Numerical Evidence of Exponential Mixing by Alternating Shear Flows

TL;DR

This paper provides numerical evidence that alternating shear flows on a 2D torus can produce exponential mixing rates, with efficiency influenced by flow durations and phase randomness.

Contribution

It demonstrates that alternating shear flows generally lead to exponential mixing, and explores how flow timing and randomness affect mixing efficiency.

Findings

Exponential mixing occurs when flows are applied for sufficiently long durations.

Shorter but not too short flow times maximize mixing rates.

Randomizing flow times or phases does not significantly improve mixing, and may slightly hinder it.

Abstract

We performed a numerical study of the efficiency of mixing by alternating horizontal and vertical shear ``wedge'' flows on the two-dimensional torus. Our results suggest that except in cases where each individual flow is applied for only a short time, these flows produce exponentially fast mixing. The observed mixing rates are higher when the individual flow times are shorter (but not too short), and randomizing either the flow times or phase shifts of the flows does not appear to enhance mixing (again when the flow times are not too short). In fact, the latter surprisingly seems to inhibit it slightly.