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Extended Stefan problem for the solidification of binary alloys in a sphere. (English) Zbl 1504.80003

Summary: 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.

MSC:

80A22 Stefan problems, phase changes, etc.
35R37 Moving boundary problems for PDEs
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
80A19 Diffusive and convective heat and mass transfer, heat flow

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