Annali di Matematica Pura ed Applicata

, Volume 146, Issue 1, pp 97–122 | Cite as

Stefan problem with a kinetic condition at the free boundary

  • A. Visintin
Article
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Summary

In the one-dimensional two-phase Stefan problem, the standard equilibrium condition θ=0 at the free boundary x=s(t) is replaced by the kinetic law
$$s'(t) = \beta \left( {\theta \left( {s\left( t \right),t} \right)} \right);$$
(1)
here β is a continuous and increasing functionR→R and β(0)=0. This introduces supercooled and superheated states. Existence of at least one solution is proved. Then (1) is replaced by
$$s'_\varepsilon (t): = \beta \left( {\theta _\varepsilon \left( {s_\varepsilon \left( t \right),t} \right)} \right) \left( {\varepsilon constant< 0} \right),$$
(2)
and it is shown that as ɛ → 0+ a subsequence of the corresponding solutions (θ ε ,s ε ) converges to a solution (θ, s) of the reduced problem, which is characterised by the free boundary condition
$$\beta \left( {\theta \left( {s\left( t \right),t} \right)} \right) = 0.$$
(3)
Then the case of a radially symmetric multidimensional system is dealt with, taking also account of the surface tension effect. Denoting by s(t) the radial co-ordinate of the free boundary, the following linearized kinetics is considered for a water ball surrounded by ice
$$ls'(t) + \frac{\lambda }{{s(t)}} = \theta \left( {s\left( t \right),t} \right), where s(t)< 0.$$
(4)
An existence result is proved for the problem obtained by coupling (4) with the heat equation.

Keywords

Boundary Condition Surface Tension Equilibrium Condition Water Ball Free Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1985

Authors and Affiliations

  • A. Visintin
    • 1
  1. 1.Istituto di Analisi Numerica del C.N.R.PaviaItalia

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