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.

"Yoshio Sone is certainly one of the world leading experts for continuum limits of the Boltzmann equation and with this book he summarizes the extensive work of his group on this subject. The text is very carefully written and even if the expansions and equations are lengthy, the results are always interpreted and discussed. ... Altogether, the book is a rich source on the theory of kinetic boundary layers in the continuum limit and it is certainly valuable for those working in fields where kinetic theory or fluid dynamics is important." -ZAMP "This book gives an impressive overview of the formidable amount of work done by the author ... [It] is very detailed, and very carefully written, aimed at a great generality of results and physical situations ... Interesting physical phenomena and engineering applications are presented throughout ... supported by numerical analysis and sometimes even by experimental demonstration. All of that makes the book appealing in various fields, including applied mathematics, physics, and engineering, especially at the level of graduate students and young researchers." -European Journal of Mechanics, B/Fluids "This book will be a useful and lasting reference point for kinetic boundary layers in the continuum limit. It isn't reading for the faint at heart-the technical skill and fearlessness of lengthy expansions of the author are visible. Yet this is a collection of a lifetime of research, conveniently collected in one place." -Mathematical Reviews "This is a comprehensive study on the relationship between...classical (continuum) fluid dynamics and...kinetic theory. In particular, the incompleteness of classical fluid dynamics in describing the time evolution of real gases is also discussed. The fluid dynamics equations are derived by the methods of asymptotic analysis from the Boltzmann system. All equations resulting from this process are classified with respect to their physical context. Applications are given in order to illustrate various interesting physical phenomena: the flows induced by temperature fields, evaporation and condensation problems, examples of the so-called ghost effect, and bifurcations of flows. In general, many applications are discussed without a sophisticated mathematical apparatus not easily accessible to engineers. On the other hand, experimental work is examined to supplement the theory, and mathematicians will certainly benefit from [the] clarity of definitions and precise physical description[s]... The book is intended for theoretical physicists, applied mathematicians, engineers, and students in mathematics." -Applications of Mathematics "The book written by Sone is in many respects excellent. It is very well structured, e.g. most chapters start with the definition of a problem, followed by the method of solution, a summary of the results and finally a number of applications. The calculations and interpretations are extremely accurate and reliable. The book makes an impression of completeness in the sense that the topics treated are presented in an exhaustive manner and the references are abundant... The book is no doubt an acquisition to the literature in the field of Kinetic Theory and Fluid Dynamics. It will prove to be an important tool for many graduate students and researchers in Applied, Mathematics, Physics and, since it also contains many engineering applications, Mechanical Engineering." -Physica B