Phase separation in the advective Cahn-Hilliard equation

TL;DR

This paper investigates how strong mixing flows can prevent phase separation in the Cahn-Hilliard equation, leading to uniform mixing instead of phase separation, with quantitative measures of mixing effectiveness.

Contribution

It introduces a criterion based on dissipation time to determine when advection suppresses phase separation in the Cahn-Hilliard model.

Findings

Strong mixing advection prevents phase separation

Exponential convergence to homogeneous state under sufficient mixing

Examples of velocity fields with small dissipation time

Abstract

The Cahn--Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn--Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection-hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection-diffusion equation.