On Stefan’s problem for spheres

Abstract
The problem of establishing an asymptotic theory to describe the inward solidification of a sphere, initially molten and at the fusion temperature, has attracted much attention recently. Notably, Riley, Smith & Poots (1974) consider the parameter λ, the ratio of latent heat to sensible heat of the substance to be large and present a two-layer analysis to describe the final temperature profile and to determine t * E , the time to complete freezing. However, the analysis breaks down in a small region just before the centre freezes, that is, for times t * E — t * = O (λ -1 e -λ½ ). This paper provides an asymptotic theory for large λ which adequately describes the final temperature profile. Moreover, the first four terms of the asymptotic expansion of the time to complete freezing have been determined. Our results may be compared with numerical studies of phase degeneration (see, for example, Tao 1967) and could also be useful in helping with questions of uniqueness.

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