Explicit solution for Stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face using Kummer functions

TL;DR

This paper derives an explicit similarity solution for a one-phase Stefan problem with position-dependent latent heat and convective boundary conditions, using Kummer functions, and analyzes its properties and limits.

Contribution

It provides a novel explicit solution for a Stefan problem with position-dependent latent heat and convective boundary, extending previous work with new analytical and numerical insights.

Findings

Existence and uniqueness of the solution are proved.

The limit behavior as heat transfer coefficient tends to infinity is analyzed.

Numerical examples validate the explicit solution and compare with general algorithms.

Abstract

An explicit solution of a similarity type is obtained for a one-phase Stefan problem in a semi-infinite material using Kummer functions. Motivated by [D.A. Tarzia, Relationship between Neumann solutions for two phase Lam\'e-Clapeyron-Stefan problems with convective and temperature boundary conditions, Thermal Sci.(2016) DOI 10.2298/TSCI 140607003T, In press], and [Y. Zhou, L.J. Xia, Exact solution for Stefan problem with general power-type latent heat using Kummer function, Int. J. Heat Mass Transfer, 84 (2015) 114-118], we consider a phase-change problem with a latent heat defined as a power function of the position with a non-negative real exponent and a convective boundary condition at the fixed face $x=0$. Existence and uniqueness of the solution is proved. Relationship between this problem and the problems already solved by Zhou and Xia with temperature and flux boundary condition is analysed. Furthermore it is studied the limit behaviour of the solution when the coefficient which characterizes the heat transfer at the fixed boundary tends to infinity. Numerical computation of the solution is done over certain examples, with a view to comparing this results with those obtained by general algorithms that solve Stefan problems.