DOCSLIB.ORG

The Stefan Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Stefan Problem 10.1090/mmono/027 TRANSLATIONS OF MATHEMATICAL MONOGRAPHS Volume 27 THE STEFAN PROBLEM by L I. RUBENSTEIN AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 02904 1971 nPOBJIEMA CTE$AH A JI. M . PYBWHU1TEMH JIaTBMMCKMM rocyAapcTBeHHH M yHMBepcMTe T MMeHw nETPA CTY^IK M BbHMCJIMTEJlbHblM UEHT P M3AaTejibCTB0 „3Baiir3He " Pural967 Translated fro m the Russian by A. D. Solomon Library o f Congress Card Number 75-168253 International Standard Book Number 0-8218-1577- 6 Copyright© 197 1 by the American Mathematical Society Printed in the United States of America All Rights Reserved May not be reproduced in any form without permission of the publishers PREFACE In recen t year s w e hav e witnesse d a n intensiv e developmen t o f th e Stefan problem , wit h whic h th e autho r ha s lon g bee n concerned . This monograp h wa s originall y intende d t o examin e th e importan t results concerned wit h th e classica l Stefa n proble m an d it s generaliza - tions, up to the present time, but thi s progra m coul d no t b e carrie d ou t completely, first of all because the literature concerned with this problem is continuousl y increasing , an d ver y importan t work , i n whic h th e Stefan problem i s treated a s a problem o f numerica l analysis , ha s bee n only partiall y publishe d an d partiall y announce d i n publication s an d at scientifi c conferences; 1* a t th e sam e tim e th e wor k planne d fo r ou r monograph was to a large extent complete . I n thi s regar d th e numerica l solution o f problem s o f Stefa n typ e i s usuall y discusse d i n monograph s in an extremely concis e manner. We will limit ourselve s t o brie f descrip - tions o f finite differenc e algorithm s whic h hav e bee n propose d fo r th e solution o f th e Stefa n proble m (Par t 2 , Chapte r VIII) ; a s a rul e w e will not give a justification o f th e algorith m bu t refe r th e reade r t o th e original literature. The autho r is , o n th e on e hand , mainl y intereste d i n the physical formulatio n o f a problem which reduces to some form of the Stefan problem (Par t 1 , Chapters I—IV) and , on the other hand , i n th e examination of general theoretical problems, i.e. problems of a qualitative character (Par t 2 , Chapter s I—VII) . I n Supplemen t 3 w e presen t illustrative examples o f th e numerica l solutio n t o problem s examine d i n the first part . I n th e first tw o supplement s w e presen t certai n know n results whic h ar e use d i n th e mai n part s o f th e book . At th e en d o f th e boo k w e includ e a bibliograph y o f th e problem . Unfortunately, man y source s wer e no t availabl e t o th e author . I t i s probable tha t man y work s exis t whic h hav e excape d hi s attention . Therefore w e mak e n o clai m o f completenes s fo r th e bibliography . The autho r take s thi s opportunit y t o expres s hi s appreciatio n t o his colleague s N . Avdoijins , M . Antimirov , A . Kuikis , H . German , We have in mind the joint work of B. M . Budak, F. P. VasiTev and A. B. Uspenskil, as presented in [15 ] and announced in their articles in a number of reports of conference s on numerica l analysi s (Mosco w Stat e Univ. , 1965) . in iv PREFACE E. Enikeeva, M. Zavorina, B. Martuzans, E. Pi^enkova and A. Skroman, who checked the accuracy o f th e numerica l solution s o f th e example s o f Supplement 3 and aide d th e autho r i n eliminatin g man y errors , a s wel l as t o E . Riekstiij s an d V . Abolinja , wh o rea d th e manuscrip t o f th e monograph. Th e autho r expresse s specia l thank