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Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary

Published online by Cambridge University Press:  17 February 2009

J. N. Dewynne ,
S. D. Howison ,
J. R. Ockendon  and
Weiqing Xie
J. N. Dewynne
Affiliation:
Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, U. K.
S. D. Howison
Affiliation:
Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, U. K.
J. R. Ockendon
Affiliation:
Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, U. K.
Weiqing Xie
Affiliation:
Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, U. K.
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Abstract

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We study the large time behaviour of the free boundary for a one-phase Stefan problem with supercooling and a kinetic condition u = −ε|⋅ṡ| at the free boundary x = s(t). The problem is posed on the semi-infinite strip [0,∞) with unit Stefan number and bounded initial temperature ϕ(x) ≤ 0, such that ϕ → −1 − δ as x → ∞, where δ is constant. Special solutions and the asymptotic behaviour of the free boundary are considered for the cases ε ≥ 0 with δ negative, positive and zero, respectively. We show that, for ε > 0, the free boundary is asymptotic to , δt/ε if < δ > 0 respectively, and that when δ = 0 the large time behaviour of the free boundary depends more sensitively on the initial temperature. We also give a brief summary of the corresponding results for a radially symmetric spherical crystal with kinetic undercooling and Gibbs-Thomson conditions at the free boundary.

Type
Research Article
Information
The ANZIAM Journal , Volume 31 , Issue 1 , July 1989 , pp. 81 - 96
Copyright
Copyright © Australian Mathematical Society 1989

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