Topology of Chaotic Mixing Patterns

TL;DR

This paper explores how topological methods can predict chaotic mixing patterns in fluid flows induced by complex rod motions, emphasizing the role of topological invariants and injection cusps.

Contribution

It introduces a topological framework to analyze and predict chaotic mixing patterns and unstable foliations in fluid flows with complex stirring device motions.

Findings

Topological invariants predict mixing behavior.

Injection cusps influence mixing efficiency.

Unstable foliations are classified by a topological index.

Abstract

A stirring device consisting of a periodic motion of rods induces a mapping of the fluid domain to itself, which can be regarded as a homeomorphism of a punctured surface. Having the rods undergo a topologically-complex motion guarantees at least a minimum amount of stretching of material lines, which is important for chaotic mixing. We use topological considerations to describe the nature of the injection of unmixed material into a central mixing region, which takes place at injection cusps. A topological index formula allow us to predict the possible types of unstable foliations that can arise for a fixed number of rods.