Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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An inequality for the coefficient $ \sigma $ of the free boundary $ s(t)=2\sigma \sqrt {t}$ of the Neumann solution for the two-phase Stefan problem


Author: Domingo Alberto Tarzia
Journal: Quart. Appl. Math. 39 (1982), 491-497
MSC: Primary 80A20; Secondary 35K20
DOI: https://doi.org/10.1090/qam/644103
MathSciNet review: 644103
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Abstract: We consider a semi-infinite body (e.g. ice), represented by $ \left( {0, + \infty } \right)$, with an initial temperature $ - c < 0$ having a heat flux $ h\left( t \right) = - {h_0}/\sqrt t \left( {{h_0} > 0} \right)$ in the fixed face $ x = 0$. If $ {h_0} > c{k_1}/\sqrt {\pi {a_1}} $ there exists a solution, of Neumann type, for the resulting two-phase Stefan problem. If we connect it with the Neumann problem (on $ x = 0$ the body has a temperature $ b > 0$ we obtain the inequality erf $ \left( {\sigma /{a_2}} \right) < \left( {{k_2}b{a_1}/{k_1}c{a_2}} \right)$ for the coefficient $ \sigma $ of the free boundary $ s\left( t \right) = 2\sigma \sqrt t $, where $ {k_i}$, and $ a_i^2$ are respectively the thermal conductivity and thermal diffusivity coefficients of the corresponding $ i$ phase $ i = 1:$ solid phase, $ i = 2:$ liquid phase). If $ {h_0} < c{k_1}/\sqrt {\pi {a_1}} $ there is no solution of the initial problem and if $ {h_0} = c{k_1}/\sqrt {\pi {a_1}} $ the problem has no physical meaning and corresponds to the case where the latent heat of fusion $ L$ tends to infinity.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/qam/644103
Article copyright: © Copyright 1982 American Mathematical Society


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