic field, the problem is solved by splitting it into odd and even sub problems with respect to y. 1.3.1. Solution of the odd problem with respect to y

The odd problem is described by the system of differential equations (1.1)-(1.3) with boundary conditions (1.24) and (1.25). To solve the problem, new functions for the velocity and pressure gradient are introduced according to (1.10). As a result, the solution of this problem is obtained in the form of convergent improper integrals:

λλλπ

dxLykkykkyxVx sinsin)coshcoshcosh(cosh11),( 21120 1

⋅⋅−⋅∆

−= ∫∞

, (1.40)

λλ

λλπ

dxLykkkykkkyxVycossin)sinhcoshsinhcosh(11),( 122211

0 1

⋅⋅−⋅∆

= ∫∞

, (1.41)

λλλπ

dxLykkkykkkHaxP sinsin)coshcoshcoshcosh(1

1222110 1

⋅⋅−⋅∆

−=∂∂

∫∞

, (1.42)

λλλλπ∂

∂ dxLykkykkHayP cossin)sinhcoshsinh(cosh 1221

0 1

⋅⋅−⋅∆

−= ∫∞

(1.43)

where 2111221 sinhcoshsinhcosh kkkkkk ⋅−⋅=∆ , (1.44)

222

221 , λµµλµµ +−=++= kk , (1.45)

The asymptotic evaluation of the solution at Ha →∞ for the odd problem:

The limits 1

/lim−=∞→ ym yP ∂∂

µ and

1/lim

−=∞→ ym xP ∂∂µ

are determined in this part of the thesis

that is of great interest for applications in Tokamak reactor:

( ) ( )⎩⎨⎧

+∞<<⋅−<<⋅⋅−

=⋅−= ∫∞

∞→ xLHaLLxxHaL

dxLHax

Pm

,2/0,2/sinsinlim 2

2

02

2

λλ

λλπ∂

∂µ

(1.46)

⎪⎩

⎪⎨

⎧

>=−

<<−=⋅−= ∫

∞

−=∞→

LxLxHa

LxHadxLHa

yP

y

m

,0,4/

0,2/cossinlim

01

λλ

λλπ∂

∂µ

(1.47)

These limits lead to important practical result, namely at large Hartmann numbers, the pressure gradient which is proportional to the square of the Hartmann number is needed for turning the flow on the angle 90 degrees, while for pumping of the fluid we need the pressure gradient which is proportional to only the first power of the Hartmann number.

24

1.3.2. Solution of the even problem with respect to y

The even problem is described by the system (1.1)-(1.3) at 2/πα = with boundary conditions (1.30)-(1.31). The solution of the problem is obtained in the form of convergent improper integrals similarly to the solution for the odd problem. In this problem )1(OPm = as Ha→∞, therefore, the principal contribution to the pressure gradient as Ha→∞ gives the odd case with respect to y.

1.3.3. Numerical results and discussion

Similarly, as it is done for the longitudinal magnetic field, the velocity field is studied numerically by using the package “Mathematica”on the basis of the obtained results. The odd problem: Velocity profiles for the x component xV are presented in Fig.1.7 for L=1 (Fig.1.7A) and for L=4 (Fig.1.7B).

0.2 0.4 0.6 0.8 1y

0.42

0.44

0.46

0.48

0.52

0.54

0.56

Vx Ha=10

x=1.01

x=1.1

x=1.5 Vhart

0.2 0.4 0.6 0.8 1y

0.46

0.48

0.52

0.54

Vx Ha=50

x=1.01

x=1.1Vhart

Fig.1.7A. Profiles of the velocity component xV for odd problem and ye eBB 0= if L=1.

(--- Vhart is the Hartmann flow).

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

Ha=10

x=1.1x=1.5

x=2

x=3

x=4Vhart

Vx

y0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

Ha=50Vx

y

x=1.5

x=1.1

x=2

x=3

x=4

Vhart

Fig.1.7B. Profiles of the velocity component xV for the odd problem and ye eBB 0= if L=4.

It can be seen from figures that xV has the M-shaped profiles only near to the entrance hole if L=1. However, even at small distance from the entrance, the flow approaches the Hartmann flow in a channel. With the increase of L, the flow slower approaches

25

the Hartmann flow. In addition, at L=4, the component xV doesn’t have M-shaped profile. The even problem: Fig.1.8 shows the profiles of the velocity component xV for the even problem for 01 ≤≤− y . In this case, the function xV is an odd function with respect to y. One can see from Fig.1.8 that the component xV differs from zero only near the entrance region. In addition, for some values of x the component xV is negative at 01 <<− y and Ha=10. However, since the fluid inflows into the channel through the hole on 1−=y , the x-component of the velocity must be positive for

01 <<− y at Ha=0. It means that there exists an opposite flow in the region in the case of transverse magnetic field. It occurs due to a vortices generated in the channel (see Fig.1.9 ).

-1 -0.8 -0.6 -0.4 -0.2y

-0.02

0.02

0.04

0.06

0.08

0.1

0.12

VxHa=10

x=1.01

x=1.1

x=1.5x=2

-1 -0.8 -0.6 -0.4 -0.2y

0.02

0.04

0.06

0.08

VxHa=50

x=1.01

x=1.1x=1.5

Fig.1.8. Velocity profiles of the x component xV for the even problem and ye eBB 0= , L=1

0.9 1.1 1.2 1.3 1.4 1.5x

-1

-0.5

0.5

1

y

Fig.1.9. Velocity field for the even problem and ye eBB 0= , L=1 at Ha=10.

26

The general problem: The solution of the general problem (1.1)-(1.3) at α=π/2 with boundary conditions (1.5), (1.7) is equal to the sum of the solutions for the odd and even problems. Fig.1.10 presents the results of calculation of the x-component of the velocity for the general problem. One can see that the profiles of the velocity component xV differ from the Hartmann flow profiles only near the entrance region.

-1 -0.5 0.5 1

0.1

0.2

0.3

0.4

0.5

0.6Ha=10Vx

y

x=1.01

x=1.1x=1.5

Vhart

-1 -0.5 0.5 1

0.1

0.2

0.3

0.4

0.5

0.6Ha=50Vx

y

x=1.01

x=1.1

Vhartm

Fig.1.10. Velocity profiles of component xV for the general problem and ye B eB 0= , L=1.

