Classical Stefan Problem: Basic Concepts, Modelling and Analysis, Volume 45 [Kietas viršelis]

(Naujas leidimas: 9780444635815)
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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. 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.

Recenzijos

"...a valuable compendium of methodologies for the Stefan problem..." J.R. Ockendon, in: (Journal of Fluid Mechanics, Vol. 520, 2004)

The Stefan Problem and its Classical Formulation
1(38)
Some Stefan and Stefan-like Problems
1(17)
Free Boundary Problems with Free Boundaries of Codimension-two
18(1)
The Classical Stefan Problem in One-dimension and the Neumann Solution
19(4)
Classical Formulation of Multi-dimensional Stefan Problems
23(16)
Two-Phase Stefan problem in multipledimensions
23(5)
Alternate forms of the Stefan condition
28(1)
The Kirchhoff's transformation
29(1)
Boundary conditions at the fixed boundary
30(3)
Conditions at the free boundary
33(1)
The classical solution
34(1)
Conservation laws and the motion of the melt
35(4)
Thermodynamical and Metallurgical Aspects of Stefan Problems
39(22)
Thermodynamical Aspects
39(8)
Microscopic and macroscopic models
39(1)
Laws of classical thermodynamics
40(2)
Some thermodynamic variables and thermal parameters
42(3)
Equilibrium temperature; Clapeyron's equation
45(2)
Some Metallurgical Aspects of Stefan Problems
47(5)
Nucleation and supercooling
47(2)
The effect of interface curvature
49(2)
Nucleation of melting, effect of interface kinetics, and glassy solids
51(1)
Morphological Instability of the Solid--Liquid Interface
52(3)
Non-material Singular Surface: Generalized Stefan Condition
55(6)
Extended Classical Formulations of n-phase Stefan Problems with n ≥ 1
61(24)
One-phase Problems
61(5)
An extended formulation of one-dimensional one-phase problem
61(2)
Solidification of supercooled liquid
63(1)
Multi-dimensional one-phase problems
64(2)
Extended Classical Formulations of Two-phase Stefan Problems
66(10)
An extended formulation of the one-dimensional two-phase problem
66(2)
Multi-dimensional Stefan problems of classes II and III
68(2)
Classical Stefan problems with n-phases, n > 2
70(3)
Solidification with transition temperature range
73(3)
Stefan problems with Implicit Free Boundary Conditions
76(9)
Schatz transformations and implicit free boundary conditions
77(4)
Unconstrained and constrained oxygen-diffusion problem (ODP)
81(4)
Stefan Problem with Supercooling: Classical Formulation and Analysis
85(44)
Introduction
85(1)
A Phase-field Model for Solidification using Landau--Ginzburg Free Energy Functional
86(8)
Some Thermodynamically Consistent Phase-field and Phase Relaxation Models of Solidification
94(10)
Solidification of Supercooled Liquid Without Curvature Effect and Kinetic Undercooling: Analysis of the Solution
104(8)
One-dimensional one-phase solidification of supercooled liquid (SSP)
104(3)
Regularization of blow-up in SSP by looking at CODP
107(2)
Analysis of problems with changes in the initial and boundary conditions in SSP
109(3)
Analysis of Supercooled Stefan Problems with the Modified Gibbs--Thomson Relation
112(17)
Introduction
112(1)
One-dimensional one-phase supercooled Stefan problems with the modified Gibbs--Thomson relation
113(2)
One-dimensional two-phase Stefan problems with the modified Gibbs--Thomson relation
115(6)
Multi-dimensional supercooled Stefan problems and problems with the modified Gibbs--Thomson relation
121(5)
Weak formulation with supercooling and superheating effects
126(3)
Superheating due to Volumetric Heat Sources: The Formulation and Analysis
129(13)
The Classical Enthalpy Formulation of a One-dimensional Problem
129(4)
The Weak Solution
133(6)
Weak solution and its relation to a classical solution
133(4)
Structure of the mushy region in the presence of heat sources
137(2)
Blow-up and Regularization
139(3)
Steady-State and Degenerate Classical Stefan Problems
142(6)
Some Steady-state Stefan Problems
