| Description |
1 online resource (xvii, 385 pages) : illustrations |
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text txt rdacontent |
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computer c rdamedia |
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online resource cr rdacarrier |
| Series |
North-Holland series in applied mathematics and mechanics, 0167-5931 ; v. 45 |
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North-Holland series in applied mathematics and mechanics ; v. 45. 0167-5931
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| Summary |
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 |
| Bibliography |
Includes bibliographical references (pages 355-380) and index. |
| Note |
Print version record. |
| Contents |
Cover -- 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 -- Chapter 2. Thermodynamical and Metallurgical Aspects of Stefan Problems -- 2.1 Thermodynamical Aspects -- 2.2 Some Metallurgical Aspects of Stefan Problems -- 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 -- 3.2 Extended Classical Formulations of Two-phase Stefan Problems -- 3.3 Stefan problems with Implicit Free Boundary Conditions -- 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 -- 4.5 Analysis of Supercooled Stefan Problems with the Modified Gibbs Thomson Relation -- Chapter 5. Superheating due to Volumetric Heat Sources: The Formulation and Analysis -- 5.1 The Classical Enthalpy Formulation of a One-dimensional Problem -- 5.2 The Weak Solution -- 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 -- Chapter 7. Elliptic and Parabolic Variational Inequalities -- 7.1 Introduction -- 7.2 The Elliptic Variational Inequality -- 7.3 The Parabolic Variational Inequality -- 7.4 Some Variational Inequality Formulations of Classical Stefan Problems -- Chapter 8. The Hyperbolic Stefan Problem -- 8.1 Introduction -- 8.2 Model I: Hyperbolic Stefan Problem with Temperature Continuity at the Interface -- 8.3 Model II: Formulation with Temperature Discontinuity at the Interface -- 8.4 Model III: Delay in the Response of Energy to Latent and Sensible Heats -- Chapter 9. Inverse Stefan Problems -- 9.1 Introduction -- 9.2 Well-posedness of the solution -- 9.3 Regularization -- 9.4 Determination of Unknown Parameters in Inverse 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 -- 10.2 One-dimensional Two-phase Stefan Problems -- 10.3 Analysis of the Classical Solutions of Multi-dimensional Stefan Problems -- Chapter 11. Regularity of the Weak Solutions of Some Stefan Problems -- 11.1 Regularity of the Weak solutions of One-dimensional Stefan Probl. |
| Subject |
Heat -- Transmission.
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Boundary value problems.
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Chaleur -- Transmission.
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Problèmes aux limites.
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heat transmission.
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SCIENCE -- Mechanics -- Thermodynamics.
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Boundary value problems
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Heat -- Transmission
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Transferência de calor.
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| Other Form: |
Print version: Gupta, S.C., Ph. D., D. Sc. Classical Stefan problem. Amsterdam ; Boston : Elsevier, 2003 0444510869 9780444510860 (DLC) 2003054657 (OCoLC)52553777 |
| ISBN |
9780444510860 |
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0444510869 |
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9780080529165 (electronic bk.) |
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008052916X (electronic bk.) |
| Standard No. |
AU@ 000054979300 |
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CHNEW 001005982 |
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DEBBG BV042315743 |
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DEBSZ 405304226 |
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DEBSZ 482353775 |
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NZ1 12433156 |
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NZ1 15191138 |
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