Explicit solutions for a two-phase unidimensional Lamé–Clapeyron–Stefan problem with source terms in both phases

Abstract

A two-phase Stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi-infinite phase-change material is studied. We assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x = 0, a constant temperature or a heat flux of the form −q 0 / √ t (q 0 > 0). The internal heat source functions are given byare functions with appropriate regularity properties, ρ is the mass density, l is the fusion latent heat by unit of mass, a 2 j is the diffusion coefficient, x is the spatial variable and t is the temporal variable. We obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x = 0. Moreover, the equivalence of the two free boundary problems is also proved. We generalize the solution obtained in [J.L. Menaldi, D.A. Tarzia, Generalized Lamé-Clapeyron solution for a one-phase source Stefan problem, Comput. Appl. Math. 12 (2) (1993) 123-142] for the one-phase Stefan problem. Finally, a particular case where β j (j = 1, 2) are of exponential type given by β j (x) = exp(−(x + d j ) 2 ) with x and d j ∈ R is also studied in details for both boundary temperature conditions at x = 0. This type of heat source terms is important through the use of microwave energy following [E.P. Scott, An analytical solution and sensitivity study of sublimation-dehydration within a porous medium with volumetric heating, J. Heat Transfer 116 (1994) 686-693]. We obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x = 0 is considered; a similar result is obtained for a heat flux condition imposed on x = 0 if an inequality for parameter q 0 is satisfied.

Relationship between Neumann solutions for two-phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions

“…Explicit solutions for Stefan-like problems can be found in [15][16][17][18][19]. A review of available analytical solutions, for a wide range of alternative boundary conditions and properties, are provided in [9].…”

Finite element schemes for a class of nonlocal parabolic systems with moving boundaries

“…Moving boundary problems occur in many physical applications involving diffusion, such as in heat transfer where a phase transition occurs, in moisture transport such as swelling grains or polymers, and in deformable porous media problems where the solid displacement is governed by diffusion (see, for example, [19,3,22,6,5]). Cavalcanti et al [7] worked with a time-dependent function a = a t, Ωt |∇u(x, t)| 2 dx to establish the solvability and exponential energy decay of the solution for a model given by a hyperbolic-parabolic equation in an open bounded subset of R n , with moving boundary.…”

Loschmidt, Stefan, and Stigler’s Law of Eponymy

“…The moving-interface problem, which was solved independently first by Lamé and Clapeyron and then by Neumann, is now known as the Stefan problem, 66 although some authors have called it the slightly more eponymously correct but out-of-order Neumann-Lamé-Clapeyron-Stefan problem, 67 or the even less accurate Lamé-Clapeyron-Stefan problem. 68 where c p is the specific heat at constant pressure and h fg is the latent heat of fusion. 69 This incarnation of the Stefan number relates the sensible heat to the latent heat, and Lock used it as a perturbation variable in an approximate asymptotic solution of the moving interface problem.…”