Stable Numerical Approximation of Two-Phase Flow with a Boussinesq--Scriven Surface Fluid

TL;DR

This paper develops and analyzes stable finite element methods for simulating two-phase Navier--Stokes flows with Boussinesq--Scriven surface fluids, incorporating surface viscosity effects and demonstrating stability through free energy inequalities.

Contribution

It introduces novel parametric finite element approximations for two-phase flows with surface viscosity and proves their stability via discrete free energy inequalities.

Findings

Numerical simulations illustrate the impact of surface viscosity.

The methods are stable and preserve a discrete free energy inequality.

Applications include various 2D and 3D flow scenarios.

Abstract

We consider two-phase Navier--Stokes flow with a Boussinesq--Scriven surface fluid. In such a fluid the rheological behaviour at the interface includes surface viscosity effects, in addition to the classical surface tension effects. We introduce and analyze parametric finite element approximations, and show, in particular, stability results for semi-discrete versions of the methods, by demonstrating that a free energy inequality also holds on the discrete level. We perform several numerical simulations for various scenarios in two and three dimensions, which illustrate the effects of the surface viscosity.