Symmetries within chaos: A route to effective mixing

Experimental studies show that simple two-dimensional time-periodic flows can produce chaotic mixing. However, the mixing is not always complete; depending on the choice of the period, there can exist large dynamic structures, called islands, that stretch and compress in a time-periodic manner but remain segregated even after long times. Obviously, a flow that contains very few islands is desired. Unfortunately, in most real systems, an analytic expression for the velocity field (or the motion) does not usually exist, making theoretical prediction of the location and size of islands virtually impossible. However, using only minimal knowledge of the gross properties of the velocity field, the flow can be analyzed in terms of its symmetries. Symmetries can be detected without reference to precise mathematical equations. Large islands are located on lines of symmetry, or in pairs on opposite sides of the line. With this knowledge, it is possible to manipulate symmetries in a systematic way so as to move an island into a region of good mixing. Applying this idea repeatedly results in a recursively generated flow history that is neither periodic nor random, but is self-similar. Mixing in these flows is efficient over the entire flow domain. In addition, the idea of recursively generated flow histories might have implications for understanding the mechanisms of turbulent flow.