Exponential Mixing by Orthogonal Non-Monotonic Shears

TL;DR

This paper introduces a new laminar mixing model using orthogonal non-monotonic shear flows, demonstrating exponential mixing in certain parameter regions and highlighting challenges in extending these results.

Contribution

The paper presents a novel two-dimensional shear flow model with proven exponential mixing properties within specific parameter windows.

Findings

Proven exponential mixing in certain parameter regions.

Identification of parameter regions with poor mixing due to elliptic islands.

Discussion on extending mixing windows and non-exponential mixing at specific parameters.

Abstract

Non-monotonic velocity profiles are an inherent feature of mixing flows obeying non-slip boundary conditions. There are, however, few known models of laminar mixing which incorporate this feature and have proven mixing properties. Here we present such a model, alternating between two non-monotonic shear flows which act in orthogonal (i.e. perpendicular) directions. Each shear is defined by an independent variable, giving a two-dimensional parameter space within which we prove the mixing property over open subsets. Within these mixing windows, we use results from the billiards literature to establish exponential mixing rates. Outside of these windows, we find large parameter regions where elliptic islands persist, leading to poor mixing. Finally, we comment on the challenges of extending these mixing windows and the potential for a non-exponential mixing rate at particular parameter values.