Fundamentals of Maxwel's Kinetic Theory of a Simple Monatomic Gas (eBook)

Front Cover 1
Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas 4
Copyright Page 5
Contents 8
Prologue 16
Acknowledgments 22
Notation 24
List of Special Symbols 26
Part A: Continuum Thermomechanics 30
Chapter I. Continuum Theories of Fluids 32
(i) Basic concepts. Field equations 32
(ii) Constitutive relations. The Navier–Stokes–Fourier theory of viscous fluids 33
(iii) Thermodynamic quantities. The Maxwell number and the caloric 36
(iv) More general constitutive assumptions and principles 37
(v) Thermodynamics: The Clausius–Duhem inequality and its special cases and generalizations 41
Chapter II. The Stokes–Kirchhoff Gas, Some of Its Peculiarities, and Some of Its Flows 48
(i) The Stokes–Kirchhoff and Euler–Hadamard theories of ideal gases 48
(ii) Some parameters that control dynamical similarity 50
(iii) The caloric of an ideal gas 51
(iv) Equilibrium 51
(v) Some particular homo-energetic flows: dilatation, affine flow, simple shearing, extension 52
Part B: Basic Structures of the Kinetic Theory 66
Chapter III. The Molecular Density, the Definitions of Gross Fields, and the Equation of Evolution 68
(i) The molecular density and the number density 68
(ii) Expectations. The thirteen basic fields 70
(iii) The higher moments 74
(iv) The retrogressors 75
(v) The equation of evolution 78
Chapter IV. Some Limits of Agreement between Kinetic Theories and Classical Fluid Mechanics 80
(i) Failure of constitutive relations in the sense of continuum mechanics 80
(ii) Limitations on the shear viscosity 81
(iii) Vanishing of the bulk viscosity 86
(iv) Disagreement between the kinetic theory and the Stokes–Kirchhoff theory for flows in which T > = 1
Chapter V. The Differential Operators of the Kinetic Theory 90
(i) The retrogressor and the retrogression reviewed 90
(ii) Differentiation along a trajectory. Mild and strong derivatives 91
(iii) Local forms of the equation of evolution 93
(iv) Moments of the strong derivative of a function 95
Chapter VI. The Dynamics of Molecular Encounters 98
(i) Binary encounters 98
(ii) Summational invariants and the Boltzmann–Gronwall theorem 100
(iii) The encounter problem and its solutions 103
(iv) The encounter operator and its properties 113
Appendix A Proof of the lemma 117
Chapter VII. The Maxwell Collisions Operator. Kinetic Constitutive Relations. The Total Collisions Operator and Bilinear Form 120
(i) The collisions operator 120
(ii) Kinetic constitutive quantities 122
(iii) Alternative forms of the collisions operator 124
(iv) The bilinear form 127
(v) Orthogonal invariance of the collisions operator and the bilinear form 130
(vi) Inconsistency of Maxwell‘s kinetic theory with Newtonian mechanics 131
Chapter VIII. Boltzmann’s Monotonicity Theorem. The Maxwellian Density. Analogues of the Caloric and Its Flux 134
(i) The Boltzmann monotonicity theorem 134
(ii) Properties of the Maxwellian density 136
(iii) Degree to which a Maxwellian expectation approximates a general one 141
(iv) The caloric of a kinetic gas: Boltzmann’s field h 145
(v) Bounds for h and for its flux s 148
(vi) Grossly determined functions, momentally determined functions 155
Part C: The Maxwell–Boltzmann Equation and Its Elementary Consequences 158
Chapter IX. The Maxwell-Boltzmann Equation. Maxwell’s Consistency Theorem and Equation of Transfer 160
Chapter X. Kinetic Equilibrium and Gross Equilibrium. Locally Maxwellian Solutions 166
Chapter XI. Boltzmann’s H-Theorem 174
(i) The formal broad H-theorem and the formal narrow H-theorem 175
(ii) Comparison and contrast of the formal H-theorem with the Clausius–Duhem inequality and the heat-bath inequality of thermomechanics 178
(iii) The concept of a solid boundary in the kinetic theory 181
(iv) The formal narrow H-theorem or the heat-bath inequality as a consequence of boundary conditions 190
(v) Traditional interpretation of the formal narrow H-theorem. The ultra-narrow trend to equilibrium. Statement of corresponding rigorous propositions 195
