Extinction behaviour for two–dimensional inward-solidification problems
Extinction behaviour for two–dimensional inward-solidification problems
复制标题
二维向内凝固问题的消光行为
DOI：
10.1098/rspa.2002.1059
复制
发表时间：
2003
期刊：
影响因子：
--
通讯作者：
D. Riley
中科院分区：
文献类型：
--
作者：
S. McCue;J. King;D. Riley
考虑流体二维区域的向内凝固问题，假设液体最初处于其熔化温度并且热量仅通过传导流动。由此产生的单相 Stefan 问题使用 Baiocchi 变换重新表述，并在 Stefan 数很大的假设下使用匹配渐近展开进行检查。对第一时间尺度的分析表明，液固自由边界在完全冻结之前有时会变成椭圆形。然而，与之前考虑的径向对称情况一样，该分析导致最终温度分布出现非物理奇点。因此需要考虑第二个时间尺度，并且结果表明，自由边界保持其形状，直到形成另一个不均匀性。最后，需要第三个（指数短的）时间尺度来解决不均匀性，它也描述了所有斯特凡数的通用灭绝行为。通过最后两个时间尺度之间的匹配，我们能够确定温度场的统一有效描述以及灭绝前有时自由边界的位置。还导出了计算完全冻结身体所需的时间以及最终冻结发生的位置的方法。
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 that 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 that the free boundary retains its shape until another non–uniformity is formed. Finally, a third (exponentially 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.