Analytical bounds on the heat transport in internally heated convection

Abstract

We obtain an analytical bound on the non-dimensional mean vertical convective heat flux $\langle w T \rangle$ between two parallel boundaries driven by uniform internal heating. We consider two configurations. In the first, both boundaries are held at the same constant temperature and $\langle wT \rangle$ measures the asymmetry of the heat fluxes escaping the layer through the top and bottom boundaries. In the second configuration, the top boundary is held at constant temperature, the bottom one is perfectly insulating, and $\langle wT \rangle$ is related to the difference between the horizontally-averaged temperatures of the two boundaries. For the first configuration, Arslan et al. (J. Fluid Mech., vol. 919, 2021, p. A15) recently provided numerical evidence that Rayleigh-number-dependent corrections to the only known rigorous bound $\langle w T \rangle \leq 1/2$ may be provable if the classical background method is augmented with a minimum principle stating that the fluid's temperature is no smaller than that of the top boundary. Here, we confirm this fact rigorously for both configurations by proving bounds on $\langle wT \rangle$ that approach $1/2$ exponentially from below as the Rayleigh number is increased. The key to obtaining these bounds is inner boundary layers in the background fields with a particular inverse-power scaling, which can be controlled in the spectral constraint using Hardy and Rellich inequalities. These allow for qualitative improvements in the analysis that are not available to standard constructions.