Analysis of mixing in three-dimensional time-periodic cavity flows

A method to locate periodic structures in general three-dimensional Stokes flows with time-periodic boundary conditions is presented and applied to mi xing cavity hows. Numerically obtained velocity fields and particle trackin g schemes are used to provide displacement and stretching fields. From thes e the location and identification of periodic points can be derived. The pr esence or absence of these periodic points allows a judgement on the qualit y of the mixing process. The technique is general and efficient, and applic able to mixing flows for which no analytical velocity field is available (t he case for all three-dimensional flows considered in this paper). Results are presented for three different mixing protocols in a three-dimensional t ime-periodic cavity flow, serving as an accessible test case for the method s developed. A major result is that periodic lines are obtained for these t hree-dimensional flows. These lines can be complex in geometry and their na ture can change along a line from hyperbolic to elliptic. They can serve as practical criteria in the optimization of three-dimensional mixing process es.