Numerical Methods for Partial Differential Equations

Wednesday, September 28, 2022 at 4:30pm to 5:30pm

MIT-Math Dept., /Room 2-449

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

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


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.

Event Type


Events By Interest


Events By Audience

MIT Community

Events By School

School of Science


Department of Mathematics
Contact Email

Add to my calendar

Recent Activity