# The Classical Stefan Problem, Volume 45

## 1st Edition

### basic concepts, modelling and analysis

**Authors:**S.C. Gupta S.C. Gupta

**eBook ISBN:**9780080529165

**Hardcover ISBN:**9780444510860

**Imprint:**JAI Press

**Published Date:**22nd October 2003

**Page Count:**404

**View all volumes in this series:**North-Holland Series in Applied Mathematics and Mechanics

## Table of Contents

Chapter 1. The Stefan Problem and its Classical Formulation

1.1 Some Stefan and Stefan-like Problems

1.2 Free Boundary Problems with Free Boundaries of Codimension-
two

1.3 The Classical Stefan Problem in One-dimension and the
Neumann Solution

1.4 Classical Formulation of Multi-dimensional Stefan Problems

1.4.1 Two-Phase Stefan problem in
multipledimensions

1.4.2 Alternate forms of the Stefan condition

1.4.3 The Kirchhoff's transformation

1.4.4 Boundary conditions at the fixed boundary

1.4.5 Conditions at the free boundary

1.4.6 The classical solution

1.4.7 Conservation laws and the motion of the melt

Chapter 2. Thermodynamical and Metallurgical Aspects of Stefan
Problems

2.1 Thermodynamical Aspects

2.1.1 Microscopic and macroscopic models

2.1.2 Laws of classical thermodynamics

2.1.3 Some thermodynamic variables and thermal
parameters

2.1.4 Equilibrium temperature; Clapeyron's equation

2.2 Some Metallurgical Aspects of Stefan Problems

2.2.1 Nucleation and supercooling

2.2.2 The effect of interface curvature

2.2.3 Nucleation of melting, effect of interface
kinetics, and glassy solids

2.3 Morphological Instability of the Solid-Liquid Interface

2.4 Non-material Singular Surface : Generalized Stefan
Condition

Chapter 3. Extended Classical Formulations of n-phase Stefan
Problems with n>1

3.1 One-phase Problems

3.1.1 An extended formulation of one-dimensional one-
phase problem

3.1.2 Solidification of supercooled liquid

3.1.3 Multi-dimensional one-phase problems

3.2 Extended Classical Formulations of Two-phase Stefan
Problems

3.2.1 An extended formulation of the one-dimensional
two-phase problem

3.2.2 Multi-dimensional Stefan problems of classes II
and III

3.2.3 Classical Stefan problems with n-phases, n> 2

3.2.4 Solidification with transition temperature range

3.3 Stefan problems with Implicit Free Boundary Conditions

3.3.1 Schatz transformations and implicit free boundary
conditions

3.3.2 Unconstrained and constrained oxygen-diffusion
problem (ODP)

Chapter 4. Stefan Problem with Supercooling : Classical
Formulation and Analysis

4.1 Introduction

4.2 A Phase-field Model for Solidification using Landau-
Ginzburg Free Energy Functional

4.3 Some Thermodynamically Consistent Phase-field and Phase
Relaxation Models of Solidification

4.4 Solidification of Supercooled Liquid Without Curvature
Effect and Kinetic Undercooling : Analysis of the
Solution

4.4.1 One-dimensional one-phase solidification of
supercooled liquid (SSP)

4.4.2 Regularization of blow-up in SSP by looking at
CODP

4.4.3 Analysis of problems with changes in the initial
and boundary conditions in SSP

4.5 Analysis of Supercooled Stefan Problems with the Modified
Gibbs-Thomson Relation

4.5.1 Introduction

4.5.2 One-dimensional one-phase supercooled Stefan
problems with the modified Gibbs-Thomson
relation

4.5.3 One-dimensional two-phase Stefan problems with the
modified Gibbs-Thomson relation

4.5.4 Multi-dimensional supercooled Stefan problems and
problems with the modified Gibbs-Thomson
relation

4.5.5 Weak formulation with supercooling and
superheating effects

Chapter 5. Superheating due to Volumetric Heat Sources:
Formulation and Analysis

5.1 The Classical Enthalpy Formulation of a One-dimensional
Problem

5.2 The Weak Solution

5.2.1 Weak solution and its relation to a classical
solution

5.2.2 Structure of the mushy region in the presence of
heat sources

5.3 Blow-up and Regularization

Chapter 6. Steady-State and Degenerate Classical Stefan
Problems

6.1 Some Steady-state Stefan Problems

6.2 Degenerate Stefan Problems

6.2.1 Quasi-static Stefan problem and its relation to
the Hele-Shaw problem

Chapter 7. Elliptic and Parabolic Variational Inequalities

7.1 Introduction

7.2 The Elliptic Variational Inequality

7.2.1 Definition and the basic function spaces

7.2.2 Minimization of a functional

7.2.3 The complementarity problem

7.2.4 Some existence and uniqueness results
concerning

elliptic inequalities

7.2.5 Equivalence of different inequality formulations
of an obstacle problem of the string

7.3 The Parabolic Variational Inequality

7.3.1 Formulation in appropriate spaces

7.4 Some Variational Inequality Formulations of
Classical Stefan Problems

7.4.1 One-phase Stefan problems

7.4.2 A Stefan problem with a quasi-variational
inequality formulation

7.4.3 The variational inequality formulation of a two-
phase Stefan problem

Chapter 8. The Hyperbolic Stefan Problem

8.1 Introduction

8.1.1 Relaxation time and relaxation models

