Stefan problems in non-cylindrical domains arising in Czochralski process of crystal growth

• Autorzy: Fukao T. , Kenmochi N. , Pawłow I.
• Języki publikacji: EN

Abstrakty

EN

In this paper we discuss a two-phase Stefan problem with convection in a non-cylindrical (time-dependent) domain. This work is motivated by phase change phenomenon arising in the Czochralski process of crystal growth. The time-dependence of domain is a mathematical description of the situation in which the material domain changes its shape with time by crystal growth. We consider the so-called enthalpy formulation for it and give its solvability, assuming that the time-dependence of the material domain is prescribed and smooth enough in time, and the convective vector is prescribed, too. Our main idea is to apply the theory of quasi-linear equations of parabolic type.

Słowa kluczowe

EN

Stefan problem; non-cylindrical domain; prescribed convection

PL

problem Stefana; domena niecylinryczna

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2003
• Tom: Vol. 32, no 2
• Strony: 201--221
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Fukao T.

• Department of Mathematics, Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan

autor

Kenmochi N.

• Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-chO, Inage-ku, Chiba, 263-8522 Japan

autor

Pawłow I.

• Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland

Bibliografia

• CROWLEY, A. B. (1983) Mathematical modelling of heat flow in Czochralski crystal pulling.
• IMA J. Appl. Math., 30, 173-189. DAMLAMIAN, A. (1977) Some results on the multi-phase Stefan problem. Comm. Partial Differential Equations, 2, 1017-1044.
• DIBENEDETTO, E. and O'LEARY, M. (1993) Three-dimensional conductionconvection problems with change of phase. Arch. Ration. Mech. Anal., 123, 99-116.
• FuKAO, T. (2002) Transmission problems for degenerate parabolic equations. "Elliptic and Parabolic Problems". World Scientific, New Jersey-LondonSingapore-Hong Kong, 103-112 .
• FUKAO, T., KENMOCHI, N. and PAWLOW, I. (2002) Transmission problems arising in Czochralski process of crystal growth. GAKUTO Internat. Ser. Math. Sci. Appl. "Mathematical Aspects of Modeling Structure Formation Phenomena", 17, Gakkotosho, Tokyo, 228-243.
• KENMOCHI, N. ( 1981) Solvability of nonlinear evolution equations with timedependent constraints and applications. Bull. Fac. Edu., Chiba Univ., 30, 1-87.
• KENMOCHI, N. and PAWLOW, I. ( 1986) A class of nonlinear elliptic-parabolic equations with time-dependent constraints. Nonlinear Analysis, 10, 1181- 1202.
• ENMOCHI, N. and PAWLOW, I. ( 1996) Controlled Czochralski crystal growth. J. Soc. Instr. Control Engin., 35, 944-950.
• LADYZHENSKAYA, 0. A., SOLONNIKOV, V. A. and URAL'TSEVA, N . N . (1968) Linear and quasilinear equations of parabolic type, 23. Translations of Mathematical Monographs, Amer. Math. Soc.
• NIEZGODKA, M. and PAWLOW, I. (1983) A generalized Stefan problem in several spaces variables. Appl. Math. Optim., 9, 193-224.
• PAWLOW, I. (2002) Three-phase boundary Czochralski model. GAKUTO Internat. Ser. Math. Sci. Appl. "Mathematical Aspects of Modeling Structure Formation Phenomena", 17. Gakkotosho, Tokyo, 203-227.
• RODRIGUES, J. F. (1994) Variational methods in the Stefan problem. Lecture Notes in Math. "Phase Transitions and Hysteresis", 1584. SpringerVerlag, Berlin/Heidelberg, 147-212.
• RODRIGUES, J. F. and YI, F. (1990) On a two-phase continuous casting Stefan problem with nonlinear flux. European J. Appl. Math., 1, 259- 278.