The growth of small droplets from a gas mixture; Results and estimates for very small radii

To test and to complement the moment method for solving kinetic equations in the space outside of a sphere that absorbs one component out of a gas mixture, we develop a solution method especially suitable for radii smaller or not much larger than a mean free path. The method is used to calculate the stationary current towards an absorbing sphere suspended in a gas mixture that is in equilibrium far from the sphere. For the simplest kinetic equation, a variant of the linear BGK-equation, and for radii below two mean free paths, this procedure gives the current towards the sphere to within an accuracy of at most a few tenths of a percent. For radii around two mean free paths, the new results agree with those of a variant of the moment method; for lower radii discrepancies of up to a few percent are found. Our new method does not work for the Klein-Kramers equation, a kinetic equation for Brownian particles. For the latter equation we obtain an estimate for the current at very low radii from the free space solution by Uhlenbeck and Ornstein. Its numerical evaluation strongly indicates an analytic behavior for small radii different from the one found in the BGK-case.