Control of mixing in fluid flows and applications to microfluidic mixing

In this thesis, we describe a framework for control and optimization of advective mixing. First, we present a multiscale measure for mixing to quantify the degree of mixedness of an advected density field. We relate this measure to the classical ergodic theoretic notion of mixing and present its basic mathematical properties such as its representation in terms of the power spectrum of the density field. Using this measure, we frame an optimal control problem for mixing in Stokes fluid flows. The velocity field is assumed to be induced by a finite set of spatially distributed force fields which can be modulated arbitrarily with time and a passive material is advected by the flow. The cost-function used is a weighted combination of the mixedness of the density field and the control effort. We derive the necessary conditions for optimality of the time modulation, which can be expressed as a two point boundary value problem. Numerical approximations to the optimal controls are computed using gradient based methods and we discuss some characteristic features of the optimal controls. Finally, we discuss the optimization of mixing in a prototype active micromixer in which the flow in a main channel is perturbed by time-periodic flows in orthogonal secondary channels.