Similarity solutions for thawing processes with a convective boundary condition

TL;DR

This paper derives similarity solutions for a one-dimensional thawing process in porous media with a convective boundary, analyzing different physical cases and establishing conditions for explicit solutions and equivalence with temperature boundary problems.

Contribution

It provides explicit similarity solutions for the Stefan problem with convective boundary conditions and analyzes the relationship with temperature boundary conditions, including conditions for their equivalence.

Findings

Explicit solutions exist under certain data inequalities.

Monotony property of the free boundary coefficient with respect to heat transfer coefficient.

Conditions for equivalence between convective and temperature boundary Stefan problems.

Abstract

Similarity solutions for a one-dimensional mathematical model for thawing in a saturated semi-infinite porous media is considered when change of phase induces a density jump and a convective boundary condition is imposed at the fixed face $x=0$. Different cases depending on physical parameters are analysed and an explicit solution of a similarity type is obtained if and only if an inequality for data is verified. Moreover, a monotony property respect to the coefficient which characterizes the heat transfer at the fixed face $x=0$ is obtained for the coefficient involved in the definition of the free boundary. Relationship between the Stefan problem with convective condition at $x=0$ considered in this paper and the Stefan problem with temperature condition at the same face studied in (Fasano-Primicerio-Tarzia, Math. Models Meth. Appl. Sci., 9 (1999), 1-10) is analized and conditions for physical parameters under which both problems became equivalents are obtained. Furthermore, an inequality to be satisfied for the coefficient which characterizes the free boundary of the problem with a temperature or a convective boundary condition at the fixed face $x=0$ is also obtained.