The Classical Stefan Problem - Basic Concepts, Modelling and Analysis

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 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