(vi) Dificulties faced in interpretation of the more general narrow H-theorem and the strict trend to equilibrium 197
(vii) Lack of interpretation for the broad H-theorem 200
Part D: Particular Molecular Models and Exact Solutions for Moments 202
Chapter XII. The Collisions Operator for Some Special Kinetic Constitutive Relations, Especially Maxwellian Molecules 204
Chapter XIII. The Pressures and the Energy Flux in a Gas of Maxwellian Molecules. Maxwell’s Relaxation Theorem and Evaluation of Viscosity and Thermal Conductivity 216
(i) General equations for the pressures and energy flux 216
(ii) Maxwell's relaxation theorem 218
(iii) Implications of Maxwell's relaxation theorem on constitutive relations in the sense of continuum mechanics 221
(iv) Maxwell’s evaluation of viscosity and thermal conductivity 222
Chapter XIV. Homo-energetic Simple Shearing of a Gas of Maxwellian Molecules 226
(i) Homo-energetic simple shearing 226
(ii) The pressures as functions of time 228
(iii) The dominant pressures and their gross determination 231
(iv) Definition and rigorous evaluation of the viscosity of the kinetic gas 232
(v) Reduced viscometric functions of the Maxwellian gas 233
(vi) Comparison of the pressures as fictions of time with their counterparts according to the Stokes–Kirchhoff theory 234
(vii) Asymptotic forms for fast shearing or rarefied gases 236
(viii) Solution for the energy flux. Instability 237
(ix) Entropy. Dissipation 241
(x) The principal solutions 243
Chapter XV. General Solution for the Pressures in Homo-energetic Affine Flows of a Gas of Maxwellian Molecules 248
(i) Affine flows in general 248
(ii) Homo-energetic dilatation 251
(iii) Homo-energetic extension, I. The general solution for the pressures 253
(iv) Homo-energetic extension, II. The principal solutions 256
(v) Homo-energetic extension, III. Asymptotic status of the Stokes–Kirchhoff solution 260
(vi) Retrospect 262
Part E: The System of Equations for the Moments 264
Chapter XVI. The General System of Equations for the Moments in a Gas of Maxwellian Molecules. Ikenberry's Theorem on the Structure of Collisions Integrals 266
(i) Explicit collisions integrals for a gas of Maxwellian molecules 266
(ii) Ikenberry’s theorem: The structure of collisions integrals 273
(iii) The general system of equations for the moments 278
Appendix A Integration formulae and the proof of Ikenberry’s theorem 280
Appendix B Multi-indices 287
Chapter XVII. Grad’s Formal Evaluation of Collisions Integrals, and His Method of Approximating the Initial-value Problem 290
(i) Grad’s expansion and equations of transfer for the Hermite coefficients 290
(ii) Contrast and comparison of Grad’s formal expansion with lkenberry’s theorem 299
(iii) Grad's method of truncation. His 13-moment system and his 20-moment system 301
(iv) Comparison of solutions of Grad’s systems with corresponding exact solutions for shearing 306
(v) The relaxation theorem for Grad's 13-moment system. Grad's derivation of Enskog's first approximation to the viscosity and the Maxwell number 307
Appendix A Conversion Formulae 310
Appendix B Exact solutions of the Maxwell–Boltzmann equation for a gas of Maxwellian molecules 313
Part F: Existence, Uniqueness, and Qualitative Behavior 322
Chapter XVIII. Existence Theory for the General Initial-value Problem. Part I: Molecules with Intermolecular Forces of Infinite Range 324
(i) Prolegomena to existence theory 324
(ii) Spatially homogeneous solutions for a gas of Maxwellian molecules: existence, uniqueness, and the trend to equilibrium 326
(iii) Estimate of the rates of approach to equilibrium 328
(iv) Retrospect 331
Chapter XIX. Convergence Theorems and the Domain of the Collisions Operator 334
(i) Preliminaries 335
(ii) Restrictions on the growth of the integrand 336
(iii) Convergence theorems 338
(iv) Inverse Kth-power molecules 345
Chapter XX. Existence Theory for the General Initial-value Problem. Part II: Place-dependent Solutions for Molecules with a Cut-off 348
(i) Integral forms of the Maxwell–Boltmann equation 348
(ii) Survey of possibly place-dependent solutions 349
(iii) A class of body forces 352
(iv) Preliminary estimates 352
