Linear Theory — Small Reynolds Numbers

Abstract

From this chapter, we start the discussion of the main topics of this monograph, i.e., the relation between kinetic theory and fluid dynamics. When a system, subject to no external force, deviates slightly from an equilibrium state at rest, the linearized Boltzmann equation (2.82) is often used to analyze the system. In this chapter, we discuss the asymptotic behavior of the solution for small Knudsen numbers of a time-independent boundary-value problem in a general domain of the linearized Boltzmann equation. Both kinds of boundaries, a simple boundary and a boundary made of the condensed phase of the gas, are considered. First, we derive the solution expressing the overall behavior of the gas in the form of a power series of the Knudsen number under the assumption that the length scale of variation of the velocity distribution function is the geometric characteristic length of the system. Its solution is reduced to solving a series of fluid-dynamic-type sets of equations (Stokes sets of equations in the linearized problem) for the component functions of the expansion of the macroscopic variables: density (or pressure), velocity, and temperature. The solution, however, cannot generally be made to satisfy the kinetic boundary condition. Thus, its correction in the neighborhood of the boundary (Knudsen-layer correction) is introduced.