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Type: Journal article
Title: Classical two-phase Stefan problem for spheres
Author: McCue, S.
Wu, B.
Hill, J.
Citation: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008; 464(2096):2055-2076
Publisher: Royal Soc London
Issue Date: 2008
ISSN: 1364-5021
Statement of
Scott W. Mccue, Bisheng Wu and James M. Hill
Abstract: The classical Stefan problem for freezing (or melting) a sphere is usually treated by assuming that the sphere is initially at the fusion temperature, so that heat flows in one phase only. Even in this idealized case there is no (known) exact solution, and the only way to obtain meaningful results is through numerical or approximate means. In this study, the full two-phase problem is considered, and in particular, attention is given to the large Stefan number limit. By applying the method of matched asymptotic expansions, the temperature in both the phases is shown to depend algebraically on the inverse Stefan number on the first time scale, but at later times the two phases essentially decouple, with the inner core contributing only exponentially small terms to the location of the solid–melt interface. This analysis is complemented by applying a small-time perturbation scheme and by presenting numerical results calculated using an enthalpy method. The limits of zero Stefan number and slow diffusion in the inner core are also noted.
Keywords: two-phase Stefan problem
large Stefan number expansion
formal asymptotics
small-time behaviour
Rights: © 2008 The Royal Society
DOI: 10.1098/rspa.2007.0315
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Mathematical Sciences publications

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