# RELATIONSHIP BETWEEN NEUMANN SOLUTIONS FOR TWO-PHASE LAMÉ-CLAPEYRON-STEFAN PROBLEMS WITH CONVECTIVE AND TEMPERATURE BOUNDARY CONDITIONS

## Main Article Content

## Abstract

We obtain for the two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material an equivalence between the temperature and convective boundary conditions at the fixed face in the case that an inequality for the convective transfer coefficient is satisfied. Moreover, an inequality for the coefficient which characterizes the solid-liquid interface of the classical Neumann solution is also obtained. This inequality must be satisfied for data of any phase-change material, and as a consequence the result given in Tarzia, Quart. Appl. Math., 39 (1981), 491-497 is also recovered when a heat flux condition was imposed at the fixed face.

## Article Details

**Thermal Science**, [S.l.], mar. 2017. ISSN 2334-7163. Available at: <http://thermal-science.tech/journal/index.php/thsci/article/view/2073>. Date accessed: 26 june 2017. doi: https://doi.org/10.2298/TSCI140607003T.

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Accepted 2017-03-13

Published 2017-03-13

## References

[2] J.R. Cannon, The one-dimensional heat equation, Addison-Wesley, Menlo Park, California, 1984.

[3] H.S. Carslaw, C.J. Jaeger, Conduction of heat in solids, Clarendon Press, Oxford, 1959.

[4] J. Crank, Free and moving boundary problem, Clarendon Press, Oxford, 1984.

[5] A. Fasano, Mathematical models of some diffusive processes with free boundary, MAT – Serie A, 11 (2005), pp. 1-128.

[6] S.C. Gupta, The classical Stefan problem. Basic concepts, modelling and analysis, Elsevier, Amsterdam, 2003.

[7] V.J. Lunardini, Heat transfer with freezing and thawing, Elsevier, London, 1991.

[8] L.I. Rubinstein, The Stefan problem, American Mathematical Society, Providence, 1971.

[9] D.A. Tarzia, Explicit and Approximated Solutions for Heat and Mass Transfer Problems with a Moving Interface, Chapter 20, in Advanced Topics in Mass Transfer, (Ed. M. El-Amin), InTech Open Access Publisher, Rijeka, 2011, pp. 439-484.

[10] A.B. Tayler, Mathematical models in applied mechanics, Clarendon Press, Oxford, 1986.

[11] D.A. Tarzia, A bibliography on moving-free boundary problems for heat diffusion equation. The Stefan problem, MAT - Serie A, 2 (2000), pp. 1-297.

[12] G. Lamé, B.P. Clapeyron, Memoire sur la solidification par refroidissement d'un globe liquide, Annales Chimie Physique, 47 (1831), pp. 250-256.

[13] D.A. Tarzia, An inequality for the coefficient σ of the free boundary s(t)=2σ(t)^(1/2) of the Neumann solution for the two-phase Stefan problem, Quart. Appl. Math., 39 (1981), pp. 491-497.

[14] H. Weber, Die partiellen Differential-Gleinchungen der Mathematischen Physik, nach Riemann's Vorlesungen, t. II, Braunwschweig, 1901.

[15] A. C. Briozzo, M.F. Natale, D.A. Tarzia, “Explicit solutions for a two-phase unidimensional Lamé-Clapeyron-Stefan problem with source terms in both phases”, J. Math. Anal. Appl., 329 (2007), pp. 145-162.

[16] M.F. Natale, D.A. Tarzia, "Explicit solutions to the two-phase Stefan problem for Storm's type materials", J. Physics A: Math. General, 33 (2000), pp. 395-404.

[17] N.N. Salva, D.A. Tarzia, “Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions”, J. Math. Anal. Appl., 379 (2011), pp. 240-244.

[18] V.R. Voller, F. Falcini, Two exact solutions of a Stefan problem with varying diffusivity, Int. J. Heat Mass Transfer, 58 (2013), pp. 80-85.

[19] V.R. Voller, J.B. Swenson, C. Paola, An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat Mass Transfer, 47 (2004), pp. 5387-5390.

[20] D.A. Tarzia, An explicit solution for a two-phase unidimensional Stefan problem with a convective boundary condition at the fixed face, MAT – Serie A, 8 (2004), pp. 21-27.

[21] S.M. Zubair, M.A. Chaudhry, Exact solutions of solid-liquid phase-change heat transfer when subjected to convective boundary conditions, Heat Mass Transfer, 30 (1994), pp. 77-81.

[22] P.M. Beckett, A note on surface heat transfer coefficients, Int. J. Heat Mass Transfer, 34 (1991), pp. 2165-2166.

[23] J. Cadwell, Y. Kwan, A brief review of several numerical methods for one-dimensional Stefan problems, Thermal Science, 13 (2009), pp. 61-72.

[24] S.D. Foss, An approximate solution to the moving boundary problem associated with the freezing and melting of lake ice, A.I.Ch.E. Symposium Series, 74 (1978), pp. 250-255.

[25] R. Grzymkowski, E. Hetmaniok, M. Pleszczynski, D. Slota, A certain analytical method used for solving the Stefan problem, Thermal Science, 17 (2013), pp. 635-642.

[26] C.L. Huang, Y.P. Shih, Perturbation solution for planar solidification of a saturated liquid with convection at the hall, Int. J. Heat Mass Transfer, 18 (1975), pp. 1481-1483.

[27] T.J. Lu, Thermal management of high power electronic with phase change cooling, Int. J. Heat Mass Transfer, 43 (2000), pp. 2245-2256.

[28] A.P. Roday, M.J. Kazmiercza, Melting and freezing in a finite slab due to a linearly decreasing free-stream temperature of a convective boundary condition, Thermal Science, 13 (2009), pp. 141- 153.

[29] N. Sadoun, E. Si-ahmed, P. Colinet, J. Legrand, On the Goodman heat-balance integral method for Stefan lie-problems, Thermal Science, 13 (2009), pp. 81-96.

[30] J. Singh, P.K. Gupta, K.N.Rai, Variational iteration method to solve movind boundary problem with temperature dependent physical properties, Thermal Science, 15 (2011), Suppl. 2, pp. S229- S239.

[31] Z. Wu, Q. Wang, Numerical approach to Stefan problem in a two-region and limited space, Heat Mass Transfer, 30 (1994), pp. 77-81.

[32] A.C. Briozzo, D.A. Tarzia, Explicit solution of a free-boundary problem for a nonlinear absorption model of mixed saturated-unsaturated flow, Adv. Water Resources, 21 (1998), pp. 713- 721.

[33] P. Boadbridge, Solution of a nonlinear absorption model of mixed saturated-unsaturated flow, Water Resources Research, 26 (1990), pp. 2435-2443.

[34] A.D. Solomon, D.G. Wilson, V. Alexiades, Explicit solutions to change problems, Quart. Appl. Math., 41 (1983), pp. 237-243.

[35] S. Roscani, E.A. Santillan Marcus, A new equivalence of Stefan’s problems for the time- fractional diffusion equation, Fractional Calculus and Appl. Anal., 17 (2014), 371-381.

[36] S.D. Roscani, D.A. Tarzia, A generalized Neumann solution for the two-phase fractional Lamé- Clapeyron-Stefan problem, Advances of Math. Sciences and Appl., Accepted, In Press.

[37] D.A. Tarzia, The two-phase fractional Lamé-Clapeyron-Stefan problem with a convective boundary condition, Forthcoming work.