(v) Glikson’s theorem 356
Chapter XXI. Existence Theory for the General Initial-value Problem. Part III: Spatially Homogeneous Solutions for Molecules with a Cut-off 364
(i) Survey of spatially homogeneous solutions 364
(ii) General results on existence and regularity 367
(iii) A modified collisions operator and its properties 376
(iv) An existence theorem for spatially homogeneous solutions 378
(v) Proof of the ultra-narrow H-theorem 385
(vi) Proof of the ultra-narrow trend to equilibrium 392
Appendix A Estimation of fourth moments 397
Part G: Grossly and Momentally Determined Solutions and the Iterative Procedures of the Kinetic Theory 402
Chapter XXII. Hilbert’s Formal Iterative Procedure for Calculating Gas-dynamic Solutions. The Assertion of Gross Causality. The Hilbert Mapping 404
(i) Hilbert’s formal iterative procedure 404
(ii) Proof of effectiveness 409
(iii) Hilbert’s assertion of gross causality 413
(iv) Properties of Hilbert's formal solutions. The Hilbert mapping 415
(v) Locally Maxwellian solutions 419
(vi) Proof that Hilbert’s solutions are grossly determined 423
(vii) Retrospect 424
Chapter XXIII. Grossly Determined Solutions. The Equations of Gross Determinism 426
(i) Gas-dynamic solutions. The importance of grossly determined solutions 427
(ii) Methods of determining gas flows 430
(iii) The Maxwell–Boltzmann equation for grossly determined solutions 432
(iv) The equations of gross determinism and properties of gross determiners 434
(v) Principles of local action and the domain of the gross determiner 436
(vi) A space of functions for the principal moment 440
(vii) Gross determiners depending upon the body force. The generalized equations of gross determinism and the equation of transfer for gross determiners 442
(viii) Gross determinism for affine flows 449
Appendix A Calculus in Banach Spaces 453
Chapter XXIV. The Method of Stretched Fields for Approximating Gross Determiners. Use of It to Obtain the Results of Enskog’s Procedure 458
(i) Enskog’s procedure 459
(ii) The method of stretched fields 463
(iii) The basic expansion of gross determiners 466
(iu) Approximate gross determiners 470
(v) The expansion coefficients 474
(vi) Derivation of the iterative system for the gross determiner when b = 0 475
(vii) Structure of the iterative system. Proof of effectiveness 481
(viii) Properties of some of the expansion coefficients 484
(ix) The formulae of Enskog, Burnett, Chapman & Cowling, and Boltzmann
(x) Extension to take account of the body force 501
(xi) Explicit results for Maxwellian molecules 508
(xii) Explicit first approximations for general molecular models 510
(xiii) Retrospect 517
Appendix A Derivation of the iterative system 517
Appendix B Computational formulae 523
Chapter XXV. The Maxwellian Iteration of Ikenberry & Truesdell
(i) Exact results to which Maxwellian iteration is applied 537
(ii) The scheme of Maxwellian iteration 541
(iii) Illustration of the idea of Maxwellian iteration, applied to an ordinary differential equation 543
(iv) The first two stages of Maxwellian iteration: The Maxwell second approximation to P and its companion for q 544
(v) Comments on the results, origin, and nature of Maxwellian iteration 547
(vi) The third stage of Maxwellian iteration 552
(vii) Proof of effectiveness 553
(viii) Example: Homo-energetic simple shearing of a gas of Maxwellian molecules 555
(ix) Atemporal Maxwellian iteration 557
(x) Use of differential iteration to generate and improve Grad’s method of truncation 566
(xi) Retrospect upon formal methods of approximation 568
Chapter XXVI. Convergence and Divergence of Atemporal Maxwellian Iteration in Flows for Which an Exact Solution Is Known. Failure of the Higher Iterates to Improve the Asymptotic Approximation 570
(i) Homo-energetic affine flows in general 571
(ii) Homo-energetic dilatation 575
(iii) Homo-energetic simple shearing' 575
(iv) Homo-energetic extension 577
(v) Failure of the classical approach to approximate solution 578
(vi) Retrospect 585
Epilogue 588
List of Works Cited 598
Index of Authors Cited 608
Index of Matters Treated 611