## Description

This volume emphasises studies related to

classical Stefan problems. The term "Stefan problem" is

generally used for heat transfer problems with

phase-changes such

as from the liquid to the solid. Stefan problems have some

characteristics that are typical of them, but certain problems

arising in fields such as mathematical physics and engineering

also exhibit characteristics similar to them. The term

``classical" distinguishes the formulation of these problems from

their weak formulation, in which the solution need not possess

classical derivatives. Under suitable assumptions, a weak solution

could be as good as a classical solution. In hyperbolic Stefan

problems, the characteristic features of Stefan problems are

present but unlike in Stefan problems, discontinuous solutions are

allowed because of the hyperbolic nature of the heat equation. The

numerical solutions of inverse Stefan problems, and the analysis of

direct Stefan problems are so integrated that it is difficult to

discuss one without referring to the other. So no strict line of

demarcation can be identified between a classical Stefan problem

and other similar problems. On the other hand, including every

related problem in the domain of classical Stefan problem would

require several volumes for their description. A suitable

compromise has to be made.

The basic concepts, modelling, and analysis of the classical

Stefan problems have been extensively investigated and there seems

to be a need to report the results at one place. This book

attempts to answer that need. Within the framework of the

classical Stefan problem with the emphasis on the basic concepts,

modelling and analysis, it tries to include some weak

solutions and analytical and numerical solutions also. The main

considerations behind this are the continuity and the clarity of

exposition. For example, the description of some phase-field

models in Chapter 4 arose out of this need for a smooth transition

between topics. In the mathematical formulation of Stefan

problems, the curvature effects and the kinetic condition are

incorporated with the help of the modified Gibbs-Thomson relation.

On the basis of some thermodynamical and metallurgical

considerations, the modified Gibbs-Thomson relation can be

derived, as has been done in the text, but the rigorous

mathematical justification comes from the fact that this relation

can be obtained by taking appropriate limits of phase-field

models. Because of the unacceptability of some phase-field models

due their so-called thermodynamical inconsistency, some consistent

models have also been described. This completes the discussion of

phase-field models in the present context.

Making this volume self-contained would require reporting and

deriving several results from tensor analysis, differential

geometry, non-equilibrium thermodynamics, physics and functional

analysis. The text is enriched with appropriate

references so as not to enlarge the scope of the book. The proofs

of propositions and theorems are often lengthy and different from

one another. Presenting them in a condensed way may not be of much

help to the reader. Therefore only the main features of proofs

and a few results have been presented to suggest the essential

flavour of the theme of investigation. However at each place,

appropriate references have been cited so that inquisitive

readers can follow them on their own.

Each chapter begins with basic concepts, objectives and the

directions in which the subject matter has grown. This is followed

by reviews - in some cases quite detailed - of published works. In a

work of this type, the author has to make a suitable compromise

between length restrictions and understandability.