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On weakly nonlinear convection in mushy layers during solidification of alloys

Published online by Cambridge University Press:  17 January 2008

B. S. OKHUYSEN  and
D. N. RIAHI
B. S. OKHUYSEN
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM87545, USA
D. N. RIAHI
Affiliation:
Department of Mathematics, 1201 West University Drive, University of Texas-Pan American, Edinburg, TX 78541-2999, USA
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Abstract

We consider the problem of weakly nonlinear buoyant convection in horizontal mushy layers with permeable mush–liquid interface during the solidification of binary alloys. We analyse the effects of several parameters of the problem on the stationary modes of convection in the form of either a hexagonal pattern or a non-hexagonal pattern such as rolls, rectangles and squares. No assumption is made on the thickness of the mushy layer, and a number of simplifying assumptions made in previous theoretical investigations of the problem are relaxed here in order to study the problem based on a more realistic model. Using both analytical and numerical methods, we determine the steady solutions for the nonlinear problem in a range of the Rayleigh number R near its critical value. Both the nonlinear basic state and variable permeability of the present problem favour hexagon-pattern convection. The results of the analyses and computations indicate that depending on the range of values of the parameters, bifurcation to hexagonal or non-hexagonal convection can be either supercritical or subcritical. However, among all the computed solutions in the particular range of values of the parameters that are most relevant to those of the experiments, only convection in the form of down-hexagons with downflow at the cell centres and upflow at the cell boundaries, was found to be realizable, in the sense that its amplitude increases with R.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 596 , 25 January 2008 , pp. 143 - 167
Copyright
Copyright © Cambridge University Press 2008

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References

Amberg, G. & Homsy, G. M. 1993 Nonlinear analysis of buoyant convection in binary solidification with application to channel formation. J. Fluid Mech. 252, 79–98.CrossRefGoogle Scholar
Anderson, D. M. & Worster, M. G. 1995 Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys. J. Fluid Mech. 302, 307–331.CrossRefGoogle Scholar
Ascher, U. M., Mattheij, R. M. & Russell, R. D. 1995 Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, 2nd edn. SIAM, Philadelphia.CrossRefGoogle Scholar
Aussillous, P., Sederman, A. J., Gladen, L. F., Huppert, H. E. & Worster, M. G. 2006 Magnetic resonance imaging of structure and convection in solidifying mushy layers. J. Fluid Mech. 552, 99–125.CrossRefGoogle Scholar
Busse, F. H. 1967 The stability of finite-amplitude convection and its relation to an extremum principal. J. Fluid Mech. 30, 625–649.CrossRefGoogle Scholar
Busse, F. H. 1978 Nonlinear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Chen, F. 1995 Experimental study of convection in a mushy layer during directional solidification. J. Fluid Mech. 293, 81–98.CrossRefGoogle Scholar
Chen, F. & Chen, C. F. 1991 Experimental study of directional solidification of aqueous ammonium chloride solution. J. Fluid Mech. 227, 567–586.CrossRefGoogle Scholar
Chen, F., Lu, J. W. & Yang, T. L. 1994 Convective instability in ammonium chloride solution directionally solidified from below. J. Fluid Mech. 276, 163–187.CrossRefGoogle Scholar
Chung, C. & Chen, F. 2000 Onset of plume convection in mushy layers. J. Fluid Mech. 408, 53–82.CrossRefGoogle Scholar
Copley, S. M., Giamei, A. F., Johnson, S. M. & Hornbecker, M. F. 1970 The origin of freckles in unidirectionally solidified casting. Metall. Trans. 1, 2193–2204.CrossRefGoogle Scholar
Emms, P. & Fowler, A. 1994 Compositional convection in the solidification of binary alloys. J. Fluid Mech. 262, 111–139.CrossRefGoogle Scholar
Fowler, A. C. 1985 The formation of freckles in binary alloys. IMA J. Appl. Maths 35, 159–174.CrossRefGoogle Scholar
Hills, R., Loper, D. & Roberts, P. 1983 A thermodynamically consistent model of a mushy zone. Q. J. Mech. Appl. Maths. 36, 505–539.CrossRefGoogle Scholar
Iooss, G & Joseph, D. D. 1980 Elementary Stability and Bifurcation Theory. Springer.CrossRefGoogle Scholar
Joseph, D. D. 1976 Stability of Fluid Motions. Springer.Google Scholar
Keller, H. B. 1976 Numerical Solution of Two Point Boundary Value Problems. SIAM, Philadelphia.CrossRefGoogle Scholar
Lage, J. 1998 The fundamental theory of flow through permeable media from Darcy to turbulence. In Transport Phenomena in Porous Media (ed. Ingham, D. & Pop, I.), pp. 1–30. Pergamon.Google Scholar
Okhuysen, B. S. 2005 Analytical and computational studies of convection in solidifying binary media. PhD thesis, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, USA.Google Scholar
Tait, S. & Jaupart, C. 1992 Compositional convection in a reactive crystalline mush and melt differentiation. J. Geophys. Res. 97, 6735–6756.CrossRefGoogle Scholar
Tait, S., Jahrling, K. & Jaupart, C. 1992 The planform of compositional convection and chimney formation in a mushy layer. Nature 359, 406–408.CrossRefGoogle Scholar
Worster, M. G. 1991 Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359.CrossRefGoogle Scholar
Worster, M. G. 1992 Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237, 649–669.CrossRefGoogle Scholar