Rayleigh's Problem at Low Mach Number According to the Kinetic Theory of Gases

Abstract

Rayleigh's problem of an infinite flat plate set into uniform motion impulsively in its own plane is studied by using Grad's equations and boundary conditions developed from the kinetic theory of gases. For a heat insulated plate and a small impulsive velocity (low Mach number), only tangential shear stress and velocity and energy (heat) flow parallel to the plate are generated, while the pressure, density, and temperature of the gas remain unchanged. Moreover, no normal velocity, normal stress, or normal energy flow is developed. Near the start of the motion the flow behaves like a "free-molecule flow", and all physical quantities are analytic functions of the flow parameters and time. The results obtained for "large time", however, add to the growing lack of confidence in the Burnett-type series expansions in powers of mean free path. Although such expansions are obtained here, they are poorly convergent and inappropriate to the problem. To replace these unsatisfactory solutions, approximate closed-form solutions valid for all values of the time are developed, which agree with the free-molecule values for small time and the classical Rayleigh solution for large time. This technique may be useful in studying more general flow problems within the framework of the kinetic theory of gases.