ADVANCES IN APPLIED MECHANICS VOLUME 7
This Page Intentionally Left Blank
ADVANCES IN APPLIED MECHANICS Editors
TH...
31 downloads
814 Views
13MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
ADVANCES IN APPLIED MECHANICS VOLUME 7
This Page Intentionally Left Blank
ADVANCES IN APPLIED MECHANICS Editors
TH.
H. L. DRYDEN
VON
K ~ R M ~ N
Managing Editor
G. KUERTI Case Institute of Technology, Cleveland, Ohio
Associate Editors
F. H.
VAN DEN
DUNCEN
L. HOWARTH
VOLUME 7
1962 ACADEMIC PRESS
NEW YORK AND LONDON
COPYRIGHT
0 1982,
BY
ACADEMIC P R E S S INC.
ALL RI GHTS R E S E R V E D
NO PART O F T H I S BOOK MAY B E R E P R O D U C E D I N A N Y FORM B Y PHOTOSTAT, MICROFILM, OR A N Y QTHER MEANS, W I T H O U T W R I T T E N PERMISSION FROM T H E P U B L I S H E R S
ACADEMIC PRESS INC. 111 FIFTHAVENUE
NEW YORK3, N.Y.
United Kingdom Edition Published by
ACADEMIC PRESS INC. (LONDON) LTD. BERKELEY S Q U A R E HOUSE,LONDON W. 1
Library of Congress Catalog Card Number: 48-8503
PRINTED IN THE UNITEDSTATES OF AMERICA
CONTRIBUTORS TO VOLUME 7 G. I. BARENBLATT, Institute of Geology and Development of Combustible Minerals of the U.S.S.R. Academy of Sciences, Moscow, U.S.S.R.I RAYMOND HIDE,Physics Department, King’s College (University of Durham), Newcastle-upon-T yne, England2 HAROLDMIRELS, Lewis Research Center, National Aeronautics and Space Administration, Cleveland, Ohios W. OLSZAK,Institute of Fundamental Technical Problems, Polish Academy of Sciences, Warsaw, Poland PAULH. ROBERTS,Physics Department, King’s College (University of Durham), Newcastle-upon-Tyne, England4 J. RYCHLEWSKI, Institute of Fundamental Technical Problems, Polish Academy of Sciences, Warsaw, Poland W. URBANOWSKI, Institute of Fwdamental Technical Problems, Polish Academy of Sciences, Warsaw, Poland
Present address: Present address: of Technology, a Present address : Present address: Wisconsin. l
a
Institute of Mechanics, Moscow State University, Moscow, USSR. Department of Geology and Geophysics, Massachusetts Institute Cambridge, Massachusetts. Aerospace Corporation, El Segundo, California. Yerkes Observatory (University of Chicago), Williams Bay,
V
This Page Intentionally Left Blank
Preface The seventh volume of Advances in Applied Mechanics includes two extensive reviews of topics in solid mechanics and an account of recent analytical results obtained in the field of hypersonic obstacle flow. A detailed presentation of the basic physical principles and problems of phenomenological magneto-hydrodynamics concludes this volume ; it may serve as an introduction into this comparatively new branch of hydrodynamics.
THE EDITORS July, 1962
Vii
This Page Intentionally Left Blank
Contents ....................... ...............................
CONTRIBUTORS TO VOLUME 7 PREFACE.
v vii
Hypersonic Flow o ~ e rSlender Bodies Associated with Power-Law Shocks
.
BY HAROLD MIRELS.Lewis Research Center National Aeronautics and Space Administration. Cleveland. Ohio I . Introduction . . . . . . . . . . . . . . . . . . . . . . . I1 Hypersonic Slender-Body Theory . . . . . . . . . . . . . I11. Flows Associated with Power-Law Shocks . . . . . . . . . IV . Flows Associated with Slightly Perturbed Power-Law Shocks V Integral Methods . . . . . . . . . . . . . . . . . . . . . VI . Validity of Self-similar Solutions . . . . . . . . . . . . . VII Further Discussion of Integral Methods . . . . . . . . . . VIII Concluding Remarks . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . Addendum . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . . . . . . .
..... . . . . . . . . . .
2 4 8 26 36 43 41 49 51 52 311
The Mathematical Theory of Equllibrium Cracks In Brittle Fracture
.
BY G. I BARENBLATT. Institute of Geology and Development of Combustible Minerals of the U.S.S.R. Academy of Sciences. Moscow. U.S.S.R.
. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Development of the Equilibrium Crack Theory . . . . . . . . . . . The Structure of the Edge of an Equilibrium Crack in a Brittle Body . . Basic Hypotheses and General Statement of the Problem of Equilibrium
I I1 I11 IV
Cracks . . . . . . . . . . . . . . . . . . . . . . . V . Special Problems in the Theory of Equilibrium Cracks VI Wedging; Dynamic Problems in the Theory of Cracks . References . . . . . . . . . . . . . . . . . . . . . . . .
.
56 62 69
....... 76 . . . . . . . . 90 . . . . . . . . 114 ....... 125
Plasticity Under Non-Homogeneous Conditions
.
.
BY W OLSZAK.J RYCHLEWSKI A N D W . URBANOWSKI. Institute Of Fundamentat Technical Problems. Polish Academy of Sciences. Warsaw
. Physical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic-plastic Non-homogeneous Plates . . . . . . . . . . . . . . .
I 11 I11 IV
ix
.
132 151 183 190
CONTENTS
X
V . Limit Analysis and Limit Design . . . . . . . . . . . . . . . . . VI Propagation of Elastic-plastic Waves in a Non-homogeneous Medium . V I I . Other Problems . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
191 201 203 206
Some Elementary Problems in Magneto-hydrudynrtmics
BY RAYMOND HIDE A N D PAULH . ROBERTS.Physics Department. King’s College (University of Durham) Newcastle.upon.Tyne. 1. England
.
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Basic Equations of Magneto-hydrodynamics . . . . . . . . . . . . . . 219 Electromagnetic and. Mechanical Effects ; Dimensionless Parameters . . . 224 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 233 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Alfvbn Waves in Systems of Finite Extent . . . . . . . . . . . . . . 261 Gravity Waves : Rayleigh-Taylor Instability . . . . . . . . . . . . . 267 Gravitational Instability: Jeans’ Criterion . . . . . . . . . . . . . . 270 IX . Steady Flow between Parallel Planes . . . . . . . . . . . . . . . . 274 X . Flow due t o an Oscillating- Plane: Rayleigh’s Problem . . . . . . . . . 286 XI . Steady Two-dimensional Inertial Flow in the Presence of a Magnetic Field . 300 Appendix A : The Hydromagnetic Energy Equation . . . . . . . . . . . . 305 Appendix B : Relativistic Magneto-Hydrodynamics . . . . . . . . . . . . . 311 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 SUBJECTINDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
11. 111. I V. V. VI . VII . VIII .
Hypersonic Flow over Slender Bodies Associated with Power-Law Shocks
BY HAROLD MIRELS Lewis Research Center+ National Aeronautics and Space Administration Cleveland. Ohio Page I . Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
I1. Hypersonic Slender-Body Theory
I11. Flows Associated with Power-Law Shocks 1. Basic Equations . . . . . . . . . . . 2. Alternative Formulations . . . . . . . Stream-function formulation . . . . Lagrangian formulation . . . . . . Sedov formulation . . . . . . . . 3. Analytic Solutions . . . . . . . . . . Blast wave . . . . . . . . . . . . Newtonian theory . . . . . . . . . “Sharp-blow’’ solution . . . . . . . Approximate solutions . . . . . . . 4 Nature of the Flow . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . Power-Law Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
IV Flows Associated with Slightly Perturbed 1 . Basic Equations . . . . . . . . . . . 2. Applications . . . . . . . . . . . . . Boundary-layer effects . . . . . . . Angle-of-attack effects . . . . . . . Effect of blunting the nose of very slender wedges and cones 3. Effect of #0 . . . . . . . . . . . . . . . . . . . . . V. Integral Methods . . . . . . . . . . . . . . . . . . . 1. Continuity Integral . . . . . . . . . . . . . . . . General case . . . . . . . . . . . . . . . . . . Hypersonic slender body approximations . . . . . Slender blunt-nosed bodies a t infinite Mach number 2. Momentum Integral . . . . . . . . . . . . . . . . General case . . . . . . . . . . . . . . . . . . Hypersonic slender body approximation . . . . . .
. .
.
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . VI . Validity of Self-similar Solutions . . . . . . . . . . . . . . . . . 1. Infinite Mach Number . . . . . . . . . . . . . . . . . . . . . . 2 . Finite Mach Number . . . . . . . . . . . . . . . . . . . . . . .
*
2
. . . . . . . . . . . . . . . . . .
Present Address: Aerospace Corporation. El Segundo. California
1
. . . . . .
4 8 8 13 13 14 16 16 16 18 19 21 23 26 26 29 29 31 33 34 36 37 37 38 39 40 40 41 43 43 46
2
HAROLD MIRELS
. . . . . . . . . . . . . . . V I I I . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Addendum.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
VII. Further Discussion of Integral Methods
References.
49 51
317
I. INTRODUCTION The steady-state equations of motion for hypersonic flow over slender bodies can be reduced to simpler form by incorporating the “hypersonicslender-body approximations” (e.g., Hayes [l] and Van Dyke [?]). The reduced equations are valid provided S 2 < 1 and ( M 6 ) - 2 is not near one, where M is the free.stream Mach number and S is a characteristic shock slope. If the streamwise coordinate is considered as time, these reduced equations are identical with the full (exact) equations for a corresponding unsteady flow in one space variable less. Forebody drag on a hypersonic slender body is equivalent to the net energy perturbation (from the undisturbed state) in the corresponding unsteady flow. Taylor [3, 41 has treated the unsteady constant-energy flow field behind the spherical “blast” wave which is generated when a finite amount of energy is released instantaneously at a point. The analysis assumes a very strong wave and is valid (for a perfect gas) until the decay of shock strength is sufficient to violate the strong shock assumptions. The problem of planar, cylindrical, and spherical blast waves was treated in a unified manner by Sakurai [5,6] and the flow-field modifications associated with more moderate shock strengths were found by a perturbation analysis. The solution for the cylindrical blast wave was obtained, independently, by Lin [7]. References [7] to [O] have pointed out that, within the framework of hypersonic slenderbody theory (in the limit (M6)-2--+0),the hypersonic flow over a bluntnosed flat plate, or circular cylinder, may be considered as the steady-state analog of the constant-energy planar, or cylindrical, blast-wave problem, respectively. The nose drag in the steady problem is equivalent to the finite energy which is instantaneously released in the blast-wave problem. The steady-flow solution is not correct near the nose (where the hypersonic slender-body approximation S2 0 , an infinite tensile stress acts a t the point 0. The shape of the deformed discontinuity surface and the distribution of normal stresses uy near the point 0 are represented in Fig. 7a. If N < 0 , then an infinite compressive stress acts at the point 0; the shape of the deformed discontinuity surface and the distribution of stresses near 0 are represented in Fig. 7b. The opposite faces of the crack overlap in this case, and it is quite evident that this case is physically unrealistic. Finally, if N = 0 , the stress acting near the contour is finite and tends to the normal stress applied at point 0 of the contour if 0 is approached. Thus the stress ay is continuous at the contour, and the opposite faces of the discontinuity surface close smoothly (Fig. 7c).
MATHEMATICAL T H E O R Y O F E Q U I L I B R I U M CRACKS
73
The investigation of the stress and strain distribution near the edge of the surface of normal discontinuity was begun by Westergaard [44, 131 and Sneddon [14, 151 and continued later by the author [40], by Williams [17], and by Irwin [45-471. In view of the character of the stress states considered in [14, 15, 45-47] results were obtained only for the case N > 0.
2. Stresses and Strains Near the Edge of an Equilibrium Crack The results obtained in the preceding section pertain to an arbitrary surface of discontinuity of normal displacement. We now show that, for an equilibrium crack, N = 0 a t all points of its contour.
FIG. a.
