Instability in the self-similar motion of a planar solidification front

Understanding the solidification process of a binary alloy is important if one is to control the microstructure obtained during the casting of metals. Whilst much work has been done on the steady-state solidification problem, despite their relevance to metallurgical applications, there is less known about non-steady solidification problems and their stability. In the paper we shall consider the non-steady solidification problem in which the planar solidification front moves in a self-similar manner, in both infinite and semi-infinite planar 1D geometries. For each geometry exact solutions are known for the resulting Stefan problem. We direct our attention to the stability of each solution, demonstrating that whilst the concentration and thermal solutions remain stable, the interface corresponding to the solidification front can develop instabilities. For each geometry, we find that there are always unstable perturbations, although we observe qualitative differences in the form of the unstable perturbations for each case. These results generalize and extend several existing studies in the literature, and throw light on the instability inherent in the non-steady solidification process.