  We compute steady planar incompressible flows and wall shapes that maximize
the rate of heat transfer (Nu) between and hot and cold walls, for a given rate
of viscous dissipation by the flow (Pe$^2$). In the case of no flow, we show
theoretically that the optimal walls are flat and horizontal, at the minimum
separation distance. We use a decoupled approximation to show that flat walls
remain optimal up to a critical nonzero flow magnitude. Beyond this value, our
computed optimal flows and wall shapes converge to a set of forms that are
invariant except for a Pe$^{-1/3}$ scaling of horizontal lengths. The
corresponding rate of heat transfer Nu $\sim$ Pe$^{2/3}$. We show that these
scalings result from flows at the interface between the diffusion-dominated and
convection-dominated regimes. We also show that the separation distance of the
walls remains at its minimum value at large Pe.
