Asymptotic solutions for the distribution function in non-equilibrium flows. Part 1. The weak shock

Abstract

This paper is the first part of an investigation of the molecular velocity distribution function in non-equilibrium flows. In this part, the general features of the distribution are discussed and illustrated by a detailed study of its asymptotic expansions in different velocity domains for a weak shock, employing a simple relaxation model for the collisions. Using the strength of the shock as a small parameter, the Chapman-Enskog distribution is derived, in a less restrictive way than in previous analyses, as the first two terms in a suitable 'inner' asymptotic expansion of the distribution in velocity space, valid only for not very high velocities. It is shown that the consideration of two further limits, called the intermediate and outer, is necessary for a complete description of the distribution in velocity space. The uniformly valid composite expansion demonstrates the slow approach to equilibrium of fast molecules. The outer solution depends on integrals over the flow and is in general 'global', in contrast to the inner solution which is essentially local; this introduces certain asymmetries on a fine scale even in a weak shock. It is shown, for example, that fast molecules moving towards the hot side accumulate by collisionless streaming, whereas those moving towards the cold side attenuate like a molecular beam and represent essentially a 'precursor' of the hot side. A simple approximation for the distribution in the precursor is derived, and found to contain, in the outer limit, a large perturbation on the local Maxwellian; this results in an approach to equilibrium like $\exp (-|x|^{\frac{2}{3}})$ on the cold side. A heuristic extension of the argument to the true Boltzmann equation leads to the result that for molecules with an interparticle potential varying as the inverse m-power of the distance, the approach to equilibrium through the precursor is like $\exp (-|x|^l)$, where l = m/(m + 2).