Rigorous scaling laws for internally heated convection at infinite Prandtl number

Abstract

We prove rigorous scaling laws for measures of the vertical heat transport enhancement in two models of convection driven by uniform internal heating at infinite Prandtl number. In the first model, a layer of incompressible fluid is bounded by horizontal plates held at the same constant temperature and convection reduces the fraction of the total dimensionless heat input per unit volume and time escaping the layer through the bottom boundary. We prove that this fraction decreases no faster than O(R−2), where R is a “flux” Rayleigh number quantifying the strength of the internal heating relative to diffusion. The second model, instead, has a perfectly insulating bottom boundary, so all heat must escape through the top one. In this case, we prove that the Nusselt number, defined as the ratio of the total-to-conductive vertical heat flux, grows no faster than O(R4). These power-law bounds improve on exponential results available for fluids with finite Prandtl number. The proof combines the background method with a minimum principle for the fluid’s temperature and with Hardy–Rellich inequalities to exploit the link between the vertical velocity and temperature available at infinite Prandtl number.