Chapter 2. MHD PROBLEM ON AN INFLOW OF A CONDUCTING FLUID INTO A CHANNEL THROUGH THE CHANNEL’S LATERAL SIDE IN THE PRESENCE OF ROTATIONAL SYMMETRY

In this part of the thesis the following two problems on an inflow of a conducting

fluid into a channel in the presence of the rotational symmetry in the geometry of the flow are considered: 1) MHD problem on an inflow of a conducting fluid into a plane channel through a round hole of finite radius in the channel’s lateral side. 2) MHD problem on an inflow of a conducting fluid into the round channel through a split of finite width in the channel’s lateral side. Section 2.1. MHD problem on an inflow of a conducting fluid into a plane channel through a round hole of finite radius in the channel’s lateral side Formulation of the problem:

The plane channel with conducting fluid is located in the region D:{ +∞≤≤ r~0 , πϕ 2~0 ≤≤ , hzh ≤≤− ~ }, ( r~ ,ϕ~ , z~ are cylindrical coordinates). A conducting fluid flows into the channel with the constant velocity zeV0 through

a round hole of finite radius R~ , located in its lateral side. It is supposed that the channel’s walls hz ±=~ are non-

0

zhz

eVV 0~ =

−=

ze e=BB 0

~ -h

hD

R~

r~

z~

R~

Fig.2.1. The geometry of the flow for the plane channel with a hole in the channel’s lateral side.

27

conducting. It is also assumed that induced streams do not flow through the hole hz −=~ , Rr ~~0 << in the region hz −<<∞− ~ . The case of the transverse external

magnetic field is considered for this problem, i.e. ze eBB 0= . The geometry of the flow

is shown in Fig.2.1. For this problem the intensity of electrical field 0=E (see [36],[37]).

We introduce dimensionless variables as in Section 1. MHD equations in cylindrical coordinates in Stokes and inductionless approximation are used (see [36],[37]). In projections onto the r and z axes these equations have the form:

021 =⋅++− rr VHaVL

rP

∂∂ , (2.1)

00 =+− zVLzP

∂∂ , (2.2)

0)(1=⋅+ r

z Vrrrz

V∂∂

∂∂ , (2.3)

where 2

2

2

2

01

zrrrL

∂∂

+∂∂

+∂∂

= , 201 /1 rLL −= , hRR /~= , (2.4)

zzrr ezrVezrVV ),(),( += is the velocity of the fluid, ),( zrP is the pressure. Boundary conditions are:

1−=z : 0=rV , ⎩⎨⎧

>≤≤

=Rr

RrVz ,0

0,1, (2.5)

1=z : 0=rV , 0=zV , (2.6)

0: →±∞→ rVr , 0/ →∂∂ rP . (2.7)

Solution of the problem:

Due to the axial symmetry of the problem with respect to r, the Hankel transform (see [2]) is used for the solution of the problem:

rdrrJVzV rr )(),(ˆ1

0

λλ ∫∞

= , drrrJVzV zz )(),(ˆ0

0 λλ ∫∞

= , drrrPJzP )(),(ˆ0

0 λλ ∫∞

= (2.8)

where )( rJ λν is the Bessel function of order ν.

Applying the Hankel transform (2.8) we obtain the system of ordinary differential equations for the Hankel transforms ),(ˆ zVr λ , ),(ˆ zVz λ , ),(ˆ zP λ . The general solution of this system for the function zV̂ has the form:

28

zkCzkCzkCzkCzVz 24132211 coshcoshsinhsinh),(ˆ +++=λ (2.9)

where λλλλ HaikHaik ⋅−=⋅+= 22

21 , . (2.10)

41 ,..,CC are arbitrary constants and 2/Ha=µ .

In order to simplify the problem and reduce the number of constants 41 CC − in (2.9), the problem is divided into two sub-problems: the odd problem with respect to z and the even problem, on considering a plane channel with two holes in its lateral sides

hz ±= in region Rr ~~0 << . 1) The odd problem (Fig.2.2): the fluid with velocities ( ) 2/0 zeV∓ flows into the channel through the both holes on hz ±=~ . The nondimensional boundary conditions are:

1±=z :

0=rV , ⎩⎨⎧

>≤≤

=Rr

RrVz ,0

0,1∓ (2.11)

0: →±∞→ rVr , 0/ →∂∂ rP . (2.12)

In this case the velocity component zV is an odd function with respect to z, therefore, the coefficients 043 == CC in Eq.(2.9). The solution of the odd problem is obtained in the form of convergent improper integrals:

λλλλ drJRJzkkzkkRzrVr )()()coshcoshcosh(cosh12

),( 1112210 1

⋅⋅⋅⋅−⋅∆

= ∫∞

, (2.13)

λλλ drJRJzkkkzkkkRzrVz )()()sinhcoshsinhcosh(12

),( 011222110 1

⋅⋅⋅−⋅∆

= ∫∞

, (2.14)

λλλλ drJRJzkkkzkkkRHarP )()()coshcoshcoshcosh(1

2 111222110 1

2

⋅⋅⋅⋅−⋅∆

⋅−=

∂∂

∫∞

, (2.15)

λλλλ∂∂ drJRJzkkzkkRHa

zP )()()sinhcoshsinh(cosh1

2 012

12210 1

⋅⋅⋅⋅−⋅∆

⋅−= ∫

∞

(2.16)

where 2111221 sinhcoshsinhcosh kkkkkk ⋅−⋅=∆ . (2.17)

222

221 , λµµλµµ +−=++= kk .

0

zhz

eVV 02

1~ =−=

ze e=BB 0

~R~

h

h−

z~

r~

R~

zhz

eVV 02

1~ −==

Fig.2.2. Plane channel with a round hole. The odd problem with respect to z.

29

Asymptotic evaluation at ∞→r and ∞→Ha for the odd problem:

In this part of the thesis the limits rrV

∞→lim , rP

r∂∂ /lim

∞→ and limits for pressure

gradient at large Hartmann numbers are obtained for the odd problem.

