Sensitivity of mixing optimization to the geometry of the initial scalar field

Abstract

In this study, we address the conceptual problem of designing a mixing device able to maintain a uniform mixing quality for di erent initial configurations of the mixture. As a case study, we consider as a mixing device a two-dimensional, piecewise steady, nonlinear flow, the sine flow. To promote mixing in the sine flow, we generate an optimized stirring protocol using the short time horizon procedure. We use the mix-norm as a measure of the mixing efficiency and as the cost function of the optimization. We assess the sensitivity to the geometry of the initial configuration of the short-time-horizon optimal protocols in terms of their mixing efficiency. We use the periodic and recursive symmetry-breaking protocols as benchmarks for our assessment. We show that the optimized protocols are generally quite insensitive to the geometry of the initial scalar field when compared to the periodic and recursive symmetrybreaking protocols. We show that the on-line optimization is essential for achieving a uniform mixing efficiency. We characterize the e ect of the switching time horizon on the sensitivity of the optimized protocols to the geometry of the initial configuration. Our results indicate that the optimization over very short time horizons could be in principle used as an on-line procedure for maintaining a uniformly high quality of mixing for di erent initial configurations of the mixture.