Optimal scalar and vector transport using branching flows

We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximizes the heat transfer between two differentially heated walls for a given power supply budget. Previous work established that heat transport cannot scale faster than 1/3-power of the power supply. We present a novel construction of three dimensional ``branching pipe flows'' for which we show the sharpness of the upper bound. After carefully examining these designs, we extract the underlying physical mechanism that makes the branching flows ``efficient,'' based on which we present a design of mechanical apparatus that, in principle, can achieve the best possible case scenario of heat transfer. In the latter part of the talk, we extend this approach to transport of a divergence-free vector. We present branching flow designs that achieve a momentum transport scaling (in Reynolds number) that aligns precisely with that of turbulent flow in a channel.