American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - B, 2004, 4 (4) : 1091-1116. doi: 10.3934/dcdsb.2004.4.1091
[1]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth. Communications on Pure & Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527
[2]

Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185
[3]

Stéphane Brull, Bruno Dubroca, Corentin Prigent. A kinetic approach of the bi-temperature Euler model. Kinetic & Related Models, 2020, 13 (1) : 33-61. doi: 10.3934/krm.2020002
[4]

Hung-Wen Kuo. Effect of abrupt change of the wall temperature in the kinetic theory. Kinetic & Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030
[5]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593
[6]

Samuel Walsh. Steady stratified periodic gravity waves with surface tension II: Global bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3287-3315. doi: 10.3934/dcds.2014.34.3287
[7]

Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153
[8]

Hyung Ju Hwang, Youngmin Oh, Marco Antonio Fontelos. The vanishing surface tension limit for the Hele-Shaw problem. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3479-3514. doi: 10.3934/dcdsb.2016108
[9]

Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217
[10]

Colette Calmelet, Diane Sepich. Surface tension and modeling of cellular intercalation during zebrafish gastrulation. Mathematical Biosciences & Engineering, 2010, 7 (2) : 259-275. doi: 10.3934/mbe.2010.7.259
[11]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109
[12]

Nataliya Vasylyeva, Vitalii Overko. The Hele-Shaw problem with surface tension in the case of subdiffusion. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1941-1974. doi: 10.3934/cpaa.2016023
[13]

Samuel Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3241-3285. doi: 10.3934/dcds.2014.34.3241
[14]

Reiner Henseler, Michael Herrmann, Barbara Niethammer, Juan J. L. Velázquez. A kinetic model for grain growth. Kinetic & Related Models, 2008, 1 (4) : 591-617. doi: 10.3934/krm.2008.1.591
[15]

Ming Chen, Meng Fan, Xing Yuan, Huaiping Zhu. Effect of seasonal changing temperature on the growth of phytoplankton. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1091-1117. doi: 10.3934/mbe.2017057
[16]

Rafael Granero-Belinchón, Martina Magliocca. Global existence and decay to equilibrium for some crystal surface models. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2101-2131. doi: 10.3934/dcds.2019088
[17]

Kazuo Aoki, Ansgar Jüngel, Peter A. Markowich. Small velocity and finite temperature variations in kinetic relaxation models. Kinetic & Related Models, 2010, 3 (1) : 1-15. doi: 10.3934/krm.2010.3.1
[18]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023
[19]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
[20]

Naoufel Ben Abdallah, Antoine Mellet, Marjolaine Puel. Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach. Kinetic & Related Models, 2011, 4 (4) : 873-900. doi: 10.3934/krm.2011.4.873

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences