Quantifying the dissipation enhancement of cellular flows

TL;DR

This paper provides quantitative bounds on how cellular flows enhance dissipation, linking flow parameters to mixing time and effective diffusivity, and introduces a probabilistic approach for analysis.

Contribution

It improves previous results by offering explicit bounds on dissipation enhancement based on flow amplitude, cell size, and diffusivity, and relates dissipation time to mixing time using probabilistic methods.

Findings

Mixing time bounded by exit time from one cell at high flow amplitude.

Effective diffusivity inversely related to flow amplitude at low flow amplitude.

Probabilistic coupling approach to analyze flow dynamics.

Abstract

We study the dissipation enhancement by cellular flows. Previous work by Iyer, Xu, and Zlato\v{s} produces a family of cellular flows that can enhance dissipation by an arbitrarily large amount. We improve this result by providing quantitative bounds on the dissipation enhancement in terms of the flow amplitude, cell size and diffusivity. Explicitly we show that the mixing time is bounded by the exit time from one cell when the flow amplitude is large enough, and by the reciprocal of the effective diffusivity when the flow amplitude is small. This agrees with the optimal heuristics. We also prove a general result relating the dissipation time of incompressible flows to the mixing time. The main idea behind the proof is to study the dynamics probabilistically and construct a successful coupling.