fast viscous convection

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Abstract
A method is described for the asymptotic study of natural convection problems in the limit of fast flow, when the dimensionless Rayleigh number R tends to infinity. The Boussinesq approximation is used, and attention is confined to a viscous limit, with the Prandtl number taken as infinite; this makes the fluid momentum negligible. The flow is nevertheless so fast that the temperature is disturbed only in thin layers. The heat equation is studied in these layers, and the effect of the buoyancy in the layers on the motion is expressed in terms of the heat flux. The method is applied to steady two-dimensional convection from a heated strip of an infinite horizontal plane, with a no-stress free-surface boundary, and with a non-slip fixed-surface boundary. The respective results for the total dimensionless heat flux are as R → ∝, where the Rayleigh number R is defined in terms of the half-width of the strip, and Г is the gamma function.

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