# The Classical Stefan Problem

**basic concepts, modelling and analysis**

**By**

- S.C. Gupta, Department of Mathematics, Indian Institute of Science, Bangalore, India
- S.C. Gupta, Department of Mathematics, Indian Institute of Science, Bangalore, India

This volume emphasises studies related toclassical Stefan problems. The term "Stefan problem" isgenerally used for heat transfer problems with phase-changes suchas from the liquid to the solid. Stefan problems have somecharacteristics that are typical of them, but certain problemsarising in fields such as mathematical physics and engineeringalso exhibit characteristics similar to them. The term``classical" distinguishes the formulation of these problems fromtheir weak formulation, in which the solution need not possessclassical derivatives. Under suitable assumptions, a weak solutioncould be as good as a classical solution. In hyperbolic Stefanproblems, the characteristic features of Stefan problems arepresent but unlike in Stefan problems, discontinuous solutions areallowed because of the hyperbolic nature of the heat equation. Thenumerical solutions of inverse Stefan problems, and the analysis ofdirect Stefan problems are so integrated that it is difficult todiscuss one without referring to the other. So no strict line ofdemarcation can be identified between a classical Stefan problemand other similar problems. On the other hand, including everyrelated problem in the domain of classical Stefan problem wouldrequire several volumes for their description. A suitablecompromise has to be made.The basic concepts, modelling, and analysis of the classicalStefan problems have been extensively investigated and there seemsto be a need to report the results at one place. This bookattempts to answer that need. Within the framework of theclassical Stefan problem with the emphasis on the basic concepts,modelling and analysis, it tries to include some weaksolutions and analytical and numerical solutions also. The mainconsiderations behind this are the continuity and the clarity ofexposition. For example, the description of some phase-fieldmodels in Chapter 4 arose out of this need for a smooth transitionbetween topics. In the mathematical formulation of Stefanproblems, the curvature effects and the kinetic condition areincorporated with the help of the modified Gibbs-Thomson relation.On the basis of some thermodynamical and metallurgicalconsiderations, the modified Gibbs-Thomson relation can bederived, as has been done in the text, but the rigorousmathematical justification comes from the fact that this relationcan be obtained by taking appropriate limits of phase-fieldmodels. Because of the unacceptability of some phase-field modelsdue their so-called thermodynamical inconsistency, some consistentmodels have also been described. This completes the discussion ofphase-field models in the present context.Making this volume self-contained would require reporting andderiving several results from tensor analysis, differentialgeometry, non-equilibrium thermodynamics, physics and functionalanalysis. The text is enriched with appropriatereferences so as not to enlarge the scope of the book. The proofsof propositions and theorems are often lengthy and different fromone another. Presenting them in a condensed way may not be of muchhelp to the reader. Therefore only the main features of proofsand a few results have been presented to suggest the essentialflavour of the theme of investigation. However at each place,appropriate references have been cited so that inquisitivereaders can follow them on their own.Each chapter begins with basic concepts, objectives and thedirections in which the subject matter has grown. This is followedby reviews - in some cases quite detailed - of published works. In awork of this type, the author has to make a suitable compromisebetween length restrictions and understandability.

View full description### Book information

- Published: October 2003
- Imprint: JAI Press (Elsevier)
- ISBN: 978-0-444-51086-0

### 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.

### Table of Contents

Chapter 1. The Stefan Problem and its Classical Formulation1.1 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 R

^{n}, n> 1

11.2.2 Regularity of the weak solutions of one-phase Stefan problems in R

^{n}, 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