Table Of ContentELECTRICALL Y INDUCED VORTICAL FLOWS
MECHANICS OF FLUIDS AND TRANSPORT PROCESSES
Editors: R. J. Moreau and G. IE. Oravas
V. BOJAREVICS, J. A. FREIBERGS, E. I. SHILOV A,
and E. V. SHCHERBININ
Institute 0/ Physics, Latvian S.S.R. Academy o/Sciences,
Riga, Salaspi/s, U.S.S.R.
Electrically Induced
Vortical Flows
Kluwer Academic Publishers
Dordrecht / Boston / London
Library of Congress Cataloging in Publication Data
Elektrovlkhrevye techenlia. Engllsh.
Electrically lnduced vortical flows / by V. BOjareV1CS ... [et
al. 1 ; edited by E.V. Shcherbinln ; translated ana edited by
V. BOjarevlcs.
p. cm. -- (Mechanics of fluids and transport processes; 9)
Translatlon of, Elektrovikhrevye techenlla.
Bibll0graphy, p.
Includes lndex.
ISBN-\3: 978-94-010-7017-1 e-ISBN-\3: 978-94-009-1163-5
001: 10.1007/978-94-009-1\63-5
1. Vortex-motlon. 2. Magnetohydrodynamlcs. 3. Eddy currents
(Electrlc) I. BOlareV1Ch, V. V. <Valdis Vladls1avovich)
II. Shcherblnln, E. V. (Eduard Vasil 'evich) III. Title.
IV. Series. V. Serles, Mechanics of fluIds and transport processes
; v. 9.
OC159.E4413 1988
532' . 0595--dc 19 88-8823
CIP
Originally published under the title' Elektro-vikhrevye techenija'
Translated by V. Bojarevic's
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands
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© 1989 by Kluwer Academic Publishers
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Contents
Preface IX
Introduction
Chapter 1 / Basic Properties of Axially Symmetric Motions in
Magnetohydrodynamics 18
1.1. Magnetohydrodynamic Equations 18
1.2. Some Facts about Orthogonal Curvilinear Coordinates 21
1.3. Differential Operators in Orthogonal Curvilinear Coordinates 24
1.4. The Most Commonly Used Rotational Coordinate Systems 25
1.5. Axisymmetric Motions 29
1.6. Relation between Stokes Stream Function and Self-Magnetic
Field of an Electric Current in Problems with Axial Symmetry 32
1.7. Feasible Schemes for Axially Symmetric Electric Current Dis
tributions 35
1.8. Magnetic Field in Axisymmetric Flow 41
1.9. Electric Field in Axisymmetric Flow 55
1.10. Full Set of Equations for Axisymmetric Motion 57
Chapter 2 / Solutions in Spherical Coordinates 62
2.1. Definition of the Class of Exact Solutions 63
2.2. Low Magnetic Reynolds Number Approximation. Electric Cur-
rent and External Magnetic Fields 67
2.3. Integral Flow Characteristics and Dimensionless Criteria 72
2.4. Review of the Class of Exact Solutions in Spherical Coordinates 74
2.5. Electrically Induced Vortex Flow in a Cone 84
2.6. Gas Flow in an Electrical Arc 91
2.7. Problems of the Nonlinear Solution 95
2.8. Landau-Squire Flows in the Presence of a Radially Diverging
Electric Current 99
v
VI Contents
2.9. Effect of the Induced Electric Current on the Flow at a Point
Current Source 104
2.10. Electrically Induced Flows at Finite Size Electrodes 107
Chapter 3 / Electrically Induced Vortex Flow at a Point Electrode and
Azimuthal Rotation 120
3.1. Integral Properties of the Flows Driven by Rotational Electro-
magnetic Forces 121
3.2. A Model Demonstrating the Effect of Viscosity 123
3.3. Flow at an Immersed Electrode 125
3.4. Asymptotic Solution for High 5 128
3.5. Electrically Induced Flow with Differential Rotation 133
3.6. Growth of Azimuthal Disturbance in the Electrically Induced
Flow at a Point Electrode 136
3.7. Intensification of Rotation in a Closed Volume 143
3.8. Mechanism of Rotation Intensification in an Axisymmetric
Vortex 148
Chapter 4 / Flows with Cylindrical Symmetry 154
4.1. External Electric Current and Magnetic Field In Cylindrical
Coordinates 154
4.2. Similarity Solutions 155
4.3. Electrically Induced Flow between Two Parallel Walls 162
4.4. Flow with Line Source in a Circular Cone 169
4.5. Magnetohydrodynamic Model of Tornado 174
4.6. EVF in a Cylindrical Container 184
4.7. Effect of Electric Current Configuration on Flow in a Cylindrical
Container 190
Chapter 5 / Periodic Electrically Induced Flows 196
5.1. Periodic Distributions of Current and Magnetic Field in Cylin
drical Coordinates 196
5.2. Integral Action of Electromagnetic Force 198
5.3. A Method Used to Construct Linear Solution of Periodic EVF
in Tubes 203
5.4. EVF in a Tube with Radial Current Supply 209
5.5. EVF in a Tube with Longitudinal Electric Current 216
5.6. EVF in an Annular Tube 224