Consider a possible state of the elastic system, which differs from the actual state of equilibrium only by a slight variation in the form of the crack contour in a small vicinity of the arbitrary point 0 (Fig. 8). The new contour is a curve that encloses the point 0 lying in the plane of the crack. This curve is tangential to the former contour of the crack at points A and B close to 0 ; everywhere else the contours of all the cracks remain unchanged. In view of the closeness of the points of tangency A and B to the point 0, the initial contour of the crack at the portion A B can be considered as straight. The distribution of normal displacements of the points of the new crack surface and the distribution of tensile stresses a t these points prior to the formation of the new crack surface are, according to the above, given, to within small quantities, by (3.10)
v = T
4(1 - y 2 ) N
E
~
Vh-y,
a,=
N
VY - 1
where N is the stress intensity factor a t the point 0. The energy released in the formation of the new crack surface, which is equal to the work required to close this new surface, is given by
74
G . I . BARENBLATT b
(3.11)
h
- 2(1 - va),N2 -
hdx
E
=
2(1 - v ~ ) T c N ~ ~ S
E
a
where 6s is the area of the projection of the new crack surface on its plane. The condition of equilibrium of the crack requires that 6 A vanishes; this together with (3.11) implies that N = 0. Thus we arrive a t a very important result characterizing the structure of cracks near their contours : 1. T h e tensile stress at the contour of a crack i s finite. 2. The o#posite faces of a crack close smoothly at its contour. I t appears, therefore, that contrary to Griffith’s conception the form of a crack near its edge is as represented in Fig. 4. Since the only acting forces at the surface of a crack near its contour are forces of cohesion, it follows from (3.9) that the tensile stress at the crack contour is equal to the intensity of forces of cohesion at the contour. In particular, if there are no forces of cohesion, the tensile stress at the crack contour is equal to zero. The condition of finiteness of stresses and smooth closing of the opposite faces a t the edges of a crack was first suggested as a hypothesis by S. A. Khristianovitch [38], to serve as a basic condition that determines the position of the crack edge. The proof of this condition given above follows [60] mainly. Formula (3.11) for the case of plane stress was first proved by Irwin [45, 461 irrespective of finiteness of stresses and smoothness of closing (see also the review by Irwin [47] and the paper by Bueckner [33]). The early paper by Westergaard [44] contains a statement concerning the absence of stress concentration at the end of a crack in brittle materials like concrete, but the condition of finiteness of stress that appears in this work was not connected with the determination of the size of the crack. We have confined ourselves here to the examination of cracks of normal discontinuity only for simplicity of treatment. Analogous reasoning, in particular the proof of finiteness of stress a t the crack edge, can be extended without any substantial changes to cover the general case in which also the tangential displacement components have a discontinuity a t the crack surface. 3. Determination of the Boundaries of Equilibrium Cracks
The conditions of finiteness of stresses and smooth closing of a crack a t its contour permit us to formulate the problem of equilibrium cracks for a given system of loads acting upon the body: for a given position of initial
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
75
cracks and a given system of forces acting upon the body, it is required to find stresses, deformations, and crack contours in the elastic body so as to satisfy the differential equations of equilibrium and the boundary conditions, and to insure finiteness of stresses and smooth closing of the opposite faces at the crack contours. We shall illustrate the solution of this problem by an elementary example of an isolated straight crack in an infinite body under all-round compressive stress q a t infinity and with concentrated forces P applied at opposite points of the crack surface (Fig. 9). The solution of the equilibrium equations satisfying the boundary conditions can be obtained by Muskhelishvili's method [18] for an arbitrary crack length 21. Stresses and displacements are expressed by formulas (3.1)-(3.3) with @(z) =
4c2 (3.12) z
='(c4 +
+).
FIG.9.
Evidently, equilibrium equations and boundary conditions do not determine the length of the crack. The distributions of stresses uy at the prolongation of the crack and normal displacement v of points of the crack surface near its edge are given by
(3.13)
Finiteness of stress and smooth closing of the crack at its ends are assured simultaneously by the condition (3.14)
1=-,
P n4
which determines the crack size under given loads P and q. Let us now attempt to determine the size 21 of an isolated straight crack in an infinite body stretched by uniform stress Po at infinity in the direction perpendicular to the crack. If the crack surface is assumed to be free of
76
G . I . BARENBLATT
stress, then one can easily show that the tensile stress a t the prolongation of the crack near its edge depends on the distance s1 as follows:
Po v 1 .
(3.15)
UY =--
V2S, ’
hence it appears that for no 1 the stress uy will be finite a t the crack end and there does not exist an equilibrium crack! This paradoxical result is due to the fact that we did not take into account the molecular forces of cohesion acting near the crack edges and thus did not completely account for the loads acting upon the body. The consideration of these forces and the definitive formulation of problems in the theory of equilibrium cracks of brittle fracture are discussed in the following section.
IV. BASICHYPOTHESESAND GENERALSTATEMENT OF EQUILIBRIUM CRACKS
THE
PROBLEM OF
1. Forces of Cohesion; Inner and Edge Regions; Basic Hypotheses
In order to construct an adequate theory of cracks of brittle fracture, it is necessary t o supplement the model of a brittle body by considering the molecular forces of cohesion acting near the edge of a crack at its surface. It is known that the intensity of forces of cohesion depends strongly on the distance. Thus, for a perfect crystal the intensity f of forces of cohesion acting between two atomic planes a t the distance y from each other is zero if y is equal to the normal intermolecular distance b. With y increasing up to about one and a half of b, the intensity f grows and reaches a very high maximum f, VETo/b E/10; after that it diminishes rapidly with further increase of y (Fig. 10). Here E is Young’s modulus, and To is the surface tension related to f ( y ) by the formula
-
N
m
r
b
The maximum intensity fm defines the theoretical strength, i.e. the strength of a solid if it were a perfect crystal. The actual strength of solids is usually several orders of magnitude lower because of defects of crystal structure. For amorphous bodies the relation between the intensity of forces of cohesion and the distance has qualitatively the same character. Data at present available, which confirm the above character of the relation between the intensity of forces of cohesion and the distance, lead
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
77
to the following conclusion. I t has long been known that the strength of thin fibers exceeds considerably that of large specimens of the same material [62, 631. Experiments carried out recently with filamentary crystals of some metals revealed an exceptionally high strength approaching the theoretical value [63]. I t is supposed that this phenomenon is due to the relatively small amount of structural defects in thin fibers and filamentary crystals. Furthermore, numerous direct measurements of the intensity of molecular forces of cohesion for glass and silica [64-661 were made recently. The
FIG.10.
FIG.11. I-inner region, ZZ-edge region.
very elegant method used in this kind of measurements is based on the application of a regenerative microbalance and was suggested and employed by B. V. Deryagin and I. I. Abrikosova [64, 651. However, these direct measurements deal with very great distances y compared to the normal intermolecular distance and thus determine only the end of the falling branch of the curve f ( y ) . A macroscopic theory for forces of cohesion at such distances was developed by E. M. Lifshitz [64] and was found in good agreement with the results of these aforementioned measurements. The relation f ( y ) , if y equals several normal intermolecular distances, seems to be beyond any strict quantitative theory and difficult for the direct experimental determination at present. A description of available attempts to estimate the relation f ( y ) at such distances and, consequently, the theoretical strength can be found in [67, 63, 681. The distance between the opposite faces of a crack varies from magnitudes of the order of the intermolecular distance near the crack edge to sometimes rather great magnitudes far from the edge. I t is therefore convenient to divide the crack surface into two parts (Fig. 11). The opposite faces in the first part - the inner region of the crack - are a great distance apart, hence their interaction is vanishingly small, and the crack surface can be considered free of stresses caused by the interaction of the opposite faces. The opposite faces of a crack in the second part are adjacent to the crack contour - the edge region of che crack - and come close to each other so that the intensity
78
G. I. BARENBLATT
of the molecular forces of cohesion acting on this part of the surface is great. Of course, the boundary between the edge and inner region of the crack surface is conventional to a certain extent. For very small cracks there may be no inner region of the crack a t all. Since the distribution of the forces of cohesion over the surface of the edge region is not known beforehand, a substantial part of the loads applied to the body is not known. I t is thus impossible to handle the problem of cracks directly in the way it was stated in Chapter 111. But the following method of solving problems of cracks is possible in principle: the distance between the opposite faces of a crack is found at each surface point as a function of the unknown distribution of forces of cohesion over the surface. Assuming the relation f ( y ) between forces of cohesion and distance as given, a relationship can be obtained which determines the distribution of forces of cohesion over the crack surface. Such an approach is not practicable. First, the relation f ( y ) is not known to a sufficient extent for a single real material. Even if it were known, the problem would constitute a very complex non-linear integral equation, the effective solution of which presents great difficulties even in the simplest cases.* Attempts were made to prescribe the distribution of forces of cohesion over the crack surface in a definite manner, but these attempts cannot be considered sufficiently well founded. For sufficiently large cracks, consideration of which is of principal interest, the difficulty connected with our lack of knowledge of the distribution of forces of cohesion can be avoided without making any definite assumptions concerning this distribution. In this case the general properties of the relation between forces of cohesion and distance allow the formulation of two basic hypotheses which not only simplify essentially the further analysis, but permit the determination of contours of cracks, although the forces of cohesion are finally altogether excluded from consideration as loads acting upon the body. First hypothesis: The width d of the edge region of a crack i s small compared to the size of the whole crack. This hypothesis is acceptable because of the rapid diminution of forces of cohesion with the increase in the distance between the opposite faces of
* In papers of M. Ya. Leonov and V. V. Panasyuk [69, 701 the relation f ( y ) is approximated by a broken line, and on the basis of this approximation a linear integral equation for the normal displacements of the crack surface points is derived. It is solved approximately, the representation of the solution being not quite successfully selected SO that the form of the crack at its end appears wedge-shaped with a finite edge angle. In fact, as was shown above, the edge angle must be zero. The shortcoming of these papers lies also in t h e application of the results obtained by the methods of mechanics of continua to cracks whose longitudinal dimensions are only of the order of several intermolecular distances.
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
79
a crack. Of course, there exist micro-cracks to which this hypothesis cannot be applied. However, as the width d of the edge region is quite small, the hypothesis is already valid for very small cracks and certainly for all macrocracks. Nevertheless, the width d is considered to be sufficiently great compared to micro-dimensions (for instance, compared to the lattice constant in a crystalline body), so that it is permissible to employ the methods of continuum mechanics over distances of the order of d.
Second hypothesis: The form of the normal section* of the crack surface i n the edge region (and consequently the local distribution of the forces of cohesion over the crack surface) does not depend on the acting loads and is always the same for a given material under given conditions (temperature, composition and pressure of the surrounding atmosphere and so on). When the crack expands, the edge region near a given point, according to the second hypothesis, moves as if it had a motion of translation, and the form of its normal section remains unchanged. This hypothesis is applicable only to those points of the crack contour where the maximum possible intensity of forces of cohesion is reached; an expansion of the crack occurs then a t this point with an arbitrarily small increase in the loads applied to the body. Equilibrium cracks, on whose contour is a t least one such point, will be called mobile-equilibrium cracks to distinguish them from immobile-equilibrium cracks which do not possess this property, i.e. do not expand with an infinitesimal increase in the load. Thus the second hypothesis and all conclusions based on it are applicable to reversible cracks as well as to irreversible equilibrium cracks, which formed a t the initial rupture of a brittle body in the process of increasing the load. I t is not applicable to cracks which result from equilibrium cracks existing at some greater load by diminishing that load; nor can it be applied, to artificial cuts made without subsequent expansion. The second hypothesis is suggested by the fact that the maximum intensity of the forces of cohesion is so very great and exceeds by several orders of magnitude the stresses which would arise under the same loads in a continuous body without a crack. Therefore it is possible to ignore the change of stress in the edge region when loads vary and, consequently, the corresponding variation of the normal sections. These two hypotheses reformulate the results of the qualitative analysis of the brittle-fracture phenomenon carried out by a number of investigators beginning with Griffith. They are the only assumptions concerning the forces of cohesion which underlie the theory presented below and appear in this explicit form in [56,571. = intersection '
with a plane normal to the crack contour.
80
G. I . BARENBLATT
2. Modulus of Cohesion
The body considered is assumed to be linearly elastic up to fracture. The elastic field in the presence of cracks can then be represented as the sum of two fields: a field evaluated without taking into account forces of cohesion and a field corresponding to the action of forces of cohesion alone. Therefore the quantity N entering in formulas (3.15) and, as was proved, equal to zero can be written as N = N o N,, where the stress intensity factor N o corresponds to the loads acting upon the body and to the same configuration of cracks without considering forces of cohesion, and the stress intensity factor N , corresponds to the same configuration of cracks and forces of cohesion only. According to the first hypothesis the width d of the edge region acted upon by forces of cohesion is small compared to the crack dimensions on the whole and, in particular, to the radius of curvature of the crack contour a t the point considered. In determining the value of N , we may thus assume that the field belongs to the configuration discussed in Section I I I , l , i.e. to an infinite body with a semi-infinite cut, with symmetrical normal stresses being applied to the surface of the cut. Hence it follows from (3.7) that
+
m
d
where G(t) is the distribution of forces of cohesion different from zero only in the edge region 0 t d . According to the second hypothesis, the distribution of forces of cohesion and the width d of the edge region at those points of the contour, where the intensity of forces of cohesion is a maximum, do not depend on the applied load; the integral in (4.2) represents then a constant characterizing the given material under given conditions. This constant will be denoted by K :
<
E ~ , the function Ro is computed from (3.2). Having uo,Ro, we compute R,,T, and, substituting in (3.6), we face the next problem of the theory of elasticity. Solving this we obtain u',&,,~~,, and so on. The above algorithm is very general, but it is seen to lead to very cumbersome computations. Its convergence in the classical case, for plates and shells, has been proved by V. M. Panferov, [ 8 Q ] .