s t o th e Recto r o f th e Stucki Institut e o f th e Latvia n Stat e University , Professo r V . A . Steinberg, an d t o th e directo r o f th e Universit y computin g center , Docent E . I . Arin , withou t whos e efficien t ai d th e monograp h coul d not hav e bee n brough t t o completion . L. RUBINSTEI N TABLE O F CONTENT S PREFACE ii i INTRODUCTION 1 1. Historica l survey 1 2. Notatio n and terminolog y 1 5 PART ONE . Physica l Problems Reducing to Problems of the Stefan Typ e CHAPTER I . DIFFUSIO N O F HEA T B Y CONDUCTIO N I N A MEDIU M WITH A CHANGE O F PHASE STAT E 1 8 1. Th e Stefan condition s 1 8 2. Freezin g o f the ground 2 0 3. Crystallizatio n o f a melt when a plate is immersed i n it . 2 5 4. Formatio n o f a continuous ingot 3 2 5. Zona l noncrucible melting o f a cylindrical rod 3 7 CHAPTER II . THERMA L DIFFUSIO N PROCESSE S I N A MEDIU M WITH VARIABL E PHAS E STAT E 3 9 1. Dissolutio n o f a gas bubble in a liquid 3 9 2. Dynamic s o f one-dimensiona l nonisotherma l evaporatio n of an ideal liquid mixture 4 3 3. Crystallizatio n o f a binary allo y 5 2 CHAPTER III . PROBLEM S O F THE THEOR Y O F FILTRATION 6 1 1. Forcin g o f a hydrauli c solutio n int o th e groun d (proble m ofVerigin) 6 1 2. Advanc e o f a water-oil interface i n an elastic regime 6 3 3. Invers e problems 7 1 CHAPTER IV . SOM E PROBLEM S O F MECHANIC S O F CONTINUOU S MEDIA THA T REDUCE T O PROBLEMS OF STEFAN TYP E ... 8 1 1. A proble m o f convectio n arisin g fro m crystallizatio n o f a supercooled melt 8 1 2. Hig h speed flow o f a soli d bod y i n a viscou s incompressibl e fluid 8 5 3. Nonstationar y flows o f a viscous plastic medium 8 9 v VI CONTENTS PART TWO . Th e Classical Stefan Proble m and its Generalizations CHAPTER I . TH E SINGLE-PHAS E STEFA N PROBLE M WIT H STRON G NONLINEARITY 9 4 1. Statemen t o f the problem. Formulation o f the basic results. 9 4 2. Reductio n to an integral equation. Theorem o f equivalence. 9 7 3. Existenc e o f the solution in the small 10 4 4. Uniquenes s of the solution 11 4 5. Stabilit y o f the solution 13 0 6. Furthe r observations 13 5 CHAPTER II . TH E CLASSICA L TWO-PHAS E STEFA N PROBLEM . TH E CASE O F A TWO-PHAS E INITIA L STAT E WIT H CONTINUOU S AGREEMENT O F BOUNDARY AN D INITIA L CONDITION S 14 1 1. Statemen t o f the problem 14 1 2. Th e first boundar y proble m o n a segmen t fo r a two-phas e initial stat e an d continuou s agreemen t o f boundar y an d initial conditions. Existence o f the solution in the large . 14 3 3. Asymptoti c behavior o f the solution to Problem B x 15 5 4. Degeneratio n o f one of the phases for t > t 0 16 1 5. Th e thir d boundar y problem . Existenc e o f th e solutio n in the large 16 4 6. Th e thir d boundar y problem . Asymptoti c behavio r o f the solution 17 0 7. Th e Cauchy-Stefan proble m 18 1 CHAPTER III . TH E INITIA L VELOCIT Y O F ADVANC E O F TH E INTER - PHASE BOUNDAR Y 190 1. Th e integral equation s o f Problem s B x an d A x withou t th e continuous agreement o f boundary and initial data 190 2. Initia l spee d o f th e interphas e boundary . Formulatio n o f the result 19 5 3. ProofofTheoremll.Thecase/xCO ) ^ 0 19 9 4. ProofofTheoremll.CaseACO ) = 0 20 5 5. Proo f o f Theorem 1 2 21 3 CHAPTER IV . SOLUTIO N O F PROBLEMS A X AND A 3 21 7 1. Th e first boundary problem (A x). Subsidiary function s . 21 7 2. Existenc e o f a solution to problem Ai 22 1 3. Th e third boundary problem (A 3) 23 1 CONTENTS vi i CHAPTER V . TH E DOUBLE-LAYE R STEFA N PROBLE M WIT H DE - GENERATION O F ON E O F TH E PHASE S A T TH E INITIA L MOMENT AN D A DISCONTINUIT Y I N TH E BOUNDAR Y AN D INITIAL CONDITION S 23 8 1.
