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Motivated by the search for sharp bounds on turbulent heat transfer as well as the design of optimal heat exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, and exits through the boundary at a fixed temperature. We seek an advecting flow to optimize this exchange. Previous work on energy-constrained cooling with a constant source has conjectured that global optimizers should resemble convection rolls; we prove one-sided bounds on energy-constrained cooling corresponding to, but not resolving, this conjecture. In the case of an enstrophy constraint, our results are more complete: we construct a family of self-similar, tree-like ‘branching flows’ whose cooling we prove is within a logarithm of globally optimal. These results hold for general space- and time-dependent source–sink distributions that add more heat than they remove. Our main technical tool is a non-local Dirichlet-like variational principle for bounding solutions of the inhomogeneous advection–diffusion equation with a divergence-free velocity. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’. 

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of the velocity fields by a Péclet number $Pe$ proportional to their root-mean-square rate of strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e. the Nusselt number $Nu$ up to $$Pe\approx 10^{5}$$ . The resulting transport exhibits a change of scaling from $$Nu-1\sim Pe^{2}$$ for $Pe<10$ in the linear regime to $$Nu\sim Pe^{0.54}$$ for $$Pe>10^{3}$$ . Optimal fields are observed to be approximately separable, i.e. products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound $${\lesssim}Pe^{6/11}=Pe^{0.\overline{54}}$$ as $$Pe\rightarrow \infty$$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh–Bénard convection are discussed.