Mixing as a correlated aggregation process

TL;DR

This paper reveals that in smooth 2D chaotic flows, mixing behaves as a correlated aggregation process influenced by fractal properties, leading to less efficient mixing compared to random aggregation.

Contribution

It demonstrates that mixing in such flows follows a correlated aggregation process governed by fractal geometry, expanding aggregation theories to new systems.

Findings

Mixing obeys a correlated aggregation process.

Correlations reduce mixing efficiency.

A single exponent characterizes the aggregation process.

Abstract

Mixing describes the process by which solutes evolve from an initial heterogeneous state to uniformity under the stirring action of a fluid flow. Fluid stretching forms thin scalar lamellae which coalesce due to molecular diffusion. Owing to the linearity of the advection-diffusion equation, coalescence can be envisioned as an aggregation process. Here, we demonstrate that in smooth two-dimensional chaotic flows, mixing obeys a correlated aggregation process, where the spatial distribution of the number of lamellae in aggregates is highly correlated with their elongation and is set by the fractal properties of the advected material lines. We show that the presence of correlations makes mixing less efficient than a completely random aggregation process because lamellae with similar elongations and scalar levels tend to remain isolated from each other. We show that correlated aggregation is uniquely determined by a single exponent which quantifies the effective number of random aggregation events. These findings expand aggregation theories to a larger class of systems, which have relevance to various fundamental and applied mixing problems.