186
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
In conclusion of this section let us mention the paper by D. D. Ivlev, [45 b], where a plastically non-homogeneous body is considered under the
conditions of so-called total plasticity, corresponding to the state of stress on the edges of the Tresca prism. 2. Axially Symmetric Problems In what follows we give a survey of the results obtained for axially symmetric problems.
A. Isotropic, thick-walled cylinder. For the non-homogeneous case this problem has been solved by W. Olszak and W. Urbanowski, [82 a] and [82 b]. These authors assume that the modulus of elasticity in shear, G,
3
Q
b
r
Q
b
r
--L
Q
r
FIG.20. Various cases of propagation of the plastic region in a non-homogeneous thickwalled cylinder.
and the yield limit for shear, K , are functions of the radius only, G = G(r), K = K ( r ) . Further assumptions are: incompressibility of the material in the elastic and plastic ranges, plane strain with respect to the axis of the cylinder, and an ideally plastic material. I t was found that the necessary condition for the plastic region to start from the interior surface of the cylinder and to expand towards its exterior surface [case (l), Fig. 201 is that the function
186
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
(3.9)
be monotonically decreasing in the interval [a, b ] , where a denotes the interior radius, and b the exterior radius. If f(r) = const, the whole of the cross section becomes plastic simultaneously [case (a)]. For f ( r )monotonically increasing, the plastic region would start from the external surface of the cylinder [case (2)]. This region would expand with increasing load p ( t ) towards the internal surface of the cylinder. In Fig. 20, J,(r) denotes the second stress invariant.
If f(r) is not monotonic in the interval [u,b], the radial distance, a t which the first plastic deformation will occur, appears somewhere inside the wall [case (3)]. Two critical values of the internal pressure p are found, the first, p,, being characteristic of the appearance of the first plastic deformations in the cylinder, the second, p,, being related to the phenomenon of full plastification of the material in the entire system considered. The authors considered in greater detail the first case (plastic zone starts from the interior surface). They proved that the first critical pressure p , (corresponding to the appearance of the first plastic deformations) is equal to (3.10)
p
K(4
-2 G(a) a,[g(u)
- g(b)],
whereas the second critical pressure (3.11)
p,
p,
= h(b) - h(a),
In the particular case when f ( r ) =
g(r) = -
{Far.
is given by h(Y) =2
5
dr.
r = const, we have
If p , < p < p,, the radius n(a < n < b) of the interface separating the two regions (elastic and plastic) may be calculated as the root of the transcendental equation (3.13)
The state of stress in the elastic (outer) region is determined by the expressions
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
187
(3.14)
where (3.15)
The state of stress in the plastic (inner) region is determined by the expressions a, = h(r) - k(a) - p , (3.16)
+ 2K(r) - 9, a, = h(r) - h(a) + K ( r ) - p . at = h(r) - h(a)
A similar problem for non-homogeneous strain-hardening characteristics has been studied by Lee Ming-hua and Pei Ming-li, [ 5 6 ] . The problem of a quasi-static motion of a thick-walled circular cylinder made of a rigidplastic material and subjected to internal pressure p has been treated by E. T. Onat, [ 8 5 ] . He obtained the relation (3.17)
where 9’ stands for the rate of internal pressure, U for the boundary velocity, Yo and Yo‘ for the yield stress and the slope of the stress-strain curve a t the yield point, respectively. It follows that the quasi-static motion can only be maintained by decreasing the pressure, if the non-dimensional rate of hardening Yo’/Yois smaller than Experimental investigations of thick-walled cylinders under internal pressure have been carried out by M. C. Steel and J. Young [118] and M. C. Steel and L. C. Eichberger [119]. The results show the appearance of an irregular strain distribution over the cross-section, although its structure was initially carefully prepared to be uniform. This phenomenon may be explained as the effect of small non-homogeneities of another type than the axially symmetrical distribution.
v%
B. Orthotropic thick-walled cylinder. This problem has been solved by W. Olszak and W. Urbanowski, [82e] and [82f]. The authors assume
188
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
that the yield limits in the radial and the axial directions are the same and constant, that is Qr = Q, = Qo = const. Similarly it is assumed that the yield limit in shear Qrt in the plane normal to the axis of the cylinder is constant, whereas the variable plastic properties are assumed to depend on the variable yield limit in the circumferential direction Qt = Qt(r). Such assumptions provide a fair approximation to the properties of thick-walled reinforced concrete pipes with circumferential reinforcement. The state of stress in the plastic region under the condition of plane strain (E, = 0) is determined by the expressions (3.18)
where
The constant C and the signs in the above expressions should be determined from the conditions a t the boundary of the region. The practical application of the result for thick-walled non-homogeneous tubes is the object of the papers by W. Olszak [73 el, [73 f], [73 j], and [73 q] where reinforced concrete tubes with variable amounts of reinforcement, such as are used in coal mines for pressures of 20 to 50 atm, are considered. C . Transversally isotropic non-homogeneous cylinder. B. R. Seth, [112], takes the yield condition in the form (3.19)
Tll - T33
+ K l k )( TI1 + T,2 + T33)
= K,(r)
where T,, >, T,, 2 T33 are the principal stresses referred to the strained frame of reference. Assuming a linear stress-strain law for large deformations and rotations he found the yield stress for tension and compression.
D. The Boussinesq problem for the semi-infinite space. The problem of a non-homogeneous linear foundation subsoil was treated by I. BabuSka,
PI. The problem of the elastic semi-space loaded by a concentrated force has been treated by K. Hruban, [38 a] and [38 b]. The physical relatibn between the stress intensity oi and the strain intensity E~ is assumed in the parabolic form E~ = (ui/K)”, K and n denoting constants. Also Poisson’s ratio is assumed to be constant. The results for various n are collected in a table [38 a]. Under the provision of an “active” straining process these solutions hold also for a non-homogeneous elastic-plastic body (on the basis of Hencky’s “deformation” theory).
PLASTICITY U N D E R NON-HOMOGENEOUS CONDITIONS
189
3. Spherically Symmetric Problems Problems of spherical symmetry are treated in the literature in a manner analogous to axially symmetric problems ; therefore results will be indicated only briefly. W. Olszak and W. Urbanowski discuss in Refs. [82 c] and [82 d] the problem of a thick-walled spherical shell with interior radius a and exterior radius b made of an incompressible material ( v = 1/2), radially non, to the action of internal and homogeneous [G = G(r),Q = e ( ~ ) ]subjected external pressures p and q, the difference I7 = p - q being a monotonically increasing function of time t. Here Q = Q ( r ) denotes the yield limit in tension. The results obtained do not differ qualitatively from those obtained by the same authors for the thick-walled cylinder. The functions describing the states of stress and strain are of course different. The case of a thick-walled transversally isotropic sphere under internal pressure was studied by B. R. Seth, [112], the assumptions being those of Section 2. Some problems of thermoplastic strain of the spherical shell have been discussed by M. Rogoziliski, [1021.
4. Torsion of Prismatic Bars
For this problem A. I. Kuznetzov considered in the first part of his paper [55 b] the state of full plasticity of the cross section, assuming that the yield limit is an arbitrary point function and formulating the characteristics for the problem. It was found that in this case the characteristics are also slip lines, but, in general, are not straight lines. The function obtained by the author, a generalization of the “sand hill surface” to the case of plastically non-homogeneous bars, may be interpreted as a characteristic surface of the wave equation with variable velocity, inversely proportional to the yield limit of the material. The slip lines are the extremum lines* on this surface, its contour lines are the trajectories of the shear stresses, and its ridges are lines (or points) of discontinuity of the stress field. I t was also found that in the particular case when the characteristics are straight lines, the yield limit varies along the normal to the contour.
* The slip line passing through the points M , and M , is a n extremum line of the integral
190
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
The author succeeded also in effectively solving the problem in the particular case of partial plasticity of a circular cross-section, if the yield limit is a function of the radius only. 6. Rotating Circular Disc
The problem of determining the angular velocity o,for which the disc becomes entirely plastic, was studied by M. Zyczkowski, [129 a], [129 b]. The assumptions are as follows: the material of the disc is perfectly plastic; the yield limit Q is a function of the radius only, Q = Q(r),similarly the specific gravity y = y ( r ) . In addition, it is assumed that the disc is fully connected (without hole) Although the author has indicated how the problem may be solved in the case of variable thickness, the results are only given, in principle, for a disc of constant thickness. Assuming that the values of the functions Q(r) and y ( r ) do not deviate considerably from their mean values and introducing the yield condition of maximum shear stress, the author obtained 1
(3.20)
0
where R denotes the exterior radius, p = r / R and g is the acceleration due to gravity. For Q(r) and y ( r ) varying in an arbitrary manner, the author indicated a method for approximate solution of the problem based on the HuberMises yield condition.
IV. ELASTIC-PLASTIC NON-HOMOGENEOUS PLATES The basic equation for elastic-plastic bending of non-homogeneous plates of arbitrary form has been established in the papers by W. Olszak and J. Murzewski [76 a]-[76 c] for arbitrary boundary conditions and for various types of the yield condition. The analysis is concerned with those particular cases of bending, for which the principal directions of stress and strain coincide with those of orthotropy. The following equations express the relations between the bending moments m1,m2 and the curvatures a1,a2 of an elastic-plastic orthotropic plate :
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
191
The elastic rigidities 8,,8, and Poisson’s ratios v1,v2, appearing in these equations, are functions of the coordinates of the point P of the middle surface of the plate. Also the plastic moduli, appearing in the coefficients x,w1,w2 may vary with the position of the point P . The latter are functional coefficients defined by
x = hcl/h,
h
=h(P)
for a plate of variable thickness
are “left-hand” limit values of principal stresses a t the distance hcz/hfrom the middle plane of the plate. The quantities x,wl,w2 may be expressed as functions of the principal curvatures a1,a2. The corresponding values of the coefficients w1,w2 are given, and the existence of the plastic potential in the “quadratic” form is assumed (here the orthotropy of the material is taken into account). Making use of the results of [76 a]-[76 c] the same authors applied in [76 a], [76 d] the general equations of elastic-plastic bending of plates to the problem of axially symmetric bending of such plates. These references contain some numerical examples : (a) elastic-plastic bending of an orthotropic ring with variable thickness, clamped along the exterior periphery and acted on uniformly by a linear load on the interior edge; this problem may be reduced to the corresponding problem of a non-homogeneous ring. (b) elastic-plastic bending of a simply-supported densely meshed circular grid-work ; this consists of radial and circumferential beams and is uniformly loaded on the central platform; such a system is adequately represented by a continuous model of a non-homogeneous elastic-plastic plate with well-determined mechanical properties. (c) a circular reinforced concrete plate with variable radial and circumferential percentage of steel reinforcement. a;l,aaCl
V. LIMITANALYSIS AND LIMITDESIGN 1. One-Dimensional Structural Elements
A. Non-homogeneity function. For the purpose of limit analysis it is convenient to distinguish between the “transverse” non-homogeneity and the axial (longitudinal) non-homogeneity of a structural element. The transverse plastic non-homogeneity corresponds to the variable yield-point distribution across the thickness of a bar. Thus this type of non-homogeneity influences both the dimensions and shape of the yield locus I; in the stress-
192
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
resultant space. However, the most important thing in this case is the influence on the shape of the yield locus. If the yield point uo* is a continuous function of the bar thickness [ao* = ao(l f ) , say, with f ( z ) # 01, then the limit state of the rectangular cross section is defined in pure bending by the following relations
+
F
(54
= M = Mo*,
Mo* = uob
j+ [l
f(z)]zd~,
-H
where h stands for the width of the bar, and 2H for its depth. This type of transverse non-homogeneity was applied to simple cases of beams by
a)
I
n FIG.21. Bending of a rectangular cross section with a piece-wise linear non-homogeneity distribution.
