# 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

## 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.

## 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**

## Ratings and Reviews

## 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