A decoupled, stable, and linear FEM for a phase-field model of variable density two-phase incompressible surface flow

TL;DR

This paper introduces a stable, linear, and decoupled finite element method for simulating variable density two-phase incompressible flows on surfaces, with applications in biomembrane research.

Contribution

The authors develop a fully discrete, linear, and decoupled FEM scheme for a complex surface flow model with variable density, ensuring stability and efficiency.

Findings

The scheme is unconditionally stable under certain discretization conditions.

Numerical results confirm the method's accuracy and efficiency.

Flow statistics depend on surface geometry in the studied examples.

Abstract

The paper considers a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids. The model allows for a non-linear dependence of fluid density on the phase-field order parameter. Driven by applications in biomembrane studies, the model is written for tangential flows of fluids constrained to a surface and consists of (surface) Navier-Stokes-Cahn-Hilliard type equations. We apply an unfitted finite element method to discretize the system and introduce a fully discrete time-stepping scheme with the following properties: (i) the scheme decouples the fluid and phase-field equation solvers at each time step, (ii) the resulting two algebraic systems are linear, and (iii) the numerical solution satisfies the same stability bound as the solution of the original system under some restrictions on the discretization parameters. Numerical examples are provided to demonstrate the stability, accuracy, and overall efficiency of the approach. Our computational study of several two-phase surface flows reveals some interesting dependencies of flow statistics on the geometry.