Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-17T07:26:30.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupled convective and morphological instability in a simple model of the solidification of a binary alloy, including a shear flow

Published online by Cambridge University Press:  26 April 2006

S. A. Forth  and
A. A. Wheeler
S. A. Forth
Affiliation:
School of Mathematics, University Walk, Bristol, BS8 1TW, UK Present address: British Aerospace, Sowerby Research Centre, FPC 266, PO Box 5, Filton, Bristol, UK.
A. A. Wheeler
Affiliation:
School of Mathematics, University Walk, Bristol, BS8 1TW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we provide a detailed description of the interaction of solutal convection and morphological instability in the presence of a model boundary-layer flow. We present a detailed investigation of the structure of the marginal surfaces in Rayleigh-number, Sekerka-number, Reynolds-number space associated with a linear stability analysis. We give mathematical arguments and physical mechanisms to explain the results and present a coherent description of this complicated situation. We identify two new modes, one convective and one morphological. We show that the oscillatory so-called ‘mixed’ modes that result from the coupling of morphological and convective modes play a central role in the unfolding of the solution structure by the shear flow. This flow has the effect of decoupling the convective and morphological modes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 236 , March 1992 , pp. 61 - 94
Copyright
© 1992 Cambridge University Press

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

Caroli, B., Caroli, C., Misbah, C. & Roulet, B. 1985 Solutal convection and morphological instability in directional solidification of binary alloys. J. Phys. Paris 46, 401–413.Google Scholar
Coriell, S. R., Cordes, M. R., Boettinger, W. J. & Sekerka, R. F. 1980 Convective and interfacial instabilities during unidirectional solidification of a binary alloy. J. Cryst. Growth 49, 13–28.Google Scholar
Coriell, S. R., McFadden, G. B., Boisvert, R. F. & Sekerka, R. F. 1984 The effect of a forced Couette flow on coupled convective and morphological instabilities during directional solidification. J. Cryst. Growth 69, 15–22.Google Scholar
Coriell, S. R., McFadden, G. B. & Sekerka, R. F. 1985 Cellular growth during directional solidification. Ann. Rev. Mater. Sci. 15, 119–145.Google Scholar
Coriell, S. R., McFadden, G. B., Voorhees, P. W. & Sekerka, R. F. 1987 Stability of a planar interface during solidification of a multicomponent system. J. Cryst. Growth 82, 295–302.Google Scholar
Davis, S. H. 1990 Hydrodynamic interactions in directional solidification. J. Fluid Mech. 212, 241–262.Google Scholar
Delves, R. T. 1968 Theory of the stability of a solid—liquid interface during growth from sitrred melts. J. Cryst. Growth 3, 4, 562–568.Google Scholar
Delves, R. T. 1971 Theory of stability of a solid—liquid interface during growth from stirred melts. II. J. Cryst. Growth 8, 13–25.Google Scholar
Forth, S. A. 1989 Morphological and hydrodynamic instabilities in unidirectional alloy solidification. Ph.D. thesis, University of Bristol, England.
Forth, S. A. & Wheeler, A. A. 1989 Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model. J. Fluid Mech. 202, 339–366.Google Scholar
Gershuni, G. Z. & Zhukhovitskii 1976 Conective Stability In Incompressible Fluids. Jerusalem: Keter.
Glicksman, M. E., Coriell, S. R. & McFadden, G. B. 1986 Interaction of flows with the crystal—melt interface. Ann. Rev. Fluid Mech. 18, 307–35.Google Scholar
Hurle, D. T. J., Jakeman, E. & Wheeler, A. A. 1982 Effect of solutal convection on the morphological stability of a binary alloy. J. Cryst. Growth 58, 163–179.Google Scholar
Hurle, D. T. J., Jakeman, E. & Wheeler, A. A. 1983 Hydrodynamic stability of the melt during solidification of a binary alloy. Phys. Fluids 26, 624–626.Google Scholar
Jenkins, D. R. 1985a Nonlinear analysis of convective and morphological instability during solidification of a dilute binary alloy. Physicochem. Hydrodyn. 6, 521–537.Google Scholar
Jenkins, D. R. 1985b Nonlinear interaction of morphological and convective instabilities during solidification of a dilute binary alloy. IMAJ Appl. Maths 35, 145–157.Google Scholar
Jenkins, D. R. 1990 Oscillatory instability in a model of directional solidification. J. Cryst. Growth 102, 481–490.Google Scholar
Keller, H. B. 1976 Numerical Solution of Two Point Boundary Value Problems. SIAM.
Langer, J. S. 1980 Instabilities and pattern formation in crystal growth. Rev. Mod. Phys. 52, 1–28.Google Scholar
McFadden, G. B., Coriell, S. R. & Alexander, J. I. D. 1988 Hydrodynamic and free boundary instabilities during crystal growth: the effect of a plane stagnation flow. Comm. Pure Appl. Maths 41, 683–706.Google Scholar
Mullins, W. W. & Sekerka, R. F. 1964 Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35, 444–451.Google Scholar
Ng, B. S. & Reid, W. H. 1980 On the numerical solution of the Orr—Somerfeld problem: asymptotic initial conditions for shooting methods. J. Comput. Phys. 38, 275–293.Google Scholar
Riley, D. S. & Davis, S. H. 1990 Long-wave interactions in morphological and convective instabilities. SIAM J. Appl. Maths 50, 420–436.Google Scholar
Rutter, J. W. & Chalmers, B. A. 1953 A prismatic substructure formed during solidification of metals. Can. J. Phys. 31, 15–39.Google Scholar
Schaeffer, R. J. & Coriell, S. R. 1982 Convective and interfacial instabilities during solidification of succinonitrile containing ethanol. In Materials Processing in the Reduced Gravity Environment of Space. (ed. G. E. Rindone), pp. 479–489. Elsevier.
Scott, M. R. & Watts, H. A. 1975 SUPORT: A computer code for two point boundary values problems via orthonormalisation. SAND 75–0198. Albuquerque: Sandia Laboratories.
Ungar, L. H. & Brown, R. A. 1984 Cellular interface morphologies in directional solidification. The one-sided model. Phys. Rev. B 29, 1367–1380.Google Scholar
Wollkind, D. J. & Segel, L. A. 1970 A nonlinear stability analysis of the freezing of a dilute binary alloy. Phil. Trans. R. Soc. Lond. A 268, 351–380.Google Scholar