1) It follows from the obtained limits rrV

∞→lim , rP

r∂∂ /lim

∞→ that the Hartmann flow takes

place at ∞→r , but the magnitude of the velocity and pressure gradient are inversely proportional to the distance from the hole. It corresponds to the conservation law of flow rate. Similar conclusions can be drawn from the numerical results (see Fig.2.4)

2) The pressure gradient at large Hartmann numbers is also analyzed, i.e.

hzrP

−=∞→∂∂

µ/lim and

hzzP

−=∞→∂∂

µ/lim are obtained ( 2/Ha=µ ). The determination of

these limits has the greatest interest for applications in reactor Tokamak:

( )( ) ( )⎩

⎨⎧

+∞<≤−≤<−

=−= ∫∞

∞→ rRrRHaRrrHa

drJRJRHa

rP

,4/0,4/)()(

2lim 22

2

0

112

λλ

λλ∂∂

µ (2.18)

2/)(~)()(2

lim0

01 rFRHadrJRJRHa

zP

⋅⋅−=⋅

−= ∫∞

∞→λ

λλλ

∂∂

µ, (2.19)

where ( )⎪⎩

⎪⎨⎧

∞<<⋅

<<−=

rRrRFrRRrRrF

rF),/;2;2/1;2/1(2/

0),/;1;2/1;2/1()(~

22

22

, (2.20)

F(α, β, γ, z) is the hypergeometric function (see [10], formulae 6.574(1), 6.574(3)).

2) The even problem (Fig.2.3): the fluid with velocity ( ) 2/0 zeV flows into the channel through the hole on hz −= and flows out with the same velocity through the hole on hz = . The dimensionless boundary conditions for the even problem are:

1±=z :

0=rV , ⎩⎨⎧

>≤≤

=Rr

RrVz ,0

0,1 (2.21)

0: →±∞→ rVr , 0/ →∂∂ rP . (2.22)

Since in this problem the velocity component zV is an even function with respect to z, the coefficients 021 == CC in Eq.(2.9). Solution of the even problem for the velocity and pressure gradient in the case of longitudinal magnetic field is obtained in the form of convergent improper integrals similarly to the solution of the odd problem .

0

ze e=BB 0

~R~

h

h−

z~

r~

R~

zhz

eVV 02

1~ =−=

zhz

eVV 02

1~ =−=

Fig.2.3.The plane channel with a round hole. The even problem with respect to z.

30

Numerical results for the transverse magnetic field and discussion

The odd problem: The results of calculations of the velocity component rV for the odd problem are shown in Fig.2.4. The component rV is an odd function with respect to z. One can see that rV has the M-shaped profiles only near the entrance hole.

0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

Ha=10Vr

z

r=1.01r=1.1

r=1.5

r=2

r=3

r=6r=4

0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

0.3 Ha=50Vr

z

r=1.01

r=1.1

r=1.5

r=2

r=3

r=6r=4

Fig. 2.4. Velocity profiles for the radial component rV for the odd problem at R=1. The even problem: Fig.2.5 shows the results of calculation of the r-component rV of the velocity for the even problem. One can see that rV differs from zero only near the entrance region. Besides, in Fig.2.5 some values of rV are positive at 10 ≤< r and Ha=10. However, since the fluid flows out through the hole at 1=z , rV must be negative for 10 << r at Ha=0. It means that there exists an opposite flow in transverse magnetic field. It occurs due to vortices generated in the channel (see Fig.2.6).

-1 -0.5 0.5 1

-0.1

-0.05

0.05

0.1

Ha=10

r

zr=1.01

r=1.1

r=1.5

r=2

-0.1

-1 -0.5 0.5 1

-0.05

0.05

0.1 Ha=50

r

z

r=1.01

r=1.1

Fig.2.5. Profiles for the velocity radial component rV for the even problem ( R=1).

0.8 1.2 1.4 1.6 1.8 2r

-1

-0.5

0.5

1

z

Fig.2.6. The velocity field for the even problem (R=1) at Ha=10.

31

The general problem. Fig.2.7 plots the results of calculation of the velocity component rV for the general problem. One can see that, similarly to the odd problem, the profiles of rV differ from the Hartmann flow profiles only near the entrance region. The magnitude of the velocity is inversely proportional to the distance from the hole.

-1 -0.5 0.5 1

0.05

0.1

0.15

0.2

0.25

0.3

0.35 Ha=10Vr

z

r=1.01r=1.1

r=1.5

r=2

r=4

r=3

-1 -0.5 0.5 1

0.05

0.1

0.15

0.2

0.25

0.3

0.35Ha=50

Vr

z

r=1.01

r=1.1

r=1.5

r=2

Fig. 2.7. Velocity profiles for the radial component rV for the general problem and

ze B eB 0= , R=1.

Section 2.2. Analytical solution for magnetohydrodynamical problem on a flow of a conducting fluid in the initial part of a circular channel.

Formulation of the problem:

A circular channel is located in region D~ :{ }∞+<<∞−<≤<≤ zRr ~,2~0,~0 πϕ . There is a split on the channel's lateral surface in region { dzdRr ~~~,2~0,~ ≤≤−<≤= πϕ }. The conducting fluid flows into the channel through this split with constant velocity

reVV 0−= . External magnetic field

ze eBB 0= is parallel to the z-axis.

The geometry of the flow is shown in Fig.2.8. The case of nonconducting walls Rr =~ is considered. It is also assumed that induced streams do not flow through the split { Rr =~ ,

dzd ~~~<<− } in region +∞<< rR ~ .

In order to introduce the dimensionless variables the radius R of the channel is used as the length scale, but the other variables are introduced similarly as it is done in Chapter 1 for the plane channel.

MHD equations in cylindrical coordinates in Stokes and inductionless approximation are used (see [36], [37]). In this problem 0=E (see [36]). The projections onto r- and z- axes have the following dimensionless form:

0

r~

z~R

reVVRr

0~ −=

=zeBB 0

~ =

d~d~−

Fig.2.8. Circular channel. The geometry of the flow.

32

0)( 21 =−+− rVHaL

rP

∂∂ , (2.23)

00 =+− zVLzP

∂∂ , (2.24)

0)(1=+ r

z rVrrz

V∂∂

∂∂ . (2.25)

The boundary conditions are:

1=r : 0=zV , ⎩⎨⎧

−∉−∈−

=),(,0),(,1

ddzddz

Vr (2.26)

where operators 1L , 0L are described by formula (2.4) and Rdd /~= .

Solution of the problem:

In order to solve system (2.23)-(2.26), the symmetry of the problem with respect to z is used. In this problem the velocity component rV and pressure P are even functions with respect to z, but the component zV is an odd function with respect to z. The problem is solved by using the Fourier cosine and Fourier sine transforms ([4]):

dzzzrVrV rc

r λπ

λ cos),(2),(0∫∞

= , dzzzrVrV zs

z λπ

λ sin),(2),(0∫∞

= ,

dzzzrPrPc λπ

λ cos),(2),(0∫∞

= .

