THE ANALYTIC STRUCTURE OF THE STATIONARY KINETIC BOUNDARY-LAYER FOR BROWNIAN PARTICLES NEAR AN ABSORBING WALL

A method for solving kinetic boundary layer problems for the (1D) Klein-Kramers equation, the kinetic equation for Brownian particles, was proposed a few years ago by Marshall and Watson for the special case of particles moving under the influence of a constant or zero external force. In this paper we apply this method to a number of classical stationary boundary layer problems for completely or partially absorbing walls. These are the Milne problem, in which a current flows towards the wall from infinity, and the albedo problem, in which particles are injected into the system at the wall with a prescribed velocity distribution. The solutions are known to be non-analytic at the wall for zero normal velocity; we pay particular attention to the nature of this singularity for several special cases. The results are compared with results obtained by approximate methods. The latter typically provide quicker and more accurate results for the distribution away from the singularity, due to the slow convergence of the series encountered in the Marshall-Watson approach. The latter is clearly superior, however, for determining the nature of the singularity and for calculating distribution functions and density profiles in its vicinity.