Fundamentals of Maxwel's Kinetic Theory of a Simple Monatomic Gas

Name: Fundamentals of Maxwel's Kinetic Theory of a Simple Monatomic Gas | Treated as a Branch of Rational Mechanics
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Treated as a Branch of Rational Mechanics

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Published on 13. February 1980

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Front Cover
Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas
Copyright Page
Contents
Prologue
Acknowledgments
Notation
List of Special Symbols
Part A: Continuum Thermomechanics
Chapter I. Continuum Theories of Fluids
(i) Basic concepts. Field equations
(ii) Constitutive relations. The Navier-Stokes-Fourier theory of viscous fluids
(iii) Thermodynamic quantities. The Maxwell number and the caloric
(iv) More general constitutive assumptions and principles
(v) Thermodynamics: The Clausius-Duhem inequality and its special cases and generalizations
Chapter II. The Stokes-Kirchhoff Gas, Some of Its Peculiarities, and Some of Its Flows
(i) The Stokes-Kirchhoff and Euler-Hadamard theories of ideal gases
(ii) Some parameters that control dynamical similarity
(iii) The caloric of an ideal gas
(iv) Equilibrium
(v) Some particular homo-energetic flows: dilatation, affine flow, simple shearing, extension
Part B: Basic Structures of the Kinetic Theory
Chapter III. The Molecular Density, the Definitions of Gross Fields, and the Equation of Evolution
(i) The molecular density and the number density
(ii) Expectations. The thirteen basic fields
(iii) The higher moments
(iv) The retrogressors
(v) The equation of evolution
Chapter IV. Some Limits of Agreement between Kinetic Theories and Classical Fluid Mechanics
(i) Failure of constitutive relations in the sense of continuum mechanics
(ii) Limitations on the shear viscosity
(iii) Vanishing of the bulk viscosity
(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
(i) The retrogressor and the retrogression reviewed
(ii) Differentiation along a trajectory. Mild and strong derivatives
(iii) Local forms of the equation of evolution
(iv) Moments of the strong derivative of a function
Chapter VI. The Dynamics of Molecular Encounters
(i) Binary encounters
(ii) Summational invariants and the Boltzmann-Gronwall theorem
(iii) The encounter problem and its solutions
(iv) The encounter operator and its properties
Appendix A Proof of the lemma
Chapter VII. The Maxwell Collisions Operator. Kinetic Constitutive Relations. The Total Collisions Operator and Bilinear Form
(i) The collisions operator
(ii) Kinetic constitutive quantities
(iii) Alternative forms of the collisions operator
(iv) The bilinear form
(v) Orthogonal invariance of the collisions operator and the bilinear form
(vi) Inconsistency of Maxwell's kinetic theory with Newtonian mechanics
Chapter VIII. Boltzmann's Monotonicity Theorem. The Maxwellian Density. Analogues of the Caloric and Its Flux
(i) The Boltzmann monotonicity theorem
(ii) Properties of the Maxwellian density
(iii) Degree to which a Maxwellian expectation approximates a general one
(iv) The caloric of a kinetic gas: Boltzmann's field h
(v) Bounds for h and for its flux s
(vi) Grossly determined functions, momentally determined functions
Part C: The Maxwell-Boltzmann Equation and Its Elementary Consequences
Chapter IX. The Maxwell-Boltzmann Equation. Maxwell's Consistency Theorem and Equation of Transfer
Chapter X. Kinetic Equilibrium and Gross Equilibrium. Locally Maxwellian Solutions
Chapter XI. Boltzmann's H-Theorem
(i) The formal broad H-theorem and the formal narrow H-theorem
(ii) Comparison and contrast of the formal H-theorem with the Clausius-Duhem inequality and the heat-bath inequality of thermomechanics
(iii) The concept of a solid boundary in the kinetic theory
(iv) The formal narrow H-theorem or the heat-bath inequality as a consequence of boundary conditions
(v) Traditional interpretation of the formal narrow H-theorem. The ultra-narrow trend to equilibrium. Statement of corresponding rigorous propositions
(vi) Dificulties faced in interpretation of the more general narrow H-theorem and the strict trend to equilibrium
(vii) Lack of interpretation for the broad H-theorem
Part D: Particular Molecular Models and Exact Solutions for Moments
Chapter XII. The Collisions Operator for Some Special Kinetic Constitutive Relations, Especially Maxwellian Molecules
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
(i) General equations for the pressures and energy flux
(ii) Maxwell's relaxation theorem
(iii) Implications of Maxwell's relaxation theorem on constitutive relations in the sense of continuum mechanics
(iv) Maxwell's evaluation of viscosity and thermal conductivity
Chapter XIV. Homo-energetic Simple Shearing of a Gas of Maxwellian Molecules
(i) Homo-energetic simple shearing
(ii) The pressures as functions of time
(iii) The dominant pressures and their gross determination
(iv) Definition and rigorous evaluation of the viscosity of the kinetic gas
(v) Reduced viscometric functions of the Maxwellian gas
(vi) Comparison of the pressures as fictions of time with their counterparts according to the Stokes-Kirchhoff theory
(vii) Asymptotic forms for fast shearing or rarefied gases
(viii) Solution for the energy flux. Instability
(ix) Entropy. Dissipation
(x) The principal solutions
Chapter XV. General Solution for the Pressures in Homo-energetic Affine Flows of a Gas of Maxwellian Molecules
(i) Affine flows in general
(ii) Homo-energetic dilatation
(iii) Homo-energetic extension, I. The general solution for the pressures
(iv) Homo-energetic extension, II. The principal solutions
(v) Homo-energetic extension, III. Asymptotic status of the Stokes-Kirchhoff solution
(vi) Retrospect