Load more

Recommended publications
  • THE STEFAN PROBLEM with ARBITRARY INITIAL and BOUNDARY CONDITIONS* By
    QUARTERLY OF APPLIED MATHEMATICS 223 OCTOBER 1978 THE STEFAN PROBLEM WITH ARBITRARY INITIAL AND BOUNDARY CONDITIONS* By L. N. TAO Illinois Institute of Technology Abstract. The paper is concerned with the free boundary problem of a semi-infinite body with an arbitrarily prescribed initial condition and an arbitrarily prescribed bound- ary condition at its face. An analytically exact solution of the problem is established, which is expressed in terms of some functions and polynomials of the similarity variable x/tin and time t. Convergence of the series solution is considered and proved. Hence the solution also serves as an existence proof. Some special initial and boundary conditions are discussed, which include the Neumann problem and the one-phase problem as special cases. 1. Introduction. Diffusive processes with a change of phase of the materials occur in many scientific and engineering problems. They are known as Stefan problems or free boundary problems. Many examples of these problems can be given, e.g. the melting or freezing of an ice-water combination, the crystallization of a binary alloy or dissolution of a gas bubble in a liquid. To find the solutions to this class of problems has been the subject of investigations by many researchers. Because of the presence of a moving boundary between the two phases, the problem is nonlinear. Various mathematical methods and techniques have been used to study free boundary problems. They have been discussed and summarized in several books [1-5] and many survey papers [6-9]. Free boundary problems have been studied since the nineteenth century by Lame and Clapeyron in 1831, Neumann in the 1860s and Stefan in 1889.
    [Show full text]
  • Analytical Solutions to the Stefan Problem with Internal Heat Generation
    Applied Thermal Engineering 103 (2016) 443–451 Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng Research Paper Analytical solutions to the Stefan problem with internal heat generation ⇑ David McCord a, John Crepeau a, , Ali Siahpush b, João Angelo Ferres Brogin c a Department of Mechanical Engineering, University of Idaho, 875 Perimeter Drive, MS 0902, Moscow, ID 83844-0902, United States b Department of Integrated Engineering, Southern Utah University, 351 W. University Blvd., Cedar City, UT 84720, United States c Departamento de Engenharia Mecânica, São Paulo State University, Ilha Solteira, SP 15385-000, Brazil highlights A differential equation modeling the Stefan problem with heat generation is derived. The analytical solutions compare very well with the computational results. The system reaches steady-state faster for larger Stefan numbers. The interface location is proportional to the inverse square root of the heat generation. article info abstract Article history: A first-order, ordinary differential equation modeling the Stefan problem (solid–liquid phase change) Received 15 February 2016 with internal heat generation in a plane wall is derived and the solutions are compared to the results Accepted 24 March 2016 of a computational fluid dynamics analysis. The internal heat generation term makes the governing equa- Available online 13 April 2016 tions non-homogeneous so the principle of superposition is used to separate the transient from steady- state portions of the heat equation, which are then solved separately. There is excellent agreement Keywords: between the solutions to the differential equation and the CFD results for the movement of both the Stefan problem solidification and melting fronts.
