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## Basic Concepts, Modelling and Analysis

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- 1st Edition, Volume 45 - October 22, 2003
- Author: S.C. Gupta
- Language: English
- Hardback ISBN:9 7 8 - 0 - 4 4 4 - 5 1 0 8 6 - 0
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 5 2 9 1 6 - 5

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 li… Read more

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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.

Chapter 1. The Stefan Problem and its Classical Formulation1.1 Some Stefan and Stefan-like Problems1.2 Free Boundary Problems with Free Boundaries of Codimension-two1.3 The Classical Stefan Problem in One-dimension and the Neumann Solution1.4 Classical Formulation of Multi-dimensional Stefan Problems1.4.1 Two-Phase Stefan problem in multipledimensions1.4.2 Alternate forms of the Stefan condition1.4.3 The Kirchhoff's transformation1.4.4 Boundary conditions at the fixed boundary1.4.5 Conditions at the free boundary1.4.6 The classical solution1.4.7 Conservation laws and the motion of the melt

Chapter 2. Thermodynamical and Metallurgical Aspects of Stefan Problems2.1 Thermodynamical Aspects2.1.1 Microscopic and macroscopic models2.1.2 Laws of classical thermodynamics2.1.3 Some thermodynamic variables and thermal parameters2.1.4 Equilibrium temperature; Clapeyron's equation2.2 Some Metallurgical Aspects of Stefan Problems2.2.1 Nucleation and supercooling2.2.2 The effect of interface curvature2.2.3 Nucleation of melting, effect of interface kinetics, and glassy solids2.3 Morphological Instability of the Solid-Liquid Interface2.4 Non-material Singular Surface : Generalized Stefan Condition

Chapter 3. Extended Classical Formulations of n-phase Stefan Problems with n>1 3.1 One-phase Problems3.1.1 An extended formulation of one-dimensional one-phase problem3.1.2 Solidification of supercooled liquid3.1.3 Multi-dimensional one-phase problems3.2 Extended Classical Formulations of Two-phase Stefan Problems3.2.1 An extended formulation of the one-dimensional two-phase problem3.2.2 Multi-dimensional Stefan problems of classes II and III3.2.3 Classical Stefan problems with n-phases, n> 23.2.4 Solidification with transition temperature range3.3 Stefan problems with Implicit Free Boundary Conditions3.3.1 Schatz transformations and implicit free boundary conditions3.3.2 Unconstrained and constrained oxygen-diffusion problem (ODP)

Chapter 4. Stefan Problem with Supercooling : Classical Formulation and Analysis4.1 Introduction4.2 A Phase-field Model for Solidification using Landau-Ginzburg Free Energy Functional4.3 Some Thermodynamically Consistent Phase-field and Phase Relaxation Models of Solidification4.4 Solidification of Supercooled Liquid Without Curvature Effect and Kinetic Undercooling : Analysis of the Solution4.4.1 One-dimensional one-phase solidification of supercooled liquid (SSP)4.4.2 Regularization of blow-up in SSP by looking at CODP4.4.3 Analysis of problems with changes in the initial and boundary conditions in SSP4.5 Analysis of Supercooled Stefan Problems with the Modified Gibbs-Thomson Relation4.5.1 Introduction4.5.2 One-dimensional one-phase supercooled Stefan problems with the modified Gibbs-Thomson relation4.5.3 One-dimensional two-phase Stefan problems with the modified Gibbs-Thomson relation4.5.4 Multi-dimensional supercooled Stefan problems and problems with the modified Gibbs-Thomson relation4.5.5 Weak formulation with supercooling and superheating effects

Chapter 5. Superheating due to Volumetric Heat Sources: Formulation and Analysis5.1 The Classical Enthalpy Formulation of a One-dimensional Problem5.2 The Weak Solution5.2.1 Weak solution and its relation to a classical solution5.2.2 Structure of the mushy region in the presence of heat sources5.3 Blow-up and Regularization

