# Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps

## TL;DR

This paper develops methods to design efficient stirrers for optimizing mixing in two-dimensional fluid flows over finite times, using area-preserving maps and spectral analysis to identify near-optimal stirring protocols.

## Contribution

It introduces a novel approach to optimize mixing protocols in 2D flows using a sequence of area-preserving maps and spectral control, including a predictive scheme for Harper maps.

## Key findings

- Single vertical shear with horizontal shearing is nearly optimal for Chirikov maps.
- A predictive scheme effectively controls Fourier spectrum in Harper maps.
- Near-optimal mixing protocols significantly enhance scalar mixing efficiency.

## 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, incompressible 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 standard 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.