Explicit solutions related to the Rubinstein binary-alloy solidification problem with a heat flux or a convective condition at the fixed face

TL;DR

This paper derives explicit similarity solutions for the Rubinstein binary-alloy solidification problem under heat flux or convective boundary conditions, establishing conditions for instantaneous solidification and solution equivalence.

Contribution

It introduces new explicit solutions for the binary-alloy solidification problem with boundary conditions, and provides necessary and sufficient data conditions for instantaneous solidification.

Findings

Solutions are explicit and similarity-based.

Conditions for instantaneous solidification are established.

All solutions are equivalent under certain data restrictions.

Abstract

Similarity solutions for the two-phase Rubinstein binary-alloy solidification problem in a semi-infinite material are developed. These new explicit solutions are obtained by considering two cases: a heat flux or a convective boundary conditions at the fixed face, and the necessary and sufficient conditions on data are also given in order to have an instantaneous solidification process. We also show that all solutions for the binary-alloy solidification problem are equivalent under some restrictions for data. Moreover, this implies that the coefficient which characterizes the solidification front for the Rubinstein solution must verify an inequality as a function of all thermal and boundary conditions.