Chapter 6. Steady-State and Degenerate Classical Stefan Problems6.1 Some Steady-state Stefan Problems6.2 Degenerate Stefan Problems6.2.1 Quasi-static Stefan problem and its relation to the Hele-Shaw problem

Chapter 7. Elliptic and Parabolic Variational Inequalities7.1 Introduction7.2 The Elliptic Variational Inequality7.2.1 Definition and the basic function spaces7.2.2 Minimization of a functional7.2.3 The complementarity problem7.2.4 Some existence and uniqueness results concerning elliptic inequalities7.2.5 Equivalence of different inequality formulations of an obstacle problem of the string7.3 The Parabolic Variational Inequality7.3.1 Formulation in appropriate spaces7.4 Some Variational Inequality Formulations of Classical Stefan Problems7.4.1 One-phase Stefan problems7.4.2 A Stefan problem with a quasi-variational inequality formulation7.4.3 The variational inequality formulation of a two-phase Stefan problem

Chapter 8. The Hyperbolic Stefan Problem8.1 Introduction8.1.1 Relaxation time and relaxation models8.2 Model I : Hyperbolic Stefan Problem with Temperature Continuity at the Interface8.2.1 The mathematical formulation8.2.2 Some existence, uniqueness and well-posedness results8.3 Model II : Formulation with Temperature Discontinuity at the Interface8.3.1 The mathematical formulation8.3.2 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 Heats8.4.1 The Clasical and the Weak FormulationsChapter 9. Inverse Stefan Problems9.1 Introduction9.2 Well-posedness of the solution9.2.1 Approximate solutions9.3 Regularization9.3.1 The regularizing operator and generalized discrepancy principle9.3.2 The generalized inverse9.3.3 Regularization methods9.3.4 Rate of convergence of a regularization method9.4 Determination of Unknown Parameters in Inverse Stefan Problems9.4.1 Unknown parameters in the one-phase Stefan problems9.4.2 Determination of Unknown parameters in the two-phase Stefan problems9.5 Regularization of Inverse Heat Conduction Problems by Imposing Suitable Restrictions on the solution9.6 Regularization of Inverse Stefan Problems Formulated as Equations in the form of Convolution Integrals9.7 Inverse Stefan Problems Formulated as Defect Minimization Problems

Chapter 10. Analysis of the Classical Solutions of Stefan Problems10.1 One-dimensional One-phase Stefan Problems10.1.1 Analysis using integral equation formulations10.1.2 Infinite differentiability and analyticity of the free boundary10.1.3 Unilateral boundary conditions on the boundary: Analysis using finite-difference schemes10.1.4 Cauchy-type free boundary conditions10.1.5 Existence of self-similar solutions of some Stefan problems10.1.6 The effect of density change10.2 One-dimensional Two-phase Stefan Problems10.2.1 Existence, uniqueness and stability results10.2.2 Differentiability and analyticity of the free boundary in the one-dimensional two-phase Stefan problems10.2.3 One-dimensional n-phase Stefan problems with n > 210.3 Analysis of the Classical Solutions of Multi-dimensional Stefan Problems10.3.1 Existence and uniqueness results valid for a short time10.3.2 Existence of the classical solution on an arbitrary time intervalChapter 11. Regularity of the Weak Solutions of Some Stefan Problems11.1 Regularity of the Weak solutions of One-dimensional Stefan Problems11.2 Regularity of the Weak solutions of Multi-dimensional Problems11.2.1 The weak solutions of some two-phase Stefan problems in R^{n}, n> 111.2.2 Regularity of the weak solutions of one-phase Stefan problems in R^{n}, n> 1Appendix A. PreliminariesAppendix B. Some Function Spaces and normsAppendix C. Fixed Point Theorems and Maximum PrinciplesAppendix D. Sobolev Spaces

- No. of pages: 404
- Language: English
- Edition: 1
- Volume: 45
- Published: October 22, 2003
- Imprint: JAI Press
- Hardback ISBN: 9780444510860
- eBook ISBN: 9780080529165

SG

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, IndiaRead *The Classical Stefan Problem* on ScienceDirect