  The mixing of binary fluids by stirrers is a commonplace procedure in many
industrial and natural settings, and mixing efficiency directly translates into
more homogeneous final products, more enriched compounds, and often substantial
economic savings in energy and input ingredients. Enhancements in mixing
efficiency can be accomplished by unorthodox stirring protocols as well as
modified stirrer shapes that utilize unsteady hydrodynamics and vortex-shedding
features to instigate the formation of fluid filaments which ultimately succumb
to diffusion and produce a homogeneous mixture. We propose a PDE-constrained
optimization approach to address the problem of mixing enhancement for binary
fluids. Within a gradient-based framework, we target the stirring strategy as
well as the cross-sectional shape of the stirrers to achieve improved mixedness
over a given time horizon and within a prescribed energy budget. The
optimization produces a significant enhancement in homogeneity in the initially
separated fluids, suggesting promising modifications to traditional stirring
protocols.
