Bounds on heat transport for internally heated convection

Abstract

Convection of a fluid between parallel plates driven by uniform internal heating is a problem where the asymptotic scaling of the mean vertical convective heat transport ⟨wT⟩ was largely unknown. This thesis proves upper bounds on ⟨wT⟩ with respect to the non-dimensional Rayleigh number R. Here R quantifies the destabilising effect of heating compared to the stabilising effect of diffusion. By the background field method, formulated in terms of quadratic auxiliary functionals, linear convex optimisation problems are constructed whose solutions provide upper bounds on ⟨wT⟩. The numerical optimisation carried out with semidefinite programming guides the mathematical analysis and subsequent proofs. The quantity ⟨wT⟩ has different physical implications based on the three thermal boundary conditions studied: perfect conductors, an insulating bottom and perfectly conducting top, and poorly conducting boundaries. In the first setup, ⟨wT⟩ quantifies the flux of heat out of the top and bottom. Whereas in the latter two cases, ⟨wT⟩ quantifies the ratio of total heat transport to the mean conductive heat transport. Critical to the proofs is the use of a minimum principle on the temperature. Finally, we also prove bounds in the scenarios of infinite Prandtl numbers and free-slip boundaries.