Exact solution for a Stefan problem with convective boundary condition and density jump

We consider the solidification of a semi‐infinite material which is initially at its liquid phase at a uniform temperature T i . Suddenly at time t > 0 the fixed face x = 0 is submitted to a convective cooling condition with a time‐dependent heat transfer coefficient of the type H ( t ) = ht –1/2 ( h > 0) The bulk temperature of the liquid at a large distance from the solid‐liquid interface is T ∞ , a constant temperature such that T ∞ < T f < T i where T f is the freezing temperature. We also consider the density jump between the two phases. We obtain that the corresponding phase‐change (solidification) process has an explicit solution of a similarity type for the temperature of both phases and the solid‐liquid interface, if and only if the coefficient h is large enough, that is where k l and α l are the conductivity and diffusion coefficients of the initial liquid phase. Moreover, when h ≤ h 0 we only have a heat conduction problem for the initial liquid phase and the corresponding change of phase does not occur. We do the mathematical analysis of a problem which was presented in S.M. Zubair – M.A. Chaudhry, Wärme‐und‐Stoffübertragung, 30 (1994), 77‐81 and we generalized the results obtained in D.A. Tarzia, MAT‐Serie A, 8 (2004), 21‐27 when the density jump between the two phases was neglected. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)