Optimal convective mixing by forced two-dimensional Stokes flows

Numerous mixing strategies in the Stokes flow regime rely on time-dependent body forces. The question of determining the required forcing function to achieve optimal mixing at a given power input remains however open. Using optimal control theory, we numerically determine general optimal mixing flows in a two-dimensional periodic geometry as truncated sums of time-modulated Fourier modes. The time-averaged power spectra of these flows are calculated to investigate the effect of scale, and demonstrate that best mixing is achieved when a wide range of scales are present in the flow. We also determine the frequency spectra of the time-modulating functions and characterize the importance of non-harmonic forcing.