Dynamics of thermal diffusion in a linear temperature field

Abstract

An exact solution to the nonlinear differential equation describing thermal diffusion (the Ludwig-Soret effect) for a binary mixture in a linear temperature field is given. The differential equation of motion for the components of the mixture is reduced to a heat diffusion equation with boundary conditions that act as unbounded sources which grow in time. The differential equation of motion is also solved in the limit where mass diffusion is neglected, showing that shocks are generated. For a temperature field of infinite extent, distributions originally localized in space move at long times with constant speed with self similar form.