Numerical Methods for Partial Differential Equations

Speaker:  Anuj Kumar  (University of California, Santa Cruz)

Title:  Optimal scalar transport using steady branching pipe flows and unsteady branching blob flows

Abstract: 

We consider the problem of "wall-to-wall optimal transport," in which we attempt to maximize the transport of a passive temperature field between hot and cold plates. Specifically, we optimize the choice of the divergence-free velocity field in the advection-diffusion equation amongst all velocities satisfying an enstrophy constraint (which can be understood as a constraint on the power required to generate the flow). Previous work established an a priori upper bound on the transport, scaling as the 1/3-power of the flow's enstrophy. Recently, Doering & Tobasco'19 constructed self-similar two-dimensional steady branching flows saturating this bound up to a logarithmic correction to scaling. This logarithmic correction appears to arise due to a topological obstruction inherent to two-dimensional steady branching flows. We present a construction of three-dimensional ``branching pipe flows" that eliminates the possibility of this logarithmic correction and therefore identifies the optimal scaling as a clean 1/3-power law. Also, using an unsteady "branching blob flow" construction, it appears that the 1/3 scaling is, in fact, optimal in two dimensions as well. We discuss the underlying physical mechanism that makes the branching flows "efficient" in transporting heat and present a design of a mechanical apparatus that can transfer heat close to the best possible scenario.