Publication 21-CNA-002

Bounds On The Heat Transfer Rate Via Passive Advection

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

Truong-Son Van
Department of Mathematical Sciences and Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh, PA 15213, USA
sonv@andrew.cmu.edu

Abstract: In heat exchangers, an incompressible fluid is heated initially and cooled at the boundary. The goal is to transfer the heat to the boundary as efficiently as possible. In this paper we study a related steady version of this problem where a steadily stirred fluid is uniformly heated in the interior and cooled on the boundary. For a given large Péclet number, how should one stir to minimize some norm of the temperature? This version of the problem was previously studied by Marcotte, Doering et al. (SIAM Appl. Math ’18) in a disk, where the authors showed that when the Péclet number, Pe, is sufficiently large one can stir the fluid in a manner that ensures the total heat is O(1/Pe). In this paper we instead study the problem on an infinite strip. By forming standard convection rolls we show that one can stir the fluid in a manner that ensures that the temperature of the hottest point is O(1/Pe^4/7), up to a logarithmic factor. The same upper bound is expected to be true for the total heat and other L^p-norms of the temperature. We do not, however, know if this is optimal in a strip and are presently unable to prove a matching lower bound.

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