Hele-Shaw Model for Melting/Freezing with Two Dendrits


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Melting/freezing process with two dendrits (or freeze “pipes”) is modelled by the complex Hele-Shaw moving boundary value problem in a doubly connected domain. The later is equivalently reduced to a couple of problems, namely, to the linear Riemann-Hilbert boundary value problem in a doubly connected domain and to evolution problem, which can be written in a form of an abstract Cauchy-Kovalevsky problem. The later is studied on the base of Nirenberg-Nishida theorem, and for the former a generalization of the Schwarz Alternation Method is proposed. By using composition of these two approaches we get the local in time solvability of this couple of problems in appropriate Banach space setting.



Edited by:

Prof. Andreas Öchsner and José Grácio




S. Rogosin and T. Vaitekhovich, "Hele-Shaw Model for Melting/Freezing with Two Dendrits", Materials Science Forum, Vol. 553, pp. 143-151, 2007

Online since:

August 2007




[1] Y. -W. Lee, R. Ananth, W.N. Gill: J. Cristal Growth. Vol. 132 (1993), p.226.

[2] J.G. Dash, Haiying Fu, J.S. Wettlaufer: Rep. Prog. Phys. Vol. 58 (1995), p.115.

[3] M.D. Kunka, M.R. Foster, S. Tanveer: Phys. Rev. E. Vol. 56 (1997), p.3068.

[4] L.M. Cummings, Yu.E. Hohlov, S.D. Howison, K. Kornev: J. Fluid Mech. Vol. 378 (1999), p.1.

[5] Yu.P. Vinogradov, P.P. Kufarev: Prikl. Mat. Mech. Vol. 12 (1948), p.181 (in Russian). (English translation: University of Delaware, Applied Mathematics Institute, Technical Report 182A, 1984).

[6] S.D. Richardson: J. Fluid Mech. Vol. 56 (1972), p.609.

[7] B. Gustafsson, A. Vasil'ev: Conformal and Potential Analysis in Hele-Shaw Cells (Birkh¨auser Verlag, Basel-Stockholm, 2005).

[8] M.M. Alimov, K.G. Kornev, G.I. Mukhamadullina: J. Appl. Math. Mech. (PMM) Vol. 58 (1994), p.873.

[9] K.G. Kornev, G.I. Mukhamadullina: Proc. R. Soc. London. Ser A. Vol. 447 (1994), p.281.

[10] M. Alimov, K. Kornev, G. Mukhamadullina: SIAM J. Appl. Math. Vol. 59, No. 2 (1998), p.387.

[11] M. Reissig, L. von Wolfersdorf: Arkiv f¨or matematik. Vol. 31, No. 1 (1993), p.101.

[12] M. Reissig, S.V. Rogosin, with an appendix of F. H¨ubner: Euro J. Appl. Math. Vol. 10 (1999), p.561.

[13] S.D. Richardson: Euro J. Appl. Math. Vol. 5 (1994), 97-122.

[14] S.D. Richardson: Phil. Trans. R. Soc. London. Vol. 354 (1996), 2513-2553.

[15] S.D. Richardson: Euro J. Appl. Math. Vol. 12, No. 5 (2001), 571-599.

[16] D. Crowdy, H. Kang: J. Nonlinear Sci. Vol. 11, No. 4 (2001), p.279.

[17] V.V. Mityushev, S.V. Rogosin: Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions: Theory and Applications (Chapman & Hall / CRC PRESS, Boca Raton - London - New York - Washington, 1999 (Monographs and surveys in pure and applied mathematics, Vol. 108).

[18] L. Nirenberg: J. Differ. Geom. Vol. 6 (1972), p.561.

[19] T. Nishida: J. Differ. Geom. Vol. 12 (1977), p.629.

[20] S.V. Rogosin, T.S. Vaithekhovich: J. Appl. Funct. Anal. (2007) (submitted).

[21] G.M. Goluzin: Geometric Theory of Functions of a Complex Variable (Providence, R. I.: American Mathematical Society, 1969).

[22] F.D. Gakhov: Boundary Value Problems (Pergamon Press, UK, 1966).