Document Type

Article

Publication Date

Summer 8-17-2017

Publication Title

SIAM J. APPLIED DYNAMICAL SYSTEMS

ISBN

1536-0040

Volume

16

Issue

3

First Page

1514

Last Page

1542

DOI

10.1137/16M1107139

Abstract

Mixing of a passive scalar in a fluid flow results from a two part process in which large gradients are first created by advection and then smoothed by diffusion. We investigate methods of designing efficient stirrers to optimize mixing of a passive scalar in a two-dimensional, nonautonomous, incom- pressible flow over a finite-time interval. The flow is modeled by a sequence of area-preserving maps whose parameters change in time, defining a mixing protocol. Stirring efficiency is measured by a negative Sobolev seminorm; its decrease implies creation of fine-scale structure. A Perron–Frobenius operator is used to numerically advect the scalar for two examples: compositions of Chirikov stan- dard maps and of Harper maps. In the former case, we find that a protocol corresponding to a single vertical shear composed with horizontal shearing at all other steps is nearly optimal. For the Harper maps, we devise a predictive, one-step scheme to choose appropriate fixed point stabilities and to control the Fourier spectrum evolution to obtain a near-optimal protocol.

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