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Interface closure in the root region of steady deep-cellular growth in directional solidification

Published online by Cambridge University Press:  06 March 2015

J.-J. XU  and
Y.-Q. CHEN
J.-J. XU
Affiliation:
School of Material Science and Engineering, USTB, Beijing, 100083, China email: xu@math.mcgill.ca Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 0B9, Canada
Y.-Q. CHEN
Affiliation:
School of Science, Tianjin Chengjian University, Tianjin, 300384, China email: yonqiang@gmail.com
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Abstract

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.

Keywords

Asymptotics beyond all ordersMultiple variables expansionQuantization conditionSteady deep-cellular growthInterface closure
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 3 , June 2015 , pp. 355 - 382
Copyright
Copyright © Cambridge University Press 2015 

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References

[1]Rutter, J. W. & Chalmers, B. (1953) A prismatic substructure formed during solidification of metals. Can. J. Phys. 31, 15–39.Google Scholar
[2]Mullins, W.W. & Sekerka, R. F. (1963) Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 34, 323–329.Google Scholar
[3]Somboonsuk, K., Mason, J. T. & Trivedi, R. (1984) Interdendritc spacing: Part (I)-(II). Metall. Tran. A 15, 967–975; 977–982.Google Scholar
[4]Pelce, P. & Pumir, A. (1985) Cell shape in directional solidification in the small Péclet number limit. J. Cryst. Growth 73, 337–342.Google Scholar
[5]Weeks, J. D. & van-Saarloos, W. (1989) Directional solidification cells with grooves for a small partition coeffcient. Phy. Rev. A 39, 2772–2775.CrossRefGoogle Scholar
[6]Ungar, L. H. & Brown, R. A. (1985), Cellular interface morphologies in directional solidification. 4. The formation of deep cells. Phys. Rev. B 31, 5931–5940.CrossRefGoogle Scholar
[7]Georgelin, M. & Pocheau, A. (2006) Shape of Growth Cells in Directional Solidifiction, Phys. Rev. E, 73, 011604.Google Scholar
[8]Weeks, J. D., van-Saarloos, W. & Grant, M. (1991) Stability and shape of cellular profiles in directional solidification: Expansion and matching methods. J. Cryst. Growth. 112, 244–282.Google Scholar
[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.Google Scholar
[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.Google Scholar
[11]Chen, Y. Q. & Xu, J. J. (2011) Global theory of steady deep-cellular growth in directional solidification, Phys. Rev. E 83, 041601.Google Scholar
[12]Xu, J. J. & Chen, Y. Q. (2011) Global stabilities, selection of steady cellular growth, and origin of side branches in directional solidification. Phys. Rev. E 83, 061605Google Scholar
[13]Saffman, P. G. & Taylor, G. I. (1958) 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.Google Scholar
[14]Xu, J. J. (1991) Global instability of viscous fingering in Hele-Shaw Cell (I) – formation of oscillatory fingers Eur. J. Appl. Math. 2, 105–132.Google Scholar
[15]Xu, J. J. (1996) Interfacial instabilities and fingering formation in Hele-Shaw Flow, IMA J. of Appl. Math. 57, 101–135.Google Scholar
[16]Xu, J. J. (1996) Interfacial wave theory for oscillatory finger's formation in a Hele–Shaw cell: A comparison with experiments. Eur. J. Appl. Math. 7, 169–199.Google Scholar
[17]Kruskal, M. & Segur, H. (1991) Asymptotics beyond all orders. Stud. Appl. Math. 85, 129–181.Google Scholar
[18]Chapman, S. J. & Vanden-Broeck, J. M. (2002) Exponential Asymptotics and Capillary Waves. SIAM J. Appl. Math. 62, 1872–1898.Google Scholar
[19]Chapman, S. J. & Vanden-Broeck, J. M. (2006) Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.Google Scholar
[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.Google Scholar
[21]Xu, J. J. (1998) Interfacial Wave Theory of Pattern Formation: Selection of Dendrite Growth and Viscous Fingering in a Hele–Shaw Flow, Springer-Verlag, Germany, Heidelberg.Google Scholar
[22]Georgelin, M. & Pocheau, A. (2006) Shape of growth cells in directional solidification. Phys. Rev. E 73, 011604.Google Scholar
[23]Xu, J. J. (1991) Interfacial wave theory of solidication–dendritic pattern formation and selection of tip velocity. Phys. Rev. A15, 43, 930–947.Google Scholar
[24]Xu, J. J. (1996) Generalized needle solutions, interfacial instabilities and pattern formation. Phys. Rev. E 53, 5031–5062.Google Scholar
[25]Chen, Y. Q., Tang, X. X. & Xu, J. J. (2009) 3D Interfacial wave theory of dendritic growth: (I)-(II). Chin. Phys. B 18, 671–685; 686–698.Google Scholar