Electrodynamics of Continua II: Fluids and Complex Media

375 Pages·1990·13.23 MB·English

by  G. A. Maugin (auth.), A. C. Eringen

Electrodynamics of Continua II
A. C. Eringen
G.A. Maugin
ElectrodynaDlics of
Continua II
Fluids and Complex Media
With 56 Illustrations
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo Hong Kong
A.C. Eringen G.A. Maugin
Princeton University Laboratoire de Modelisation
Princeton, N.J. 08544 en Mecanique
U.S.A. Universite Pierre et Marie
Curie et C.N.R.S.
75252 Paris 05
France
Library of Congress Cataloging in Publication Data
Eringen, A. Cerna!.
Electrodynamics of continua / A.C. Eringen, G.A. Maugin.
p. cm.
Includes bibliographical references.
Contents: I. Foundations and solid media - 2. Fluids and complex media.
I. Fluid mechanics. 2. Electrodynamics. 3. Magnetohydrodynamics. 4. Continuum
mechanics. I. Maugin, G. A. (Gerard A.), 1944-
II. Title.
QCI51.E&5 1989
537.6-dc20 89-21880
CIP
Printed on acid-free paper
© 1990 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1990
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The use of general descriptive names, trade names, trademarks, etc., in this publication, even if
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by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
987 6 543 2 1
ISBN-13: 978-1-4612-7928-0 e-ISBN -13: 978-1-4612-3236-0
DOl: 10.1007/978-1-4612-3236-0
Preface to Volume II
The first volume of Electrodynamics of Continua was devoted mainly to the
development of the theory of, and applications to, deformable solid media. In
the present volume we present discussions on fluid media, magneto hydro-
dynamics (MHD) (Chapter 10), electrohydrodynamics (Chapter 11), and
media with more complicated structures. Elastic ferromagnets (Chapter 9) and
ferrofluids (Chapter 12) require the inclusion of additional degrees of free-
dom, arising frQm spin-lattice interactions and supplementary balance
equations.
With the discussion of memory-dependent materials (Chapter 13) and
nonlocal electromagnetic theory (Chapter 14), we account for the nonlocal
effects arising from motions and fields of material points, at past times and at
spatially distant points. Thus, the damping of electromagnetic elastic waves,
photoelasticity, and streaming birefringence are the subjects of Chapter 13.
Nonlinear constitutive equations developed here, and in Chapter 14, are
fundamental to the field of nonlinear optics and nonlinear magnetism. The
content of these chapters is mostly new and is presently in the development
stage. However, they are included here in the hope that they will stimulate
further research in these important fields.
Volume II is self-contained and can be studied without the help of Volume
I. However, certain prerequisites are necessary. In order to provide quick
access to the basic equations and the underlying physical ideas, we have
included a section (Section 9.0) in Chapter 9, where the constitutive equations
for electromagnetic fluids are also presented. This section serves as a founda-
tion for the fluid media discussed in Chapters 10 and 11. Basic equations and
the underlying physical ideas, necessary for each chapter, are presented at the
beginning of each chapter.
TIle second volume extends the development of the first volume to richer
and newer grounds. Because of space limitations, and the logical development
and continuity of the text, recent developments in mixtures, semiconductors,
superconductivity, nonlinear optics, and electronic theories could not be
included.
It can be said that the electrodynamics of continua touches every aspect of
VI Preface to Volume II
the world of physics. In this regard, the present volume hopes to stimulate
certain aspects. This volume may be used as a basis for several graduate
courses in engineering schools, applied mathematics, and physics departments.
It also contains fresh ideas and directions for further research. Ferromag-
netism and plasticity, memory-dependent materials as applied to polymers,
nonlinear optics, and the nonlocal theory developed in Chapter 14 are can-
didates for deeper research, penetrating into microscopic and atomic scale
phenomena.
Nonlocal theory (Chapter 14) is still in its infancy. However, it is a parallel
discipline to the well-developed field oflattice dynamics. It has the advantage
that it can be used to discuss physical phenomena in intermediate scales
between microscopic and atomic dimensions.
Electromagnetic theory properly falls into the domain of the theory of
relativity. Consequently, we have included a chapter (Chapter 15) on this
subject to close this volume.
