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Onset of convection in two layers of a binary liquid

Published online by Cambridge University Press:  18 March 2010

G. B. McFADDEN ,
S. R. CORIELL  and
P. A. LOTT
G. B. McFADDEN*
Affiliation:
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
S. R. CORIELL
Affiliation:
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
P. A. LOTT
Affiliation:
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
*
†Email address for correspondence: mcfadden@nist.gov
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Abstract

We perform linear stability calculations for horizontal bilayers of a two-component fluid that can undergo a phase transformation, taking into account both buoyancy effects and thermocapillary effects in the presence of a vertical temperature gradient. Critical values for the applied temperature difference across the system that is necessary to produce instability are obtained by a linear stability analysis, using both numerical computations and small wavenumber approximations. Thermophysical properties are taken from the aluminum–indium monotectic system, which includes a liquid–liquid miscibility gap. In addition to buoyant and thermocapillary modes of instability, we find an oscillatory phase-change instability due to the combined effects of solute diffusion and fluid flow that persists at small wavenumbers. This mode is sensitive to the ratio of the layer depths, and for certain layer depths can occur for heating from either above or below.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 647: Dedicated to Professor S. H. Davis on his 70th birthday , 25 March 2010 , pp. 105 - 124
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Andereck, C. D., Colovas, P. W., Degen, M. M. & Renardy, Y. Y. 1998 Instabilities in two-layer Rayleigh–Benard convection: overview and outlook. Int. J. Engng Sci. 36, 1451–1470.CrossRefGoogle Scholar
Berg, R. F. & Moldover, M. R. 1989 Critical exponent for the viscosity of four binary liquids. J. Chem. Phys. 89, 3694–3704.CrossRefGoogle Scholar
Bräuer, P. & Müller-Vogt, G. 1998 Measurements of aluminum diffusion in molten gallium and indium. J. Cryst. Growth 186, 520–527.CrossRefGoogle Scholar
Busse, F. H. 1989 Fundmanetals of thermal convection. In Mantle Convection: Plate Tectonics and Global Dynamics (ed. Peltier, W.), pp. 23–95. Gordon & Breach.Google Scholar
Busse, F. & Schubert, G. 1971 Convection in a fluid with two phases. J. Fluid Mech. 46, 801–812.CrossRefGoogle Scholar
Coriell, S. R. & McFadden, G. B. 1994 Morphological stability. In Handbook of Crystal Growth 1. Fundamentals, Part B: Transport and Stability (ed. Hurle, D. T. J.), pp. 785–857. North-Holland.Google Scholar
Coriell, S. R., Mitchell, W. F., Murray, B. T., Andrews, J. B. & Arikawa, Y. 1997 Analysis of monotectic growth: infinite diffusion in the L2-phase. J. Cryst. Growth 179, 647–657.CrossRefGoogle Scholar
Coughanowr, C. A. 1989 Solid-state electrochemistry and thermodynamic assessment in the Al-In-Sb system. PhD thesis, Department of Chemical Engineering, University of Florida, Gainesville, FL.Google Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19, 403–435.CrossRefGoogle Scholar
Davis, S. H. 1994 Effects of fluid flow on morphological stability. In Handbook of Crystal Growth 1. Fundamentals, Part B: Transport and Stability (ed. Hurle, D. T. J.), pp. 859–897. North-Holland.Google Scholar
Davis, S. H. 2001 Theory of Solidification. Cambridge University Press.CrossRefGoogle Scholar
Derby, B. & Favier, J. J. 1983 A criterion for the determination of monotectic structure. Acta Metall. 31 (7), 1123–1130.CrossRefGoogle Scholar
