Singular behaviour of a rarefied gas on a planar boundary

Abstract

Singular behaviour of a rarefied gas on a planar boundary is clarified on the basis of the Boltzmann equation. The thermal transpiration between two parallel plates is taken as a specific example. First, the flow velocity is shown to behave like xlnx in the vicinity of the boundary, where x is a distance from the boundary. This implies a logarithmic divergence of the flow velocity gradient as x→0. Then, such a spatial singularity is shown to induce a similar singularity of the velocity distribution function (VDF) with respect to ζn on the boundary, where ζn is a normal component of the molecular velocity to the boundary. Moreover, the spatial singularity is shown to be quantitatively related to the discontinuity of the VDF on the boundary at ζn=0. These macroscopic and microscopic singularities should be observed generally in a rarefied gas on a planar boundary.