Optimal mixing and optimal stirring for fixed energy, fixed power or fixed palenstrophy flows
We consider passive scalar mixing by a prescribed divergence-free velocity vector field in a periodic box and address the following question: Starting from a given initial inhomogeneous distribution of passive tracers, and given a certain energy budget,power budget or finite palenstrophy budget, what incompressible flow field best mixes the scalar quantity? We focus on the optimal stirring strategy recently proposed byLinet al.(2011), which subsequently determine the flow field that instantaneously optimizes the depletion of the H−1mix-norm. In this work we bridge some of the gap between the best available a priori analysis and simulation results. After recalling some previous analysis we present a new explicit example establishing finite-time perfect mixing with a finite energy constraint on the stirring flow. On the other hand, using techniques pioneered by Yudovich in proving uniqueness of solutions to the 2-d Euler equations, we establish that if the flow is constrained to have constant palenstrophy, then the H−1mix-norm decays at most∼e−ct2. Finite-time perfect mixing is thus certainly ruled out when too much cost is incurred by small scale structures in the stirring. Direct numerical simulations suggest the impossibility of finite-time perfect mixing for flows with fixed power constraint and we conjecture an exponential lower bound on the H−1mix-norm in this case. We also discuss some related problems from other areas of analysis, which are similarly suggestive of an exponential lower bound for the H−1mix-norm
|Work Title||Optimal mixing and optimal stirring for fixed energy, fixed power or fixed palenstrophy flows|
|License||In Copyright (Rights Reserved)|
|Deposited||February 24, 2021|
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