5.7. Periodic EVF in a Longitudinal Magnetic Field 228
5.8. Longitudinal Magnetic Field Effect on Integral Features of EVF 233
5.9. Nonlinear Interaction of Periodic EVF with Through-Flow 236
5.10. Electrically Induced Flow in a Loosely Coiled Tube 240
Contents VII
Chapter 6 / Bodies in a Current-Carrying Fluid 245
6.1. Effect of Potential Forces on a Body in a Current-Carrying Fluid 245
6.2. Effect of the Rotational Electromagnetic Forces on Axisym-
metric Bodies 249
6.3. Flow at a Stationary Sphere 257
6.4. Drag of a Sphere in the Flow of Current-Carrying Fluid 259
6.5. Flows at Spheroids 266
6.6. Discharge between Electrodes of Hyperboloidal Form 269
6.7. Flow at a Cone with an Electric Current Source in the Apex 274
6.8. Motion of a Sphere with a Current-Source 277
Chapter 7 / Heat and Mass Transfer in Electrically Induced Vortical
Flows 282
7.1. Equations of Heat and Mass Transfer, and the Nondimensional
Numbers 282
7.2. Mass Transfer from a Stationary Spherical Particle in Current-
Carrying Fluid 287
7.3. Mass Transfer from a Translating Spherical Particle in a Current-
Carrying Fluid 291
7.4. Mass Transfer from a Stationary Sphere in a Longitudinal Mag-
netic Field 294
7.5. Heat and Mass Transfer in a Cylindrical Container 296
7.6. Thermal Convection in Electrically Induced Flows 307
Chapter 8 / Experimental Investigations of EVF and Applications 311
8.1. Electroslag Welding 311
8.2. Electroslag Remelting 324
8.3. Electric-Arc Furnaces 333
8.4. Hydrodynamics of Furnaces with Multiple Electrodes 335
8.5. Electrical Jet Thrusters 341
8.6. Induction Channel Furnaces 346
8.7. Electrically Induced Flows in a Flat Layer between Ferromagnet-
ic Masses 352
8.8. Electrolytic Aluminium Production 358
References 365
Index 379
Preface
Every scientific subject probably conceals unexplored or little investigated
strata, which may show up at the proper time when favourable conditions
coincide (practical demands, a circle of scientists prepared to recognize the
novelty and capable of giving impetus to the development of a new theory, etc.).
Something like this occurred in early seventies for magnetohydrodynamics,
which at the time was considered to be a relatively complete branch of hydro
dynamics with no apparent broad, unexplored areas.
It was unexpectedly realized that, in addition to the traditional methods of
affecting an electrically conducting medium, there is yet another way, one which
subsequently lead to a new direction in magnetohydrodynamics. In the Soviet
scientific literature this direction has been termed 'electrically induced vortex
flows', the essence of which are hydrodynamic effects due to the interaction of
an electric current passing through the fluid with its own magnetic field.
It cannot be said that this direction was created ex nihilo: individual studies
related to the flows driven in a current-carrying medium in the absence of
external magnetic fields appeared in the sixties; in the thirties the flows them
selves were known to take place within electrical arcs; and yet the first observa
tions on the behaviour of liquid current-carrying conductors were made at the
beginning of this century.
The growing number of publications in the seventies, of course, could not be
explained by professional interest alone. The subject was 'lucky' (and this
significantly stimulated the development of the investigations), because it was
obvious from the very beginning that the studies were closely related to high
current technologies. Moreover, the theory of electrically induced flows had to
be developed, since it was necessary for electro-technology, in particular, for
electrometallurgy.
Electrically induced flows have evolved in a separate direction, which is not
only due to the specific electromagnetic force generation; the distinguishing
feature of these flows is the rotational character of the driving electromagnetic
force, i.e. the flows can be induced even with homogeneous (zero) boundary
conditions for the velocity field.
A situation in which an electric current density is nonuniform in space (if we
IX
x Preface
restrict ourselves to uniform material properties) is typical of the rotational
electromagnetic force. Hence, the electrically induced flows should be investi
gated, first of all, as spatial flows.