Gale, W. F. & Totemeier, T. C. (Ed.) 2004 Smithells' Metals Reference Book. Elsevier Butterworth-Heinemann.Google Scholar
Glicksman, M. E., Coriell, S. R. & McFadden, G. B. 1986 Interaction of flows with the crystal-melt interface. Annu. Rev. Fluid Mech. 18, 307–335.CrossRefGoogle Scholar
Huang, A. & Joseph, D. D. 1992 Instability of the equilibrium of a liquid below its vapour between horizontal heated plates. J. Fluid Mech. 242, 235–247.CrossRefGoogle Scholar
Hunt, J. D. & Lu, S. Z. 1994 Crystallisation of eutectics, monotectics, and peritectics. In Handbook of Crystal Growth 2. Bulk Crystal Growth, Part B: Growth Mechanisms and Dynamics (ed. Hurle, D. T. J.), pp. 1111–1166. North-Holland.Google Scholar
Jackson, K. A. & Hunt, J. D. 1966 Lamellar and rod eutectic growth. Trans. Metal. Soc. AIME 236 (8), 1129–1142.Google Scholar
Johnson, D. & Narayanan, R. 1998 Marangoni convection in multiple bounded fluid layers and its application to materials processing. Phil. Trans. R. Soc. Lond. A 356, 885–898.CrossRefGoogle Scholar
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Springer.Google Scholar
Keller, H. B. 1976 Numerical Solution of Two-Point Boundary Value Problems. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, Course of Theoretical Physics, 2nd edn., vol. 6. Butterworth-Heinemann.Google Scholar
Lupis, C. H. P. 1983 Chemical Thermodynamics of Materials. North-Holland.Google Scholar
McFadden, G. B. & Coriell, S. R. 2009 Onset of oscillatory convection in two liquid layers with phase change. Phys. Fluids 21, 0341017.CrossRefGoogle Scholar
McFadden, G. B., Coriell, S. R., Gurski, K. F. & Cotrell, D. L. 2007 a Onset of convection in two liquid layers. J. Res. NIST 112, 271–281.CrossRefGoogle Scholar
McFadden, G. B., Coriell, S. R., Gurski, K. F. & Cotrell, D. L. 2007 b Onset of convection in two liquid layers with phase change. Phys. Fluids 19, 104109.CrossRefGoogle Scholar
Merkwitz, M. & Hoyer, W. 1999 Liquid–liquid interfacial tension in the demixing metal systems Al-Pb and Al-In. Z. Metallkd. 90, 363–370.Google Scholar
Nepomnyashchy, A. A., Verlarde, M. G. & Colinet, P. 2002 Interfacial Phenomena and Convection. Chapman & Hall/CRC.Google Scholar
Powell, M. J. D. 1970 A hybrid method for nonlinear equations. In Numerical Methods for Nonlinear Algebraic Equations (ed. Rabinowitz, P.), pp. 87–114. Gordon & Breach.Google Scholar
Sakurai, S., Tschammer, A., Pesch, W. & Ahlers, G. 1999 Convection in the presence of a first-order phase change. Phys. Rev. E 60, 539–550.CrossRefGoogle ScholarPubMed
Schatz, M. F. & Neitzel, G. P. 2001 Experiments on thermocapillary instabilities. Annu. Rev. Fluid Mech. 33, 93–127.CrossRefGoogle Scholar
Scott, M. R. & Watts, H. A. 1977 Computational solution of linear two-point boundary-value problems via orthonormalization. SIAM J. Numer. Anal. 14, 40–70.CrossRefGoogle Scholar
Stöcker, C. & Ratke, L. 1999 A new ‘Jackson–Hunt’ model for monotectic composite growth. J. Cryst. Growth 203 (4), 582–593.CrossRefGoogle Scholar
Stöcker, C. & Ratke, L. 2000 Monotectic composite growth with fluid flow. J. Cryst. Growth 212 (1), 324–333.CrossRefGoogle Scholar
Tiller, W. A., Jackson, K. A., Rutter, J. W. & Chalmers, B. 1953 The redistribution of solute atoms during the solidification of metals. Acta Metall. 1, 428–437.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Vandevender, W. H. & Kaskell, K. H. 1970 The SLATEC mathematical subroutine library. SIGNUM Newsl. 17, 16 (As of December 2009, the SLATEC library is available at http://www.netlib.org/slatec. The program SNSQ was written by K. Hiebert and is based on an algorithm of Powell 1970).Google Scholar