Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A note on the existence of a waiting time for a two-phase Stefan problem


Authors: Domingo Alberto Tarzia and Cristina Vilma Turner
Journal: Quart. Appl. Math. 50 (1992), 1-10
MSC: Primary 35R35; Secondary 35K05
DOI: https://doi.org/10.1090/qam/1146619
MathSciNet review: MR1146619
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Abstract: We consider a slab, represented by the interval $ 0 < x < {x_0}$, at the initial temperature $ {\theta _0} = {\theta _0}\left( x \right) \ge 0\left( {or {\phi _0} = {\phi _0}\left( x \right) \ge 0} \right)$ having a heat flux $ q = q\left( t \right) > 0$ (or convective boundary condition with a heat transfer coefficient $ h$) on the left face $ x = 0$ and a temperature condition $ b\left( t \right) > 0$ on the right face $ x = {x_0}$ ($ x_{0}$ could be also $ + \infty $, i.e., a semi-infinite material). We consider the corresponding heat conduction problem and assume that the phase-change temperature is $ {0^ \circ }C$.


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  • [1] D. G. Aronson, The porous medium equation, Nonlinear Diffusion Problems (A. Fasano and M. Primicerio, eds.), Lecture Notes in Math., Vol. 1224, Springer-Verlag, Berlin, 1986, pp. 1-46 MR 877986
  • [2] A. B. Bancora and D. A. Tarzia, On the Neumann solution for the two-phase Stefan problem including the density jump at the free boundary, Lat. Am. J. Heat Mass Transfer 9, 215-222 (1985)
  • [3] J. R. Cannon, The One-dimensional Heat Equation, Addison-Wesley, Menlo Park, California, 1967
  • [4] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1959 MR 959730
  • [5] A. Fasano and M. Primicerio, General free boundary problems for the heat equation. I, J. Math. Anal. Appl. 57, 694-723 (1977) MR 0487016
  • [6] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964 MR 0181836
  • [7] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, 1967 MR 0219861
  • [8] B. Sherman, General one-phase Stefan problems and free boundary problems for the heat equation with Cauchy data prescribed on the free boundary, SIAM J. Appl. Math. 20, 555-570 (1971) MR 0287193
  • [9] A. D. Solomon, V. Alexiades, and D. G. Wilson, The Stefan problem with a convective boundary condition, Quart. Appl. Math. 40, 203-217 (1982) MR 666675
  • [10] A. D. Solomon, D. G. Wilson, and V. Alexiades, Explicit solutions to change problems, Quart. Appl. Math. 41, 237-243 (1983) MR 719507
  • [11] D. A. Tarzia, Sobre el caso estacionario del problema de Stefan a dos fases, Math. Notae 28, 73-89 (1980)
  • [12] D. A. Tarzia, An inequality for the coefficient $ \sigma $ of the free boundary $ s\left( t \right) = 2\sigma \sqrt t $ of the Neumann solution for the two-phase Stefan problem, Quart. Appl. Math. 39, 491-497 (1981-82) MR 644103
  • [13] D. A. Tarzia, An inequality for the constant heat flux to obtain a steady-state two-phase Stefan problem, Engineering Analysis 5, 177-181 (1988). See also On heat flux in materials on free boundary problems: Theory and applications, Irsee/Bavaria, 11-20 June 1987, Res. Notes in Math., No. 186, Pitman, London, 1990, pp. 703-709.
  • [14] D. A. Tarzia, The two-phase Stefan problem and some related conduction problems, Reuniões em Matemática Aplicada e Computacão Científica, Vol. 5, SBMAC-Soc. Brasileira Mat. Apl. Comput., Gramado, 1987

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DOI: https://doi.org/10.1090/qam/1146619
Article copyright: © Copyright 1992 American Mathematical Society

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