Numerical approximation of a phase-field surfactant model with fluid flow

TL;DR

This paper develops efficient, unconditionally energy-stable numerical schemes for simulating interfacial dynamics with surfactants in multiphase flows, accurately capturing effects like droplet deformation and collision.

Contribution

It introduces semi-explicit, decoupled time marching schemes for a complex phase-field surfactant model with fluid flow, with proven unconditional energy stability.

Findings

Schemes are accurate and efficient in 2D and 3D simulations.

Surfactant concentration influences droplet deformation and coalescence.

Numerical results demonstrate the effectiveness of the proposed methods.

Abstract

Modelling interfacial dynamics with soluble surfactants in a multiphase system is a challenging task. Here, we consider the numerical approximation of a phase-field surfactant model with fluid flow. The nonlinearly coupled model consists of two Cahn-Hilliard-type equations and incompressible Navier-Stokes equation. With the introduction of two auxiliary variables, the governing system is transformed into an equivalent form, which allows the nonlinear potentials to be treated efficiently and semi-explicitly. By certain subtle explicit-implicit treatments to stress and convective terms, we construct first and second-order time marching schemes, which are extremely efficient and easy-to-implement, for the transformed governing system. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of phase-field variables, velocity and pressure are fully decoupled. We further establish a rigorous proof of unconditional energy stability for the first-order scheme. Numerical results in both two and three dimensions are obtained, which demonstrate that the proposed schemes are accurate, efficient and unconditionally energy stable. Using our schemes, we investigate the effect of surfactants on droplet deformation and collision under a shear flow, where the increase of surfactant concentration can enhance droplet deformation and inhibit droplet coalescence.