A. S. Grigoryev, [31], assuming that f ( z ) = f ( - z ) . Similarly in the case of simultaneous bending and tension or compression, the parametric equation of the yield locus is simply H
I n
The elimination of the parameter zo from these equations yields the corresponding interaction curve, Thus the plastic interaction curve or, in general, the plastic interaction surface F is a function of the transverse non-homogeneity . If the non-homogeneity function is a step function, the corresponding yield locus can be obtained in a similar way. P. G. Hodge, [35 b], applied this approach to derive approximate interaction curves for a homogeneous material. For a rectangular cross section and a piece-wise linear non-homogeneity distribution as shown in Fig. 21 a the interaction curve given in Fig. 21 b
PLASTICITY U N D E R NON-HOMOGENEOUS C O N D I T I O N S
193
has been obtained in Ref. [35b]. The material is supposed to obey the Coulomb-Tresca yield condition. In Fig. 21 n and m stand for the properly defined dimensionless axial force n and bending moment m. The concept of step-wise transverse non-homogeneity is therefore useful in the linearization of limit analysis problems. The influence of non-homogeneity on the shape of the yield locus is evident. The behaviour of structures made of imperfectly plastic material was studied by J. Heyman, [33 d].
‘t
FIG. 22. Interaction surface for simultaneous bending, compression and shear of a step-wise transversally non-homogeneous element.
A composite structural element made of materials characterized by distinct yield properties (e.g. reinforced concrete) can also be considered as step-wise transversally non-homogeneous. For such a case the yield locus in simultaneous bending, compression, and shear has been derived by A. Sawczuk and M. Janas, (1061. The octant of the interaction surface is shown in Fig. 22. The details can be found in the reference. The difference between the interaction curve in the m,n-plane for a homogeneous and a non-homogeneous material becomes apparent simply by comparison of Figs. 21 and 22. Thus the non-homogeneity influences
194
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
the solution of limit-analysis problems because it influences the shape of the interaction surface F . Thc case, when only the size but not the shape of the yield locus varies with the longitudinal coordinate of a bar, corresponds to the axial (or longitudinal) non-homogeneity. To this group belong, for example, the cases of plastic beams of variable cross sections. Solutions of practical
FIG.23. Load carrying capacity of circular non-homogeneous arch with concentrated load applied a t the arch center.
problems concerning the “variable rigidity” can be found in standard textbooks on the limit analysis of beams and frames (cf. [3] and 135 c]), since it appears that the methods of analysis are the same as in the case of constant size of the yield locus. B. Arches. The load carrying capacity of circular arches of transverse non-homogeneous cross sections was studied by A. Sawczuk and M. Janas, [106]. For a simple concentrated load applied at the arch center results have been compared with the experimentally obtained values. The comparison is shown in Fig. 23.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
195
2. Plates
A number of papers are concerned with the influence of non-homogeneity on the phenomena accompanying the exhaustion of load carrying capacity of different types of plates and shells. These are based on the general theory of plastically non-homogeneous bodies (cf. [73 k]-[73 13). In most cases the extremum theorems are used; in the kinematical approach, an upper bound of the load intensity, which cannot be lower than the actual intensity of the ultimate load, is determined or a lower bound of this intensity is determined by considering the statically admissible fields of internal forces. Statical methods, often giving valuable insight, but generally involving more complicated procedures, were applied to the analysis of systems exhibiting plastic non-homogeneity coupled with orthotropy in the papers by J. Murzewski, [65], and A. Sawczuk, [lo5 a] and [lo5 b]. The problems connected with the determination of the field of internal forces for non-homogeneous plates were treated on the basis of the HuberMises and Coulomb-Tresca yield conditions. As an example let us mention Ref. [lo5 a] by A. Sawczuk, who studies the problem of choosing the type of non-homogeneity so that the circular plate, subjected to axially symmetric load, passes into the fully plastic state a t once over the entire region. If the value of the limit moment is denoted by N ( r ) and the shear force by Tv(r),then the necessary condition may be expressed by means of the two inequalities
(5.3)
N>rT,,
dN dr
-
> 1) mechanical effects will not be noticeable if the field is too weak. A useful measure of the relative magnetic field strength is S, where (3.11)
s2 = (+B2/p)l(4pU2)= B2IppU2,
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
229
which is the ratio of the magnetic energy to the kinetic energy of ordered mass motion of the system, if B is a typical field strength. Another parameter, (3.12)
where 2, is the average pressure in the fluid, measures the magnetic energy in terms of the thermal energy (i.e. kinetic energy of random molecular motion). Note that for a perfect gas, (2sz/y.S2)1/2 is equal to U divided by the speed of sound, which is, of course, the Mach number. is usually called “beta” by plasma physicists and S-2 is sometimes called “dynamic beta”. In the case of an incompressible fluid (for which the Mach number is zero) vanishes and is therefore a redundant parameter. In order to classify hydromagnetic phenomena in incompressible fluids we distinguish two general cases, R > 1, corresponding, respectively, to weak and strong coupling between u and B. Because in the weak coupling case the externally impressed field B, is hardly affected by u, we can take B = B,. Except in the cases of certain especially simple flow fields and geometrical configurations of the boundaries, when E may practically annul (u x B) (see (2.15)),
s-2
s
j
(3.13)
N
a(u x B,).
Hence, the magnetic term in the body force (see (2.4)) (3.14)
This force retards the motion and degrades the kinetic energy through the agency of ohmic heating. If S is such that the so-called Hartmann number (3.15)
M
B , L ( U / ~ Y=) S(RR’)l12 ~/~
is much greater than unity, the dissipative force represented by (3.14) is more powerful than viscous friction ; otherwise viscosity is largely responsible for energy dissipation. In the field of a powerful electromagnet, waves on the surface of mercury are attenuated a t a rate much in excess of that due to viscosity acting alone and this effect is readily demonstrated. Some insight into the subtle interplay between viscous and magnetic forces is to be gained from analysing hydromagnetic flow between parallel planes as we shall see when this problem is discussed in Ch. IX. In contrast to the case of weak coupling ( R > 1) the fluid effectively takes on “elastic” properties, the importance of which depends on S. When S is small these “elastic” forces are insignificant. (However, in the case of turbulent flow S may not remain small indefinitely because in general the lines of magnetic force will increase in length as they move with the fluid [22]. As a result B will increase a t the expense of U
230
RAYMOND HIDE AND PAUL H. ROBERTS
until a state of equipartition has been attained, and oscillations about this equilibrium state may occur.) When S is large, however, the magnetic field dominates. Because of the elastic properties of the magnetic lines of force oscillations can occur in which the inertia is provided by the fluid and the restoring force by the field. These oscillations give rise t o hydromagnetic waves which travel with velocity V = B/(,up)'12 (see Sec. V.2), which were first discovered theoretically by AlfvCn [23]. When R >> 1, the Lundquist number,
K
(3.16)
aLB(p/p)'12 = S R ,
which is a magnetic Reynolds number based on the AlfvCn speed, is the appropriate parameter determining the degree of mechanical coupling between the field and the motion. 3. Two-dimensional Theorem: Analogy with the Proudman-Taylor Result for Rotating Fluids In view of the number of papers in the recent literature which deal with nearly uniform flow in a nearly uniform magnetic field [24, 25, 261, it is instructive to consider whether any general statements can be made about b where B, is supposed to be a uniform magnetic such flows. Let B = B, field parallel to the z axis, and b 1 and S2RR' >> 1 (see (3.3), (3.4), (3.11)),the right-hand side of (3.25) can be ignored, and to a first approximation (3.26)
a2
- (w,j) = 0. a22
S2>> 1 implies that the AlfvQ speed is very much greater than U,. S2RR'>> 1 is equivalent to requiring that the Hartmann number (3.15) should be much greater than unity; this is a weaker requirement than that of high conductivity ( R >> 1) and low viscosity (R' >> 1). Equations (3.25) and (3.26) are independent of the form of El and n1,. When I7n1= 0, that is, when the total pressure, fil B,b,/p, is constant, but El # 0, it can be shown that
+
(3.27)
a more restrictive condition than (3.26), which it replaces. When, in addition to VITl = 0, we require that El = 0, (3.28)
232
RAYMOND HIDE A N D PAUL H. ROBERTS
where I and I I designate, respectively, components perpendicular and # 0 and parallel to the direction of the magnetic field. Finally, when El = 0, it may be shown that
onl
(3.29) The foregoing results are reminiscent of the so-called Proudman-Taylor theorem governing slow, steady hydrodynamical flow of an inviscid, homogeneous, uniformly rotating (non-conducting) fluid [ 2 7 ] . This flow is twodimensional, having no variation in the direction of SZ, the basic rotation vector, Coriolis forces being the dynamical constraints operating in this case. If u is the flow velocity relative to a uniformly rotating frame of reference, (2.2) still holds provided 2pSZ x II is added t o the left-hand side, and centrifugal effects are included in F (see (2.5)). On taking the curl of the resulting equation, if SZ = (0, 0, Q), then, remembering that p is assumed unifom and, since u = 0, j = 0, (3.30)
2 Q a ~ I a z= vP(cur1 u )
+ curl (u x curl u ) ,
and in the limit of small viscosity (more precisely when the Ekman number (v/2RLz)”2 is very small) and slow relative flow (small Rossby number, UILR), where U is a typical flow speed and L a characteristic length, we have the result (3.31)
aulaz = 0.
Some writers have erroneously concluded that there is an exact parallel between the hydromagnetic case and the rotating fluids case. According to (3.26) and (3.31) this is not so. Although the Proudman-Taylor theorem has been amply verified by experiment, and work on the dynamics of rotating fluids now forms a large and fascinating chapter of hydrodynamics, the experimental verification of the hydromagnetic two-dimensional theorem has not been given. It is instructive to consider flows which satisfy the condition aulaz = 0. In the case of the flow caused by the uniform motion in the z direction of a solid object immersed in a fluid of indefinite extent, a whole column of fluid extending from the object to infinity in both upstream and downstream directions partakes of the motion of the object and in consequence, the total energy of the flow is infinite. In the absence of the constraint that au1a.z = 0 the total energy is finite (e.g. potential flow). Hence, while the latter flow can be set up from rest in a finite time by the application of finite forces, the former cannot. This can be important when one considers the mathematical uniqueness of solutions of steady state problems, and care has to be exercised in the interpretation of such solutions. As Stewartson [28]
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
233
has emphasized the solution, if unique, should be the limit as t+ bo and v + 0 (and/or A + 0 ) of some time dependent solution for a real fluid (v # 0, A # 0 ) irrespective of the order in which one proceeds to these limits. The danger of such pitfalls can be reduced by working not with t , v and A, but with the appropriate dimensionless parameters which measure their values (see above).
IV. BOUNDARY CONDITIONS* 1. Methods of Deriving Boundary Conditions
In many situations, two media ( M , and M,) of almost uniform composition are separated by a relatively thin layer Y (of thickness 1, say) in which there is a rapid and continuous transition between the two states. Also, some of the physical variables (such as, for example, the normal component D, of electric displacement D) change rapidly in 9.I t is clear that, rather than treat 9as being of finite thickness, it would be theoretically simpler to treat it as an abrupt interface S of zero thickness. However the basic differential equations break down on such an interface since some quantities (e.g. 6) are unbounded and likewise some derivatives ( e g the normal derivative of D,) do not exist. Nevertheless it is necessary to establish relations between the fields on either side of S before the problem can be solved uniquely. These are called “boundary conditions”. The form they take depends on the relative magnitude of 1 compared with other length scales of the system. In a viscous fluid, 1 must be small compared with the boundarylayer thickness, but, if viscosity is ignored, the boundary layer must be thought of as being contained in Y no matter how small 1 is. Similarly in a fluid of finite conductivity, 1 must be small compared with the thickness of the electromagnetic boundary layer (in which the eddy currents flow) but, if resistivity is ignored, the boundary layer must be thought of as being contained in Y no matter how small 1 is. The boundary conditionsmay be derived in one of two equivalent ways. Either the basic differential equations are integrated across Y before the limit 1+ 0 is taken, or the integral equations (from which the differential forms were originally derived) are used. We will adopt the latter approach. Displacement currents are retained in the first instance, but are neglected subsequently in taking the hydromagnetic approximations. We use the following notation. We will define (locally Cartesian) coordinates x+, on S and locate a point Q near S by its shortest distance x, to S Owing to the length of this section, in which it was felt desirable to discuss boundary conditions in more detail than is customary, the principal results are given in boxed equations.
