×

Interface closure in the root region of steady deep-cellular growth in directional solidification. (English) Zbl 1383.82014

Summary: The present paper investigates the mechanism of interface closure in the root region of the solutions for steady deep-cellular growth. This phenomenon is determined by a transcendentally small factor beyond all orders. It is found that the root region comprises three inner-inner regions; the inner system in the root region has a simple turning point, whose presence generates the so-called trapped-waves mechanism, which is responsible for the interface closure at the bottom of root. The quantization condition derived from the trapped-waves mechanism yields the eigenvalue that determines the location of interface closure and its dependence on the interfacial energy and other physical parameters.

MSC:

82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
35B40 Asymptotic behavior of solutions to PDEs
76D27 Other free boundary flows; Hele-Shaw flows
81S10 Geometry and quantization, symplectic methods
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rutter, J. W.; Chalmers, B., A prismatic substructure formed during solidification of metals., Can. J. Phys., 31, 15-39, (1953) · doi:10.1139/p53-003
[2] Mullins, W. W.; Sekerka, R. F., Morphological stability of a particle growing by diffusion or heat flow., J. Appl. Phys., 34, 323-329, (1963) · doi:10.1063/1.1702607
[3] Somboonsuk, K., Mason, J. T. & Trivedi, R. (1984) Interdendritc spacing: Part (I)-(II). Metall. Tran. A15, 967-975; 977-982. doi:10.1007/BF02644688 · doi:10.1007/BF02644688
[4] Pelce, P.; Pumir, A., Cell shape in directional solidification in the small Péclet number limit., J. Cryst. Growth, 73, 337-342, (1985) · doi:10.1016/0022-0248(85)90310-0
[5] Weeks, J. D.; Van-Saarloos, W., Directional solidification cells with grooves for a small partition coeffcient., Phy. Rev. A, 39, 2772-2775, (1989) · doi:10.1103/PhysRevA.39.2772
[6] Ungar, L. H.; Brown, R. A., Cellular interface morphologies in directional solidification. 4. The formation of deep cells, Phys. Rev. B, 31, 5931-5940, (1985) · doi:10.1103/PhysRevB.31.5931
[7] Georgelin, M.; Pocheau, A., Shape of Growth Cells in Directional Solidifiction,, Phys. Rev. E, 73, 011604, (2006) · doi:10.1103/PhysRevE.73.011604
[8] Weeks, J. D.; Van-Saarloos, W.; Grant, M., Stability and shape of cellular profiles in directional solidification: Expansion and matching methods., J. Cryst. Growth., 112, 244-282, (1991) · doi:10.1016/0022-0248(91)90928-X
[9] Caroli, B., Caroli, C. & Roulet, B. (1991) Instability of planar solidification fronts. In: Godreche, C. (editor), Solids Far from Equilibrium, Cambridge University Press, Cambridge, New York, pp. 155-296.
[10] Billia, B. & Trivedi, R. (1993) Pattern formation in crystal growth. In: Hurle, D. T. J. (editor), Handbook of Crystal Growth, Vol. 1: Fundamentals, Part B: Transport and Stability, Elsevier Science Publishers, Norh-Holland, Amsterdam, pp. 872-1008.
[11] Chen, Y. Q.; Xu, J. J., Global theory of steady deep-cellular growth in directional solidification,, Phys. Rev. E, 83, 041601, (2011)
[12] Xu, J. J.; Chen, Y. Q., Global stabilities, selection of steady cellular growth, and origin of side branches in directional solidification, Phys. Rev. E, 83, 061605, (2011)
[13] Saffman, P. G.; Taylor, G. I., The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid., Proc. R. Soc. Lond. Ser. A., 245, 312-329, (1958) · Zbl 0086.41603 · doi:10.1098/rspa.1958.0085
[14] Xu, J. J., Global instability of viscous fingering in Hele-Shaw Cell (I) - formation of oscillatory fingers, Eur. J. Appl. Math., 2, 105-132, (1991) · Zbl 0727.76047 · doi:10.1017/S0956792500000437
[15] Xu, J. J., Interfacial instabilities and fingering formation in Hele-Shaw Flow,, IMA J. of Appl. Math., 57, 101-135, (1996) · Zbl 0868.35145 · doi:10.1093/imamat/57.2.101
[16] Xu, J. J., Interfacial wave theory for oscillatory finger’s formation in a Hele-Shaw cell: A comparison with experiments., Eur. J. Appl. Math., 7, 169-199, (1996) · Zbl 0863.76023
[17] Kruskal, M.; Segur, H., Asymptotics beyond all orders., Stud. Appl. Math., 85, 129-181, (1991) · Zbl 0732.34047
[18] Chapman, S. J.; Vanden-Broeck, J. M., Exponential Asymptotics and Capillary Waves., SIAM J. Appl. Math., 62, 1872-1898, (2002) · Zbl 1034.76008 · doi:10.1137/S003613990038116X
[19] Chapman, S. J.; Vanden-Broeck, J. M., Exponential asymptotics and gravity waves., J. Fluid Mech., 567, 299-326, (2006) · Zbl 1177.76044 · doi:10.1017/S0022112006002394
[20] Berry, M. V. (1991) Asymptotics, superasymptotics, hyperasymptotics. In: Segur, H., Tanveer, S. & Levine, H. (editors), Asymptotics Beyond All Orders, NATO ASI Series. Vol. 284, Plenum, Amsterdam, pp. 1-14. doi:10.1007/978-1-4757-0435-8 · doi:10.1007/978-1-4757-0435-8
[21] Xu, J. J., Interfacial Wave Theory of Pattern Formation: Selection of Dendrite Growth and Viscous Fingering in a Hele-Shaw Flow, (1998), Springer-Verlag: Springer-Verlag, Germany, Heidelberg · Zbl 0901.76001
[22] Georgelin, M.; Pocheau, A., Shape of growth cells in directional solidification., Phys. Rev. E, 73, 011604, (2006) · doi:10.1103/PhysRevE.73.011604
[23] Xu, J. J., Interfacial wave theory of solidication – dendritic pattern formation and selection of tip velocity., Phys. Rev. A15, 43, 930-947, (1991) · doi:10.1103/PhysRevA.43.930
[24] Xu, J. J., Generalized needle solutions, interfacial instabilities and pattern formation., Phys. Rev. E, 53, 5031-5062, (1996) · Zbl 1262.60043
[25] Chen, Y. Q., Tang, X. X. & Xu, J. J. (2009) 3D Interfacial wave theory of dendritic growth: (I)-(II). Chin. Phys. B18, 671-685; 686-698. doi:10.1088/1674-1056/18/2/047 · doi:10.1088/1674-1056/18/2/047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.