Flowing a fluid is a familiar and efficient way to cool: fans cool electronics, water cools nuclear reactors, and the atmosphere cools the Earth. In this talk, we discuss a class of problems from fluid dynamics concerning the design of incompressible wall-bounded flows achieving optimal rates of heat transport for a given flow intensity budget. Guided by a perhaps unexpected connection between this optimal design problem and various “energy-driven pattern formation” problems from materials science, we construct flows achieving nearly optimal rates of heat transport in their scaling with respect to the intensity budget. The resulting flows share striking similarities with self-similar elastic wrinkling patterns, such as can be seen in the shape of a hanging drape or nearby the edge of a torn plastic sheet. They also remind of (carefully designed versions) of the complex multi-scale patterns seen in turbulent fluids. Nevertheless, we prove that in certain cases natural buoyancy-driven convection is not capable of achieving optimal rates of cooling. This is joint work with Charlie Doering.