Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model

TL;DR

This paper derives an exact similarity solution for a one-dimensional two-phase Stefan problem with a convective boundary condition and a mushy zone model, analyzing its relation to temperature boundary problems and convergence properties.

Contribution

It provides a novel exact similarity solution for a complex Stefan problem with a convective boundary and mushy zone, extending previous work and analyzing solution convergence.

Findings

Exact similarity solution derived under specific data restrictions.

Solution with convective boundary converges to temperature boundary solution as heat transfer coefficient increases.

The results extend and improve previous related studies.

Abstract

A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia, Compt. Appl. Math., 9 (1990), 201-211.