a1 Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA
a2 Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
a3 Department of Mathematics, Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA
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 → ∞ 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.
(Received September 04 2010)
(Revised October 30 2010)
(Accepted January 12 2011)
(Online publication March 10 2011)