We study the extended Stefan problem which includes constitutional supercooling for the solidification of a binary alloy in a finite spherical domain. We perform an asymptotic analysis in the limits of large Lewis number and small Stefan number which allows us to identify a number of spatio-temporal regimes signifying distinct behaviours in the solidification process, resulting in an intricate boundary layer structure. Our results generalise those present in the literature by considering all time regimes for the Stefan problem while also accounting for impurities and constitutional supercooling. These results also generalise recent work on the extended Stefan problem for finite planar domains to spherical domains, and we shall highlight key differences in the asymptotic solutions and the underlying boundary layer structure which result from this change in geometry. We compare our asymptotic solutions with both numerical simulations and real experimental data arising from the casting of molten metallurgical grade silicon through the water granulation process, with our analysis highlighting the role played by supercooling in the solidification of binary alloys appearing in such applications.

中文翻译：

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球体中二元合金凝固的扩展 Stefan 问题

我们研究了扩展的 Stefan 问题，其中包括在有限球域中二元合金凝固的组成过冷。我们在大 Lewis 数和小 Stefan 数的限制下进行了渐近分析，这使我们能够识别出许多表示凝固过程中不同行为的时空状态，从而产生复杂的边界层结构。我们的结果通过考虑 Stefan 问题的所有时间机制，同时考虑杂质和组成过冷，概括了文献中存在的那些。这些结果也概括了最近关于有限平面域到球形域的扩展 Stefan 问题的工作，我们将强调渐近解和由几何变化引起的底层边界层结构的关键差异。我们将我们的渐近解与数值模拟和通过水粒化过程铸造熔融冶金级硅产生的真实实验数据进行比较，我们的分析强调了过冷在此类应用中出现的二元合金凝固中所起的作用。