Applying the Fourier transforms to (2.23)-(2.26) we get the system of ordinary differential equations for the functions ),( λrV c

r , ),( λrV sz , ),( λrPc . Eliminating

functions szV , cP from this system, we obtain the differential equation for c

rV :

.0)(~2)~( 2221

20 =++− c

rc

rrc

rrr VHaVLVLLrd

d λλλ (2.27)

where rd

dr

L r +=1~

0 , drd

rdrdLr

12

2

+= , 211~r

LL rr −= .

The following formula is proved in the thesis:

[ ]crrr

crrr VLLVLL

rdd

110~~)~( = . (2.28)

This fact allowed us to simplify the left-hand side of Eq.(2.27) and transform it into a product of two linear differential operators of the second order for function ),( λrV c

r :

,0)~)(~( 221

211 =−− c

rrr VkLkL (2.29)

where λλλλ iHakiHak −=+= 22

21 , . (2.30)

33

As a result, the solution of Eq.(2.27), bounded at 0=r , is obtained in the form:

))) 212111 rkICrkIC(r,V сr (+(=λ (2.31)

where 21,...,CC are arbitrary constants that are determined from the boundary conditions and )(1 zI is the modified Bessel function of the first kind of order 1.

As a result, we obtain the solution of the problem for the velocity and pressure gradient in the form of convergent improper integrals:

[ ] λλλλπ

dzrkIcrkIcVr cos)()()()(2

0212111∫

∞

+= , (2.32)

[ ] λλλλλ

πdzrkIkcrkIkczrVz

sin)()()()(2),(0

20221011∫∞

+−= , (2.33)

[ ] λλλλπ∂

∂ dzrkIcrkIcrP cos)()(~)()(~2

0212111∫

∞

−= , (2.34)

[ ] λλλλπ∂

∂ dzrkIkcrkIkciHazP sin)()()()(2

020221011∫

∞

−−= , (2.35)

where )()(1)(),()(1)( 10122021 kIkAckIkAc λλλλ∆

−=∆

= ,

)()()(~),()()(~2211 λλλλλλ cHaiHaccHaiHac ⋅+⋅=⋅−⋅= ,

)()()()(,sin2)( 1120221101 kIkIkkIkIkdA −=∆=λλ

πλ , 21 , kk are given by (2.30).

Numerical results for the circular channel:

The results of numerical calculation of zV by means of formula (2.33) are presented in the thesis for different Hartmann numbers. In Fig.2.9 the profiles of velocity component zV are shown for Ha=10 and Ha=50. It follows from calculations that at Ha ≥20 the velocity component zV has the M-shaped profile at the entrance of the channel. The explanation of appearance of these profiles is given in the thesis.

0.2 0.4 0.6 0.8 1r

1

2

3

4Vz Ha=10

z=1.1

z=1.2z=2

Vp

Fig.2.9. Circular channel. The velocity profiles at d=1. (Vp is the Poiseuille flow).

0.2 0.4 0.6 0.8 1r

1

2

3

4Vz

z=1.1

z=2

z=3

z=5

z=8

VpHa=50

34

The asymptotic evaluation of the solution at z→∞ and Ha →∞ :

The asymptotic solutions for z→∞ and for Hartmann large numbers are obtained for this problem.

1) asymptotic solution for velocity component zV at z→∞ gives the Poiseuille flow in a circular channel. Consequently, this limit may be used for calculation of length L∞ on which the flow approaches the Hartmann flow. 2) The greatest interest for applications is the limit

1)/(lim

=∞→∂∂

rHarP , that was obtained

in the thesis and has the form:

⎩⎨⎧

−∉<<−

=∂∂

= ).,(,0,2

1 ddzdzdHa

rP

r

(2.36)

It follows from (2.36) that at large Hartmann numbers, the pressure gradient which is proportional to the square of the Hartmann number is needed for turning the flow on the angle 90 degrees in the circular channel.

The rP ∂∂ / at 1=r and finite value of Hartmann number ( 0≠Ha and ∞≠Ha ) is also obtained in the thesis in the form:

∫∞

=

−⎢⎢⎣

⎡

−−+

−+++=

∂∂

0 21101120

2110112022

1 )()()()()()()()(2

kIkIiHakIkIiHakIkIiHakIkIiHa

HaiHarP

r λλλλ

λπ

] λλ

λλλ dzL cossin2 2− . (2.37)

The integral (2.37) converges in ordinary sense and it is available for the calculation by using package “Mathematica”. In the thesis the results of calculation by means formula (2.37) at 0=z and formula (2.36) are compared and the relative error between numerical results by formulae (2.36) and (2.37) is studied.

Chapter 3. THE DEPENDENCE BETWEEN THE BOUNDARY VELOCITY PROFILE OF THE FLUID AND THE FULL PRESSURE FORCE AT THE ENTRANCE OF REGION

In this part of the thesis we study the dependence between the profile of the

boundary velocity of the fluid and the full pressure force at the entrance region by means of analytical solution of two hydrodynamic problems and two MHD problems on an inflow of a viscous fluid into a half-space through a plane split or through a round hole. It is shown that the full pressure force at the entrance region is equal to infinity if the uniform velocity profile is given at the entry into a half-space, consequently, such problem is physically unrealistic, but if the profile of boundary velocity is given as a parabolic function (continuous function), the full pressure force at the entrance region has a finite value and the problem is physically realistic.

35

Section 3.1. The dependence between the boundary velocity profile of the fluid and the full pressure force at the entrance of region in hydrodynamic problems

3.1.1. The problem on a plane jet flowing into a half-space through a split of finite width

We consider a half-space with viscous fluid, located in the region D:{ +∞<< y~0 , +∞<<∞− zx ~,~ }. This half-space is bounded by the impermeable plate in the plane 0~ =y with a split in the region { LxL ≤≤− ~ , +∞<<∞−= zy ~,0~ }. Viscous fluid flows into the half-space through the split with given velocity

yexVV ⋅⋅= )(~0 ψ . The geometry of the

flow is shown in Fig.3.1. Dimensionless variables are introduced by using L-half width of the split as the scale of length, 0V as the scale of the velocity, LV /0ρν as the scale of pressure. We use the nondimensional Navier-Stokes equations in the Stokes approximation. In projections on the x- and y- axis, the problem is described by the system (1.1)-(1.3) at 0=Ha .

The dimensionless boundary conditions are:

0=y : 0=xV ,⎩⎨⎧

− ∉− ∈

= )1;1(x,0)1;1( x),(

x

Vy

ψ; 0,:22 →∞→+ yx VVyx . (3.1)

To study the dependence of the full pressure force at the entrance of the half-space on the inlet velocity profile, the following two cases for the inlet velocity of the fluid at the entrance of the region are considered:

1) The profile of the inlet velocity is given as a constant function: yeV ⋅= 1 . Then 1)( =xψ ; 2) The profile of the inlet velocity is given as a parabolic function yexV )1( 2−= .

