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Electrodynamics of Continua II: Fluids and Complex Media PDF

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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 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec- tion with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood 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

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