Solution to the 1D Stefan problem using the unified transform method

Dokoza T,D Plümacher,C Jegust,M Smuda,M Oberlack

doi:10.1088/1751-8121/ac1b79

Dokoza T, D Plümacher + Show 3 more

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https://doi.org/10.1088/1751-8121/ac1b79

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Abstract

In this paper the one-dimensional two-phase Stefan problem is studied analytically leading to a system of non-linear Volterra-integral-equations describing the heat distribution in each phase. For this the unified transform method has been employed which provides a method via a global relation, by which these problems can be solved using integral representations. To do this, the underlying partial differential equation is rewritten into a certain divergence form, which enables to treat the boundary values as part of the integrals. Classical analytical methods fail in the case of the Stefan problem due to the moving interface. From the resulting non-linear integro-differential equations the one for the position of the phase change can be solved in a first step. This is done numerically using a fix-point iteration and spline interpolation. Once obtained, the temperature distribution in both phases is generated from their integral representation.

Highlights

Many important heat conduction problems occurring in science, nature and engineering involve a phase change due to melting or freezing
The underlying partial differential equation is rewritten into a certain divergence form, which enables to treat the boundary values as part of the integrals
The test case of a one-phase Stefan problem developed in [9] was used to prove the accuracy of the present solution scheme in section 5.1 by comparing with the results provided by the author

Summary

Introduction

Many important heat conduction problems occurring in science, nature and engineering involve a phase change due to melting or freezing. These are special cases of moving phase boundary problems, in which the evolution of the interface itself is unknown and . The most common example for such an interface problem is the Stefan problem, treating the melting of ice. The most common example for such an interface problem is the Stefan problem, treating the melting of ice It is an interface problem for the heat equation, a parabolic partial differential equation and usually referred to as free boundary problem. The Stefan problem was first described in 1891 by Jozef Stefan and describes the temperature distribution in a fluid undergoing a phase change from its solid to its liquid phase due to a heat introduction [2]. Some historical notes on Stefan’s modeling of ice melting can be taken from [3]

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