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SUMMARY:Optimizing scalar transport using branching flows - Anuj Kumar\, U
 niversity of California Santa Cruz
DTSTART:20221024T120000Z
DTEND:20221024T130000Z
UID:TALK185138@talks.cam.ac.uk
CONTACT:9780
DESCRIPTION:We consider the problem of "wall-to-wall optimal transport\," 
 in which we attempt to maximize the transport of a passive temperature fie
 ld between hot and cold plates. Specifically\, we are interested in the de
 sign of forcing in the forced Navier--Stokes equation that maximizes this 
 transport for a given power supply budget. One can equivalently formulate 
 this problem as the design of a divergence-free flow field that maximizes 
 scalar transport under an enstrophy constraint (which can be understood as
  a constraint on the power supply). Previous work established that the tra
 nsport cannot scale faster than 1/3-power of the power supply. Recently\, 
 Tobasco & Doering (Phys. Rev. Lett. vol.118\, 2017\, p.264502) and Doering
  & Tobasco (Comm. Pure Appl. Math. vol.72\, 2019\, p.2385--2448) construct
 ed self-similar two-dimensional steady branching flows saturating this upp
 er bound up to a logarithmic correction to scaling. This logarithmic corre
 ction appears to arise due to a topological obstruction inherent to two-di
 mensional steady branching flows. We present a construction of three-dimen
 sional "branching pipe flows" that eliminates the possibility of this loga
 rithmic correction and for which the corresponding passive scalar transpor
 t scales as a clean 1/3-power law in power supply. Our flows resemble prev
 ious numerical studies of the three-dimensional wall-to-wall problem by Mo
 toki\, Kawahara & Shimizu (J. Fluid Mech. vol.851\, 2018\, p.R4). However\
 , using an unsteady branching flow construction\, it appears that the 1/3 
 scaling is also optimal in two dimensions. This unsteady flow design chall
 enges the general belief that steady flows are optimal for transporting he
 at in the family of all incompressible flows. After carefully examining th
 ese designs\, we extract the underlying physical mechanism that makes the 
 branching flows "efficient." We present the relevance of branching in natu
 rally occurring buoyancy-driven flows and discuss if these flows are optim
 al for transporting a scalar. We also present a design of mechanical appar
 atus\, which in principle\, can achieve the best possible case scenario of
  heat transfer.\n
LOCATION:MR5\, CMS
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