Extinction behaviour for two–dimensional inward-solidification problems

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