Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

TL;DR

This paper models heat transfer in a semi-infinite, variable cross-section material using a two-phase Stefan problem for a generalized heat equation, employing similarity solutions and fixed point theorems.

Contribution

It introduces a novel approach to solving the two-phase Stefan problem with nonlinear thermal coefficients using similarity methods and proves solution existence and uniqueness.

Findings

Derived temperature solutions for both phases.

Established existence and uniqueness of solutions.

Identified free boundaries for phase interfaces.

Abstract

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component is considered. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modelled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the solution is provided by using the fixed point Banach theorem.