Then )1()( 2xx −=ψ .

In this problem the velocity component yV and pressure P are even functions with respect to x, but the component xV is an odd function with respect to x. Therefore, the problem is solved by using of Fourier cosine and Fourier sine transforms:

L-LyexVV ⋅= )(~

0

x~

y~

Fig.3.1. Plate with a plane split. The geometry of the flow.

36

dxxyxVyV ycy λ

πλ cos),(2),(

0∫∞

= , dxxyxVyV xsx λ

πλ sin),(2),(

0∫∞

=

dxxyxPyPc λπ

λ cos),(2),(0∫∞

= (3.2)

As a result, the solution of the problem is obtained in the form of improper integrals:

λλλψλπ

λ xdyeV yy cos)(ˆ)1(2

0∫∞

− += , λλλψλπ

λ dxyeV yx sin)(ˆ2

0∫∞

−= (3.3)

λλλψλπ

λ dxeP y cos)(ˆ220∫∞

−= . (3.4)

where dxxxV cy λψπλλψ cos)(/2)0,()(ˆ

1

0∫== . (3.5)

The full pressure force at the entrance into the half-space is equal to

∫∫ ∫∫∞

=

∞−

==

⎭⎬⎫

⎩⎨⎧

==0

1

0 00

1

0

sin)(ˆ/24cos)(ˆ/242 λλλψπλλλψλπ λ xddxxdedxPFy

yoy

(3.6)

1) The fluid flows into a half-space through the split with velocity ye1V ⋅= .

In this case, the integrals (3.3)-(3.4) are evaluated analytically by using Laplace transform and it’s properties. As a result, the solution of the problem is obtained in the form of elementary functions. The integral for the full pressure force at the entrance into half-space is:

λλ

λπ

dF ∫∞

=0

2sin8 (3.7)

This integral diverges and, consequently, full pressure is ∞=F .

2) The fluid flows into the half-space through the split with velocity y2 e)x(1V −=

In this case, the solution of the problem for velocity and pressure is also obtained in the form of elementary functions. The full pressure at the entrance of the half-space has the form:

∫∞ −

=0

3 sincossin16 λλλ

λλλπ

dF (3.8)

Integral in (3.8) converges and it is equal to F=1/2.

37

3.1.2. The problem on a round jet flowing into a half-space through a round hole of finite radius.

In this part of the thesis the problem similar to the one considered in previous part of the thesis is solved for the case where fluid flows into the half-space 0~ ≥z through a round hole of finite radius in the plane 0~ =z located in region

{ Rr ≤≤ ~0 , πϕ 20 <≤ , 0~ =z } with given velocity zerVV ⋅⋅= )(~0 ϕ . ( r~ ,ϕ~ , z~ are the

cylindrical coordinates). The dimensionless quantities are the same as in §3.1.1, only the radius of the hole R is used as the length scale. We use the nondimensional Navier-Stokes equations in polar coordinate system in the Stokes approximation. In projections on the r- and z- axis these equations have the form of Eqs.(2.1)-(2.4) with Ha=0.

The dimensionless boundary conditions are:

⎩⎨⎧

><

= =1,01),(

:0zrrr

Vz

ϕ, 0=rV ; 0,:22 →∞→+ zr VVzr (3.9)

In order to solve the problem, the Hankel transform (2.8) is used. The solution of this problem has the form:

λλλλλϕ λ drJzeV zz ∫

∞− +=

00 )()1()(ˆ , λλλϕλ λ drJezV z

r ∫∞

−=0

12 )()(ˆ (3.10)

λλλϕλ λ drJeP z∫∞

−=0

02 )()(ˆ2 , ∫==

1

00 )()(),0(ˆ)(ˆ drrrJrVz λϕλλϕ . (3.11)

The full pressure force at the entrance into the half-space has the form:

rdrdrJrdrPFz ∫ ∫∫

⎭⎬⎫

⎩⎨⎧

==∞

=

1

0 00

21

00

)()(ˆ42 λλλϕλππ ∫∞

=0

1 )()(ˆ4 λλλϕλπ dJ (3.12)

Similarly to the previous paragraph, two cases are considered:

1) The fluid flows into the half-space through the hole with velocity ze1V ⋅= .

In this case, the function 1)( =rϕ and the solution of the problem for the velocity and pressure is obtained in the form of convergent improper integrals. It is proved in the thesis that the integral (3.12) for the full pressure force in the entrance hole diverges.

2) The fluid flows into the half-space through the hole with velocity ze)r(1V 2−= .

The function )(rϕ is 21)( rr −=ϕ in (3.12) for the parabolic boundary velocity. The solution for the velocity and pressure is also obtained in the form of convergent improper integrals. It is shown in the thesis that integral (3.12) converges in this case and the full pressure at the entrance into half-space is ( ) 3/16 π=F .

38

Section 3.2. The dependence between the boundary velocity profile of the fluid and the full pressure force at the entrance of region for MHD problems

In this part we consider the MHD analogue of hydrodynamic problems considered in the first part of the Chapter 3.

3.2.1. The MHD problem on a plane jet flowing fluid into a half-space through a plane split of finite width

We solve the problem considered in §3.1.1 on an inflow of conducting fluid into a half-space 0~ >y with a given velocity )(0 xV ψ through a split located in the region { 0~ =y , LxL ≤≤− ~ , +∞<<∞− z~ }, but with the presence of external magnetic field

that is parallel to the y-axis, i.e. ye eBB 0

~ = . Dimensionless MHD equations in the Stokes and inductionless approximation have the form (1.1)-(1.3) with 2/πα = . Boundary conditions for the problem have the form (3.1). The problem is solved by using the Fourier cosine and Fourier sine transforms (3.2). The full pressure force in the entrance of the half-space is obtained in the form:

∫ ==

1

00

2 dxPFy

= ∫∞

− +0

221 sin)(ˆ8 λλ

λλψµλπ d . (3.13)

where dxxx∫=1

0

cos)(/2)(ˆ λψπλψ .

Similarly to Section 3.1.1, we consider two cases:

1) the fluid flows into the half-space through the split with velocity ye1V ⋅= .

2) the fluid flows into the half-space through the split with velocity y2 e)x(1V ⋅−= .

