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# Volume 45. The Classical Stefan Problem

## 1st Edition

**basic concepts, modelling and analysis**

**basic concepts, modelling and analysis**

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 som

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 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.

- No. of pages:
- 404

- Language:
- English

- Copyright:
- © 2003

- Published:
- 22nd October 2003

- Imprint:
- JAI Press

- Print ISBN:
- 9780444510860

- Electronic ISBN:
- 9780080529165

Professor S.C. Gupta obtained his DSc degree from the Indian Institute of Science in Bangalore, India. In 1997 he retired from the Indian Institute of Science in Bangalore. 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 in international scientific journals such as International J. of Heat and Mass Transfer, Quart. of Applied Mathematics. AIAA J., Int. J. of Eng. Sci., and Computational Methods in Applied Math. and Eng.

Department of Mathematics, Indian Institute of Science, Bangalore, India

Department of Mathematics, Indian Institute of Science, Bangalore, India

The book is well organized, so that, in spite of its complexity, the exposition is seemingly effortless and reading is greatly facilitated by a very judicions mixing of phemenomenology and mathematics. A book like this, bridging physics and mathematics in a concrete and readable way, was very much needed. Prof. Gupta's book fulfills that task nicely."

Antonio Fasano (Firenze) in: Zentralblatt MATH Database 1931 - 2005.

Antonio Fasano (Firenze) in: Zentralblatt MATH Database 1931 - 2005.