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Approximation techniques for the Stefan problem Katz, Hart Victor

Abstract

The macroscopic description of matter undergoing a phase change (the Stefan Problem) can be formulated as a set of coupled, non-linear partial differential equations. For the case of one space dimension, the thesis develops three approximation methods to solve these equations. (a) Asymptotic Expansions With the Green's functions to suit the given boundary conditions, the system can be transformed into a set of integral equations. For the case where the initial phase grows without limit as т → ∞ ,the large т expansions for the integrals are calculated and the large т behaviour of the interphase boundary found.(b) Perturbation Expansions When the latent heat of fusion of a material is large relative to the heat content of that material, a uniformly valid perturbation expansion in a parameter related to the ratio of these heats is possible. The first few terms of the expansions for the temperature distribution and the position of the interphase boundary are calculated and found to be in good agreement with the few known exact solutions and numerically calculated solutions. (c) Numerical Techniques Rather than use a traditional finite difference scheme, only time is discretized and an analytic expression for an approximate temperature is found for each time step. This method gives good results for the temperatures over the time intervals required by the physics of the problem. Finally, the method is applied to describe the freezing of a shallow lake.

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