Unsteady-state slip of a gas near an infinite plane with diffusion-mirror reflection of molecules

Abstract

On the basis of a “two-point” approximation for the distribution function, an approximate analytical solution is obtained to the Rayleigh problem describing (for an arbitrary moment of time) the unsteady-state slip of a rarefied gas near a surface with diffusion-mirror reflection of molecules. In the solution of the problem it was postulated that the characteristic value of the macroscopic velocity of the gas is small in comparison with the speed of sound. An approximate analogy is established with the propagation of free vibrations in an electrical line of infinite length. An investigation is made of the limiting transition σ→ 0 (σ is the fraction of mirror-reflected particles) in an exact solution of the Boltzmann problem. It is shown that, with σ ≪ 1, the rate of slip for any given moment of time is determined from the hydrodynamic equations of motion. An investigation is made of the nonsingularity of the limiting transitions (t→∞, σ→ 0; σ→ 0, t→ ∞) with determination of the rate of thermal slip.