  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.
