Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity

Natale, María F.; Tarzia, Domingo A.

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 79-99 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity with a constant temperature or a heat flux condition of the type $-q_{0}/\sqrt{t}$ ( $q_{0}>0$ ) at the fixed face $x=0$ . We obtain in both cases sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for $t\geq t_{0}>0$ with $t_{0}$ an arbitrary positive time. These explicit solutions are obtained through the unique solution of an integral equation with the time as a parameter.

Si studia un problema di Stefan a una fase per un materiale semi-infinito con un coefficiente di conduttività termica dipendente dalla temperatura e con una condizione di temperatura costante o un flusso di calore del tipo $-q_{0}/\sqrt{t}$ ( $q_{0}>0$ ) sulla faccia fissa $x=0$ . Si ottengono, in entrambi i casi, condizioni sufficienti per i dati in modo da avere una rappresentazione parametrica della soluzione di tipo similarità per $t\geq t_{0}>0$ con $t_{0}$ un tempo positivo arbitrario. Queste soluzioni esplicite sono ottenute attraverso l’unica soluzione di una equazione integrale dove il tempo è un parametro.

Publié le : 2006-02-01

MR 2204902 | Zbl 1118.80005

@article{BUMI_2006_8_9B_1_79_0,
     author = {Mar\'\i a F. Natale and Domingo A. Tarzia},
     title = {Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {9-A},
     year = {2006},
     pages = {79-99},
     zbl = {1118.80005},
     mrnumber = {2204902},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2006_8_9B_1_79_0}
}

Natale, María F.; Tarzia, Domingo A. Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity. Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006) pp. 79-99. http://gdmltest.u-ga.fr/item/BUMI_2006_8_9B_1_79_0/

Bibliographie

[1] Alexiades, V. - Solomon, A. D., Mathematical modeling of melting and freezing processes, Hemisphere - Taylor & Francis, Washington (1983).

[2] I. Athanasopoulos - G. Makrakis - J. F. Rodrigues (EDS.), Free Boundary Problems: Theory and Applications, CRC Press, Boca Raton (1999). | MR 1702131

[3] Barber, J. R., An asymptotic solution for short-time transient heat conductionbetween two similar contacting bodies, Int. J. Heat Mass Transfer, 32, No 5 (1989),943-949.

[4] Barry, D. A. - Sander, G. C., Exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption, Water Resources Research, 27, No 10 (1991), 2667-2680.

[5] Bluman, G. - Kumei, S., On the remarkable nonlinear diffusion equation, J. Math. Phys., 21 (1980), 1019-1023. | MR 574874 | Zbl 0448.35027

[6] Briozzo, A. C. - Natale, M. F. - Tarzia, D. A., Determination of unknown thermal coefficients for Storm’s-type materials through a phase-change process, Int. J.Non-Linear Mech., 34 (1999), 324-340. | MR 1658930

[7] Broadbridge, P., Non-integrability of non-linear diffusion-convection equationsin two spatial dimensions, J. Phys. A: Math. Gen., 19 (1986), 1245-1257. | MR 844454 | Zbl 0617.35122

[8] Broadbridge, P., Integrable forms of the one-dimensional flow equation for unsaturated heterogeneous porous media, J. Math. Phys., 29 (1988), 622-627. | MR 931465 | Zbl 0651.76041

[9] Cannon, J. R., The one-dimensional heat equation, Addison - Wesley, Menlo Park (1984). | MR 747979 | Zbl 0567.35001

[10] Carslaw, H. S. - Jaeger, J. C., Conduction of heat in solids, Oxford University Press, London (1959). | MR 959730 | Zbl 0972.80500

[11] J. M. Chadam - H. Rasmussen H. (EDS.), Free boundary problems involving solids, Pitman Research Notes in Mathematics Series 281, Longman, Essex (1993). | MR 1216392

[12] Coelho Pinheiro, M. N., Liquid phase mass transfer coefficients for bubbles growing in a pressure field: a simplified analysis, Int. Comm. Heat Mass Transfer,27, No 1 (2000), 99-108.

[13] Crank, J., Free and moving boundary problems, Clarendon Press, Oxford (1984). | MR 776227 | Zbl 0547.35001

[14] J. I. Diaz - M. A. Herrero - A. Liñan - J. L. Vazquez (Eds.), Free boundary problems: theory and applications, Pitman Research Notes in Mathematics Series323, Longman, Essex (1995).

