Extinction behaviour for two-dimensional inward-solidification problems

The problem of the inward solidification of a two-dimensional region of fluid is considered, it being assumed that the liquid is initially at its fusion temperature and that heat flows by conduction only. The resulting one-phase Stefan problem is reformulated using the Baiocchi transform and is examined using matched asymptotic expansions under the assumption that the Stefan number is large. Analysis on the first time-scale reveals the liquid-solid free boundary becomes elliptic in shape at times just before complete freezing. However, as with the radially symmetric case considered previously, this analysis leads to an unphysical singularity in the final temperature distribution. A second time-scale therefore needs to be considered, and it is shown the free boundary retains it shape until another non-uniformity is formed. Finally, a third (exponentionally-short) time-scale, which also describes the generic extinction behaviour for all Stefan numbers, is needed to resolve the non-uniformity. By matching between the last two time-scales we are able to determine a uniformly valid description of the temperature field and the location of the free boundary at times just before extinction. Recipes for computing the time it takes to completely freeze the body and the location at which the final freezing occurs are also derived