Contents (Volume II)
Preface to Volume II . v
CHAPTER 9
Elastic Ferromagnets. 437
9.0. An Overview of Basic Equations 437
9.l. Scope of the Chapter. 443
9.2. Model of Interactions 444
A. Gyroscopic Nature of the Spin Density. 445
B. Spin-Lattice Model ofInteractions. 446
9.3. Balance Equations 449
A. Global Balance Equations 449
B. Local Balance Equations 450
C. The Clausius-Duhem (C-D) Inequality 452
D. Boundary Conditions 453
9.4. Constitutive Theory . 453
A. Saturated Ferromagnetic Elastic Insulators 453
B. Free Energy 456
C. Correspondence Between the Microscopic Model and the
Continuous Representation . 458
D. Infinitesimal Strains . 460
E. Centro symmetric Cubic Crystals 461
F. Uniaxial Crystals. 463
G. Elementary Dissipative Processes 464
H. Small Fields Superposed on a Constant Bias Magnetic Field 466
9.5. Resume of Basic Equations . 469
9.6. Coupled Magnetoelastic Waves in Ferromagnets . 472
A. Preliminary Remarks 472
B. Plane Harmonic Waves. 474
C. Damping of Magnetoelastic Waves. 482
D. Magnetoelastic Faraday Effect . 484
9.7. Applications of the Magnon-Phonon Coupling 487
A. Pumping and Temporal Magnon-Phonon Conversion 487
B. Drift-Type Amplification of Magnetoelastic Waves 490
viii Contents (Volume II)
9.8. Other Works. 490
A. Continuum Descriptions of Ferromagnetic Deformable Bodies. 490
B. Wave Propagation . 491
C. Ferrimagnetic Deformable Bodies. 492
Problems. 497
CHAPTER 10
Magnetohydrodynamics . 502
10.1. Scope of the Chapter 502
10.2. Basic Equations of Electromagnetic Fluids 503
10.3. Magnetohydrodynamic Approximation 507
10.4. Perfect Magnetohydrodynamics 512
A. Field Equations. 512
B. "Frozen-In" Fields . 513
C. Bernoulli's Equation in Magnetohydrodynamics. 514
D. Kelvin's Circulation Theorem in Magnetohydrodynamics 515
E. Alfven Waves 515
F. Generalized Hugoniot Condition . 516
10.5. Incompressible Viscous Magnetohydrodynamic Flow 518
A. Magnetohydrodynamic Poiseuille Flow 518
B. Magnetohydrodynamic Couette Flow. 520
10.6. One-Dimensional Compressible Flow. 521
10.7. Shock Waves in Magnetohydrodynamics. 525
A. Classification of Magnetohydrodynamic Shock Waves 526
B. Shock Structure . 530
10.8. Magnetohydrodynamic Equilibria. 530
10.9. Equilibrium of Magnetic Stars. 533
10.10. Magnetohydrodynamic Stability . 537
A. The Energy Method 537
B. Equilibrium States and Perturbations. 539
C. Quantities Conserved in the Perturbation 540
D. Elementary Perturbations . 540
E. Change in the Energy Integrals 543
F. Application to the Linear Pinch 545
Problems. 547
CHAPTER II
Electrohydrodynamics 551
11.1. Scope of the Chapter 551
11.2. Field Equations. 552
11.3. Charge Relaxation . 554
1 i.4. Stability Condition . 554
11.5. Helmholtz and Bernoulli Equations 555
A. Generalization of the Helmholtz Equation 555
B. Vorticity Generation in a Space-Charge-Loaded Electric Field. 556
C. Generalization of Bernoulli's Equations 556
11.6. Equilibrium of a Free Interface. 557
Contents (Volume II) IX
11.7. Effect of Free Charges at an Interface. 558
11.8. Electrohydrodynamic Stability. 561
11.9. Electrohydrodynamic Flow in a Circular Cylindrical Conduit 567
11.10. Electrogasdynamic Energy Converter. 569
Problems. 573
CHAPTER 12
Ferrofluids . 574
12.1. Scope of the Chapter 574
12.2. Constitutive Equations of Ferromagnetic Fluids. 575
12.3. Theory of Ferrofluids 577
A. Equilibrium Constitutive Equations . 577
B. Nonequilibrium Constitutive Equations 579
C. Balance Laws 580
12.4. Existence and Stability of a Constant Magnetization in aMoving
Ferrofluid 582
12.5. Ferrohydrodynamic Approximation . 585
