Abstract
Bounds are derived for the space–time averaged temperature 〈T〉 of a fluid layer in the Boussinesq approximation between fixed-temperature horizontal boundaries subject to uniform heating H throughout the volume. The analysis is carried out for both finite and infinite Prandtl number fluids. While the average temperature 〈T〉∼H in the purely conductive state, convection enhances the heat transport beyond static conduction reducing the temperature. Lower bounds to the average temperature of the layer scale with the magnitude of the imposed heat flux, with one scaling exponent for the arbitrary Prandtl number case and another for the infinite Prandtl number model. Specifically, it is proven here that at large heating rates where convection is important, 〈T〉⩾c1H2/3 for finite Prandtl number fluids and 〈T〉⩾c2H5/7 for infinite Prandtl number fluids. Explicit prefactors c1 and c2 for the scaling bounds are computed as well.
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