We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximizes the transport of a passive temperature between two differentially heated walls for a given power supply budget. Previous work established that the transport cannot scale faster than 1/3-power of the power supply. Recently, Doering & Tobasco (CPAM'19) constructed self-similar two-dimensional steady branching flows, saturating this upper bound up to a logarithmic correction to scaling. We present a construction of three-dimensional "branching pipe flows" that eliminates the possibility of this logarithmic correction and for which the corresponding passive scalar transport scales as a clean 1/3-power law in power supply. However, using an unsteady branching flow construction, it appears that the 1/3 scaling is also optimal in two dimensions. 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.
Finally, we present an application of branching flows to a study concerning the nonuniqueness of trajectories. After the theory of DiPerna–Lions'89, a question remained, whether there are continuous Sobolev vector fields such that the trajectories are not unique on a set of positive measures. Recently, an answer to this question was partially given, see, for example, Bruè, Colombo and De Lellis'21 and Fefferman et al.'21. Borrowing a few ideas from our study on optimal scalar transport, we present an explicit construction of a vector field that fully resolves this question.