Bounds on heat transfer by incompressible flows between balanced sources and sinks

Internally heated convection involves the transfer of heat by fluid motion between a distribution sources and sinks. Focusing on balanced case where total added matches taken away sinks, we obtain \emph{a priori} bounds minimum mean thermal dissipation $\langle |\nabla T|^2\rangle$ as measure inefficiency transport. In advective limit, our scale with inverse kinetic energy flow. The constant in this scaling law depends source--sink distribution, explain both pair examples involving oscillatory or concentrated heating cooling, via general asymptotic variational principle for optimizing Key to analysis is solution pure advection equation, which do find extreme cellular `pinching' flows. When flow obeys momentum bound re-expressed terms flux-based Rayleigh number $Ra$ yielding T|^2\rangle\geq CRa^{-\alpha}$. power $\alpha$ $0, 2/3$ $1$ depending arrangement sinks relative gravity.