    [Show full text]
  • Beyond the Classical Stefan Problem
    Beyond the classical Stefan problem Francesc Font Martinez PhD thesis Supervised by: Prof. Dr. Tim Myers Submitted in full fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics in the Facultat de Matem`atiques i Estad´ıstica at the Universitat Polit`ecnica de Catalunya June 2014, Barcelona, Spain ii Acknowledgments First and foremost, I wish to thank my supervisor and friend Tim Myers. This thesis would not have been possible without his inspirational guidance, wisdom and expertise. I appreciate greatly all his time, ideas and funding invested into making my PhD experience fruitful and stimulating. Tim, you have always guided and provided me with excellent support throughout this long journey. I can honestly say that my time as your PhD student has been one of the most enjoyable periods of my life. I also wish to express my gratitude to Sarah Mitchell. Her invaluable contributions and enthusiasm have enriched my research considerably. Her remarkable work ethos and dedication have had a profound effect on my research mentality. She was an excellent host during my PhD research stay in the University of Limerick and I benefited significantly from that experience. Thank you Sarah. My thanks also go to Vinnie, who, in addition to being an excellent football mate and friend, has provided me with insightful comments which have helped in the writing of my thesis. I also thank Brian Wetton for our useful discussions during his stay in the CRM, and for his acceptance to be one of my external referees. Thank you guys. Agraeixo de tot cor el suport i l’amor incondicional dels meus pares, la meva germana i la meva tieta.
    [Show full text]
  • A Two–Sided Contracting Stefan Problem
    A Two–Sided Contracting Stefan Problem Lincoln Chayes and Inwon C. Kim April 2, 2008 Department of Mathematics, UCLA, Los Angeles, CA 90059–1555 USA Abstract: We study a novel two–sided Stefan problem – motivated by the study of certain 2D interfaces – in which boundaries at both sides of the sample encroach into the bulk with rate equal to the boundary value of the gradient. Here the density is in [0, 1] and takes the two extreme values at the two free boundaries. It is noted that the problem is borderline ill–posed: densities in excess of unity liable to cause catastrophic behavior. We provide a general proof of existence and uniqueness for these systems under the condition that the initial data is in [0, 1] and with some mild conditions near the boundaries. Applications to 2D shapes are provided, in particular motion by weighted mean curvature for the relevant interfaces is established. Key words and phrases: Free boundary problems, Stefan equation, Interacting particle systems. AMS subject classifications: 35R35 and 82C22 1 Introduction A while ago, one of us – in a collaborative effort, [4] – investigated a number of physical problems leading to the study of certain one–dimensional Stefan–problems with multiple boundaries. We will begin with an informal (i.e. classical) description of the latter starting with the single boundary case. The problem is an archetype free boundary problem, here the boundary is denoted by L(t). In the region L ≤ x, there is a density ρ(x, t) which is non–negative (and may be assumed identically zero for x < L) and which satisfies the diffusion equation with unit constant ∆ρ = ρt (1.1) subject to Dirichlet conditions, with vanishing density, at x = L: ρ(L(t), t) ≡ 0.
    [Show full text]
  • Arxiv:1707.00992V1 [Math.AP] 4 Jul 2017
    OBSTACLE PROBLEMS AND FREE BOUNDARIES: AN OVERVIEW XAVIER ROS-OTON Abstract. Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, Geometric Measure Theory, and Harmonic Analysis. The most important mathematical challenge in this context is to understand the structure and regularity of free boundaries. In this paper we provide an invitation to this area of research by presenting, in a completely non-technical manner, some classical results as well as some recent results of the author. 1. Introduction Many problems in Physics, Industry, Finance, Biology, and other areas can be described by PDEs that exhibit apriori unknown (free) interfaces or boundaries. These are called Free Boundary Problems. A classical example is the Stefan problem, which dates back to the 19th century [54, 35]. It describes the melting of a block of ice submerged in liquid water. In the simplest case (the one-phase problem), there is a region where the temperature is positive (liquid water) and a region where the temperature is zero (the ice). In the former region the temperature function θ(t; x) solves the heat equation (i.e., θt = ∆θ in f(t; x): θ(t; x) > 0g), while in the other region the temperature θ is just zero. The position of the free boundary that separates the two regions is part of the problem, and is determined by an extra boundary condition on such interface 2 (namely, jrxθj = θt on @fθ > 0g).
