The influence of temperature gradients on the kinetic boundary layer problem for a condensing droplet

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state; the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.