Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T23:48:17.776Z Has data issue: false hasContentIssue false
  • English
  • Français

On long-wave morphological instabilities in directional solidification

Published online by Cambridge University Press:  26 September 2008

A. C. Skeldon ,
G. B. McFadden ,
M. D. Impey ,
D. S. Riley ,
K. A. Cliffe ,
A. A. Wheeler  and
S. H. Davis
A. C. Skeldon
Affiliation:
Department of Theoretical Mechanics, University of Nottingham, University Park, Nottingham NG7 2RD, UK
G. B. McFadden
Affiliation:
National Institute of Science and Technology, Gaithersburg, Maryland 20899, USA
M. D. Impey
Affiliation:
Inter a Information Technologies Ltd, Chiltern House, 45 Station Road, Henley-on-Thames, Oxon RG9 1 AT, UK
D. S. Riley
Affiliation:
Department of Theoretical Mechanics, University of Nottingham, University Park, Nottingham NG7 2RD, UK
K. A. Cliffe
Affiliation:
AEA Technology, B424.4Harwell Laboratory, Didcot, Oxon OXII ORA, UK
A. A. Wheeler
Affiliation:
Faculty of Mathematical Studies, University of Southampton, Southampton S09 5NH, UK
S. H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A binary liquid that undergoes directional solidification is susceptible to morphological instabilities which cause the solid/liquid interface to change from a planar to a cellular state. This paper presents a numerical study of a class of long-wave equations that describe the evolution of interface morphology. We find new bifurcation points, new solution branches, and the existence of inverted hexagonal nodes and cells.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 6 , December 1995 , pp. 639 - 652
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Mullins, W. W. & Sekerka, R. F. 1964 Stability of a planar interface during directional solidification of a dilute binary alloy. J. Appl. Phys. 35, 444–451.CrossRefGoogle Scholar
[2] Langer, J.S. 1980 Instabilities and pattern formation in crystal growth. Rev. Modern Phys. 52, 1–28.CrossRefGoogle Scholar
[3] Wollkind, D. J. & Segel, L. A. 1970 A nonlinear stability analysis of the freezing of a dilute binary alloy. Phil. Trans. Roy. Soc. London A 268, 351–380.Google Scholar
[4] Riley, D. S. & Davis, S. H. 1990 Long-wave morphological instabilities in the directional solidification of a dilute binary mixture. SIAM J. Appl. Math. 50, 420–436.CrossRefGoogle Scholar
[5] Sivashinsky, G. I. 1983 On cellular instability in the solidification of a dilute binary alloy. Physica D 8, 243–248.CrossRefGoogle Scholar
[6] Brattkus, K. & Davis, S. H. 1988 Cellular growth near absolute stability. Phys. Rev. B 38, 11452–11460.CrossRefGoogle ScholarPubMed
[7] Wollkind, D. J., Sriranganathan, R. & Oulton, D. B. 1984 Interfacial patterns during plane front alloy solidification. Physica D 12, 215–240.CrossRefGoogle Scholar
[8] Riley, D. S. & Winters, K. H. 1989 Modal exchange mechanisms in Lapwood convection. J. Fluid Mech. 204, 325–358.CrossRefGoogle Scholar
[9] McFadden, G. B., Boisvert, R. F. & Coriell, S. R. 1987 Nonplanar interface morphologies during unidirectional solidification of a binary alloy. II Three dimensional computations. J. Crystal Growth 84, 371–388.CrossRefGoogle Scholar
[10] Impey, M. D., Riley, D. S. & Wheeler, A. A. 1993 Bifurcation analysis of cellular interfaces in the unidirectional solidification of a dilute binary alloy. SIAM J. Appl. Math. 53, 78–95.CrossRefGoogle Scholar
[11] Skeldon, A. C, Cliffe, K. A. & Riley, D. S. On the computation of hexagon-roll interactions using a finite element method. In preparation.Google Scholar
[12] Gottlieb, D., Hussaini, M. Y. & Orszag, S. A. 1984 Theory and applications of spectral approximations. In Spectral Methods for Partial Differential Equations, ed. Voigt, R. G., Gottlieb, D. & Hussaini, M. Y., SIAM, Philadelphia, pp. 1–54.Google Scholar
[13] Powell, M. J. D. 1970 A hybrid method for nonlinear equations. In Numerical Methods for Nonlinear Algebraic Equations, ed. Rabinowitz, P., Gordon and Breach.Google Scholar
[14] SLATEC Common Math Library, Package 181-CY001–00, Energy Science and Technology Software Center, P.O. Box 1020, Oak Ridge, TN 37381, USA.Google Scholar
[15] Buzano, E. & Golubitsky, M. 1983 Bifurcation on the hexagonal lattice and the planar Bénard problem. Phil. Trans. R. Soc. Lond. A 308, 617–667.Google Scholar
[16] Wheeler, A. A. & Winters, K. H. 1989 On a finite element method for the calculation of steady cellular interfaces in the one-sided model of solidification. Comm. Appl. Numer. Meth. 5, 309–320.CrossRefGoogle Scholar
[17] Ungar, L. H. & Brown, R. A. 1985 Cellular interface morphologies in directional solidification. IV. The formation of deep cells. Phys. Rev. B 31, 5931–5940.CrossRefGoogle ScholarPubMed
[18] Kessler, D. A. & Levine, H. 1989 Steady-state cellular growth during directional solidification. Phys. Rev. A 39, 3041–3052.CrossRefGoogle ScholarPubMed
[19] Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260.CrossRefGoogle Scholar
[20] Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.CrossRefGoogle Scholar
[21] Scanlon, J. W. & Segel, L. A. 1967 Finite amplitude cellular convection induced by surface tension. J. Fluid Mech. 30, 149–162.CrossRefGoogle Scholar
[22] Humphreys, L. B., Heminger, J. A. & Young, G. W. 1990 Morphological stability in a float zone. J. Crystal Growth 100, 31–50.CrossRefGoogle Scholar
[23] Brattkus, K. 1990 Oscillatory instabilities in cellular solidification. Appl. Mech. Rev. 43, S56–S58.CrossRefGoogle Scholar
[24] Hyman, J. M., Novick-Cohen, A. & Rosenau, P. 1988 Modified asymptotic approach to modeling a dilute-binary-alloy solidification front. Phys. Rev. B 37, 7603–7608.CrossRefGoogle ScholarPubMed
[25] Riley, D. S. & Davis, S. H. 1990 Long-wave interactions in morphological and convective instabilities. IMA J. Appl. Math. 45, 267–285.CrossRefGoogle Scholar
[26] Wheeler, A. A. 1991 A strongly-nonlinear analysis of the morphological instability of a freezing binary alloy: Solutal convection, density change, and nonequilibrium effects. IMA J. Appl. Math. 47, 173–192.CrossRefGoogle Scholar