    [Show full text]
  • A Stochastic Stefan-Type Problem Under First-Order Boundary Conditions
    Submitted to the Annals of Applied Probability 2018 Vol. 28, No. 4, 2335–2369 DOI: 10.1214/17-AAP1359 A STOCHASTIC STEFAN-TYPE PROBLEM UNDER FIRST-ORDER BOUNDARY CONDITIONS By Marvin S. Muller¨ ∗,† ETH Z¨urich Abstract Moving boundary problems allow to model systems with phase transition at an inner boundary. Motivated by problems in eco- nomics and finance, we set up a price-time continuous model for the limit order book and consider a stochastic and non-linear extension of the classical Stefan-problem in one space dimension. Here, the paths of the moving interface might have unbounded variation, which in- troduces additional challenges in the analysis. Working on the dis- tribution space, It¯o-Wentzell formula for SPDEs allows to transform these moving boundary problems into partial differential equations on fixed domains. Rewriting the equations into the framework of stochas- tic evolution equations and stochastic maximal Lp-regularity we get existence, uniqueness and regularity of local solutions. Moreover, we observe that explosion might take place due to the boundary inter- action even when the coefficients of the original problem have linear growths. Multi-phase systems of partial differential equations have a long history in applications to various fields in natural science and quantitative finance. Recent developments in modeling of demand, supply and price formation in financial markets with high trading frequencies ask for a mathematically rigorous framework for moving boundary problems with stochastic forcing terms. Motivated by this application, we consider a class of semilinear two- phase systems in one space dimension with first order boundary conditions at the inner interface.
    [Show full text]
  • Jfr Mathematics for the Planet Earth
    SOME MATHEMATICAL ASPECTS OF THE PLANET EARTH José Francisco Rodrigues (University of Lisbon) Article of the Special Invited Lecture, 6th European Congress of Mathematics 3 July 2012, KraKow. The Planet Earth System is composed of several sub-systems: the atmosphere, the liquid oceans and the icecaps and the biosphere. In all of them Mathematics, enhanced by the supercomputers, has currently a key role through the “universal method" for their study, which consists of mathematical modeling, analysis, simulation and control, as it was re-stated by Jacques-Louis Lions in [L]. Much before the advent of computers, the representation of the Earth, the navigation and the cartography have contributed in a decisive form to the mathematical sciences. Nowadays the International Geosphere-Biosphere Program, sponsored by the International Council of Scientific Unions, may contribute to stimulate several mathematical research topics. In this article, we present a brief historical introduction to some of the essential mathematics for understanding the Planet Earth, stressing the importance of Mathematical Geography and its role in the Scientific Revolution(s), the modeling efforts of Winds, Heating, Earthquakes, Climate and their influence on basic aspects of the theory of Partial Differential Equations. As a special topic to illustrate the wide scope of these (Geo)physical problems we describe briefly some examples from History and from current research and advances in Free Boundary Problems arising in the Planet Earth. Finally we conclude by referring the potential impact of the international initiative Mathematics of Planet Earth (www.mpe2013.org) in Raising Public Awareness of Mathematics, in Research and in the Communication of the Mathematical Sciences to the new generations.
    [Show full text]
  • Handbook.Pdf
    1 4 3 Conference program Monday, November 30th 9:30: Welcome 10:00: Plenar section. Chair: I. A. Taimanov (Assembly Hall, Laboratory Building) 12:20: Coffee break (Lobby of Assembly Hall, Laboratory Building) 13:20: Section «Inverse problems and integral geometry» Chair: M. I. Belishev (Assembly Hall, Laboratory Building) Section «Nonlinear PDEs and integrability» Chair: A. K. Pogrebkov (202, Laboratory Building) 15:40: Coffee break (Lobby of Assembly Hall, Laboratory Building) 16:40: Section «Inverse problems and integral geometry» Chair: M. I. Belishev (Assembly Hall, Laboratory Building) Section «Nonlinear PDEs and integrability» Chair: A. K. Pogrebkov (202, Laboratory Building) Tuesday, December 1st 10:00: Plenar section. Chair: S. I. Kabanikhin (Assembly Hall, Laboratory Building) 12:20: Coffee break (Lobby of Assembly Hall, Laboratory Building) 13:20: Section «Inverse problems of mathematical physics» Chair: V. A. Sharafutdinov (BioPharm Building) Section «Nonlinear PDEs and dynamical systems» Chair: J.-C. Saut (BioPharm Building) Section «Numerical methods» (school for young scientists) Chair: A. A. Shananin (202, Laboratory Building) 14:30: Coffee break (112, 118, BioPharm Building) 14:50: Section «Inverse problems of mathematical physics» Chair: V. A. Sharafutdinov (BioPharm Building) Section «Nonlinear PDEs and dynamical systems» Chair: J.-C. Saut (BioPharm Building) Section «Numerical methods» (school for young scientists) Chair: A. A. Shananin (202, Laboratory Building) 17:00 Cultural program Wednesday, December 2nd 10:00: Plenar section. Chair: R.Novikov (BioPharm Building) 11:45: Coffee break (112, 118, BioPharm Building) 12:00: Introductory lecture for students «Inverse scattering and applications» by R. G. Novikov (BioPharm Building) 14:00: Coffee break (112, 118, BioPharm Building) 15:00: Section «Modeling in Continuum Mechanics and Inverse Problems using HPC» Chair: I.
    [Show full text]
  • Optimal Control of a Stefan Problem with Gradient-Based Methods in Fenics
    Optimal Control of a Stefan Problem with Gradient-Based Methods in FEniCS Master Thesis submitted to the Faculty of Mathematics at Otto-von-Guericke-Universit¨atMagdeburg in accordance with the requirements for the degree Master of Science authored by Bjorn¨ Baran born at December 08, 1989 in Flensburg, university course Mathematics. February 4, 2016 Supervised at Max Planck Institute for Dynamics of Complex Technical Systems by Dr. Jan Heiland Contents Notation V Symbols VI Abbreviations X 1. Introduction 2 1.1. Motivation . .2 1.2. Prior Work . .2 1.3. Research Goals . .4 1.4. Outline . .4 2. Two-Phase Stefan Problem 6 2.1. Domain . .6 2.2. Heat Equation . .7 2.3. Mesh Movement . .9 2.4. Navier{Stokes Equations . .9 3. Optimization 11 3.1. Optimal Control Problem . 11 3.1.1. Forward System . 13 3.1.2. Cost Functional . 14 3.2. Derivation of the Adjoint System . 14 3.2.1. Lagrange Functional . 15 3.2.2. Adjoint Equations . 16 3.2.3. Backward System . 33 3.3. Gradient Method . 34 3.3.1. Projected Gradient Method . 35 3.3.2. Line Minimization Algorithm . 37 3.3.3. Treatment of Control Constraints . 38 4. Implementation and Discretization 39 4.1. Spatial Discretization . 40 4.2. Weak Formulations . 40 5. Numerical Examples 49 5.1. General Setting . 49 5.2. Stabilizing to a Flat Position . 50 5.3. Stabilizing to a Reachable Flat Position . 59 5.4. Moving to a Reachable Position . 64 { II { Contents 6. Conclusions 69 7. Perspectives 70 A. Appendix 71 A.1. Interface Normal Representation .
