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Energy stability theory for free-surface problems: buoyancy-thermocapillary layers

Published online by Cambridge University Press:  19 April 2006

Stephen H. Davis  and
George M. Homsy
Stephen H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University
George M. Homsy
Affiliation:
Department of Chemical Engineering, Stanford University
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Abstract

Energy stability theory has been formulated for two-dimensional buoyancy–thermocapillary convection in a layer with a free surface. The theory yields a critical Rayleigh number RE for which R < RE is a sufficient condition for stability of the layer. RE emerges from the variational formulation as an eigenvalue of a nonlinear system of Euler–Lagrange equations. For the case of small capillary number (large mean surface tension) explicit values are obtained for RE. The analogous linear-theory results for this case are obtained in terms of a critical Rayleigh number RL. These are compared. It is found that the existence of the deformable interface can lead to a stabilization relative to the case of a planar interface. This result is explained in physical terms. The energy theory is then generalized to include general flow problems having three-dimensional disturbances, non-Newtonian bulk fluids and general interfacial mechanics such as surface viscosity and elasticity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 98 , Issue 3 , 12 June 1980 , pp. 527 - 553
Copyright
© 1980 Cambridge University Press

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References

Davis, S. H. 1969 J. Fluid Mech. 39, 347.
Davis, S. H. & von Kerczek, C. 1973 Arch. Ration. Mech. Anal. 52, 112.
Davis, S. H. & Segel, L. A. 1963 Not. Am. Math. Soc. 10, 496.
Dussan, V. E. B. 1975 Arch. Ration. Mech. Anal. 57, 363.
Dussan, V. E. B. 1979 Ann. Rev. Fluid Mech. 11, 371.
Dussan, V. E. B. & Davis, S. H. 1974 J. Fluid Mech. 65, 71.
Gummerman, R. J. & Homsy, G. M. 1974 A.I.Ch.E. J. 20, 981.
Jeffreys, H. 1951 Quart. J. Mech. Appl. Math. 4, 283.
Joseph, D. D. 1965 Arch. Rat. Mech. Anal. 20, 59.
Joseph, D. D. 1976 Stability of Fluid Motions, vol. I. Springer.
Joseph, D. D. & Hung, W. 1971 Arch. Rat. Mech. Anal. 44, 1.
Kayser, W. V. & Berg, J. C. 1973 J. Fluid Mech. 57, 739.
Moeckel, G. 1975 Arch. Rat. Mech. Anal. 57, 255.
Nield, D. A. 1964 J. Fluid Mech. 19, 341.
Nield, D. A. 1977 J. Fluid Mech. 81, 513.
Pellew, A. & Southwell, R. V. 1940 Proc. Roy. Soc. A 176, 312.
Rayleigh, Lord 1879 Proc. Lond. Math. Soc. 10, 4.
Scriven, L. E. & Sternling, C. V. 1964 J. Fluid Mech. 19, 321.
Smith, K. A. 1966 J. Fluid Mech. 24, 401.
Zeren, R. W. & Reynolds, W. C. 1972 J. Fluid Mech. 53, 305.