Optimal stirring strategies for passive scalar mixing

Abstract

We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a quantitative measure of mixing we adopt the $H^{-1}$ norm of the scalar fluctuation field, equivalent to the (square-root of the) variance of a low-pass filtered image of the tracer concentration field. First we establish that this is a useful gauge even in the absence of molecular diffusion: its vanishing as $t --> \infty$ is evidence of the stirring flow's mixing properties in the sense of ergodic theory. Then we derive absolute limits on the total amount of mixing, as a function of time, on a periodic spatial domain with a prescribed instantaneous stirring energy or stirring power budget. We subsequently determine the flow field that instantaneously maximizes the decay of this mixing measure---when such a flow exists. When no such `steepest descent' flow exists (a possible but non-generic situation) we determine the flow that maximizes the growth rate of the $H^{-1}$ norm's decay rate. This local-in-time optimal stirring strategy is implemented numerically on a benchmark problem and compared to an optimal control approach using a restricted set of flows. Some significant challenges for analysis are outlined.