Title: Optimal heat transfer and optimal exit times,
Abstract: A heat exchanger can be modeled as a closed domain containing an
incompressible fluid. The moving fluid has a temperature distribution obeying
the advection-diffusion equation, with zero temperature boundary conditions at
the walls. Starting from a positive initial temperature distribution in the
interior, the goal is to flux the heat through the walls as efficiently as
possible. Here we consider a distinct but closely related problem, that of the
integrated mean exit time of Brownian particles starting inside the domain.
Since flows favorable to rapid heat exchange should lower exit times, we
minimize a norm of the exit time. This is a time-independent optimization
problem that we solve analytically in some limits, and numerically otherwise.
We find an (at least locally) optimal velocity field that cools the domain on a
mechanical time scale, in the sense that the integrated mean exit time is
independent on molecular diffusivity in the limit of large-energy flows.