The Linearized Boltzmann Equation

Abstract

The Hilbert and Chapman-Enskog methods are perturbation procedures for solving the Boltzmann equation on the basis of the assumption of a small Knudsen number ; other procedures based on the assumption of a large Knudsen number will briefly be described later (Chapter VIII, Section 3). The above two procedures are valid in the so-called near-continuum (or slip) regime (Kn → 0) and in nearly-free regime (Kn → ∞). They are both based upon a specific assumption on the order of magnitude of the Knudsen number. Accordingly, the intermediate regime (the so-called transition region) remains untouched by the above procedures because it cannot be described in terms of either a higher-order continuum theory or of small corrections to a picture of essentially noninteracting particles. A treatment of the transition regime requires the full use of the Boltzmann equation (or, at least, sufficiently accurate models of the latter). As a consequence, if we want to investigate the transition regime, we have either to give up the idea of using perturbation methods, or look for some other parameter, different from the Knudsen number, to be regarded as small under suitable conditions.