On the initial-value problem in the kinetic theory of gases

Abstract

The relaxation of an initially non-uniform gas to equilibrium is studied within the framework of the kinetic theory of gases. The macroscopic gas properties are taken to depend on one spatial dimension as well as the time. The amplitude of the non-uniformity is assumed to be small with a length scale large compared with the mean free path, and the Krook model of the Boltzmann collision integral is employed.By applying multi-time scale perturbation methods to this reduced problem, uniformly valid analytical solutions for the macroscopic velocity, density and temperature are obtained. The macroscopic equations appropriate to each stage of the relaxation process are obtained in a straightforward and unambiguous manner. The distribution function obtained is shown to be a re-expansion of the Chapman–Enskog solution of the Krook equation, with additional terms accounting for the relaxation of the initial conditions to a near equilibrium form. The results indicate that the uniformly valid frst approximation to the macroscopic velocity, density and temperature can be obtained from the Navier–Stokes equations, but that no purely macroscopic set of equations will suffice for the determination of higher approximations.