The same result, as in the previous section for hydrodynamic problems, is obtained in this part of thesis, i.e. in the first case the integral for the full pressure force at the entrance into a half-space diverges and in second case this integral converges and, consequently, the flow with parabolic boundary velocity profile is physically realistic.

The asymptotic solution at ∞→Ha is also derived for the second problem when fluid flows into the half-space with velocity given as a parabolic function.

0lim =∞→ xHaV , )1()1(lim 2 xxVyHa

−−=∞→

η , (3.14)

)1()1(221coscossin24lim 2

03 xxHaxdHaP

Ha−−⋅=⎟

⎠⎞

⎜⎝⎛ −

= ∫∞

∞→η

πλλ

λλλλ

ππ. (3.15)

HadHaFHa π

λλλ

λλλππ

234sincossin28lim

04 =⎟

⎠⎞

⎜⎝⎛ −

= ∫∞

∞→, (3.16)

where )(xη is the Heaviside step function.

39

3.2.2. The MHD problem on a round jet flowing into a half-space through a round hole of finite radius.

In this part of the thesis we study the dependence between the profile of the

boundary velocity of the fluid and the full pressure force at the entrance region for the problem, considered in §3.1.2, on an inflow of a conducting fluid into a half-spice

0~ >z through a round hole of finite radius R, but in the presence of external magnetic

field ze eBB 0

~ = that is perpendicular to the plane. Dimensionless MHD equations in cylindrical coordinate system in Stokes and

inductionless approximation have the form (2.1)-(2.3) with 2/πα = . The boundary conditions are (3.9). For the solution of the problem the Hankel transform (2.8) with respect to r is used.

It is shown in the thesis that the integral for full pressure force diverges for the case where the conducting fluid flows into half-space with velocity ze1V ⋅= , therefore, the flow in this case is physically unrealistic. If the conducting fluid flows into half-space with the parabolic velocity z

2 e)r(1V −= , then this integral converges, consequently, this flow is physically realistic.

The new asymptotic solution at ∞→Ha also is obtained for the problem for the case of parabolic boundary velocity profile.

Chapter 4. ANALYTICAL SOLUTION OF THE MHD PROBLEM ON THE INFLUENCE OF A CROSS FLOW ON A MAIN FLOW IN THE INITIAL PART OF A PLANE CHANNEL

In this chapter of the thesis the analytical solution is obtained for the problem on

a MHD flow of conducting fluid in a plane channel in the presence of a cross flow. Namely, we consider the problem on a MHD flow in the initial part of a plane channel if the conducting fluid flows into the channel through a split in one channel’s lateral side and flows out through a split in its other lateral side in the presence of a main flow in the channel. The influence of the cross flow on the main flow in the channel is studied. The results of previous chapter are taken into account on solving the problem, namely, the velocity at the entrance of the channel is given as a parabolic function. The problem is solved both for a transverse and longitudinal magnetic field in Oseen and inductionless approximation. The dependence of the length of the initial part on the Reynolds and Hartman numbers is analyzed numerically. The field of perturbation velocity is also studied for different Hartmann and Reynolds numbers.

Section 4.1. Formulation of the problem

The plane channel with flowing conducting fluid is located in the region D:{ ∞+<<∞−<≤− zxhyh ~,~;~ }. There are two splits in region LxL ~~~ ≤≤− on the channel's lateral sides hy ±=~ . The conducting fluid with the constant velocity

40

( ) yent exLVV 22 ~~~~ −= flows into the channel through the split in the channel’s wall at hy −=~ and flows out with the same velocity through the split at hy =~ .

Two cases are considered: 1) the external magnetic field is y

e eBB 0= (transverse magnetic field)

2) the external magnetic field xe eBB 0= (longitudinal magnetic field).

The velocity of a main flow in the channel at a sufficient distance from the entrance region )y(V ~~

∞ depends on the external magnetic field. In the transverse magnetic field Hartman flow exists at the sufficient distance from the entrance region of the channel, i.e. )y(V)y(VV Hartmx

~~~~~lim~ == ∞±∞→. The

geometry of the flow for the transverse magnetic field is shown in Fig.4.1. For the longitudinal magnetic field, the Poiseuille flow takes place at a sufficient distance from the entrance region, i.e. )y(V)y(VV px

~~~~~lim~ == ∞±∞→.

In order to solve the problem, we define a new function, so-called the perturbation velocity

yyxxnew e)y,x(Ve)y,x(VVVV ~~~~~~~~~ +=−= ∞ . (4.1)

We introduce dimensionless variables as in Chapter 1, except we use 0V - the magnitude of the average velocity of the fluid at the cross section of the channel as the velocity scale. The dimensionless MHD equations in inductionless and Oseen approximation for the new velocity of the fluid newV and pressure of P are:

( ) BBnewnewnewx e)eV(HaVPVe ××+∇+−∇=∇ 22Re , (4.2)

0=∇ newV , (4.3)

where jyaixaa ⋅∂∂+⋅∂∂=∇ // , yaxaa yx ∂∂+∂∂=∇ // , Be is the unit vector of the external magnetic field, ν/Re 0 LV= is the Reynolds number that describes the ratio of the inertial force to the viscous force.

0

( ) yenthy

y exLVV 22 ~~~~−=

−=y

e e=BB 0

-h

hHartmV

L~− L~

( ) yenthy

y exLVV 22 ~~~~−=

=

x~

y~

HartmV

Fig.4.1. Plane channel with cross flow.

The geometry of the flow for transverse magnetic field.

41

The boundary conditions are

( )⎩⎨⎧

−∈−−∉

==±=),(,

),(,0,0:1 22 LLxxLV

LLxVVy

entyx (4.4)

0,0: →→±∞→ yx VVx , (4.5)

where hLL /~= , 0/~ VVV entent = , L~2 is the width of the split in the channel’s wall.

