Mathematical analysis of thermal diffusion shock waves

Thermal diffusion, also known as the Ludwig-Soret effect, refers to the separation of mixtures in a temperature gradient. For a binary mixture the time dependence of the change in concentration of each species is governed by a nonlinear partial differential equation in space and time. Here, an exact solution of the Ludwig-Soret equation without mass diffusion for a sinusoidal temperature field is given. The solution shows that counterpropagating shock waves are produced which slow and eventually come to a halt. Expressions are found for the shock time for two limiting values of the starting density fraction. The effects of diffusion on the development of the concentration profile in time and space are found by numerical integration of the nonlinear differential equation.