Bounds on the mixing enhancement for a stirred binary fluid

TL;DR

This paper derives bounds on how effectively stirring can homogenize a phase-separating binary fluid described by the Cahn-Hilliard equation, supported by numerical simulations of model flows.

Contribution

It introduces theoretical bounds on composition fluctuations during stirring, linking homogenization efficiency to flow characteristics in binary fluids.

Findings

Bounds on composition fluctuations derived from the Cahn-Hilliard model

Comparison of theoretical bounds with numerical simulations

Assessment of stirring protocols' effectiveness in homogenization

Abstract

The Cahn-Hilliard equation describes phase separation in binary liquids. Here we study this equation with spatially-varying sources and stirring, or advection. We specialize to symmetric mixtures and time-independent sources and discuss stirring strategies that homogenize the binary fluid. By measuring fluctuations of the composition away from its mean value, we quantify the amount of homogenization achievable. We find upper and lower bounds on our measure of homogenization using only the Cahn-Hilliard equation and the incompressibility of the advecting flow. We compare these theoretical bounds with numerical simulations for two model flows: the constant flow, and the random-phase sine flow. Using the sine flow as an example, we show how our bounds on composition fluctuations provide a measure of the effectiveness of a given stirring protocol in homogenizing a phase-separating binary fluid.