Slow decay of concentration variance due to no-slip walls in chaotic mixing

TL;DR

This paper investigates how no-slip walls in chaotic mixing cause a slow, algebraic decay of concentration variance, highlighting the role of parabolic points and unstable manifolds in persistent mixing patterns.

Contribution

It reveals the impact of parabolic periodic points at walls on decay rates and introduces a 1-D model that captures the slow decay behavior in viscous flows.

Findings

Decay is algebraic due to parabolic points at walls.

Unstable manifolds of parabolic points dominate persistent patterns.

A universal decay law is derived based on particle approach rate to walls.

Abstract

Chaotic mixing in a closed vessel is studied experimentally and numerically in different 2-D flow configurations. For a purely hyperbolic phase space, it is well-known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincare section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1-D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.