The extinction problem for three-dimensional inward solidification
The one-phase Stefan problem for the inward solidification of a three-dimensional body of liquid that is initially at its fusion temperature is considered. In particular, the shape and speed of the solid-melt interface is described at times just before complete freezing takes place, as is the temperature field in the vicinity of the extinction point. This is accomplished for general Stefan numbers by employing the Baiocchi transform. Other previous results for this problem are confirmed, for example the asymptotic analysis reveals the interface ultimately approaches an ellipsoid in shape, and furthermore, the accuracy of these results is improved. The results are arbitrary up to constants of integration that depend physically on both the Stefan number and the shape of the fixed boundary of the liquid region. In general it is not possible to determine this dependence analytically; however, the limiting case of large Stefan number provides an exception. For this limit a rather complete asymptotic picture is presented, and a recipe for the time it takes for complete freezing to occur is derived. The results presented here for fully three-dimensional domains complement and extend those given by McCue et al. [Proc. R. Soc. London A 459 (2003) 977], which are for two dimensions only, and for which a significantly different time dependence occurs.
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|Item Type:||Journal Article|
|Keywords:||Stefan problem, extinction problem, inward solidification, matched asymptotic expansions, large Stefan number|
|Subjects:||Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > OTHER MATHEMATICAL SCIENCES (019900) > Mathematical Sciences not elsewhere classified (019999)|
|Divisions:||Past > QUT Faculties & Divisions > Faculty of Science and Technology|
|Copyright Owner:||Copyright 2005 Springer|
|Copyright Statement:||The original publication is available at SpringerLink http://www.springerlink.com|
|Deposited On:||21 Jul 2008 00:00|
|Last Modified:||09 Jun 2010 13:01|
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