Abstract

Convective flows driven by internal sources of heat present many confounding aspects that have attracted increasing attention over the last few years. One fundamental question that remains largely open is how the mean vertical convective heat flux depends on the non-dimensional control parameters of the flow — namely, the Prandtl number Pr and a “flux” Rayleigh number R measuring the strength of internal heating. This talk presents new parameter-dependent rigorous upper bounds on the mean vertical convective heat flux for a uniformly heated layer of fluid confined between by two horizontal plates. For fluids with a finite Prandlt number, the upper bound asymptotes to a constant value from below as R is raised, and it does so at an exponential rate that depends on the thermal boundary conditions (isothermal vs insulating). In the infinite-Pr limit, instead, the bound displays a slower power-law increase. These bounds are proven using a modern formulation of the background method of Doering & Constantin, augmented with a minimum principle for the temperature of the fluid.