Convective heat transport in a fluid layer of infinite Prandtl number: upper bounds for the case of rigid lower boundary and stress-free upper boundary

摘要

We present the theory of the multi-α-solutions of the variational problem for the upper bounds on the convective heat transport in a heated from below horizontal fluid layer with rigid lower boundary and stress-free upper boundary. A sequence of upper bounds on the convective heat transport is obtained. The highest bound Nu = 1 + (1/6)R~(1/3) is between the bounds Nu = 1 + 0.152R~(1/3) for the case of a fluid layer with two rigid boundaries and Nu = 1 + 0.3254R~(1/3) for the case of a fluid layer with two stress-free boundaries. As an additional result of the presented theory we obtain small corrections of the boundary layer thicknesses of the optimum fields for the case of fluid layer with two rigid boundaries. These corrections lead to systematically lower upper bounds on the convective heat transport in comparison to the bounds obtained in [5].