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. Within the framework of the
classical Stefan problem with the emphasis on the basic concepts,
modelling and analysis, it tries to include some weak
solutions and analytical and numerical solutions also. The main
considerations
behind this are the continuity and the clarity of
exposition. For example, the description of some phase-field
models in Chapter 4 arose
out of this need for a smooth transition
between topics. In the mathematical formulation of Stefan
problems, the curvature effects and
the kinetic condition are
incorporated with the help of the modified Gibbs-Thomson relation.
On the basis of some thermodynamical and
metallurgical
considerations, the modified Gibbs-Thomson relation can be
derived, as has been done in the text, but the rigorous
mathematical
justification comes from the fact that this relation
can be obtained by taking appropriate limits of phase-field
models. Because of the
unacceptability of some phase-field models
due their so-called thermodynamical inconsistency, some consistent
models have also been described.
This completes the discussion of
phase-field models in the present context.
Making this volume self-contained would require reporting
and
deriving several results from tensor analysis, differential
geometry, non-equilibrium thermodynamics, physics and functional
analysis.
The text is enriched with appropriate
references so as not to enlarge the scope of the book. The proofs
of propositions and theorems
are often lengthy and different from
one another. Presenting them in a condensed way may not be of much
help to the reader. Therefore
only the main features of proofs
and a few results have been presented to suggest the essential
flavour of the theme of investigation.
However at each place,
appropriate references have been cited so that inquisitive
readers can follow them on their own.
Each chapter
begins with basic concepts, objectives and the
directions in which the subject matter has grown. This is followed
by reviews - in some
cases quite detailed - of published works. In a
work of this type, the author has to make a suitable compromise
between length restrictions
and understandability.
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 τ → 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
Bibliographic & ordering Information Hardbound, 404 pages, publication date: OCT-2003
ISBN-13: 978-0-444-51086-0
ISBN-10: 0-444-51086-9
Imprint: JAI Price:Order form EUR 100 GBP 66.50 USD 114
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basic concepts, modelling and analysis
