Analysis of Mixing in a 2D Drop with Time-Periodic Boundary Forcing

We carry out a detailed analysis on the model problem of Nadim \& Miraghaie [Bull.\ Am.\ Phys.\ Soc., {\bf 49}, 188 (2004)], which consists of a 2D circular drop driven by a tangential stress applied at its boundary giving rise to a pair of circulating flows in each half of the drop that are periodically reoriented. We characterize the resulting chaotic flow by computing the Lyapunov Exponents (LE) and their Finite-Time counterparts (FTLE) for all initial positions within the drop. We calculate the mean and variance of the FTLEs for a wide range of switching times, and identify the optimal switching time for efficient mixing. At some non-optimal switching times, the drop domain contains a mixing region and non-mixing islands which are associated with a bimodal distribution of FTLEs. For a certain switching time, we identify a group of 4 points (which form a square in the drop) that are permuted by the flow and return to their original positions after 4 switching periods. The space-time trajectories of these points, which can be regarded as virtual ``stirring rods,'' form braids when the flow is chaotically mixing. Calculation of braiding factors associated with different patterns of switching enables us to assess their mixing efficacy. [Supported by Fletcher Jones Fellowships/CCMS]