  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 P\'eclet 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 used matched asymptotics to show that when the P\'eclet
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 confirm their results
with rigorous proofs, and also provide an almost matching lower bound. For
simplicity, we work on the infinite strip instead of the unit disk and the
proof uses probabilistic techniques.