Note that solving the problem on a flow of viscous fluid in the initial part of the channel by using the Oseen approximation, it is usually assumed that the velocity and the pressure of the fluid are given at the entrance of the channel (see [44], [37]). In our opinion, these boundary conditions overdetermine the problem and it is sufficient to prescribe only the velocity at the entrance of the channel. In the thesis the problem is solved with the assumption that only the velocity of the fluid is given at the entrance of the channel. Section 4.2. The solution of the problem for the transverse magnetic field

Projecting equations (4.2), (4.3) onto the x- and y- axis and taking into account that yB ee = , we obtain:

xxxx VHa

yV

xV

xP

xV 2

2

2

2

2

Re −∂∂

+∂∂

+∂

−=∂

∂∂

, (4.6)

2

2

2

2

ReyV

xV

yP

xV yyy

∂

∂+

∂

∂+

∂−=

∂

∂

∂, (4.7)

0=∂

+∂

yV

xV yx

∂∂ (4.8)

In order to solve this system with boundary conditions (4.4)-(4.5) we use the complex Fourier transform with respect to x (1.9). We took also into account that the velocity component yV is an odd function with respect to y. The solution of the problem for transverse magnetic field is obtained in the form of convergent improper integrals:

λλπ

λ deVykkkykkkkkiyxV xientx ⋅⋅−⋅

⋅∆⋅

= ∫∞

∞−

ˆ)sinhsinhsinhsinh(21),( 211122

21 , (4.9)

λπ

λ deVykkkykkkyxV xienty ⋅

∆⋅−⋅

= ∫∞

∞−

ˆcoshsinhcoshsinh21),( 211122 , (4.10)

42

( ) λλπ

λ deVykkDykkDkikxP xi

ent ⋅⋅−⋅⋅∆

−=

∂∂

∫∞

∞−

ˆsinhsinhsinhsinh21

21112221 , (4.11)

λλπ

λ deVykkkDykkkDkkyP xi

ent ⋅⋅−⋅⋅∆

−=

∂∂

∫∞

∞−

ˆ)coshsinhcoshsinh(21

2121121221 , (4.12)

where 211122 coshsinhcoshsinh kkkkkk ⋅−⋅=∆ . (4.13)

12

1 Dk += λ , 22

2 Dk += λ , (4.14)

⎟⎠⎞

⎜⎝⎛ −= LLLV

V entent λ

λλ

πλλ cossin4

)(ˆ2

, (4.15)

( )2

4ReRe 2222

1HaHaiHai

D++++

=λλ , ( )

24ReRe 2222

2

HaHaiHaiD

++−+=

λλ .

Numerical results for the transverse magnetic field

On the basis of the obtained solution, the length of the initial part of the channel initL ,

where the x component of the velocity ),( yxV of fluid in the channel differs from the velocity of the Hartman flow

HartmV by less than 1%, is analyzed numerically. The dependence of the length of the initial part initL on the Reynolds number for different Hartmann numbers is shown in Fig.4.2. As one can see from the picture, at fixed Hartmann number the increase of Reynolds number leads to the increase in the length of the initial part initL . Besides, initL decreases as the Hartmann number grows. It means that with the increase in the intensity of magnetic field, the flow of conducting fluid in channel faster approaches Hartmann flow.

Fig.4.3 presents the results of calculation of the x-component xV of the

perturbation velocity newV by means of formula (4.9). One can see from Fig.4.3 that for some values of x the component xV is positive. It means that in region 10 << y there exists an opposite flow in transverse magnetic field. It happens because the cross-flow generates vortices in the channel. The vector field of perturbation velocity newV is shown in Fig.4.4 for Ha=20 and Re=35.

10 20 30 40 50 60Re

2

3

4

5

LinitHa=6

Ha=8

Ha=10

Ha=12Ha=14Ha=18Ha=22Ha=30Ha=40Ha=50

10

Fig.4.2. The dependence of the length of the initial part of the channel initL on Re at y=-0.5 for y

e eBB 0= . (L=1, 1=entV )

43

0.2 0.4 0.6 0.8 1

0.02

0.04

0.06

Ha=8,Re=5Vx

yx=1.001

x=1.01

x=1.1

x=1.2x=1.5x=1.6

x=1.7

x=2

0.2 0.4 0.6 0.8 1

-0.2

-0.15

-0.1

-0.05

0.05

0.1

Ha=8, Re=35Vx

y

x=1.001x=1.01

x=2x=1.7

x=1.5x=1.6

x=1.1

x=1.2

0.2 0.4 0.6 0.8 1

0.01

0.02

0.03

0.04Ha=20, Re=5Vx

y

x=1.001

x=1.01

x=1.1

x=1.2

x=1.5

x=1.6x=1.7

x=2 0.2 0.4 0.6 0.8 1

0.02

0.04

0.06

0.08

0.1

0.12Ha=20, Re=35Vx

y

x=1.001x=1.01

x=1.1x=1.2

x=1.5

x=1.7

x=2

x=1.6

Fig.4.3. Profiles for the x component ),( yxVx of the perturbation velocity newV for y

e eBB 0= .

1.5 2 2.5 3x

-1

-0.5

0.5

1

y Ha=20, Re=35

Fig.4.4. The vector field of perturbation velocity newV for 30 ≤< x and y

e eBB 0= .

44

Section 4.3. The solution of the problem for the longitudinal magnetic field.

Projecting Eqs. (4.2)-(4.3) onto the x- and y- axis and taking into account that xB ee = , the problem takes the form:

2

2

2

2

ReyV

xV

xP

xV xxx

∂∂

+∂∂

+−=∂

∂∂∂ , (4.16)

yyyy VHa

yV

xV

yP

xV 2

2

2

2

2

Re −∂

∂+

∂

∂+−=

∂

∂

∂∂ , (4.17)

0=+y

Vx

V yx

∂∂

∂∂

, (4.18)

with the boundary conditions

( )⎩⎨⎧

−∈−−∉

==±=),(,),(,0

,0:1 22 LLxxLVLLx

VVyent

yx (4.19)

0,0: →→±∞→ yx VVx , (4.20)

The problem is also solved by using complex Fourier transform (1.9). As a result, the solution of the problem is obtained in the form of improper convergent integrals similar to the integrals for the case of transverse magnetic field. Numerical results for the longitudinal magnetic field

Fig.4.5 presents the results of calculation of the length of the initial part of the

channel initL for different Hartman numbers. In the present problem the initial part of the channel is the part where the x-component of the velocity (x,y)V differs from the Poiseuille flow pV by less than 1%. One can see that initL increases at growing Hartmann number. Besides, the increase in the Reynolds number leads to the increase in the length of the initial part at fixed Hartmann numbers. The dependence of the value initL on Reynolds number becomes linear at large Hartmann numbers.

Fig.4.6 presents the profiles of the perturbation velocity newV , but Fig.4.7 shows the vector field of perturbation velocity. As it can be seen from Fig.4.6, the opposite flow doesn’t appear in the channel in the case of longitudinal field, because the cross

10 20 30 40 50 60Re

5

10

15

20

25

30

Linit

Ha=4

Ha=5Ha=6Ha=7Ha=8Ha=9Ha=10Ha=12Ha=14Ha=16Ha=18Ha=20

Fig.4.5. The dependence of the length of the initial part initL on Re at y=0.5 for x

e eBB 0= (L=1, 1=entV )

45

flow doesn’t generate vortices in the channel in this case (see Fig.4.7). In the channel’s entrance region the component xV of the velocity newV has more pronounced M-shaped profiles for small x than in the case of transverse magnetic field.

