Analytical Methods for Problems of Molecular Transport

Contents; Table of Tables; Table of Figures; Preface; Acknowledgments;

Chapter 1. The General Description of a Rarefied Gas: 1. Some Introductory Remarks; 2. Density and Mean Motion; 3. The Distribution Function of Molecular Velocities; 4. Mean Values ofFunctions of Molecular Velocities; 5. Transport of Molecular Properties; 6. The Pressure Tensor; 7. The Hydrostatic Pressure; 8. The Amount of Heat; 9. The Kinetic Temperature; 10. The Equation of State for a Perfect Gas; 11. The Thermal Flux Vector; 12. Summary; Problems; References;

Chapter 2. The Boltzmann Equation; 1. Derivation of the Boltzmann Equation; 2. The Moment Equations; 3. Another Form of the Moment Equations; 4. The Equations for a Continuum Medium; 5. Molecular Encounters; 6. The Relative Motion of Two Molecules; Problems; References;

Chapter 3. The Collision Operator; 1. The Differential and Total Scattering Cross Sections; 2. The Statistics of Molecular Encounters; 3. The Transformation of Some Integrals; Problems; References;

Chapter 4. The Uniform Steady-State of a Gas; 1. The Boltzmann H-Theorem; 2. The Maxwellian Velocity Distribution; 3. The Mean Free Path of a Molecule; Problems; References;

Chapter 5. The Non-Uniform State for a Simple Gas; 1. Expansion in Powers of a Small Parameter; 2. The First Approximation; 3. A General Formal Solution for the Second Correction; 4. The Transformation of the Non-Homogeneous Term; 5. The Second Approximation; 6. The First-Order Chapman-Enskog Solution for Thermal Conduction; 7. The First-Order Chapman-Enskog Solution for Viscosity; 8. The Thermal Conductivity and Viscosity Coefficients; 9. The First-Order Approximation for Arbitrary Intermolecular Potential; 10. The Second-Order Approximation for Arbitrary Intermolecular Potential; Problems; References;

Chapter 6. Regimes of Rarefied Gas Flows; 1. The Knudsen Number; 2. A General Analysis of the Different Gas Flow Regimes; 3. The Boundary Conditions; 4. The Boundary Dispersion Kernel; 5. Features of the Boundary Conditions for Small Knudsen number; Problems; References;

Chapter 7. The Free-Molecular Regime; 1. The Free-Molecular Distribution Function; 2. The Force on a Particle in a Uniform Gas Flow; 3. Calculation ofMacroscopic Values in the Free-Molecular Regime; 4. Thermophoresis of Particles in the Free-Molecular Regime; 5. Condensation on a Spherical Droplet; 6. Non-Stationary Gas Flows; Problems; References; 

Chapter 8. Methods of Solution of Planar Problems; 1. Maxwell's Method; 2. Loyalka's Method; 3. The Half-Range Moment Method; 4. Features of the Boundary Conditions for the Moment Equations; 5. Solution of the Thermal-Creep Problem by the Half-Range Moment Method; 6. Influence of the Boundary Models on the Thermal-Creep Coefficient; Problems; References;

Chapter 9. The Variational Method for the Planar Geometry; 1. Another Form of the Boltzmann Equation; 2. The Variational Technique for the Slip-Flow Problem; 3. Discussion of the Slip-Flow Results; 4. The Variational Solution for the Thermal-Creep Problem; 5. Discussion of the Thermal-Creep Results; 6. Slip-Flow and Temperature-jump Coefficients for the Lennard-Jones (6-12); Potential Model; Problems; References;

Chapter 10. The Slip-Flow Regime; 1. Basic Equations; 2. The Spherical Drag Problem; 3. The Thermal Force Problem; Problems; References;

Chapter 11. Boundary Value Problems for All Knudsen Numbers; 1. The Moment Equations in Arbitrary Curvilinear Coordinates; 2. The Two-Sided Maxwellian Distribution Functions; 3. Moments of Discontinuous Distribution Functions; 4. Analytical Expressions for the Bracket Integrals; 5. Boundary Conditions for Moment Equations; 6. Thermal Conduction from a Heated Sphere; 7. Method of the 'Smoothed' Distribution Function; 8. The Polynomial Expansion Method; 9. Solution of One Classic Transport Problem. 10. A Simplification of Moment Systems for Curvilinear Problems. 11. The Torque Problem; Problems; References;

Chapter 12. Boundary Slip Phenomena in a Binary Gas Mixture; 1. The First-Order Chapman-Enskog Approximation for a Binary Gas Mixture; 2. The Transport Coefficients for a Binary Gas Mixture; 3. The Second-Order Chapman-Enskog Approximation for a Binary Gas Mixture; 4. Analytical Methods of Solution for Planar Boundary Value Problems Involving Binary Gas Mixtures; 5. The Slip Coefficients for a Binary Gas Mixture; 6. Discussion of the Slip Coefficient Results; Problems; References;

Appendix 1. Bracket Integrals for the Planar Geometry; 1. Bracket Integrals Involving Two Sonine Polynomials; 2. Bracket Integrals Containing Several Components of Molecular Velocity; 3. Bracket Integrals Containing Two Discontinuous Functions; 4. Bracket Integrals Containing One Discontinuous Function; References;

Appendix 2. Bracket Integrals for Curvilinear Geometries; 1. The Special Function of the First Kind for the Spherical Geometry; 2. The Special Function of the Second Kind for the Spherical Geometry; 3. The Special Function of the First Kind for the Cylindrical Geometry; 4. The Special Function ofthe Second Kind for the Cylindrical Geometry; 5. Approximate Expressions for the Special Functions; References;

Appendix 3. Bracket Integrals for Polynomial Expansion Method; 1. Calculation of the Bracket Integrals of the First Kind; 2. Analytical Expressions for the Bracket Integrals of the Second Kind; References;

Appendix 4. The Variational Principle for Planar Problems; 1. Some Definitions and Properties for Integral Operators; 2. The Variational Principle; References;

Appendix 5. Some Definite Integrals; 1. Some Frequently Encountered Integrals; 2. Some Integrals Encountered in Boundary Problems; 3. Some Integrals Connected with the Second-Order Chapman-Enskog Solution; 4. Some Integrals Connected with Non-Linear Transport Problems;

Appendix 6. Omega-Integrals for Second-Order Approximation;

References; Author Index ; Subject Index