Diffusion-limited mixing by incompressible flows

TL;DR

This paper investigates how diffusion influences mixing in incompressible flows, revealing that diffusion can limit long-term mixing efficiency due to a length scale constraint, especially under energy constraints.

Contribution

It provides numerical and analytical evidence that diffusion can hinder mixing in optimal incompressible flows, challenging the common perception of diffusion as purely beneficial.

Findings

Diffusion has negligible impact on enstrophy-bounded optimal flows.

In energy-constrained flows, increased diffusion reduces mixing rates.

Identifies a limiting length scale affecting long-term mixing performance.

Abstract

Incompressible flows can be effective mixers by appropriately advecting a passive tracer to produce small filamentation length scales. In addition, diffusion is generally perceived as beneficial to mixing due to its ability to homogenise a passive tracer. However we provided numerical evidence that, in the case where advection and diffusion are both actively present, diffusion produces nearly neutral or even negative effects by limiting the mixing effectiveness of incompressible optimal flows. This limitation appears to be due to the presence of a limiting length scale given by a generalised Batchelor length. This length scale limitation in turn affects long-term mixing rates. More specifically, we consider local-in-time flow optimisation under energy and enstrophy flow constraints with the objective of maximising mixing rate performance. We observe that, for enstrophy-bounded optimal flows, the strength of diffusion has no impact on the long-term mixing rate performance. For energy-constrained optimal flows, however an increase in the strength of diffusion decreases the mixing rate. We provide analytical lower bounds on mixing rates and length scales achievable under related constraints (point-wise bounded speed and rate-of-strain) by extending the work of Z. Lin et al. (Journal of Fluid Mech., 2011) and C.-C. Poon (Comm. in Partial Differential Equations, 1996).