Balanced Heat Transport and Optimal Cooling

Abstract

In this dissertation, we study the problem of designing an optimal incompressible fluid flow to achieve the most efficient heat transfer between a pair of balanced heat source and sink. We obtain a priori bounds on the heat transfer in terms of various measures of bulk flow intensity, such as the mean kinetic energy. A key question in our analysis is how the choice of source–sink function should enter into the optimization. We begin in Chapter III by presenting our results on the two-dimensional version of this problem from [SFT23], where we obtained a general bound on heat transfer in terms of a Hardy norm of the source–sink function, and introduced a ‘pinching’ flow design to saturate the scaling law of this bound for a pair of point-like sources and sinks. We discuss the implications of our bounds for physical, buoyancy-driven internally heated convection, which is a cousin of the more famous Rayleigh–B´enard problem. Chapter IV extends our bounds on heat transfer in various directions, to higher dimensions as well as to a dumbbell-shaped domain with a pair of point source and sink separated by a thin neck. Chapter V improves our analysis of the point source–sink problem by finding the optimal prefactor in our bound on heat transfer, for a subclass of radially symmetric heat profiles. Interestingly, we find that the bulk properties of the fluid flow do not enter into the leading order optimization of heat transfer in this setup. Instead, the heat transfer is set first and foremost by the particular choice of pinching flow profile nearby the sources and sinks. We end with a short conclusion section that provides some suggestions for future work.