In this chapter we illustrate some of the ideas of radiation transport and hydrodynamics coupled with radiation transport by means of a small selection of examples. As described in the introduction, the challenging applications of the theory are left for the technical literature, and the problems presented here have been chosen for their simplicity or pedagogical value.

Marshak wave and evaporation fronts

The classic example of nonlinear radiation diffusion is the Marshak wave, first discussed by Marshak (1958). It is a self-similar thermal wave, treated with the thermal diffusion approximation, for a material with a constant specific heat and for which the Rosseland mean opacity varies as a power of the temperature. Hydrodynamic motion is ignored. This assumption is unrealistic, but is made for simplicity. This “thermal wave” is not a wave in the sense we used earlier; it does not come from a hyperbolic system of PDEs, and the dispersion relation does not yield wave speeds ω/k, etc. It is a wave in the sense that there is a characteristic structure, in this case a sharp temperature front, that moves through the material in the course of time, of which the shape remains fairly constant. The propagation law is not distance α time, as expected for a hyperbolic system, but distance α time½ instead, owing to its diffusion nature.

A thorough discussion of how the thermal diffusion solution to this problem compares with transport solutions is given in Mihalas and Mihalas (1984), Section 103.