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November  2004, 4(4): 1091-1116. doi: 10.3934/dcdsb.2004.4.1091

Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions

Jian-Jun Xu 1, and  Junichiro Shimizu 2,

1. 

Department of Mathematics, McGill University, Montreal QC H3A 2K6, Canada
2. 

National Space Development Agency of Japan (NASDA), Tsukuba Space Center, Tsukuba, Japan

Received  February 2003 Revised  January 2004 Published  August 2004

The present paper is concerned with disc-like crystal growth from a pure undercooled melt. We obtained the uniformly valid asymptotic solution for the basic state in the limit of the aspect ratio $\delta = b/L$ << $1$ , in terms of the matched asymptotic expansion method. The solution obtained under the present model shows that the growth of the top/bottom interface of the disc is very slow, dominated by the kinetic effect, while the growth of its side-interface is much faster and is dominated by heat diffusion mechanism, with negligible effects by the surface tension and the kinetic attachment. Furthermore, we performed the linear stability analysis for the basic state at the early stage of growth. It is found that the system allows two discrete sets of unstable eigen-modes over the side-interface: the axi-symmetric eigen-modes and non-axi-symmetric eigen-modes. The onset of the instability is when the thickness of the disc reaches at a critical value $b_c$. The axi-symmetric eigen-modes are found can be further distinguished as the A-modes, anti-symmetric and the S-modes, symmetric about the central plane. The most dangerous axi-symmetric modes is a well isolated base mode $A_0$ with the index ($n = 0$). This mode is responsible for the formation of anti-symmetric pattern about the central plane of the disc, which is observed at the early stage of growth. We have compared the theoretical predictions with the available experimental data of ice-disc growth obtained by Shimada and Furukawa. It is found that both are in good agreements.
Keywords: undercooling temperature., kinetic attachment, disc-like crystal growth, Matched asymptotic expansion, surface tension.
Mathematics Subject Classification: 78M35, 82C24, 82C2.
Citation: Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1091-1116. doi: 10.3934/dcdsb.2004.4.1091
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