Comparison of Coarsening Dynamics for the Cahn--Hilliard and Burgers--Cahn--Hilliard Equations

TL;DR

This paper investigates how coupling phase separation with fluid flow in a Burgers--Cahn--Hilliard system affects coarsening rates, comparing uncoupled and coupled dynamics through numerical and analytical methods.

Contribution

It provides a detailed comparison of coarsening dynamics between uncoupled Cahn--Hilliard and coupled Burgers--Cahn--Hilliard systems, including well-posedness analysis.

Findings

Coupling alters coarsening rates significantly.

Numerical and analytical methods agree on coarsening behaviors.

Established global well-posedness of the coupled system.

Abstract

We consider coarsening dynamics associated with a Burgers--Cahn--Hilliard system modeling a two-phase flow in one space dimension. Our emphasis is on the effect that coupling between the phase and fluid dynamics has on coarsening rates, and on the mechanisms driving this effect. We start with a detailed examination of coarsening dynamics for the uncoupled Cahn--Hilliard equation, comparing numerically generated rates with two analytic methods, and then we consider how these dynamics are affected by appropriate coupling with a viscous Burgers equation. In order to keep the analysis as self-contained as possible, we establish the global well-posedness of the system under consideration.