The Classical Stefan Problem
basic concepts, modelling and analysis
By S.C. Gupta
This volume emphasises studies related toclassical Stefan problems. The term "Stefan problem" isgenerally used for heat transfer problems with phasechanges 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 phasefieldmodels 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 GibbsThomson relation.On the basis of some thermodynamical and metallurgicalconsiderations, the modified GibbsThomson 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 phasefieldmodels. Because of the unacceptability of some phasefield modelsdue their socalled thermodynamical inconsistency, some consistentmodels have also been described. This completes the discussion ofphasefield models in the present context.Making this volume selfcontained would require reporting andderiving several results from tensor analysis, differentialgeometry, nonequilibrium 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.
Hardbound, 404 Pages
Published: October 2003
Imprint: Jai Press (elsevier)
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.
Contents
 Chapter 1. The Stefan Problem and its Classical Formulation
1.1 Some Stefan and Stefanlike Problems
1.2 Free Boundary Problems with Free Boundaries of Codimension two
1.3 The Classical Stefan Problem in Onedimension and the Neumann Solution
1.4 Classical Formulation of Multidimensional Stefan Problems
1.4.1 TwoPhase 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 SolidLiquid Interface
2.4 Nonmaterial Singular Surface : Generalized Stefan Condition
Chapter 3. Extended Classical Formulations of nphase Stefan Problems with n>1
3.1 Onephase Problems
3.1.1 An extended formulation of onedimensional one phase problem
3.1.2 Solidification of supercooled liquid
3.1.3 Multidimensional onephase problems
3.2 Extended Classical Formulations of Twophase Stefan Problems
3.2.1 An extended formulation of the onedimensional twophase problem
3.2.2 Multidimensional Stefan problems of classes II and III
3.2.3 Classical Stefan problems with nphases, 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 oxygendiffusion problem (ODP)
Chapter 4. Stefan Problem with Supercooling : Classical Formulation and Analysis
4.1 Introduction
4.2 A Phasefield Model for Solidification using Landau Ginzburg Free Energy Functional
4.3 Some Thermodynamically Consistent Phasefield 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 Onedimensional onephase solidification of supercooled liquid (SSP)
4.4.2 Regularization of blowup 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 GibbsThomson Relation
4.5.1 Introduction
4.5.2 Onedimensional onephase supercooled Stefan problems with the modified GibbsThomson relation
4.5.3 Onedimensional twophase Stefan problems with the modified GibbsThomson relation
4.5.4 Multidimensional supercooled Stefan problems and problems with the modified GibbsThomson 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 Onedimensional 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 Blowup and Regularization
Chapter 6. SteadyState and Degenerate Classical Stefan Problems
6.1 Some Steadystate Stefan Problems
6.2 Degenerate Stefan Problems
6.2.1 Quasistatic Stefan problem and its relation to the HeleShaw 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 Onephase Stefan problems
7.4.2 A Stefan problem with a quasivariational 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 wellposedness 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 Wellposedness 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 onephase 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 Onedimensional Onephase 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 finitedifference schemes
10.1.4 Cauchytype free boundary conditions
10.1.5 Existence of selfsimilar solutions of some Stefan problems
10.1.6 The effect of density change
10.2 Onedimensional Twophase Stefan Problems
10.2.1 Existence, uniqueness and stability results
10.2.2 Differentiability and analyticity of the free boundary in the onedimensional twophase Stefan problems
10.2.3 Onedimensional nphase Stefan problems with n > 2
10.3 Analysis of the Classical Solutions of Multidimensional 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 Onedimensional Stefan Problems
11.2 Regularity of the Weak solutions of Multidimensional Problems
11.2.1 The weak solutions of some twophase Stefan problems in Rn, n> 1
11.2.2 Regularity of the weak solutions of onephase 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