Relationship among solutions for three-phase change problems with Robin, Dirichlet, and Neumann boundary conditions

TL;DR

This paper derives explicit solutions for a three-phase Stefan problem with different boundary conditions and reveals their equivalences under certain data relationships, enhancing understanding of phase change processes.

Contribution

It introduces a unified approach to relate solutions of three-phase Stefan problems with Robin, Dirichlet, and Neumann boundary conditions, under specific data conditions.

Findings

Explicit solutions for the three-phase Stefan problem were obtained.

Established equivalences among solutions with different boundary conditions.

Provided insights into phase change behaviour under varying boundary conditions.

Abstract

This study investigates the melting process of a three-phase Stefan problem in a semi-infinite material, imposing a convective boundary condition at the fixed face. By employing a similarity-type transformation, the problem is reduced to a solvable form, yielding a unique explicit solution. The analysis uncovers significant equivalences among the solutions of three different three-phase Stefan problems: one with a Robin boundary condition, another with a Dirichlet boundary condition, and a third one with a Neumann boundary condition at the fixed face. These equivalences are established under the condition that the problem data satisfy a specific relationship, providing new insights into the behaviour of phase change problems under varying boundary conditions.