One-phase Stefan problem with temperature-dependent thermal conductivity and a boundary condition of Robin type

• Autorzy: Briozzo A. C. , Natale M. F.

Identyfikatory

DOI

10.1515/jaa-2015-0009

• Języki publikacji: EN

Abstrakty

EN

We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity with a boundary condition of Robin type at the fixed face x = 0. We obtain sufficient conditions for data in order to have a parametric representation of the solution of similarity type for t ≥ t0 > 0 with t0 an arbitrary positive time. This explicit solution is obtained through the unique solution of an integral equation with the time as a parameter.

Słowa kluczowe

EN

Stefan problem; free boundary problem; phase-change process; nonlinear thermal conductivity; similarity solution

PL

problem Stefana; problem granic swobodnych; proces zmiany fazy; nieliniowa przewodność cieplna; rozwiązanie automorficzne

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2015
• Tom: Vol. 21, nr 2
• Strony: 89--97
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Briozzo A. C.

• Universidad Austral and CONICET, Paraguay 1950, 2000 Rosario, Argentina

autor

Natale M. F.

• Universidad Austral, Paraguay 1950, 2000 Rosario, Argentina

Bibliografia

• [1] V. Alexiades and A. D. Solomon, Mathematical Modeling of Melting and Freezing Processes, Taylor & Francis, Washington, 1983.
• [2] D. A. Barry and G. C. Sander, Exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption, Water Resources Research 27 (1991), no. 10, 2667–2680.
• [3] G. Bluman and S. Kumei, On the remarkable nonlinear diffusion equation, J. Math Phys. 21 (1980), 1019–1023.
• [4] A. C. Briozzo and M. F. Natale, On a non-classical non-linear moving boundary problem for a diffusion convection equation, Int. J. Non-Linear Mech. 47 (2012), 712–718.
• [5] P. Broadbridge, Non-integrability of non-linear diffusion-convection equations in two spatial dimensions, J. Phys. A Math. Gen. 19 (1986), 1245–1257.
• [6] P. Broadbridge, Integrable forms of the one-dimensional flow equation for unsaturated heterogeneous porous media, J. Math. Phys. 29 (1988), 622–627.
• [7] J. R. Cannon, The One-Dimensional Heat Equation, Addison–Wesley, Menlo Park, 1984.
• [8] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 1959.
• [9] J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984.
• [10] J. I. Diaz, M. A. Herrero, A. Liñan and J. L. Vazquez, Free Boundary Problems: Theory And Applications, Pitman Research Notes in Math. 323, Longman, Harlow, 1995.
• [11] A. Fasano and M. Primicerio, Nonlinear Diffusion Problems, Lecture Notes in Math. 1224, Springer, Berlin, 1986.
• [12] A. S. Fokas and Y. C. Yortsos, On the exactly solvable equation St = [(βS + γ)-2Sx]x + α(βS + γ)-2Sx occurring in two-phase flow in porous media, SIAM J. Appl. Math. 42 (1982), no. 2, 318–331.
• [13] N. Kenmochi, Free Boundary Problems: Theory and Applications. Vol. I–II, GAKUTO Internat. Ser. Math. Sci. Appl. 13–14, Gakkotosho, Tokyo, 2000.
• [14] J. H. Knight and J. R. Philip, Exact solutions in nonlinear diffusion, J. Engrg. Math. 8 (1974), 219–227.
• [15] G. Lamé and B. P. Clapeyron, Memoire sur la solidification par refroidissement d’un globe liquide, Ann. Chimie Phys. 47 (1831), 250–256.
• [16] V. J. Lunardini, Heat Transfer with Freezing and Thawing, Elsevier, Amsterdam, 1991.
• [17] M. F. Natale and D. A. Tarzia, Explicit solutions to the one-phase Stefan problem with temperature-dependent thermal conductivity and a convective term, Internat. J. Engrg. Sci. 41 (2003), 1685–1698.
• [18] M. F. Natale and D. A. Tarzia, Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 9 (2006), 79–99.
• [19] M. F. Natale and D. A. Tarzia, The classical one-phase Stefan problem with temperature-dependent thermal conductivity and a convective term, in: Workshop on Mathematical Modelling of Energy and Mass Transfer Processes, and Applications, MAT Ser. A Conf. Semin. Trab. Mat. 15, Universidad Austral, Departamento de Matemática, Rosario (2008), 1–16.
• [20] R. Philip, General method of exact solution of the concentration-dependent difiusion equation, Aust. J. Phys. 13 (1960), 1–12.
• [21] C. Rogers and P. Broadbridge, On a nonlinear moving boundary problem with heterogeneity: Application of reciprocal transformation, Z. Angew. Math. Phys. 39 (1988), 122–129.
• [22] G. C. Sander, I. F. Cunning, W. L. Hogarth and J. Y. Parlange, Exact solution for nonlinear nonhysteretic redistribution in vertical soil of finite depth, Water Resources Research 27 (1991), 1529–1536.
• [23] D. A. Tarzia, A Bibliography on Moving – Free Boundary Problems for the Heat-Diffusion Equation. The Stefan and Related Problems, MAT Ser. A Conf. Semin. Trab. Mat. 2, Universidad Austral, Departamento de Matemática, Rosario, 2000.
• [24] P. Tritscher and P. Broadbridge, A similarity solution of a multiphase Stefan problem incorporating general non-linear heat conduction, Int. J. Heat Mass Transfer 37 (1994), no. 14, 2113–2121.