{"product_id":"9780080529165","title":"The Classical Stefan Problem: basic concepts, modelling and analysis","description":"This volume emphasises studies related to\u003cbr\u003eclassical Stefan problems. The term \"Stefan problem\" is\u003cbr\u003egenerally used for heat transfer problems with \u003cbr\u003ephase-changes such\u003cbr\u003eas from the liquid to the solid. Stefan problems have some\u003cbr\u003echaracteristics that are typical of them, but certain problems\u003cbr\u003earising in fields such as mathematical physics and engineering\u003cbr\u003ealso exhibit characteristics similar to them. The term\u003cbr\u003e''classical\" distinguishes the formulation of these problems from\u003cbr\u003etheir weak formulation, in which the solution need not possess\u003cbr\u003eclassical derivatives. Under suitable assumptions, a weak solution\u003cbr\u003ecould be as good as a classical solution. In hyperbolic Stefan\u003cbr\u003eproblems, the characteristic features of Stefan problems are\u003cbr\u003epresent but unlike in Stefan problems, discontinuous solutions are\u003cbr\u003eallowed because of the hyperbolic nature of the heat equation. The\u003cbr\u003enumerical solutions of inverse Stefan problems, and the analysis of\u003cbr\u003edirect Stefan problems are so integrated that it is difficult to\u003cbr\u003ediscuss one without referring to the other. So no strict line of\u003cbr\u003edemarcation can be identified between a classical Stefan problem\u003cbr\u003eand other similar problems. On the other hand, including every\u003cbr\u003erelated problem in the domain of classical Stefan problem would\u003cbr\u003erequire several volumes for their description. A suitable\u003cbr\u003ecompromise has to be made.\u003cbr\u003eThe basic concepts, modelling, and analysis of the classical\u003cbr\u003eStefan problems have been extensively investigated and there seems\u003cbr\u003eto be a need to report the results at one place. This book\u003cbr\u003eattempts to answer that need. Within the framework of the\u003cbr\u003eclassical Stefan problem with the emphasis on the basic concepts,\u003cbr\u003emodelling and analysis, it tries to include some weak\u003cbr\u003esolutions and analytical and numerical solutions also. The main\u003cbr\u003econsiderations behind this are the continuity and the clarity of\u003cbr\u003eexposition. For example, the description of some phase-field\u003cbr\u003emodels in Chapter 4 arose out of this need for a smooth transition\u003cbr\u003ebetween topics. In the mathematical formulation of Stefan\u003cbr\u003eproblems, the curvature effects and the kinetic condition are\u003cbr\u003eincorporated with the help of the modified Gibbs-Thomson relation.\u003cbr\u003eOn the basis of some thermodynamical and metallurgical\u003cbr\u003econsiderations, the modified Gibbs-Thomson relation can be\u003cbr\u003ederived, as has been done in the text, but the rigorous\u003cbr\u003emathematical justification comes from the fact that this relation\u003cbr\u003ecan be obtained by taking appropriate limits of phase-field\u003cbr\u003emodels. Because of the unacceptability of some phase-field models\u003cbr\u003edue their so-called thermodynamical inconsistency, some consistent\u003cbr\u003emodels have also been described. This completes the discussion of\u003cbr\u003ephase-field models in the present context.\u003cbr\u003eMaking this volume self-contained would require reporting and\u003cbr\u003ederiving several results from tensor analysis, differential\u003cbr\u003egeometry, non-equilibrium thermodynamics, physics and functional\u003cbr\u003eanalysis. The text is enriched with appropriate\u003cbr\u003ereferences so as not to enlarge the scope of the book. The proofs\u003cbr\u003eof propositions and theorems are often lengthy and different from\u003cbr\u003eone another. Presenting them in a condensed way may not be of much\u003cbr\u003ehelp to the reader. Therefore only the main features of proofs\u003cbr\u003eand a few results have been presented to suggest the essential\u003cbr\u003eflavour of the theme of investigation. However at each place,\u003cbr\u003eappropriate references have been cited so that inquisitive\u003cbr\u003ereaders can follow them on their own.\u003cbr\u003eEach chapter begins with basic concepts, objectives and the\u003cbr\u003edirections in which the subject matter has grown. This is followed\u003cbr\u003eby reviews - in some cases quite detailed - of published works. In a\u003cbr\u003ework of this type, the author has to make a suitable compromise\u003cbr\u003ebetween length restrictions and understandability. \u003cbr\u003e\u003cbr\u003e","brand":"Elsevier Science","offers":[{"title":"Default Title","offer_id":47119232794864,"sku":"9780080529165","price":126.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0737\/7593\/9824\/files\/9780080529165_p0.jpg?v=1763638209","url":"https:\/\/shop-qa.barnesandnoble.com\/products\/9780080529165","provider":"Barnes \u0026 Noble (DEV)","version":"1.0","type":"link"}