234
RAYMOND HIDE AND PAUL H . ROBERTS
and the coordinates xq,x, of the foot of the normal from Q to S. We take the normal vector n = n(x,,xs) to be directed from M , into MI and the coordinates x,,xq,x, to be right-handed in that (alphabetical) order. It is convenient to introduce an abbreviation DIV A defined by DIV A = div A - [(n * grad) A] * n, (4.1) aAq/axq aA,/ax,. Consider a point P of S: let it be xq = 0, x , = 0 for convenience. The integrations we will perform are of two types: (i) Integrations over the interior dV and surface dS of a “penny-shaped” disk. The radius Y of the penny is small compared to the radii of curvature of S at P and small compared to all the physical length scales except,
+
T” / MI
/
Mz
FIG1.
FIG.2.
+
possibly, the boundary layer thicknesses. Its thickness I, 1, is small compared to Y . Its top surface lies in x, = 1, and its bottom surface in x, = - I, initially. Since the disk is considered to be fixed in space and the boundary may be moving (with velocity U , in the direction of its normal, say), a t a later time dt, the top surface lies in x, = 1, - U,&, and the bottom surface in x, = - I , - U,dt (see Fig. 1). (ii) Integrations over the surface d S and round the perimeter dr of a rectangle. The plane of the rectangle will be taken to be either x, = 0 or xq = 0 and, in the former case, the rectangle is defined by its intersections with x, = I,, xq = 0, x, = - I, and xq = M(>> 1, 12) (see Fig. 2). Again M is small compared with the radii of curvature and all physical length scales except, possibly, the boundary layer thickness.
+
2. The Electromagnetic Boundary Conditions (a) Normal component of B.
Apply the equation (cf. (2.13)) (44
I
Beds =0
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
236
to the penny-shaped disk. We find
(4.3)
&'[B,(') - B,(')]
+
5
B * dS
+ O(73) = 0.
b
Here
(4.4)
B,(l) = limit B,,(Zl,O,O),
B,(')
= limit
Zl+O
B,,(- Z,,O,O)
l,+O
and b denotes the curved surface of the disk. Now B is everywhere bounded; thus the integral over b can be made arbitrarily small compared with the remaining terms of (4.3)by choosing I , and I, sufficiently small. Hence (4.3) gives
(The left-hand side is an abbreviation for B,,(') - BL').) (b) Normal component of D. Apply the equation (cf. (2.14))
I
Dads
5
=
6dV
to the penny-shaped disk. The argument is essentially as in case (a), and gives
(4.7) where
7 = limit dV+O
~
'I
6dV
7C7'
is the surface charge density on S: dimensions coulomb/m2. (c) Normal component of j. From considerations of conservation of electric charge, we find
(4.9)
5
at
In applying this to the penny-shaped disk, we must recognize that (in the relativistic case) currents arising from convection of charge and (in the zeroresistivity case) eddy currents will flow in an infinitely thin layer a t the
236
RAYMOND HIDE AND PAUL H. ROBERTS
interface. Denoting this surface current by J = J(x,,x,) we can easily show that j * dS
(4.10)
=
- m2DIV J + O ( y 8 ) .
b
Thus, by (4.9), we have (4.11)
The first term on the right-hand side is negligible if displacement currents are ignored; the second if eddy currents are ignored. (d) Tangential components of E and H. Apply the equation (cf. (2.12)) (4.12)
to an elementary rectangle of the type described in the first section. If the rectangle lies in the plane x, = 0, we find that the value of the left-hand side of (4.12) is
Hence, since the right-hand side of (4.12) only differs from M[E,I12 by a second-order quantity, we have
I [E,]12 = U"[BSI,2. I
(4.14)
Similarly
I [ESl12 = - U,[B,]12.1
(4.15)
According to (4.14) and (4.15) the tangential components of E, measured in a frame of reference which is locally moving with S, are continuous. Similarly, from the equation (cf. (2.11')), (4.16)
$
1
H . dr = (j + aD/at) dS
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
237
we find (4.17)
I [Hq]12=
If p and
E
(4.18) (4.19)
Js
-
Ufi[Os]i2,
[Hs]12 =
- Jq
+
un
[Oq]i2.]
are continuous across S, (4.14), (4.15) and (4.17) require that (1 - un2/c2)[Es]12= - PunJs,
(1 - Ufi2/c2)[Eq]i2 = - PUnJq,
(1 - un2/c2)[Hs1i2= - J q .
(1 - Un2/c2) [Hq]I2= J s ,
If displacement currents are neglected, (4.18) and (4.19) reduce to (4.20) (4.21)
[Hs1i2 = - Jq,
[Hq1I2 = J s t
[Eq1I2= - P u n J q ,
[Es1i2 = - PUUnJs.
Equation (4.20) may be combined with (4.5) to give (4.22)
[HIl2 = J x n.
This completes the set of electromagnetic boundary conditions. They are still valid if differentiated with respect to xq or x s or differentiated with respect to t , following the motion of the boundary. They are therefore not independent. For example, if we differentiate (4.7) with respect to t following the motion of the boundary, or if we differentiate (4.6) with respect to t before applying it to the disk, we find (4.23)
-
U,, [DIV DIl2 = a7 - a
at
Differentiating the first of (4.17) with respect to xs and the second with respect to xq and subtracting, we find (4.24)
[(curlH),,Il2 = - DIV J
+ U,, [DIV DIl2.
By subtracting (4.23) and (4.24), and using (2.11’), we recover (4.11). Similarly, from (4.5) we find (4.25)
and from (4.14) and (4.15) we find, by differentiating with respect to x, and xq respectively, (4.26)
[(curl E)n]i2
=-
U,, [DIV BIl2.
Thus, using (2.12), we see that (4.25) and (4.26) are equivalent.
238
RAYMOND HIDE AND PAUL H. ROBERTS
In the zero resistivity case, (4.7) and (4.17) do not restrict the solutions in M , and M , : they merely serve to determine q and J. Also, since E = - u x B in this case, the two conditions which do restrict the solutions, namely (4.14) and (4.15), may be written: (4.27)
[(un -
Un)BqIi2= Bn[uq]i2,
[(un
- Un)Bsl,’ = Bn[21s112*
In the finite resistivity case, no eddy currents flow in 9 and the only surface current is that due to displacement of surface charge q, i.e. (4.28)
J
= +q[u(’)
+ u ( ~ -) ] &qn(n [u(’)+ u ( ~ ) ] ) . *
If displacement currents are ignored, J is negligibly small. Otherwise (4.28) determines J and, by (4.14), (4.15) and (4.17), four restrictive boundary conditions. 3. The Mechanical Boundary Conditions
(a) Normal component of u. Apply the equation (cf. 2. 1) (4.29)
at
to the penny-shaped disk. Assuming that p remains bounded everywhere within the disk (i.e. excluding fictitious mass surface densities), the righthand side of (4.29) may be written
and (4.29) gives
i.e. (4.32)
There are two main possibilities: either S is the contact surface between two “immiscible” media, i.e. media unrelated physically or chemically, or M , and M , are composed of the “same” fluid in two different thermodynamic or
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
239
chemical states, the fluid particles crossing S from one to the other ( e g S is a shock front or detonation front). In the former case, (4.32)simplifies to give (4.33)
(b) Tangential components of u. If we define the vorticity by (4.34)
w
= curl u,
we have, by Stokes' theorem, (4.35)
If we apply (4.35) to the rectangular circuits of Sec. IV.1 we find (4.36)
[21J12 =
w,,
[U5I12 =
-
w,,
0=
w,
Here W is the surface vorticity, dimensions m.sec-l. Its rBle may be clarified if we compare the viscous and electromagnetic boundary layers as v - 0 and A + 0. . If v # O ( A # 0 ) the viscous (electromagnetic) boundary layer is of finite thickness, but may be the seat of tangential shears (currents) which are so large that the vorticity (current density) integrated across the boundary layer does not vanish but tends to a limit W(J) as v + O ( A --+ 0). Then, in the case v = O ( A = 0 ) ,as we have seen, the viscous (electromagnetic) boundary layer is entirely contained within a, our rectangular path of integration shown in Figure 2. x contains a finite integrated vorticity W (current density J) no matter how small 1 may be. If however v # O ( A # O),a contains none of the viscous (electromagnetic) boundary layer as the limit I -,0 is taken, and the rectangular path of integration contains zero integrated vorticity (current) and we must, therefore set W = O(J = 0 ) , i.e. in this case we have (4.37)
(c) Normal components of the stress tensor. Let p:;"' be the total stress tensor in the fluid, that is the sum of the mechanical stress tensor and the electromagnetic stress tensor p,",:"'. The equation of conservation of momentum (cf. (2.2), (2.4), and Appendix A) may be written
pyh
(4.38)
Here EE x B is the momentum of the electromagnetic field and is negligible when displacement currents are negligible, as is the electrostatic part of Pfj"'. Apply (4.38) in the integrated form
240
RAYMOND HIDE AND PAUL H. ROBERTS
5
(4.39)
fit?t"'& *7
7
"5
- - - [PU at
to the penny-shaped disk.
+ E ( E x B ) ] d V-
I
pUiUjdSj
It is easily shown that
(4.40)
a5 z5 at
at
2 at
1
[pu
+ EE x B],dV
[pu
+ EE x B],dV = - nrzU, [
= - nrzU, [
+
p ~ , ,] ~nr2U, ~ [sE,B, - EE,B,],~ 0 ( r 3 ) ,
+
p ~- ~nr2U,B, ] ~ [~ E E , ] , ~
+
nrzU, [ E E , B ~ I , ~W 3 ) , [PU
+ EE x B],dV = - nrzU, [
+
p ~ ~~ G] Y ~~ U~ , B , [ E -E ~ ] ~ ~
+
. I
nr2U,[~E,Bq]12 O(r3). It therefore follows that
+ U,[EE,Bs - EEsB,11', - Un)11' + UnBn [~Esll'[~EnBsll', - [pus(u, - Un)]i' - Un&[~Eq11' + Un[~E,Bq11'.
WF'I1' = - [p%& (4.41)
total Ppnq 11 = total 2 [fins 11 =
-
U,)11'
[puq(Hn
These can easily be expressed in an alternative form which involves $yh, J and 7. We will suppose, for analytical simplicity, that E and ,u are continuous across S. The last term on the right-hand side of the first of (4.41) may be written
+
+
+
+ +Uw(Bs(l)+ Bs"))[~Eql,',
iUn&(EP(l) Eq('))[BSI1' - +U,E(E,(') ES('))[B,],'
(4.42)
- iUn(Bq(l) Bq"))[~Esl,'
which, on using (4.14), (4.15) and (4.17), is equal to
+
MEq(') Eq(2)) [EqI1'
+ M E s ( ' )+ El')) [EsI1' +
241
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
Similar results hold for the remaining equations of (4.41) and we find ~
~=
+
+
- [pun(un ~ 1 ~ n ) 1 ~1 2 qi(En(1) 2 - [ ~ ~ 9 ( u-nUn)l,2
[f~,",~= li~
+ q&(E9'')+
+ (arx
~ ~ ( 2 ) )
Eq''))
+ ~(z))),,,
+(~(1)
+ (J' x &(B(')+ B'"))9,
[pFhi1z = - [ p u , ( ~ ,- u n ) 1 1 2 + v+(E,(~) + EP)) + (J' x
+
i(~(1)
B(S)))~,
(4.44)
+
where J' = J qU,n = (qU,,J,,J,). In the absence of displacement currents, all these results assume simpler forms: the terms with E and B do not appear in (4.40) and the terms with q do not appear in (4.44). The method we have chosen to derive (4.44) may seem unnecessarily elaborate, but it introduces the mean fields (E(') E('))/2 and (B(') B(2))/2 appearing in the final result in a natural and unforced way. In the case in which M , and M , are inviscid, the mechanical stress tensor is diagonal and, if the fluids are "immiscible", the last two equations of (4.44) give
+
(4.45) B,J
=-
+ U,n
x {(E(l)
x B('))
+
+ (E(2)+ U,,n x B(2))}
If the fluids are perfectly conducting, (4.45) gives (4.46)
+
IBn[J - i q ( ~ ( ' ) u@))
+ qU,,nIl2 = 0.1
If B, is zero, (4.27) and (4.46) are identically true, and the first of (4.44) requires the total pressure P to be continuous across S. If B, is not zero, u is continuous by (4.27) and J = q(u - U,n) by (4.46). I t follows by (4.14), (4.15), (4.17) that B and E are continuous across S, and that therefore q and J are zero. Also, the first of (4.44) requires that the pressure is continuous across S. Summarizing we have: (a) If B, # 0, E, B, u and p are continuous, q and J are zero. These conditions are not independent and are all satisfied by making u and p continuous. (b) If B, = 0, u, and P must be continuous. 4. Small Departures from a Steady State
We are often faced by the question of whether a certain steady state is stable or not. A necessary condition for stability is that the system be stable against infinitesimal perturbations. Thus, denoting by B,, E,, the magnetic field, electric field, . . . in the steady system, we examine b, E = E, e, . . . . If the amplitudes of solutions of the form B = B, these perturbations grow in time, we conclude that the system is unstable.
