Control of fluid mixing using entropy methods

Abstract

We consider the problem of motion planning for fluid mixing. The control problem consists of determining the flow that will mix best among the ones achieved by composing purely horizontal and vertical shears on a two dimensional torus. This is a prototypical problem for many practical fluid flow applications, and describes the basic stretching and folding mechanisms involved in mechanical mixing. Our approach is to study the mixing process using the ergodic theoretic concepts of strong mixing and entropy. In a more general context, we develop the necessary tools for an ergodic theory of sequences of transformations. A number of generalizations of previous results obtained by the authors are presented. In particular, the maximum entropy optimization problem is posed and solved here not only on the sequence of the shear flows to be applied but also on the shape of the functions involved. The solution of the problem turns out to have a direct physical interpretation linking some facts in ergodic theory and fluid dynamics. Other general results, such as the characterization of mixing properties for sequences under a suitable convergence assumption, are also applied to our physical problem.