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 ${0^ \circ }C$.
- D. G. Aronson, The porous medium equation, Nonlinear diffusion problems (Montecatini Terme, 1985) Lecture Notes in Math., vol. 1224, Springer, Berlin, 1986, pp. 1–46. MR 877986, DOI https://doi.org/10.1007/BFb0072687
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)
J. R. Cannon, The One-dimensional Heat Equation, Addison-Wesley, Menlo Park, California, 1967
- H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, 2nd ed., Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988. MR 959730
- A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. I, J. Math. Anal. Appl. 57 (1977), no. 3, 694–723. MR 487016, DOI https://doi.org/10.1016/0022-247X%2877%2990256-6
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- 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 (1971), 555–570. MR 287193, DOI https://doi.org/10.1137/0120058
- A. D. Solomon, V. Alexiades, and D. G. Wilson, The Stefan problem with a convective boundary condition, Quart. Appl. Math. 40 (1982/83), no. 2, 203–217. MR 666675, DOI https://doi.org/10.1090/S0033-569X-1982-0666675-6
- A. D. Solomon, D. G. Wilson, and V. Alexiades, Explicit solutions to phase change problems, Quart. Appl. Math. 41 (1983/84), no. 2, 237–243. MR 719507, DOI https://doi.org/10.1090/S0033-569X-1983-0719507-5
D. A. Tarzia, Sobre el caso estacionario del problema de Stefan a dos fases, Math. Notae 28, 73–89 (1980)
- Domingo Alberto Tarzia, 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, Quart. Appl. Math. 39 (1981/82), no. 4, 491–497. MR 644103, DOI https://doi.org/10.1090/S0033-569X-1982-0644103-2
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.
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
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
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)
J. R. Cannon, The One-dimensional Heat Equation, Addison-Wesley, Menlo Park, California, 1967
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1959
A. Fasano and M. Primicerio, General free boundary problems for the heat equation. I, J. Math. Anal. Appl. 57, 694–723 (1977)
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, 1967
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)
A. D. Solomon, V. Alexiades, and D. G. Wilson, The Stefan problem with a convective boundary condition, Quart. Appl. Math. 40, 203–217 (1982)
A. D. Solomon, D. G. Wilson, and V. Alexiades, Explicit solutions to change problems, Quart. Appl. Math. 41, 237–243 (1983)
D. A. Tarzia, Sobre el caso estacionario del problema de Stefan a dos fases, Math. Notae 28, 73–89 (1980)
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)
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.
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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Article copyright:
© Copyright 1992
American Mathematical Society