Explicit solutions for the Solomon-Wilson-Alexiades's mushy zone model with convective or heat flux boundary conditions

TL;DR

This paper derives explicit solutions for a one-phase mushy zone Stefan problem with convective or heat flux boundary conditions, providing conditions for solution existence and equivalence to temperature boundary problems.

Contribution

It extends the Solomon-Wilson-Alexiades mushy zone model by providing explicit solutions and necessary conditions for problems with convective or heat flux boundary conditions.

Findings

Explicit solutions for the mushy zone problem are obtained.

Necessary and sufficient conditions for solution existence are established.

Equivalence between boundary condition types is demonstrated.

Abstract

We complete the Solomon-Wilson-Alexiades's mushy zone model (Letters Heat Mass Transfer, 9 (1982), 319-324) for the one-phase Lam\'e-Clapeyron-Stefan problem. We obtain explicit solutions when a convective or heat flux boundary condition is imposed on the fixed face for a semi-infinite material. We also obtain the necessary and sufficient condition on data in order to get these explicit solutions. Moreover, when these conditions are satisfied the two problems are equivalents to the same problem with a temperature boundary condition on the fixed face and therefore an inequality for the coefficient which characterized one of the two free interfaces is also obtained.