...
+
+
242
RAYMOND HIDE AND PAUL H. ROBERTS
The perturbations will, in general, involve a motion of the interface S separating MI and M,, and, of course, B, E,.. . must satisfy on S the boundary conditions we have derived in the preceding sections. However, it is generally more convenient to apply conditions at So, the position of the interface before the system is perturbed. For this reason, we will briefly discuss the problem of translating the boundary conditions at S to equivalent conditions at So. Erect coordinates xq,xs on So, as before. Let the equation of S be (4.47)
xn = E ( x q , x s J ) *
Let P be the point (t,xq,xs) of S and Po the point (O,xqrxS),of So. The magnetic field at P is, by supposition,
+
B ( P ) = Bo(trXq,Xs)
b(EtXq,%),
(4.48)
to first order. The direction of the unit normal to S at P is given by (4.49)
N
= (1,
- allax,, - aEjax,).
Hence the normal component B N ( P )of B at P is
The unperturbed steady state satisfied the boundary conditions (4.5). Hence Bm(P0) is continuous. Thus, since B N ( P )must be continuous also, we require (4.51)
where we have used double brackets to emphasize that the condition is to be satisfied on So and not on S. To first order, N, Q = (a(/ax,, 1, 0) and S = (@/ax,, 0, 1) form a righthanded triad of vectors at P on s. By (4.14) we see
i.e. to first order (4.63)
ELEMENTARY. PROBLEMS IN MAGNETO-HYDRODYNAMICS
243
On expanding these quantities in terms of their values on So, we find
or
Since E, refers to a steady state, U,, = 0 and so, by (4.14), [ [Eoq]]12= 0. Further, by (2.12), curl E, = 0. Thus (4.55) may be written (4.56) again expressing that the tangential electric field, in a frame moving locally with S, is continuous. A similar result holds for eos. The other boundary conditions of Sections IV.2 and IV.3 may be transformed into conditions on So in exactly the same way. 6. Boundary Conditions at a Solid Insulating Surface
In determining the flow past insulating (or poorly conducting) surfaces a t high Reynolds and magnetic Reynolds numbers, it is often convenient to assume that, except in a thin boundary layer near the insulating surfaces, the viscosity and resistivity are negligible. The problem is then divided into two parts. First, the structure of the boundary layer is determined and the “jump conditions” across it are discovered. Second a solution of the equations governing the flow in the main body of fluid is chosen to satisfy these jump conditions. In the present section we will illustrate the first of these processes (see [28]). Consider the steady flow in a sufficiently small region of the boundary layer to be laminar and in the xq direction, say. Since the boundary layer is thin, we may consider that B and 11 vary much more rapidly with x, than with xq or x,; i.e. ajax, >> a/&,, alax,. We may also assume that B, is approximately constant and u, zero in the layer. I t follows that the basic equations (3.1) and (2.2) reduce to (4.57) (Here x, is measured out of the fluid.) The solution to these is
244
RAYMOND HIDE AND PAUL H. ROBERTS
(4.58)
Bq = A
+ C exp
P'xn/(h)1/2,
(4.59)
where (4.60) (4.59) satisfies the condition that ztq vanishes at the surface of the insulator Also (4.58) and (4.59) show that the thickness of the boundary layer is of the order of ( A I J ) ~ / and ~ / V that , the changes in B, and ug across it are x, = 0.
(4.61)
i.e. [ztg]12
(4.62)
);(
= B, 1 P'
1/2
[B,112
or more generally, (4.63)
When ( 2 1 ~is) large, as it is as a rule (except invery tenuous media, see Table 1) we may often (cf. Ch. XI) replace (4.63) by (4.64)
[n x BIl2 = 0.
However, in the general case it is important to realize that even though the limit R - 00, R'+ ce has been taken in the main body of the fluid, it is still necessary to specify R ' / R = A/v (cf. (3.10)).
V. PLANE WAVES 1 . The General Effect of a Magnetic Field
In the absence of a magnetic field, an ideal fluid cannot transmit shear waves. I t can, however, transmit compressional waves, and these travel with the same velocity in all directions. In the presence of a field, the situation is radically different. We have seen in Ch. I11 that the lines of
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
245
force are “frozen” in a perfectly conducting fluid; that is, particles of the fluid lying on P particular line of force at one time lie on that line of force for all times. Following Faraday and Maxwell we may picture the lines of force as mutually repelling elastic strings, whose tension per unit area of cross-section is B2/2p. By the arguments of Ch. 111, their mass per unit length and per unit area of cross-section is p. Thus, if they are stretched and then released, a transverse wave will travel down them with a velocity of vc(B2/2p)/p] = B / ( 2 p ~ p ) ~Thus, / ~ . in the presence of a field, a conducting fluid can transmit a shear wave in the direction of the field. These waves are often called “AlfvCn waves” after their discoverer AlfvCn [23]. A more precise argument, which makes allowance for the repulsion of neighbouring lines of force, shows that (neglecting displacement currents) their velocity V is actually greater than that derived in the approximate argument above. It is
v = B(pup)-”? The repulsion between neighbouring lines of force has an important effect on the transmission of compressional waves. Since this magnetic force effectively increases the pressure driving sound waves across the field, the velocity of compressional waves travelling across the field is enhanced. Moreover the velocity of compressional waves is no longer the same in all directions. In fact it will be shown that in a direction making an angle 8 with B their velocity is (neglecting displacement currents)
if their amplitude is infinitesimal. (a,,= velocity of sound for zero B ; see Sec. 11.4.) The velocity in the direction of the field is unaffected by the field, since the field does not resist the motion of particles of fluid along the lines of force. There are many accounts of the properties of AlfvCn waves in the nonrelativistic case: we mention a few: [l, 3, 4, 5, 23, 29, 30, 311. The relativistic case is treated in [32] an& the rotating case in [21, 33, 341. The influence of a magnetic field on sound waves is discussed in, among other places, [31, 351. The effects of dissipation upon the propagation are described in some of these references and also in [l, 4, 361. Magneto-hydrodynamic shock waves are studied in [37,38,39]. In Sec. V.2 below we discuss transverse AlfvCn waves and touch briefly on the effect of Coriolis forces and displacement current upon their propagation. In Sec. V.3 the effect of a magnetic field on sound waves in a perfectly conducting inviscid fluid is considered. In Sec. V.4 the effects of finite conductivity and viscosity on the results of Sec. V.3 are discussed, and shock waves travelling perpendicular to the magnetic field are considered.
246
RAYMOND HIDE AND PAUL H . ROBERTS
2. Alfvkn Waves
Consider a perfectly conducting inviscid fluid of infinite spatial extent pervaded by a uniform magnetic field B,. Let a disturbance be generated in this fluid, and let the magnetic field then be B
(5.3)
= B,
+ b.
First suppose that the fluid is incompressible; more precisely, if L and z are a length and a time characteristic of the disturbance, we consider that L / t ( 2 ~ V / n i l ) - ~the ' ~ ,mode is damped aperiodically in a decay time of
(6.13)
-&-
="[
2 v2 1
+
V W + } ] 742s
1)
#
while, if s < (2aV/nil)-1/2, the mode is damped periodically with a time constant of decay of
z = 8a2/ln2(2s+ 1)2.
(6.14)
These results (for s = 0, 2a = I, v = 0 ) agree with (5.63). If the walls have infinite conductivity, (6.11) requires (6.15)
2anlV
V(1
+nW2)
- asni,
s =1,2..
..
264
RAYMOND HIDE AND PAUL H. ROBERTS
If s > 2aV/n1, the mode is damped aperiodically with a decay time of (6.16)
otherwise (if s time of
< 2aV/n1), the
mode is damped periodically with a decay
The limit in which the conductivity of the fluid is infinite and all dissipation takes place in the walls is particularly interesting. Then for both A' = 0 and A' = 0 0 , the waves are not damped at all and there must be some intermediate value for which dissipation is a maximum and for which the maximum decay time of the wave is least. For 1 = 0, (6.11)may be written as coth (vzu/V)= - (Vz/~1')1/2
(6.18)
or as tanh (na/V)= - (~1'/Vz)l/~.
(6.19)
For brevity we shall write (6.20)
an
= V(- x
fiy) = a ( -
x fi y ) / t 3 ,
and (6.21)
= (av/a')l/2= (22/t3)1'2,
(Note:
tl = a2/A, t 2=
$/A',
t3= a/V.)
We will consider only that mode for which x is least. For small x, (6.18) gives by successive approximation
(6.22)
= 0.56419
= 1.57080
x - 0.40528 x2 - 0.11063 x3 - 0.09886 x6 + . . .,
+ 0.56419 x + 0.11063 x3 - 0.06744 x4 - 0.09886 2' + . . . .
(6.23)
For large
x, (6.19) gives by
successive approximation
265
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
x-' + 0.50000 xF2 - 1.26260x-3 - 1.68961x-' + . . . ,
= 1.25331
(6.24)
= 3.14159 -
1.25331
x-'
+
..
- 1 . 2 6 2 6 0 ~ - ~2 . 0 9 4 4 0 ~ - ~ + 1.68961~-~+ .
(6.25) TABLE2
x
x
Y
x
x
0 0.003 0.01 0.03 0.1 0.3 0.6 1.0 1.3 1.4 1.5 1.55 1.6
0 0.00169 0.00562 0.01674 0.05436 0.15029 0.26082 0.34863 0.37801 0.38255 0.38500 0.38554 0.38566
7~12= 1.57080 1.57249 1.57644 1.58772 1.62723 1.74034 1.90853 2.11898 2.25771 2.29954 2.33908 2.35800 2.37635
1.65 1.7 1.75 1.8 1.9 2 3 4 7 10 30 100 300
0.38541 0.38481 0.38390 0.38271 0.37957 0.37559 0.3 1752 0.26242 0.16512 0.11906 0.04118 0.01248 0.004 17 0
m
Y
2.39415 2.41140 2.4281 1 2.44428 2.47509 2.50394 2.70773 2.81812 2.95984 3.01522 3.09977 3.12906 3.13742 3.14159 = n
Returning to the original non-dimensionless units, we see that the time constant of the standing waves is (6.26)
266
RAYMOND HIDE AND PAUL H. ROBERTS
< < <
1, we must, in the present context, regard the mantle of the Earth as being a good electrical conductor.
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
267
VII. GRAVITYWAVES: RAYLEIGH-TAYLOR INSTABILITY
1. Introduction: Choice of Model
In general, the surface waves travelling along the interface between two conducting fluids will bend the lines of force of any magnetic field present 151. The reaction of the lines of force to this bending will affect the propagation of the surface waves. In this section, we consider the simplest case of two semi-infinite immiscible, incompressible fluids of densities p1 and p2 separated, in the undisturbed equilibrium state, by an infinite plane horizontal interface. We suppose the fluid of density p2 lies above the fluid of density pl. For simplicity, we ignore viscosity and treat the fluids as perfect electrical conductors. In the absence of a field, there is a discontinuity of tangential fluid velocity a t the interface between the fluids. For example, consider a wave travelling in the (horizontal) x-direction. By continuity of fluid mass, the fluid particles in the troughs are moving in the positive x-direction and those on the crests in the negative x-direction. But the crests in the waves of one fluid are the troughs in the waves of the other. Hence, there is a discontinuity of velocity across the interface between them. Clearly, in the presence of a field which has a non-zero component normal to the interface, this discontinuity of velocity implies the existence of a discontinuity in the tangential components of E,and this in turn implies the existence of a surface current, across which the tangential components of R are discontinuous. But this is impossible since it would imply a discontinuity of tangential stress, giving rise to infinite acceleration. I t follows, therefore, that the perfect fluid approximation is incompatible with gravity wave solutions of the type that arise in the absence of a magnetic field. More precisely, if o is the frequency of the wave and k its horizontal wave number, there are no such solutions for which 1 and are negligible, compared to both V 2 / w and u / k 2 . In a real fluid, there can be no discontinuities of 11 and B, but these quantities change rapidly within a boundary layer separating the two fluids. The relationship between the net change in B across this boundary to the net change in u can only be found by studying the structure of the boundary layer itself. This is found to depend, in an essential way, upon v/1. We will not enter into a full discussion of this problem here because it is treated in full elsewhere [41; see particulary 9 I11 A, B of this paper]. Instead we shall consider the simpler case [42] in which the prevailing (uniform) field B, is everywhere tangential to the interface and in the z-direction (say) and thus avoid the foregoing complications. We shall take the upward vertical to be in the y direction, and initially let the density be a general function of y . I)
268
RAYMOND HIDE AND PAUL H. ROBERTS
2. Solution of Model Problem
The basic equations of the problem are (2.2), (3.8), (2.3), 2.13)
au
(7.1) -
at
1 + (u - grad)u - (B PP
aB
-
at
(7.2)
+ (u
*
div B
(7.3)
*
grad)B = -
grad)B - (B * grad)u = 0, = 0,
DplDt = 0,
where 1, is a unit vector upwards and g is the acceleration due to gravity. The steady state which satisfies these equations is (7.4)
u = 0, B
= Bo = constant,
p
=
Po = constant - g
5
pay.
