Kinetic Theory and Fluid Dynamics

This monograph is intended to provide a comprehensive description of the rela tion between kinetic theory and fluid dynamics for a time-independent behavior of a gas in a general domain. A gas in a steady (or time-independent) state in a general domain is considered, and its asymptotic behavior for small Knudsen numbers is studied on the basis of kinetic theory. Fluid-dynamic-type equations and their associated boundary conditions, together with their Knudsen-layer corrections, describing the asymptotic behavior of the gas for small Knudsen numbers are presented. In addition, various interesting physical phenomena derived from the asymptotic theory are explained. The background of the asymptotic studies is explained in Chapter 1, accord ing to which the fluid-dynamic-type equations that describe the behavior of a gas in the continuum limit are to be studied carefully. Their detailed studies depending on physical situations are treated in the following chapters. What is striking is that the classical gas dynamic system is incomplete to describe the behavior of a gas in the continuum limit (or in the limit that the mean free path of the gas molecules vanishes). Thanks to the asymptotic theory, problems for a slightly rarefied gas can be treated with the same ease as the corresponding classical fluid-dynamic problems. In a rarefied gas, a temperature field is di rectly related to a gas flow, and there are various interesting phenomena which cannot be found in a gas in the continuum limit.

1 Introduction. - 2 Boltzmann Equation. - 2. 1 Velocity distribution function and macroscopic variables. - 2. 2 Boltzmann equation. - 2. 3 Conservation equations. - 2. 4 Maxwell distribution (Equilibrium distribution). - 2. 5 Mean free path. - 2. 6 Boundary condition. - 2. 7 H theorem. - 2. 8 Model equation. - 2. 9 Nondimensional expressions I. - 2. 10 Nondimensional expressions II. - 2. 11 Linearized Boltzmann equation. - 2. 12 Boltzmann equation in the cylindrical and spherical coordinate systems. - 3 Linear Theory Small Reynolds Numbers. - 3. 1 Problem. - 3. 2 Grad Hilbert solution and fluid-dynamic-type equations. - 3. 3 Stress tensor and heat-flow vector of the Grad Hilbert solution. - 3. 4 Analysis of the Knudsen layer. - 3. 5 Slip condition and Knudsen-layer correction. - 3. 6 Determination of macroscopic variables. - 3. 7 Discontinuity of the velocity distribution function and S layer. . - 3. 8 Force and mass and energy transfers on a closed body. - 3. 9 Viscosity and thermal conductivity. - 3. 10 Summary of the asymptotic theory. - 3. 11 Applications. - 4 Weakly Nonlinear Theory Finite Reynolds Numbers. - 4. 1 Problem. - 4. 2 S solution. - 4. 3 Fluid-dynamic-type equations. - 4. 4 Knudsen-layer analysis. - 4. 5 Slip condition and Knudsen layer. - 4. 6 Determination of macroscopic variables. - 4. 7 Rarefaction effect. - 4. 8 Force and mass and energy transfers on a closed body. - 4. 9 Summary of the asymptotic theory and a comment on a time-dependent problem. - 4. 10 Applications. - 5 Nonlinear Theory I Finite Temperature Variations and Ghost Effect. - 5. 1 Problem. - 5. 2 SB solution. - 5. 3 Fluid-dynamic-type equations. - 5. 4 Knudsen layer and slip condition. - 5. 5 Determination of macroscopic variables. - 5. 6 Ghost effect: Incompleteness of the system of the classical gas dynamics. - 5. 7 Half-space problem of evaporationand condensation. - 6 Nonlinear Theory II - Flow with a Finite Mach Number around a Simple Boundary. - 6. 1 Problem. - 6. 2 Hilbert solution. - 6. 3 Viscous boundary-layer solution. - 6. 4 Knudsen-layer solution and slip condition. - 6. 5 Connection of Hilbert and viscous boundary-layer solutions. . - 6. 6 Recipe for construction of solution. - 6. 7 Discussions. - 7 Nonlinear Theory III Finite Speed of Evaporation and Condensation. - 7. 1 Problem. - 7. 2 Hilbert solution. - 7. 3 Knudsen layer. - 7. 4 Half-space problem of evaporation and condensation. - 7. 5 System of equations and boundary conditions in the continuum limit. - 7. 6 Generalized kinetic boundary condition. - 7. 7 Boundary-condition functions $$
h_1 \left( {M_n } \right), h_2 \left( {M_n } \right), F_s \left( {M_n , \overline M _t , {T \mathord{\left/
{\vphantom {T {T_w }}} \right.
\kern-\nulldelimiterspace} {T_w }}} \right)
$$ and $$
F_b \left( {M_n , \overline M _t , {T \mathord{\left/
{\vphantom {T {T_w }}} \right.
\kern-\nulldelimiterspace} {T_w }}} \right)
$$. - 7. 8 Applications. - 8 Bifurcation of Cylindrical Couette Flow with Evaporation. - 8. 1 Problem. - 8. 2 Solution type I. - 8. 3 Solution type II. - 8. 4 Bifurcation diagram and transition solution. - 8. 5 Discussions for the other parameter range. - 8. 6 Concluding remark and supplementary comment. - A Supplementary Explanations and Formulas. - A. 1 Formal derivation of the Boltzmann equation from the Liouville equation. - A. 3 Derivation of the Stokes set of equations. - A. 4 Golse s theorem on a one-way flow. - A. 6 Viscosity and thermal conductivity. - A. 9 Equation for the Knudsen layer and Bardos s theorem. - A. 10 The boundary condition for the linearized Euler set of equations. - B Spherically Symmetric Field of Symmetric Tensor. - B. 1 Problem. - B. 3. 1 Preparation. - B. 3. 3 Summary. - B. 4Applications. - B. 4. 2 Axially symmetric field. - C Kinetic-Equation Approach to Fluid-Dynamic Equations. - C. 1 Introduction. - C. 2 Exact kinetic-equation approach. - C. 3 Discussion on numerical systems.