[15] Fasano, A. - Primicerio, M. (Eds.), Nonlinear diffusion problems, Lecture Notes in Math., N. 1224, Springer Verlag, Berlin (1986). | MR 877985

[16] Fokas, A. S. - Yortsos, Y. C., On the exactly solvable equation $S_{t}=[(\beta S+\gamma)^{-2}S_{x}]_{x}+\alpha(\beta S+g)^{-2}Sx$ occurring in two-phase flow in porous media, SIAM J. Appl. Math., 42, No 2 (1982), 318-331. | MR 650227

[17] N. Kenmochi (Ed.), Free Boundary Problems: Theory and Applications, I,II, Gakuto International Series: Mathematical Sciences and Applications, Vol. 13, 14, Gakkotosho, Tokyo (2000). | MR 1793016

[18] Knight, J. H. - Philip, J. R., Exact solutions in nonlinear diffusion, J. Engrg. Math., 8 (1974), 219-227. | MR 378593 | Zbl 0279.76043

[19] Lamé, G. - Clapeyron, B. P., Memoire sur la solidification par refroidissementd’un globe liquide, Annales Chimie Physique, 47 (1831), 250-256.

[20] Lunardini, V. J., Heat transfer with freezing and thawing, Elsevier, Amsterdam (1991)

[21] Munier, A. - Burgan, J. R. - Gutierrez, J. - Fijalkow, E. - Feix, M. R., Group transformations and the nonlinear heat diffusion equation, SIAM J. Appl. Math., 40,No 2 (1981), 191-207. | MR 610415 | Zbl 0468.35051

[22] Natale, M. F. - Tarzia, D. A., Explicit solutions to the two-phase Stefan problemfor Storm-type materials, J. Phys. A: Math. Gen., 33 (2000), 395-404. | MR 1747160 | Zbl 0986.35138

[23] Natale, M. F. - Tarzia, D. A., Explicit solutions to the one-phase Stefan problemwith temperature-dependent thermal conductivity and a convective term, Int. J.Engng. Sci., 41 (2003), 1685-1698. | MR 1985327 | Zbl 1211.35285

[24] Philip, R., General method of exact solution of the concentration-dependent diffusion equation, Australian J. Physics, 13 (1960), 1-12. | MR 118858 | Zbl 0135.31802

[25] Polyanin, A. D. - Dil’Man, V. V., The method of the «carry over» of integral transforms in non-linear mass and heat transfer problems, Int. J. Heat Mass Transfer, 33, No 1 (1990), 175-181

[26] Rogers, C., Application of a reciprocal transformation to a two-phase Stefan problem, J. Phys. A: Math. Gen., 18 (1985), 105-109. | MR 783186

[27] Rogers, C., On a class of moving boundary problems in non-linear heat condition: Application of a Bäcklund transformation, Int. J. Non-Linear Mech., 21 (1986), 249-256. | MR 893762 | Zbl 0615.35044

[28] Rogers, C. - Broadbridge, P., On a nonlinear moving boundary problem with heterogeneity: application of reciprocal transformation, Journal of Applied Mathematics and Physics (ZAMP), 39 (1988), 122-129. | MR 928584 | Zbl 0661.35087

[29] Sander, G. C. - Cunning, I. F. - Hogarth, W. L. - Parlange, J. Y., Exact solution fornonlinear nonhysteretic redistribution in vertical soli of finite depth, Water Resources Research, 27 (1991), 1529-1536.

[30] Tarzia, D. A., 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), 491-497. | MR 644103

[31] Tarzia, D. A., A bibliography on moving - free boundary problems for the heat-diffusion equation. The Stefan and related problems, MAT-Serie A #2 (2000) (with 5869 titles on the subject, 300 pages). See www.austral.edu.ar/MAT-SerieA/2(2000)/ | MR 1802028

[32] Tritscher, P. - Broadbridge, P., A similarity solution of a multiphase Stefan problem incorporating general non-linear heat conduction, Int. J. Heat Mass Transfer, 37, No 14 (1994), 2113-2121. | Zbl 0926.76119