    [Show full text]
  • Explicit Solutions to Phase-Change Problems and Applications
    2016 Computational Mathematics Annual Program Review Arlington, VA (USA), August 8-11, 2016 EXPLICIT SOLUTIONS TO PHASE-CHANGE PROBLEMS AND APPLICATIONS Domingo Alberto Tarzia Grant AFOSR – SOARD FA 9550-14-1-0122 CONICET - Depto. Matemática, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina. E-mail: [email protected] Key Words: Free boundary problems, Lamé-Clapeyron-Stefan problem, Phase-change problems, Heat- Diffusion equation, Fractional diffusion, Caputo fractional derivative, Explicit solutions, Neumann solution, Unknown thermal coefficients, Over-specified boundary condition, Mushy region, Convective boundary condition. Mathematics Subject Classification 2010: 26A33, 35C05, 35R11, 35R35, 80A22. 2 ABSTRACT We obtain explicit solutions (in a closed form) for the following one-dimensional phase- change problems: • Two-phase solidification process with a convective boundary condition (T. Thermal Sci (2016), In Press); • One-phase solidification process with a mushy region and a convective boundary condition (T., J. Appl. Math. (2015)); • One-phase solidification process with a mushy region and a heat flux boundary condition (T., J. Appl. Math. (2015)); • Determination of one unknown thermal coefficient through a solidification problem with an over-specified boundary condition (Ceretani-T., Math. Problems in Eng. (2015)); AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 3 • Simultaneous determination of two unknown thermal coefficients through a solidification problem with an over-specified boundary condition (Ceretani-T., JP Journal Heat Mass Transfer (2016)); • Two-phase fractional Stefan problem with a temperature boundary condition (Roscani-T., Adv. Math. Sci. Appl. (2014)); • Two-phase fractional Stefan problem with a heat flux boundary condition (Roscani- T., In progress); • Determination of one unknown thermal coefficient through a one-phase fractional Stefan problem with an over-specified boundary condition (T.
    [Show full text]
  • Asymptotic Convergence of the Stefan Problem to Hele-Shaw
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 2, Pages 609{634 S 0002-9947(00)02739-2 Article electronically published on October 23, 2000 ASYMPTOTIC CONVERGENCE OF THE STEFAN PROBLEM TO HELE-SHAW FERNANDO QUIROS´ AND JUAN LUIS VAZQUEZ´ Abstract. We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as t !1, and give the precise asymptotic growth rate for the radius. 1. Introduction A Stefan problem is a mathematical model for describing the melting of a body of ice in contact with a region of water. In one-phase Stefan problems, the temperature of the ice is supposed to be maintained at 0◦C. Hence, the unknowns are the temperature distribution in the water and the shape of the interface (free boundary) between ice and water, see [Ru], [Me]. This problem leads to a mathematical model posed in one, two or three space dimensions. The mathematical problem is naturally generalized to arbitrary integer space dimension n ≥ 1 and is still called the Stefan problem. The Hele-Shaw problem is a two-dimensional mathematical model describing the movement of a viscous fluid confined in a narrow cell between two parallel plates, [ST], [EJ].
    [Show full text]
  • Feedback Control of the Two-Phase Stefan Problem, with an Application to the Continuous Casting of Steel
    49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA Feedback control of the two-phase Stefan problem, with an application to the continuous casting of steel Bryan Petrus, Joseph Bentsman, Member, IEEE, and Brian G. Thomas Abstract—A full-state feedback control law is derived that Since these goals can be achieved via temperature stabilizes the two-phase Stefan problem with respect to a regulation, an automated control strategy based on the steel reference solution using control of the Neumann boundary temperature would be greatly beneficial to the industry. condition. Stability and convergence are shown via a Lyapunov However, most casters use open-loop methods to control the functional on the error system with moving boundaries. A flow rates of the water sprays that cool the surface. A key second control law is also derived, for which stability is proved and convergence is conjectured due to the clearly convergent obstacle to the design of such control is the strong inherent simulation results. A simple Dirichlet controller is also nonlinearity of the solidification evolution equations. The considered, and is used to design a boundary-output-based many results in recent years for control of linear or even estimator that, in combination with full-state feedback semi-linear distributed parameter systems are not applicable. controllers, yields a plausible output feedback control law with Previous work on control of processes governed by the boundary sensing and actuation. The performance of the Stefan problem can be generally divided into three control laws is demonstrated using numerical simulation.
    [Show full text]