In the slightly perturbed state, write (7.5)
B
= Bo
+ b,
f~= $0
+ p’,
p = PO
+
~
‘
8
and neglect the squares and products of b,u,p’, and p‘ wherever they occur. We will henceforth omit primes. We then find from (7.1) to (7.3)
(7.7) (74
div b
= 0,
where
and 5 is the displacement of the fluid particle from its equilibrium position: i.e. (7.10)
u = ayat,
to first order. I t satisfies (7.11)
d i v 5 = 0.
269
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
By the equation of continuity, P = - EYaPolaY.
(7.12)
Thus, by (7.6), (7.7) and (7.11), we have (7.13)
a% ~
at2
1
= - -grad Po
6
Bo2_ a2c_ g +_ + -Eyly. aP0
PPO az2
Po ay
Suppose now that pa is a step-function (7.14)
Others of the physical variables will also have a step-function behaviour across y = 0, but, by integrating (7.11) across an infinitesimal path crossing y = 0, we see that (7.15)
[EY112 =
0.
As in Ch. IV, this notation means (7.16)
[tYl12
= limit
t y- limit Ey.
y++o
y+-0
Similarly, from (7.13) (7.17)
[all2= f Y t Y [P112 = g M P 2 - P l ) .
Supposing that all quantities are proportional to exp i(Zx we find, by (7.13) and (7.15),
+
I ~ Z-
wt),
(7.18)
where m12 = -
(7.19)
m22 =
(Z2
+
f12),
- (22 + n2),
Wm,< 0, Wm2> 0.
From the form of (7.18) and (7.19) it is evident that the amplitude of the waves dies exponentially with distance from the interface, the scale length of the attenuation being unaffected by the presence of a field. Condition (7.17) requires (7.20)
270
RAYMOND HIDE AND PAUL H. ROBERTS
where
+
(7.21)
VSZ= 2BOZ/P(fl Pz).
In the absence of gravity, (7.20) corresponds to A l f v h waves travelling along the interface with a velocity V s appropriate to the mean density (pl p z ) / 2 and with an amplitude which dies exponentially with distance into each media. Gravity waves travelling in the x-direction are unaffected by the field and it follows that the criterion for instability ( p z > PI) is unaltered by the presence of a field. The phase velocity of gravity waves travelling in the z-direction is increased by the field to
+
(7.22)
In astrophysical circumstances, the magnetic field may be associated with a strong rotation Q. It is well known that, if 0 is parallel to g, it tends to inhibit instability 1431. If S2 is perpendicular to g it promotes the instability of surface waves travelling in the direction of n and stabilizes waves travelling perpendicular to Jz and g. In the case in which B, is perpendicular to g and 51,it can be easily shown that, if 2QV > g, all waves travelling in the direction of B, are stable no matter what their wavelength and no matter what the difference in densities of the media may be: however, waves in the S2 direction are always unstable. In the case in which B, is parallel to S2 and the heavier fluid lies on top, waves in the direction perpendicular to SZ and g are stable provided their wave number is less that Q 2 ( p z - pl)/g(pz pl), and it is likely that a sufficiently high viscosity or resistivity would stabilize them completely. Waves travelling in a direction parallel to n are stable provided their wave number n exceeds g(Pz - P l ) / ~ S 2 ( P Z Pl).
+
+
VIII. GRAVITATIONAL INSTABILITY : JEANS' CRITERION
A problem of astronomical interest is that of establishing the physical conditions under which gravitational condensations of matter will arise in a large mass of gas [44]. Jeans [45] considered this problem first and put forward the criterion that the size L, of the condensation must exceed a certain value L j usually called the Jeans' wavelength:
where a, is the local speed of sound and p,, is the local density of the gas. Because of the importance of Coriolis forces and hydromagnetic forces in
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
27 1
cosmical physics, subsequent writers have extended Jeans’ analysis to include their effect, and have shown, in fact, that Jeans’ criterion is independent of them [46]. Because the exact circumstances under which Jeans’ method of analysis is acceptable are not immediately obvious in some of the treatments which have been given, we shall enumerate these circumstances before deriving their results. Imagine a large mass of gas of typical dimension Lo, in which the velocity is uo, satisfying
Dug = - grad Po
ZT where
Po is the
pressure and
@j0
+ po grad @,
is the gravitational potential satisfying
V2@o = - 4nGpo.
(8.3)
In order to discuss the gravitational stability of this motion, we consider small perturbations about the state characterized by po,uo,@o,~o.Let @, Po pl, the new values of these quantities be po pl, uo ul, Q0 respectively. By (8.2) and the equation of motion for (ug ul), we have, to first order,
+
+
+
+
+
gradp, - pograd G1 (8.4)
Jeans effectively considered a system in free motion under gravity, i.e. grad Po = 0, so that the right-hand side of (8.4) could be set equal to zero. If to is the shortest time scale associated with the basic motion and tlthat associated with the perturbation, when (8.5) we find
Thus (8.4) reduces to
(8.7)*
po au, = - grad 9,
at
+ po gradQ1.
Similarly, using (8.5), the equation of continuity (2.1) gives aPl - - po div u,. at ‘Writers who have followed Jeans’ approach have used (8.7) as a starting point.
272
RAYMOND HIDE AND PAUL H. ROBERTS
These, together with
and the appropriate thermodynamic relations, are the equations governing the perturbation. Jeans simplified the problem still further by considering one-dimensional isentropic disturbances in the z-direction, say, having the harmonic form exp i(Kz - ol). He found that their frequency is
(8.12)
[
-w_- &a, 1 2n
Ll
(EL)2]1/2,
where L , = (2n/k)is their wavelength. For L,< L j , w is real; in this case, the disturbances are propagated in the z directions with phase velocity o L J Z n , which reduces to a, when L , L,, o is imaginary and the disturbance is aperiodic and increases exponentially with time; after a time of t,, where (8.13)
tl
=“?[($Y a0
- 1/2 - 11
,
its amplitude has increased by a factor of e. Clearly L, = L,.
tl is
infinite when
These results can be understood by the following rough argument. Suppose a slight condensation occurs in the gas in a region 9i? of typical dimension L,. Because of this condensation, any two halves of 92 attract each other with a gravitational force of the order of (8.15)
Thus, an increase in pressure of approximately (8.16)
is required to act across the interface between them, in order to prevent further condensation. If this exceeds the increase aO2pl in gas pressure caused by the condensation, the region 9 will condense further, i.e. the medium is gravitationally unstable if
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
273
(8.17)* GL12PoPlln> UO2Pl. This rough analysis makes it clear why Coriolis forces and a uniform magnetic fields do not affect Jeans’ criterion. The magnetic field does not influence and is not influenced by the motion of particles along the lines of force. Thus, provided the length scale of the magnetic field is large compared to L,, condensations of W along the lines of force are unaffected by the field. Similarly, in a rotating system, condensations along the lines of S2, the angular velocity, are not affected. In rederiving Jeans’ criterion for a conducting gas in the presence of a uniform magnetic field B,, we follow the notation of Sec. V.3, adding the effects of self-gravitation. We omit suffixes 1 hereafter. We find that (5.37), (5.39), (5.40) and (5.41) still hold. However,
replaces (5.38) and (neglecting displacement currents),
(8.19) replaces (5.47). Thus the expressions (5.48) for the phase velocitias are replaced by
(8.20)
where 0 is the angle between B, and the direction of propagation of the wave. When these wave velocities are complex, the amplitude of the wave increases without limit, i.e. the system is unstable for such waves. By (8.20), it is seen that this happens when (8.17) is obeyed. This, therefore, remains the criterion for instability, even in the presence of a field. In fact, for waves travelling in the direction of the field, (8.19) separates into (cf. (5.43))
* Since the physical argument leading to (8.17) is independent of the detailed model chosen, and is almost certainly correct, the conclusion, based on (8.13). is acceptable. However, Jeans’ model is probably not the best approach, since there are uncertainties in the value of t o that should be employed. The most serious difficulty is that the value of t oimplied by (8.2) with grad Po = 0, never satisfies (8.5) when tl is given by (8.13). These difficulties have not been discussed by writers who start with (8.7), but have led other writers (cf. e. g. McCrea, [44; to consider alternative models to avoid the present inconsistencies.
274
RAYMOND HIDE AND PAUL H. ROBERTS
As in the discussion of Sec. V . 3 , the vanishing of the second bracket is uninteresting, and we find (8.22)
(g-
a2
ao2-
a22
1
- 4nGpo
B = 0,
showing that waves travelling in the direction of B, are unaffected by the field. On the other hand, for waves travelling perpendicularly to the field, the criterion for instability is
(8.23)
k(ao2
+ V2)ll2< (4nGpo)'12,
proving that the field stabilizes these waves.
IX. STEADYFLOW BETWEEN PARALLEL PLANES In this section we consider steady laminar flow, in the x-direction (see Fig. 7), of an incompressible conducting fluid along a rigid pipe of length c and of rectangular cross-section 2a x 2b, in the special case when
C
-
I
FIG.7.
a a, it is not necessary to be specific about the side walls because no boundary conditions on the differential equation governing the flow field remote from y = & 6 result from considerations of conditions near y = f b. It is immaterial, for instance, whether the situation is regarded as the limiting case of flow along a pipe of rectangular cross-section when alb tends to zero, a remaining finite, or of that of flow along a pipe of annular cross-section as the mean radius of curvature, (rl rz)/2, tends to infinity, the width (r, - rl) of the annular remaining finite and equal to 2a. However, when M # 0, although direct frictional effects of the side walls in y = f b can still be ignored, because j is in the y direction, the manner in which the current circuit is completed via the side walls and through conductors (if any) external to the fluid has to be specified. Otherwise, there are insufficient boundary conditions ta determine the mathematical problem uniquely.
+
If the side walls are not in electrical contact with one another outside the fluid, the total current,
2 76
(9.5)
RAYMOND H I D E AND PAUL H. ROBERTS
I =(j,dz -a
(ampere per unit length in the x-direction) must vanish. Hence, regions of positive j , will have to join with regions of negative j , via regions in the fluid near y = f b in which current flows parallel to the z-axis. Electric charges present on the side walls at y = f b are associated with an electric field E having a y-component only within the fluid.
Conductance /unit N a / b)
lenqfh
FIG.8.
A t the opposite extreme we have the case corresponding to perfect electrical contact outside the fluid between the side walls in y = f b. Then E , must vanish because otherwise I would be infinite. E , vanishes in the annulus problem; otherwise the line integral of E around (say) a circle parallel to both walls would not vanish, and according to Faraday’s law of induction this is inconsistent with the supposition that the system is steady. In general, if the side walls are connected externally via a conductor having conductance N-l[aa/b] per unit length in the x-direction (see Fig. 8) where N is a dimensionless parameter, by Ohm’s law applied to the external circuit,
5
b
(9.6)
N-l(aa/b) E , d y
+ I = 0,
-b
(see (9.5)). In deriving (9.6) use has been made of the fact that E , is independent not only of x and y, but of z also (see equation (9.9b) below). Now make further use of the fact that E , is independent of y and thus simplify (9.6) to (9.7)
2aaE,
+ N I = 0.
277
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
The two extreme cases considered above correspond, respectively, to N + 00, so that I = 0, E , # 0, and N = 0 so that E , = 0, I # 0. Having discussed the boundary conditions, apply the equations of Ch. 11. By (2.11)
by (2.12) (9.9a,b) and by (2.13) (9.10)
dB* = o ;
whence
~
dz
B,
= B,,
b, = 0.
