Numerical, perturbation and approximate solutions

to Stefan problems

Dr Sarah Mitchell
Mathematics Applications Consortium for Science and Industry
Department of Mathematics and Statistics, University of Limerick

Thursday 30 March 2017, 4pm
ADB-1020, Áras De Brún

Stefan problems, which describe the melting or solidification of a material, occur in a wide variety of natural and industrial applications. Mathematically, these problems represent a particular kind of boundary value problem where the phase boundary moves with time, and its location is not known a priori. To solve the problem numerically we describe using finite difference methods with increased accuracy and correct initialisation. Although the numerical solution of phase-change problems is well documented, there are still unresolved issues regarding the start-up of a computation for a region that initially has zero thickness, as well as how to determine the position of the moving boundary thereafter. A combined analytical and numerical approach is described, which eliminates completely the ad-hoc treatment of the starting solution often used. We also describe a popular "large Stefan number" perturbation expansion to give an approximate solution and also discuss the application of heat balance integral methods (HBIMs) to phase-change problems. This was originally used for analysing boundary layers, but can be applied to Stefan problems, where it has made the most impact since very few exact solutions exist. To conclude the talk we describe some applications which have made use of the above solution techniques - namely solidification of metals, removal of mass from an object by vaporization (known as ablation) and solvent diffusion in glassy polymers.

This is a joint seminar with the School of Mathematics, Statistics and Applied Mathematics.