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Onset of convection in a variable-viscosity fluid

Published online by Cambridge University Press:  20 April 2006

Karl C. Stengel ,
Dean S. Oliver  and
John R. Booker
Karl C. Stengel
Affiliation:
Geophysics Program, University of Washington, Seattle, WA 98195, U.S.A.
Dean S. Oliver
Affiliation:
Geophysics Program, University of Washington, Seattle, WA 98195, U.S.A.
John R. Booker
Affiliation:
Geophysics Program, University of Washington, Seattle, WA 98195, U.S.A.
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Abstract

The Rayleigh number R, in a horizontal layer with temperature-dependent viscosity can be based on the viscosity at T0, the mean of the boundary temperatures. The critical Rayleigh number Roc for fluids with exponential and super-exponential viscosity variation is nearly constant at low values of the ratio of the viscosities at the top and bottom boundaries; increases at moderate values of the viscosity ratio, reaching a maximum at a ratio of about 3000, and then decreases. This behaviour is explained by a simple physical argument based on the idea that convection begins first in the sublayer with maximum Rayleigh number. The prediction of Palm (1960) that certain types of temperature-dependent viscosity always decrease Roc is confirmed by numerical results but is not relevant to the viscosity variations typical of real liquids. The infinitesimal-amplitude state assumed by linear theory in calculating Roc does not exist because the convection jumps immediately to a finite amplitude at R0c. We observe a heat-flux jump at R0c exceeding 10% when the viscosity ratio exceeds 150. However, experimental measurements of R0c for glycerol up to a viscosity ratio of 3400 are in good agreement with the numerical predictions when the effects of a temperature-dependent expansion coefficient and thermal diffusivity are included.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 120 , July 1982 , pp. 411 - 431
Copyright
© 1982 Cambridge University Press

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