Two-dimensional solidification and melting in potential flows

Journal of Fluid Mechanics

Volumn 378, Issue , 1999, Pages 1-18

  Cummings, Linda M ^a   Hohlov, Yuri E ^b   Howison, Sam D ^a   Kornev, Konstantin ^c  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

^c INSTITUTE FOR PROBLEMS IN MECHANICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY VALUE PROBLEMS; CRYSTAL GROWTH; CRYSTALS; HYDRODYNAMICS; MELTING; SOLIDIFICATION; TWO DIMENSIONAL; VELOCITY;

HYDRODYNAMIC FLOW;

FLOW OF FLUIDS;

EID: 0032641671     PISSN: 00221120     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0022112098003188     Document Type: Article

References (43)

1. ALEKSANDROV, I. A. 1976 Parametric Continuations in Theory of Univalent Functions. Nauka, Moscow.
2. ANANTH, R. & GILL, W. N. 1989 Dendritic growth of an elliptical paraboloid with forced convection in the melt. J. Fluid Mech. 208, 575-593.
3. ANANTH, R. & GILL, W. N. 1991 Self-consistent theory of dendritic growth with convection. J. Crystal Growth 108, 173-189.
4. BENAMAR, M., BOUISSOU, P. & PELCE, P. 1988 An exact solution for the shape of a crystal growing in a forced flow. J. Crystal Growth 92, 97-100.
5. BENSIMON, D., KADANOFF, L. P., LIANG, S. D., SHRAIMAN, B. I. & TANG, C. 1986 Viscous flows in two dimensions. Rev. Mod. Phys. 58, 977-999.
6. BOUISSOU, P., PERRIN, B. & TABELING, P. 1989 Influence of an external flow on dendritic crystal growth. Phys. Rev. A 40, 509-512.
7. BOUSSINESQ, M. J. 1905 J. Math. Pure Appl. 1, 285.
8. BRATTKUS, K. & DAVIS, S. H. 1988 Flow-induced morphological instabilities: stagnation-point flows. J. Crystal Growth 89, 423-427.
9. BRENER, E. A. & MELNIKOV, V. I. 1991 Pattern selection in two-dimensional dendritic growth. Adv. Phys. 40, 53-97.
10. CUMMINGS, L. J. 1996 Free boundary models in viscous flow. D. Phil. Thesis, University of Oxford.
11. DASH, J. G., HAIYING FU, WETTLAUFER, J. S. 1995 The premelting of ice and its environmental consequences. Rep. Prog. Phys. 58, 115-167.
12. DASH, S. K. & GILL, W. N. 1984 Forced-convection heat and momentum-transfer to dendritic structures (parabolic cylinders and paraboloids of revolution). Intl J. Heat Mass Transfer 27, 1345-1356.
13. DAVIS, S. H. 1990 Hydrodynamic interactions in directional solidification. J. Fluid Mech. 212, 241-262.
14. DAVIS, P. J. 1974 The Schwarz Function and its Applications. Carus Math. Monographs 17, Math. Assoc. of America.
15. GLICKSMAN, M. E., CORIELL, S. R. & MCFADDEN, G. B. 1986 Interaction of flows with the crystal-melt interface. Ann. Rev. Fluid. Mech. 18, 307-335.
16. GOLDSTEIN, M. E. & REID, R. I. 1978 Proc. R. Soc. Lond. A 364, 45.
17. HOHLOV, Yu. E., HOWISON, S. D., HUNTINGFORD, C., OCKENDON, J. R. & LACEY, A. A. 1994 A model for non-smooth free boundaries in Hele-Shaw flows. Q. J. Mech. Appl. Maths 47, 107-128.
18. HOWISON, S. D. 1986 Fingering in Hele-Shaw cells. J. Fluid Mech. 167, 439-453.
19. HOWISON, S. D. 1988 Similarity solutions to the Stefan problem and the binary alloy problem. IMA J. Appl. Maths 40, 147-161.
20. HOWISON, S. D. 1992 Complex variable methods in Hele-Shaw moving boundary problems. Eur. J. Appl. Maths 3, 209-224.
21. IVANTSOV, G. P. 1947 The temperature field around a spherical, cylindrical, or point crystal growing in a cooling solution. Dokl. Akad. Nauk. SSSR 58, 567 (in Russian).
22. KELLY, E. D. & HINCH, E. J. 1997 Numerical simulation of sink flow in the Hele-Shaw cell with small surface tension Eur. J. Appl. Maths 8, 533-551.
23. KESSLER, D. A., KOPLIK, J. & LEVINE, H. 1988 Pattern selection in fingered growth phenomena. Adv. Phys. 37, 255-339.
24. KOPF-SILL, A. R. & HOMSY, G. M. 1987 Narrow fingers in a Hele-Shaw cell. Phys. Fluids 30, 2607-2609.
25. KORNEV, K. & MUKHAMADULLINA, G. 1994 Mathematical theory of freezing for flow in porous media. Proc. R. Soc. Lond. A 447, 281-297.
26. KUNKA, M. D., FOSTER, M. R. & TANVEER, S. 1997 Dendritic crystal growth for weak undercooling. Phys. Rev. E 56, 3068-3100.
27. LACEY, A. A. 1982 Moving boundary problems in the flow of liquid through porous media. J. Austral. Math. Soc. B 24, 171-193.
28. LANGER, J. S. 1986 In Direction in Condensed Matter Physics. Memorial Volume in Honor of S. K. Ma (ed. G. Grinstein & G. Masenko), p. 165. World Scientific.
29. LEE, Y.-W., ANANTH, R. & GILL, W. N. 1993 Selection of a length scale in unconstrained dendritic growth with convection in the melt. J. Crystal Growth 132, 226-230.
30. MAKSIMOV, V. A. 1965 Izv. Acad. Nauk SSSR, Mekh. 4, 41.
31. MAKSIMOV, V. A. 1976 Prikl. Matem. Mech. 40, 264.
32. MILLAR, R. F. 1989 An inverse moving boundary problem for Laplace's equation. In Proc. Workshop on Inverse Problems and Imaging (ed. G. F. Roach). London: Pitman-Longman.
33. MINEEV-WEINSTEIN, M. B. & PONCE-DAWSON, S. 1994 Class of nonsingular exact solutions for Laplacian pattern formation. Phys. Rev. E 50, 24-27.
34. MULLINS, W. W. & SEKERKA, R. F. 1963 J. Appl. Phys. 34, 323.
35. MULLINS, W. W. & SEKERKA, R. F. 1964 J. Appl. Phys. 35, 444.
36. MUSKHELISHVILI, N. I. 1968 Singular Integral Equations, 3rd edn. Nauka, Moscow.
37. PAMPLIN, B. R. 1980 Crystal Growth, 2nd edn. Pergamon.
38. POMMERENKE, CHR. 1973 Univalent Functions. Vandenhoeck and Ruprecht, Göttingen.
39. RICHARDSON, S. 1972 Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609-618.
40. SIEGEL, M. & TANVEER, S. 1996 Singular perturbation of smoothly evolving Hele-Shaw solutions. Phys. Rev. Lett. 76, 419-422.
41. SIEGEL, M., TANVEER, S. & DAI, W. 1996 Singular effects of surface tension in evolving Hele-Shaw flows. J. Fluid Mech. 323, 201-236.
42. WIJNGAARDEN, L. VAN 1966 Proc. Koninkl. Nederl. Akad. Wet. 69, 263.
43. Xu, J.-J. 1993 Uniformly valid asymptotic solution for steady dendrite growth in external flow. J. Crystal Growth 128, 219-223.