Part E: The System of Equations for the Moments
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
(i) Explicit collisions integrals for a gas of Maxwellian molecules
(ii) Ikenberry's theorem: The structure of collisions integrals
(iii) The general system of equations for the moments
Appendix A Integration formulae and the proof of Ikenberry's theorem
Appendix B Multi-indices
Chapter XVII. Grad's Formal Evaluation of Collisions Integrals, and His Method of Approximating the Initial-value Problem
(i) Grad's expansion and equations of transfer for the Hermite coefficients
(ii) Contrast and comparison of Grad's formal expansion with lkenberry's theorem
(iii) Grad's method of truncation. His 13-moment system and his 20-moment system
(iv) Comparison of solutions of Grad's systems with corresponding exact solutions for shearing
(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
Appendix A Conversion Formulae
Appendix B Exact solutions of the Maxwell-Boltzmann equation for a gas of Maxwellian molecules
Part F: Existence, Uniqueness, and Qualitative Behavior
Chapter XVIII. Existence Theory for the General Initial-value Problem. Part I: Molecules with Intermolecular Forces of Infinite Range
(i) Prolegomena to existence theory
(ii) Spatially homogeneous solutions for a gas of Maxwellian molecules: existence, uniqueness, and the trend to equilibrium
(iii) Estimate of the rates of approach to equilibrium
(iv) Retrospect
Chapter XIX. Convergence Theorems and the Domain of the Collisions Operator
(i) Preliminaries
(ii) Restrictions on the growth of the integrand
(iii) Convergence theorems
(iv) Inverse Kth-power molecules
Chapter XX. Existence Theory for the General Initial-value Problem. Part II: Place-dependent Solutions for Molecules with a Cut-off
(i) Integral forms of the Maxwell-Boltmann equation
(ii) Survey of possibly place-dependent solutions
(iii) A class of body forces
(iv) Preliminary estimates
(v) Glikson's theorem
Chapter XXI. Existence Theory for the General Initial-value Problem. Part III: Spatially Homogeneous Solutions for Molecules with a Cut-off
(i) Survey of spatially homogeneous solutions
(ii) General results on existence and regularity
(iii) A modified collisions operator and its properties
(iv) An existence theorem for spatially homogeneous solutions
(v) Proof of the ultra-narrow H-theorem
(vi) Proof of the ultra-narrow trend to equilibrium
Appendix A Estimation of fourth moments
Part G: Grossly and Momentally Determined Solutions and the Iterative Procedures of the Kinetic Theory
Chapter XXII. Hilbert's Formal Iterative Procedure for Calculating Gas-dynamic Solutions. The Assertion of Gross Causality. The Hilbert Mapping
(i) Hilbert's formal iterative procedure
(ii) Proof of effectiveness
(iii) Hilbert's assertion of gross causality
(iv) Properties of Hilbert's formal solutions. The Hilbert mapping
(v) Locally Maxwellian solutions
(vi) Proof that Hilbert's solutions are grossly determined
(vii) Retrospect
Chapter XXIII. Grossly Determined Solutions. The Equations of Gross Determinism
(i) Gas-dynamic solutions. The importance of grossly determined solutions
(ii) Methods of determining gas flows
(iii) The Maxwell-Boltzmann equation for grossly determined solutions
(iv) The equations of gross determinism and properties of gross determiners
(v) Principles of local action and the domain of the gross determiner
(vi) A space of functions for the principal moment
(vii) Gross determiners depending upon the body force. The generalized equations of gross determinism and the equation of transfer for gross determiners
(viii) Gross determinism for affine flows
Appendix A Calculus in Banach Spaces
Chapter XXIV. The Method of Stretched Fields for Approximating Gross Determiners. Use of It to Obtain the Results of Enskog's Procedure
(i) Enskog's procedure
(ii) The method of stretched fields
(iii) The basic expansion of gross determiners
(iu) Approximate gross determiners
(v) The expansion coefficients
(vi) Derivation of the iterative system for the gross determiner when b = 0
(vii) Structure of the iterative system. Proof of effectiveness
(viii) Properties of some of the expansion coefficients
(ix) The formulae of Enskog, Burnett, Chapman & Cowling, and Boltzmann
(x) Extension to take account of the body force
(xi) Explicit results for Maxwellian molecules
(xii) Explicit first approximations for general molecular models
(xiii) Retrospect
Appendix A Derivation of the iterative system
Appendix B Computational formulae
Chapter XXV. The Maxwellian Iteration of Ikenberry & Truesdell
(i) Exact results to which Maxwellian iteration is applied
(ii) The scheme of Maxwellian iteration
(iii) Illustration of the idea of Maxwellian iteration, applied to an ordinary differential equation
(iv) The first two stages of Maxwellian iteration: The Maxwell second approximation to P and its companion for q
(v) Comments on the results, origin, and nature of Maxwellian iteration
(vi) The third stage of Maxwellian iteration
(vii) Proof of effectiveness
(viii) Example: Homo-energetic simple shearing of a gas of Maxwellian molecules
(ix) Atemporal Maxwellian iteration
(x) Use of differential iteration to generate and improve Grad's method of truncation
(xi) Retrospect upon formal methods of approximation
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
(i) Homo-energetic affine flows in general
(ii) Homo-energetic dilatation
(iii) Homo-energetic simple shearing'
(iv) Homo-energetic extension
(v) Failure of the classical approach to approximate solution
(vi) Retrospect
Epilogue
List of Works Cited
Index of Authors Cited
Index of Matters Treated

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Fundamentals of Maxwel's Kinetic Theory of a Simple Monatomic Gas

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