Kinetic Theory and Fluid Dynamics

1. Front Matter

   Pages i-xi

   PDF

2. Introduction

   • Yoshio Sone

   Pages 1-4

3. Boltzmann Equation

   • Yoshio Sone

   Pages 5-26

4. Linear Theory — Small Reynolds Numbers

   • Yoshio Sone

   Pages 27-83

5. Weakly Nonlinear Theory — Finite Reynolds Numbers

   • Yoshio Sone

   Pages 85-121

6. Nonlinear Theory I — Finite Temperature Variations and Ghost Effect

   • Yoshio Sone

   Pages 123-165

7. Nonlinear Theory II — Flow with a Finite Mach Number around a Simple Boundary

   • Yoshio Sone

   Pages 167-201

8. Nonlinear Theory III — Finite Speed of Evaporation and Condensation

   • Yoshio Sone

   Pages 203-233

9. Bifurcation of Cylindrical Couette Flow with Evaporation and Condensation

   • Yoshio Sone

   Pages 235-255

10. Back Matter

    Pages 257-353

    PDF

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.

• Analysis
• Boltzmann equation
• Theoretical physics
• fluid dynamics
• kinetic theory
• model
• molecular gas dynamics
• rarefied gas (dynamics)
• statistical mechanics
• partial differential equations

"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 classifiedwith 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

• Kyoto University, Sakyo-ku Kyoto, Japan

  Yoshio Sone

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