It is well known in general hydrodynamics that the solution of the equations
for spatial nonlinear viscous flows is a difficult problem. Even the contemporary
achievements in numerical techniques and computer design have not resolved
the problem of reliable solutions for spatial flows. The study of electrically
induced spatial flows therefore necessarily involve certain simplifications, the
most significant of which is the assumption of axial symmetry for all the
quantities governing and characterizing the fluid flow.
The assumption of axial symmetry for physical fields, on the one hand,
retains a spatial distribution while, on the other these quantities are described
with reference to only two spatial coordinates. Of course, the transformation
from the three-dimensional description of electrically induced flows to the two
dimensional one significantly increases the amount of information which can be
extracted by analytical and numerical solutions of this specific class of flow
equations.
The assumption of axial symmetry made it necessary to consider the general
properties of axisymmetric velocity, electric current, magnetic and electric
fields. One of the properties is the possibility of representing the solenoidal
vector fields by their meridional and azimuthal components. The meridional
components can be expressed by introducing a common concept of stream
function (hydrodynamic, electric, magnetic), which permits us to describe each
of the vector fields by the use of two scalar quantities: the relevant stream
function and the azimuthal component of the vector field. This reduces the
number of functions to be determined.
In general hydrodynamics different methods are used to simplify the full
Navier-Stokes equations. The same simplifications may be used in the study of
electrically induced flows. Thus, many qualitative features of the electrically
induced flows can be explained on the basis of the Stokes approximation,
linearizing the equations of motion. However, the nonlinear terms in the
equation of motion, which complicate the solution substantially, could lead to
fundamentally new results that are unattainable in the linear statement of the
problem. As an example of the nonlinear effect we may mention the nonlinear
interaction of meridional electrically induced flows with an imposed azimuthal
rotation leading to an increase of the rotation energy and also to a significant
change in the meridional flow configuration.
From among the approximations that are totally unacceptable for the charac
terisation of electrically induced flows we shall mention only the following two.
First, the approximation of quasi-rigid motion widely applied in engineering
MHO is absolutely meaningless. The electrically induced flow problems are
primarily hydrodynamic problems. Moreover, in spite of the fact that the
electromagnetic force drives the flow and determines the flow configuration, the
integral flow characteristics may be in conflict with the integral electromagnetic
force. As an example, consider the periodic electrically induced flows in tubes,
Preface Xl
where the electrically induced flow rate is directed oppositely to the integral
electromagnetic force.
The second doubtful approximation is the assumption of inviscid fluid flow.
Such an approximation is unacceptable since the rotational electromagnetic
force constantly generates a vorticity in the fluid which, to avoid an infinite
growth, must be balanced by viscous or turbulent dissipation. In the absence of
dissipation, as was shown by Shercliff in a particular problem, the velocity field
inevitably contains a singularity.
As far as purely magneto hydrodynamic approximations are concerned, an
electrodynamic approximation is widely used in the book. This approximation
neglects not only the magnetic field deformation by the fluid motion (which is
the basis of the widely adopted noninduction approximation in MHD), but also
that electric current induced by the fluid motion. In other words, the electro
magnetic force is constructed by taking into account only the electric current
supplied by an external source of e.mJ. and its self-magnetic field. Yet if an
electrically induced flow takes place in an additional external magnetic field,
then the additional electromagnetic force is assumed to result from the interac
tion of this field with the unperturbed electric current. Such approximation is
motivated by a significant reduction of computational efforts without the loss of
generality.
The authors were primarily interested in the scope of physical effects occur
ring within a current-carrying fluid. Therefore we have tried to give a physical
interpretation to each result obtained, which, of course, do not exclude other
interpretations. We are also sure that the theory of electrically induced flows
contains more unsolved problems than solved ones. Several of these are mene
tioned in this book.
The book has been written on the basis of the monograph Electro-vortex
Flows (in Russian) by the same authors, published in 1985, publishing house
'Zinatne', Riga (USSR). A significant part of the contents (and practically all the
experimental material) is composed from the studies by the authors and their
colleagues in the Laboratory of Magnetic Hydraulics, the Institute of Physics,
Latvian SSR Academy of Sciences. Equally important contribution to the
development of electrically induced flow theory have been made by a great
number of scientists from different countries (mostly from England and the
USA). The English edition of the book was prepared by E. V. Shcherbinin and
V. Bojarevics; the latter, in addition, translated the book in English (and he is
totally responsible for the imperfect English in this book).
As far as it is known to the authors, this book is the first monographic
publication in English to present the fundamentals of electrically induced flow
theory and to classify the material concerning electrically induced flows.
We are aware that there will be reasons enough for criticism, and we would
greatly appreciate any constructive comments concerning both the contents of
the book and the exposition of the material.
THE AUTHORS