12.6. Some General Theorems in Ferrohydrodynamics 587
A. Generalization of the Helmholtz Equation 587
B. Generalization of the Bernoulli Equation. 588
12.7. Ferrohydrostatics 589
A. Equilibrium of a Free Surface . 589
B. Energy Conversion . 590
12.8. Ferrohydrodynamic Flow of Nonviscous Fluids. 591
A. Preliminary Remarks 591
B. Steady Two-Dimensional Source Flow 593
12.9. Simple Shear of a Viscous Ferrofluid . 596
12.10. Stagnation-Point Flow ofa Viscous Ferrofluid 598
12.11. Interfacial Stability of Ferrofluids 603
12.12. Other Problems in Ferrofluids . 608
Problems. 609
CHAPTER 13
Memory-Dependent Electromagnetic Continua. 611
13.1. Scope of the Chapter 611
13.2. Constitutive Equations. 612
13.3. Thermodynamics of Materials with Continuous Memory 613
13.4. Quasi-Linear and Linear Theories. 620
A. Quadratic Memory Dependence 621
B. Finite-Linear Theory 622
C. Linear Theory 624
D. Linear Isotropic Materials . 627
E. General Polynomial Constitutive Equations . 629
13.5. Rigid Bodies. 630
A. Continuous Memory 630
B. Polynomial Constitutive Equations 631
x Contents (Volume II)
13.6. Dispersion and Absorption. 632
13.7. A Simple Atomic Model 634
13.8. Free Motion of an Electron Under Magnetic Field 637
13.9. Electromagnetic Waves in Memory-Dependent Solids 641
13.10. Electromagnetic Waves in Isotropic Viscoelastic Materials. 647
13.11. Nonlinear Atomic Models for Polarization . 652
13.12. COI}~titutive Equations of Birefringent Viscoelastic Materials 657
A. Rate-Dependent Materials. 659
B. Linear, Continuous Memory of Strains 660
13.13. Propagation of Waves in Birefringent Viscoelastic Materials 661
13.14. Photoviscoelasticity. 666
Problems. 673
CHAPTER 14
Nonlocal Electrodynamics of Elastic Solids 675
14.1. Scope of the Chapter 675
14.2. Constitutive Equations. 677
14.3. Thermodynamics 679
14.4. Linear Theory . 682
14.5. Material Symmetry. 686
14.6. Nature of Nonlocal Moduli 688
14.7. Nonlocal Rigid Solids . 693
14.8. Electromagnetic Waves. 694
14.9. Point Charge. 696
14.10. Rigid Magnetic Solids 696
14.11. Superconductivity . 699
14.12. Piezoelectric Waves. 702
14.13. Infrared Dispersion and Lattice Vibrations 704
14.14. Memory-Dependent Nonlocal Electromagnetic Elastic Continua 707
14.15. Linear Nonlocal Theory for Electromagnetic Elastic Solids. 710
14.16. Natural Optical Activity 712
14.17. Anomalous Skin Effects. 713
Problems. 715
CHAPTER 15
Relativistic Electrodynamics of Continua. 716
15.1. Scope ofthe Chapter 716
15.2. Space-Time, Notation 717
A. Space-Time. 717
B. Special Relativity 718
C. General Relativity 718
D. Inertial Frames and Rest Frame 720
E. Proper Time, Timelikeness . 721
F. Space and Time Decomposition 722
G. Antisymmetric Tensors and Axial Four-Vectors. 724
Contents (Volume II) Xl
15.3. Relativistic Kinematics of Continua 725
A. Motion, Strain Tensors. 725
B. Relativistic Rate of Strain . 727
C. Contravariant Convective Time Derivative 728
15.4. Covariant Formulation of Maxwell's Equations in Matter 729
A. Electromagnetic Fields . 729
B. Integral Formulation of Maxwell's Equations 731
C. Four-Vector Formulation of Maxwell's Equations 733
15.5. Relativistically Invariant Balance Laws 734
15.6. Electromagnetic Interactions with Matter 738
15.7. Thermoelastic Electromagnetic Insulators 741
15.8. Electromagnetic Fluids. 743
A. General Nondissipative Case . . 743
B. Linear Electromagnetic Constitutive Equations 744
C. Elementary Dissipative Processes . 745
D. Relativistic Perfect Magnetohydrodynamics . 746
15.9. Further Problems in the Relativistic Electrodynamics of Continua. 747
Problems. 748
References 753
Index II