142(1)
Degenerate Stefan Problems
143(5)
Quasi-static Stefan problem and its Relation to the Hele-Shaw problem
146(2)
Elliptic and Parabolic Variational Inequalities
148(48)
Introduction
148(1)
The Elliptic Variational Inequality
149(21)
Definition and the basic function spaces
149(3)
Minimization of a functional
152(1)
The complementarity problem
153(3)
Some existence and uniqueness results concerning elliptic inequalities
156(10)
Equivalence of different inequality formulations of an obstacle problem of the string
166(4)
The Parabolic Variational Inequality
170(4)
Formulation in appropriate spaces
170(4)
Some Variational Inequality Formulations of Classical Stefan Problems
174(22)
One-phase Stefan problems
174(13)
A Stefan problem with a quasi-variational inequality formulation
187(4)
The variational inequality formulation of a two-phase Stefan problem
191(5)
The Hyperbolic Stefan Problem
196(28)
Introduction
196(3)
Relaxation time and relaxation models
197(2)
Model I: Hyperbolic Stefan Problem with Temperature Continuity at the Interface
199(8)
The mathematical formulation
199(4)
Some existence, uniqueness and well-posedness results
203(4)
Model II: Formulation with Temperature Discontinuity at the Interface
207(9)
The mathematical formulation
207(2)
The existence and uniqueness of the solution and its convergence as τ → 0
209(7)
Model III: Delay in the Response of Energy to Latent and Sensible Heats
216(8)
The classical and the weak formulations
216(8)
Inverse Stefan Problems
224(47)
Introduction
224(1)
Well-posedness of the solution
225(6)
Approximate solutions
229(2)
Regularization
231(13)
The regularizing operator and generalized discrepancy principle
231(3)
The generalized inverse
234(1)
Regularization methods
235(8)
Rate of convergence of a regularization method
243(1)
Determination of Unknown Parameters in Inverse Stefan Problems
244(5)
Unknown parameters in the one-phase Stefan problems
245(3)
Determination of unknown parameters in the two-phase Stefan problems
248(1)
Regularization of Inverse Heat Conduction Problems by Imposing Suitable Restrictions on the solution
249(3)
Regularization of Inverse Stefan Problems Formulated as Equations in the form of Convolution Integrals
252(5)
Inverse Stefan Problems Formulated as Defect Minimization Problems
257(14)
Analysis of the Classical Solutions of Stefan Problems
271(51)
One-dimensional One-phase Stefan Problems
272(28)
Analysis using integral equation formulations
272(14)
Infinite differentiability and analyticity of the free boundary
286(5)
Unilateral boundary conditions on the fixed boundary: Analysis using finite-difference schemes
291(4)
Cauchy-type free boundary conditions
295(3)
Existence of self-similar solutions of some Stefan problems
298(1)
The effect of density change
299(1)
One-dimensional Two-phase Stefan Problems
300(15)
Existence, uniqueness and stability results
300(11)
Differentiability and analyticity of the free boundary in the one-dimensional two-phase Stefan problems
311(1)
One-dimensional n-phase Stefan problems with n > 2
312(3)
Analysis of the Classical Solutions of Multi-dimensional Stefan Problems
315(7)
Existence and uniqueness results valid for a short time
315(6)
Existence of the classical solution on an arbitrary time interval
321(1)
Regularity of the Weak Solutions of Some Stefan Problems
322(16)
Regularity of the Weak solutions of One-dimensional Stefan Problems
322(7)
Regularity of the Weak solutions of Multi-dimensional Stefan Problems
329(9)
The weak solutions of some two-phase Stefan problems in Rn, n > 1
329(4)
Regularity of the weak solutions of one-phase Stefan problems in Rn, n > 1
333(5)
Appendix A. Preliminaries 338(7)
Appendix B. Some Function Spaces and Norms 345(4)
Appendix C. Fixed Point Theorems and Maximum Principles 349(2)
Appendix D. Sobolev Spaces 351(4)
Bibliography 355(26)
Captions for Figures 381(2)
Subject Index 383


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.