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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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