Now combine (9.8) with (2.15) and make use of the fact that u, to find that (9.11a, b, c)
- - - paE,;
dz
db, - = p a [ E y - u, B,]; dz
=
0 =pu[E,
(u,,O,O)
+ u,b,].
At this point, we introduce the assumption that i, = 0, whence, by (9.8), b y is constant and this constant must vanish because by cannot be discontinuous at the side walls a t y = 3 b. Hence, by (9.8), (9.12) by (9.9), (9.11a) and (9.11~)
E = (O,E,,O),
(9.13)
where, by (9.9b) E , is independent of z, and by (9.10) (9.14)
B
= B,
+ b = (b,,O,B,).
Now combine (9.14) and (9.12) with the equation of motion, (2.2). The z-component leads to (9.15) if gravitational effects are ignored (i.e. a@/az = 0, see (2.5)); this is the hydrostatic pressure gradient which has to be offset by stresses in the side
278
RAYMOND HIDE AND PAUL H. ROBERTS
walls a t y = & b. The only other component of (2.2) of interest, the xcomponent, leads to d2u, 1 db, O = P o + p ~ ~ + - B ,u O d z
(9.16)
- t
where, as noted above, - Po is the imposed pressure gradient along the pipe, being equal to the pressure drop between the two ends of the pipe divided by the length c. Eliminate db,/dz between (9.11b) and (9.16), whence (9.17)
0
=
[Po
+ OBOE,] + pv d2uz -- aBo2ux. a22
'"n 0.6
0.4-
0.2 , M , 10
0
20
,
,
30
FIG.9.
Because the term in square brackets is independent of z , this equation can be integrated to give (9.18) where [ E z/a, and the no-slip boundary condition (9.3) has been introduced to evaluate the constant of integration. Now we must eliminate E , by making use of (9.7). First observe that by (9.12) and (9.llb), (9.19)
. IY =
cosh MC
Po aE,Bo coshMC
6{(7 )
-
('- cosh M )}'
279
E L E M E N T A R Y P R O B L E M S I N MAGNETO-HYDRODYNAMICS
so that
Hence, (9.7) leads to
where U , is the value of U when M = 0 (see (9.4)). E , vanishes when N = 0 ; otherwise, E , is always positive and opposes 11 x B,, as expected. The variation of E , with M when N = 00 is illustrated by Fig. 9. Now substitute for E , in (9.18) and find
+
cosh MC
2(N 1) M ( M N tanh M )
(9.22)
+
In order to study the limiting forms of (9.22) first note that cosh x = 1
+ -21 x 2
;
sinh x
+x X
(9.23) 1 cosh x = - e x 2
;
tanh x 1 1 - 2e-2x
;
sinh x coth x
1
+-ex
2
A
1 $- 2e-2x
x>> 1.
.
When M is close to zero, (9.2) is a close approximation to zt, irrespective of N . For the variation of u, with t at other values of M , see Fig. 10, which illustrates two cases, corresponding to N = 0 and N = 0 0 . (As in Figs. 12 and 14, the profile on only one side of the plane of symnietry is given.) In both cases, increasing M results in reduced u, everywhere, but the reduction is more pronounced near [ = 0 than elsewhere. This has the effect of flattening the velocity profile. Evidently, these effects are much more pronounced when N = 0 than when N = m, a result which is due to the lesser restriction on the current flow in the former case than in the latter. The average flow velocity is (9.24)
u = uO M3 2 ( M +( NN t a ’)n h M ) [M - tanh M I , -
+
280
RAYMOND HIDE AND PAUL H. ROBERTS
(see (0.4)); the variation of CT with M for N = 0 and N in Fig. 11. Observe that, according to (9.21) and (9.24), when N (9.25)
E,
= 00
is plotted
=m
= B,U.
Case (b):N-O
FIG.10.
1.0
E L E M E N T A R Y PROBLEMS I N MAGNETO-HYDRODYNAMICS
FIG.11
FIG.12.
281
282
RAYMOND HIDE AND PAUL H . ROBERTS
The current density (9.26)
-
I-
+ +
(1 1/N) coshM5 (1/N tanh M / M ) cosh M
is plotted for a number of typical cases in Fig. 12. j , is always negative on 5 = 0, and is negative everywhere when N = 0. When N # 0, there are regions near 5 = & 1 in which j , is positive; the higher the value of M the thinner these regions become, and when M = oa, this return current in the positive direction flows in a sheet of zero thickness at the wall.
FIG.13.
In Fig. 13 the variation of i,(O) with M is plotted for N = 0 and N = 00. It is noteworthy that - j,(O) approaches the value Po/Boas M - 00 and is 90% of this asymptotic value when M = 2.5 in the case N = 0, and when M = 4.2 in the case N = 00. - Po/Bois just that value of j , required for a static balance between the force j x B, and the impressed pressure gradient along the channel. The induced magnetic field b, can be found by integrating (9.26) with respect to 5 (see (9.12)), giving
where A is a constant of integration. further information.
To evaluate this constant requires
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
283
By a simple application of Ampere's circuital law, (9.28)
bX(1) - b*(- 1 ) = PI,
(see (9.5) and (9.27)),and as I , given by (9.29)
I
=-
tanh M
BaaE, ~
tanh M
N
(see (9.7) and (9.21)),only vanishes when N = 00, b , ( l ) is not in general equal to b x ( - 1 ) . Because bx must be continuous everywhere, and outside
Case (a): N-w
.20I
0 1.0
z/a
0.6
0.4
0
0.2
0
0.2
0.4
% Z I I I I l O0 z/a 1.0
3.6 0.6
FIG. 14.
the fluid, b , must be uniform, if bu and bl stand respectively for the uniform values of b, outside the fluid in > 1 and [ < - 1 , (9.30)
bu = b x ( +
I),
61 = b x ( -
1))
whence, by (9.28), (9.31)
b, - bi = pI
which shows that generally the system is not symmetrical in all respects about the plane 2' = 0.
284
RAYMOND HIDE AND PAUL H. ROBERTS
The constant A depends on the properties of the external circuit, conductance per unit length N-l(ua/b). Let this conductor be made up of two components in parallel, one in the upper space mainly a t >> 1 and the
0
5
1 0
15
20
30
asymptotic
1.0-
0
25
, 5
1 0
15
FIG.16.
20
25
M
value
, 30
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
286
other in the lower space mainly at 5 1. If the criterion is not satisfied, the total current in the y direction need not be zero. Thus, care must be exercised in interpreting the results of idealized problems of the kind discussed in this section. When the moving surface is that of a solid conductor rather than that of an insulator, currents will be induced in this conductor and the behaviour of the system will be modified significantly. It is then necessary to specify carefully the relative motion between the source of the main magnetic field and the vibrating solid. In the literature there has been a certain lack of clarity on this point and some errors have been made. The difficulty seems related to the correct application of the law of induction which is often incompletely dealt with in standard texts of electromagnetism. The subtleties of this point have been considered by a few writers (see Sec. 11.3). Since (10.16) and (10.17) are linear, we seek solutions of the form (zc,b) oc exp (iwt - qz/L),
(10.22)
(L= (V/O)’/~),
where W ( q )> 0, by (10.18) and (10.20). On substitution, we find
(i - q 2 ) ( i - pq2) = uq2.
(10.23)
When u
= 0,
the roots of this equation are
(10.24)
q1 corresponds to Stokes solution in which the shear wave is attenuated in . corresponds to electromagnetic skin currents which a distance ( 2 ~ / w ) ’ / ~q2 are attenuated in a distance ( 2 2 / ~ ) ’ ’(although ~, since u = 0, they are not excited in this case). (10.23) has two roots q1 and q2 which tend uniformly to (10.21) as u-+ 0. We can therefore, without ambiguity, term them the “velocity mode” and “magnetic mode”, respectively. If u # 0 both these modes are excited. Let (10.25)
By equation (10.16), g, is related to f, by (10.26)
(i - P4l2k1 = - Y41flt
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
and a similar equation relates g, and and (10.20), we have (10.27)
fl
(10.28)
291
Also, by the conditions (10.18)
f2.
+ f z = 1, g2 = 0.
g1+
From (10.23) it follows that (10.29)
= ip-
QlQ2
112.
Solving equations (10.26, 27, 28) for f i , f 2 , g, and g, and using (10.29) as a means of simplifying the final results, we find (10.30)
fl
VF41 - 4 2
=
[1 (10.31)
g1 =
+ VPI (41 -
’
fz
VP42 - 41
=
[1
42)
+ VPI
(42
- 41) ’
1
- g2 = -
V F 11 + VPI(41 - 4 2 )
*
From these results,
and
Note that, on the plane itself, (10.34) and e”t
(10.35)
E,(z = 0) = (UoBo) 1 ~
+ va’
Thus, in the limit 8 - 0, there is a surface current on the plane, while the electric field there is given by (10.36)
E
=-
Uo x Bo.
In the limit p- 0 0 , both current and electric field tend to zero. of the force driving the plane is the average, The mean rate of work over a cycle, of (10.37)
P=-vp[U$]
E = O
9
292
RAYMOND HIDE AND PAUL H. ROBERTS
and, by (10.25),
+ +Uoe-ioo”,
U ( Z = 0)= +UOeiog
(10.38)
so that (10.40)
P
=
(pyu,21: L
-H(f1ql
+ f2q2) + periodic terms.
Hence, using equations (10.29) and (10.30) and averaging, (10.41)
3. Discussion of Some Limiting Cuses
Having given the formal solution, we now present the results in a number of limiting cases. We are interested in low, moderate and high conductivity ( p >> 1, p = 1, p 1 for mercury, u/P is the appropriate measure of Bo. As 50/w,frequencies of vibration as low as a few cycles per second would be needed to produce any marked effect. In the kilocycle region the sound speed would be reduced by something of the order of one per cent. The situation should be rather more favourable if liquid sodium were used, because then it would be possible to work at much higher frequencies.
-
4. Ra yleigh’s Problem
Instead of forcing the insulating plane z = 0 to oscillate, we will now simply suppose that a t time 1 = 0 it is jerked into uniform motion with velocity U,, in the direction of the x-axis. In the absence of a magnetic field, this problem was first considered by Rayleigh [48] and is sometimes named after him. The simplest method of recovering his result is by the method of Laplace transforms (equivalent to Heaviside’s operational method), We replace a/at by p , (10.17) then gives
where the superimposed bar distinguishes transformed from untransformed quantities. This must be solved in conjunction with the boundary conditions (cf. (10.18)) (10.43)
n(z =)..
= 0,
n(z = 0)= uo/fi.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
297
Thus (10.44)
Inverting the Laplace transform (see, for example, [52], p. 354, No. 29) (10.45)
where erfc x is the complement of the error function: m
(10.46)
In the case in which the magnetic field is non-zero, the operational solution may be easily derived by replacing io by p in the analysis of Sec. X.2. For example, by (10.25) and (10.30), we have
where
The method we use to invert (10.47) is a simple extension of a method employed by Roberts [36] to solve a modified form of the present problem. Express u(t) in the form
(10.49)
where
Then (see, for example, [52], p. 354, No. 27)
298
RAYMOND HIDE A N D PAUL H . ROBERTS
(10.51)*
{(AWl
+ ( v l k l - j11~2sz)es~z},
- v1/2sZ)esle
= O,(P)
+ %(P)’ say,
where v1 and v2 are the parts of v involving exp slz and exp szz, respectively, and
where ,8 = ~ V ( A V ) ’ / ~/ ( v). A I t follows that
Now, by elementary methods (see, for example, [52], p. 353, Nos. 7 , 8)
(10.54)
and also (see, for example, [52], p. 356, No. 53)
* The analytical advantages of this transformation are somewhat offset by the apparent dimensional inconsistencies it introduces. The reader should therefore take heed that, since O(p’/i) is related to a ( p ) by (10.51), the direct Laplace inversion of ti($) by a Bromwich integral involving ePt must lead to a function v ( t ) in which 1 has the dimension (time)’/%. (See 10.57 and 10.60 below.)
299
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
(10.56)
['P V (li(P" 2 -
"1'
- P2)
esla N
where I, and I , are modified Bessel functions of the first kind of orders zero and unity, respectively. Thus
where 1 6 = V(A1/2 - y1/2) (t' - zv- '/2),
(10.58) (10.59)
5
= p [(t- t ' ) 2
+ (A- 112 + Y- l l Z ) Z ( t - t ' ) ] l / Z .
In an exactly similar fashion
if
t