Shock wave structure in highly rarefied flows

Abstract

The Boltzmann equation is written in terms of two functions associated with the gain and loss of a certain type of molecule due to collisions. Its integral form is then applied to the problem of normal shock structure, and an iteration technique is used to determine the solution. The first approximation to the velocity distribution function of the Chapman-Enskog sequence, which leads to the Navier-Stokes equations, is used to initiate the iteration scheme. Expressions for the distribution function and the flow parameters pertinent to the first iteration are derived and show that the B-G-K model results can be obtained as a special case. This model is found to be valid in the continuum regime only, and is consequently limited to the study of strong shocks. In the present treatment the iteration is carried out on the distribution function and the analysis indicates that the method is equally valid for variations in both Mach and Knudsen numbers. Finally, the results of the first approximation are simplified, and expressed in a form suitable for numerical computation, and the range of their validity is discussed. The method should be equally suitable for other flow problems of linear or nonlinear nature.