Kinetic Theory and Fluid Dynamics

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.