Optimizing mixing in channel flows: kinematic aspects associated with secondary flows in the cross-section

We consider an Eulerian predictor of optimal mixing appropriate for steady, three-dimensional channel flow of the type that is commonly used in a variety of microfluidic mixing applications. This Eulerian indicator is applied to a kinematic model of the channel flow that allows one to control the flow structure of the secondary flows and the number of distinct secondary flows. The cases of two and three distinct secondary flows are considered and the Eulerian indicator predicts the optimal flow structure in the two secondary flow case and in the three secondary flow case that result in the best mixing. Moreover, it is shown that the case of three secondary flows mixes "optimally" (i.e., faster and more completely) than the case of two secondary flows. An explanation of this difference is given in terms of the Eulerian flow structure.