0.2 0.4 0.6 0.8 1

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

Ha=8, Re=5

Vx

y

x=1.01x=1.1x=1.2

x=1.5

x=2

x=2.5

x=3

x=4

x=50.2 0.4 0.6 0.8 1

-0.4

-0.3

-0.2

-0.1

Ha=8, Re=35

Vx

y

x=1.01x=1.1

x=1.2

x=1.5

x=2

x=3x=4

x=5

x=2.5

0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

Ha=20, Re=5

Vx

y

x=1.01x=1.1x=1.2x=1.5

x=2

x=2.5x=3

x=4

x=5

0.2 0.4 0.6 0.8 1

-0.8

-0.6

-0.4

-0.2

Ha=20, Re=35

Vx

y

x=1.01x=1.1x=1.2

x=1.5

x=2x=2.5

x=3

x=4x=5

Fig.4.6. Profiles for x component ),( yxVx of the perturbation velocity newV for x

e eBB 0= .

2 3 4 5 6x

-1

-0.5

0.5

1

y Ha=20, Re=35

Fig.4.7. The vector field of the perturbation velocity newV for xe eBB 0= .

46

CONCLUSIONS

The present thesis is devoted to the theoretical study of new magnetohydrodynamical (MHD) problems on a flow of a conducting fluid in the initial part of a channel at the condition that fluid flows into the channel through the split of finite width or through a hole of finite radius on the channel’s lateral side. The following problems are solved in the thesis.

1) MHD problems on an inflow of a conducting fluid into a plane channel through the channel’s lateral side have been solved analytically. Two cases are considered: fluid flows into a plane channel through a split of finite width on the channel’s lateral side and through a round hole of finite radius. The problems are solved by dividing them into two subproblems: even and odd problems with respect to the axis perpendicular to the channel. The cases of longitudinal and transverse magnetic fields have been studied in details. The problem on an inflow of conducting fluid into a channel through a round hole is solved only for the case of transverse magnetic field. The exact analytical solutions of the problems have been obtained in Stokes and inductionless approximation in the form of convergent improper integrals. On the basis of the obtained analytical solutions the velocity field has been analyzed numerically. For the case when fluid flows into the channel through the split, it is shown that the effect of the longitudinal magnetic field is more pronounced than the effect of transverse magnetic field. It is also shown that in a strong longitudinal magnetic field flows mostly occur along the wall with the split. As the Hartmann number increases, the layer of the flow is getting narrower and the velocity increases in this layer. The Poiseuille flow takes place at a large distance from the entrance split. Moreover, when the Hartmann number grows the flow in the channel slowly approaches Poiseuille flow. For the transverse magnetic field, the flow in the channel differs from the Hartmann flow only near the entrance region and Hartman flow is approached very quickly also for small Hartmann numbers. In addition, asymptotic solutions of the problems at large Hartmann numbers have been obtained for the transverse magnetic field and important practical result has been obtained for this problem. It says that at large Hartmann numbers, a pressure gradient which is proportional to the square of the Hartmann number is needed for turning the flow through an angle 090 , while for pumping the fluid we need a pressure gradient which is proportional to only the first power of the Hartmann number.

2) The analytical solution has been also obtained for the MHD problem on an inflow of conducting fluid into a circular channel through a split of finite width on the lateral side of the channel. The problem is solved only for the longitudinal magnetic field in Stokes and inductionless approximation. The solution is also obtained in the form of convergent improper integrals. The velocity field is studied numerically on the basis of the obtained analytical solutions. M-shaped velocity profiles in the entrance region at large Hartman numbers are obtained and the physical explanation of these velocity profiles is given in the thesis. It is also shown that when the

47

Hartmann number grows the flow in the channel slowly approaches Poiseuille flow. Asymptotic solution of the problem at large Hartmann numbers is also obtained.

3) Another new MHD problem is studied analytically in this work, namely, the problem on an influence of cross flow on the main flow in an infinitely long plane channel in the presence of a strong magnetic field. The problem is solved in Oseen and inductionless approximation both for the longitudinal and transverse magnetic field by using the Fourier transform providing that only velocity of fluid is prescribed at the entrance of the channel. The solution of the problem is obtained in the form of convergent improper integrals. The field of perturbation velocity is analyzed numerically for different Hartmann and Reynolds numbers. Numerical calculations give the M-shaped velocity profile at the initial part of the channel at large Hartmann numbers. The dependence of the length of the initial part on the Hartman number is also studied. It is shown that the increase in the Reynolds number leads to the increase in the length of the initial part. Besides, in the transverse magnetic field the flow of conducting fluid in channel faster approaches Hartmann flow with the increase in the intensity of magnetic field, but in longitudinal magnetic field the flow in the channel slowly approaches the Poiseuille flow. Moreover, the cross-flow in the transverse magnetic field generates vortices in the channel.

4) The dependence of the full pressure force in the entrance region on the profile of the inlet velocity in this region is studied analytically by means of analytical solution of two hydrodynamic problems and two MHD problems on an inflow of a viscous fluid (or conducting fluid for MHD problems) into a half-space through a plane split of finite width or through a round hole of finite radius. Both the case of a uniform inlet velocity profile and the case of a parabolic velocity profile are considered in the thesis. It is shown in this part of the thesis that the full pressure force at the entrance region is equal to infinity if the profile of the boundary velocity is uniform and, consequently, such problem is physically unrealistic. But if the profile of the boundary velocity is a parabolic function, the full pressure force at the entrance region has a finite value and the problem is physically realistic. The problems are solved in the Stokes approximation (and inductionless approximation for MHD problems). In the case of a plane split the solution of the hydrodynamic problem is obtained in terms of elementary functions, but in the case of round hole the solution is obtained in the form of convergent integrals containing Bessel functions. For the MHD problems the solutions are obtained in the form of improver convergent integrals. The new asymptotic solutions for MHD problems are obtained at large Hartmann number ( ∞→Ha ) for the case of parabolic profile of the inlet velocity.

Methods of integral transforms, namely, Fourier transform and Hankel transform are used in the thesis for solving of the problems. Numerical calculations are performed using the package “Mathematica 5.0”.

48

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