8.2 Model I : Hyperbolic Stefan Problem with Temperature
Continuity at the Interface

8.2.1 The mathematical formulation

8.2.2 Some existence, uniqueness and well-posedness
results

8.3 Model II : Formulation with Temperature Discontinuity at
the Interface

8.3.1 The mathematical formulation

8.3.2 The existence and uniqueness of the solution and
its convergence as &tgr; → 0

8.4 Model III : Delay in the Response of Energy to Latent and
Sensible Heats

8.4.1 The Clasical and the Weak Formulations

Chapter 9. Inverse Stefan Problems

9.1 Introduction

9.2 Well-posedness of the solution

9.2.1 Approximate solutions

9.3 Regularization

9.3.1 The regularizing operator and generalized
discrepancy principle

9.3.2 The generalized inverse

9.3.3 Regularization methods

9.3.4 Rate of convergence of a regularization method

9.4 Determination of Unknown Parameters in Inverse Stefan
Problems

9.4.1 Unknown parameters in the one-phase Stefan
problems

9.4.2 Determination of Unknown parameters in the two-
phase Stefan problems

9.5 Regularization of Inverse Heat Conduction Problems by
Imposing Suitable Restrictions on the solution

9.6 Regularization of Inverse Stefan Problems Formulated as
Equations in the form of Convolution Integrals

9.7 Inverse Stefan Problems Formulated as Defect Minimization
Problems

Chapter 10. Analysis of the Classical Solutions of Stefan
Problems

10.1 One-dimensional One-phase Stefan Problems

10.1.1 Analysis using integral equation formulations

10.1.2 Infinite differentiability and analyticity of the
free boundary

10.1.3 Unilateral boundary conditions on the
boundary: Analysis using finite-difference
schemes

10.1.4 Cauchy-type free boundary conditions

10.1.5 Existence of self-similar solutions of some
Stefan problems

10.1.6 The effect of density change

10.2 One-dimensional Two-phase Stefan Problems

10.2.1 Existence, uniqueness and stability results

10.2.2 Differentiability and analyticity of the free
boundary in the one-dimensional two-phase Stefan
problems

10.2.3 One-dimensional n-phase Stefan problems with
n > 2

10.3 Analysis of the Classical Solutions of Multi-dimensional
Stefan Problems

10.3.1 Existence and uniqueness results valid for a
short time

10.3.2 Existence of the classical solution on an
arbitrary time interval

Chapter 11. Regularity of the Weak Solutions of Some Stefan
Problems

11.1 Regularity of the Weak solutions of One-dimensional
Stefan Problems

11.2 Regularity of the Weak solutions of Multi-dimensional
Problems

11.2.1 The weak solutions of some two-phase Stefan
problems in Rn, n> 1

11.2.2 Regularity of the weak solutions of one-phase
Stefan problems in Rn, n> 1

Appendix A. Preliminaries

Appendix B. Some Function Spaces and norms

Appendix C. Fixed Point Theorems and Maximum Principles

Appendix D. Sobolev Spaces

Bibiography

Captions for Figures

Subject Index

## 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.

## Details

- No. of pages:
- 404

- Language:
- English

- Copyright:
- © JAI Press 2003

- Published:
- 22nd October 2003

- Imprint:
- JAI Press

- eBook ISBN:
- 9780080529165

- Hardcover ISBN:
- 9780444510860

## Reviews

The book is well organized, so that, in spite of its complexity, the exposition is seemingly effortless and reading is greatly facilitated by a very judicions mixing of phemenomenology and mathematics. A book like this, bridging physics and mathematics in a concrete and readable way, was very much needed. Prof. Gupta's book fulfills that task nicely."

Antonio Fasano (Firenze) in: Zentralblatt MATH Database 1931 - 2005.

## About the Authors

### S.C. Gupta Author

Professor S.C. Gupta retired in 1997 from the Department of Mathematics, Indian Institute of Science, Bangalore, India. He holds a PhD in Solid Mechanics and a DSc in “Analytical and Numerical Solutions of Free Boundary Problems.” His areas of research are inclusion and inhomogeneity problems, thermoelasticity, numerical computations, analytical and numerical solutions of free boundary problems and Stefan problems. He has published numerous articles in reputed international journals in many areas of his research.

### Affiliations and Expertise

Professor (Retired), Department of Mathematics, Indian Institute of Science, Bangalore, India

### S.C. Gupta Author

Professor S.C. Gupta retired in 1997 from the Department of Mathematics, Indian Institute of Science, Bangalore, India. He holds a PhD in Solid Mechanics and a DSc in “Analytical and Numerical Solutions of Free Boundary Problems.” His areas of research are inclusion and inhomogeneity problems, thermoelasticity, numerical computations, analytical and numerical solutions of free boundary problems and Stefan problems. He has published numerous articles in reputed international journals in many areas of his research.

### Affiliations and Expertise

Professor (Retired), Department of Mathematics, Indian Institute of Science, Bangalore, India