An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition

Abstract

In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a generalized modified error function which is defined as the solution to a nonlinear ordinary differential problem of second order. It is proved that the latter has a unique non-negative bounded analytic solution when the parameter on which it depends assumes small positive values. Moreover, it is shown that the generalized modified error function is concave and increasing, and explicit approximations are proposed for it. Relation between the Stefan problem considered in this article with those with either constant thermal conductivity or a temperature boundary condition is also analysed.

A Note on Corrections in Approximation of the Modified Error Function

“…Most of the results obtained here appear to be new and resolve the lack of monotonicity of successive corrections in the approximation of MEF appearing in [18]. The results derived here may be useful for the researchers working in the field of Stefan problems e.g., in the approximations of generalized error function introduced by Ceretani et al in their recent work [21].…”

Auxiliary functions in the study of Stefan-like problems with variable thermal properties

“…Similar approaches to those introduced by Sunderland and collaborators were followed to find exact similarity solutions in the cases when non-Dirichlet boundary conditions are prescribed at x = 0 or when the physical domain is allowed to move, see e.g. [7][8][9]. In all cases it was assumed that α = β > 0, or α = 0 and β > 0.…”

Explicit solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms

“…where δ and p are given non-negative constants, k 0 = k (θ f ) and c 0 = c (θ f ) are the reference thermal conductivity and the specific heat coefficients, respectively. Some other models involving temperature-dependent thermal conductivity can also be found in [3,4,11,13,23,24,26,27,36,38,40,41,42,43,44,45].…”