Numerical investigation of the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations

TL;DR

This study investigates the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard model for binary-fluid flows using adaptive finite-element methods, deriving analytical solutions and exploring the asymptotic behavior as interface thickness diminishes.

Contribution

It introduces new analytical expressions for droplet oscillations and employs adaptive refinement to analyze the sharp-interface limit in binary-fluid flow models.

Findings

Optimal scaling relation between interface thickness and mobility parameter identified.

Deviations from optimal scaling affect the convergence to sharp-interface solutions.

Adaptive finite-element method enables exploration of very small interface thicknesses.

Abstract

In this article, we study the behavior of the Abels-Garcke-Gr\"un Navier-Stokes-Cahn-Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. We consider this so-called sharp-interface limit in the setting of the classical oscillating-droplet problem. To provide reference limit solutions, we derive new analytical expressions for small-amplitude oscillations of a viscous droplet in a viscous ambient fluid in two dimensions. We probe the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations by means of an adaptive finite-element method, in which the refinements are guided by an a-posteriori error-estimation procedure. The adaptive-refinement procedure enables us to consider diffuse-interface thicknesses that are significantly smaller than other relevant length scales in the droplet-oscillation problem, allowing an exploration of the asymptotic regime. For two distinct modes of oscillation, we determine the optimal scaling relation between the diffuse-interface thickness parameter and the mobility parameter. Additionally, we examine the effect of deviations from the optimal scaling of the mobility parameter on the approach of the diffuse-interface solution to the sharp-interface solution.