A Cut Finite Element Method for two-phase flows with insoluble surfactants

TL;DR

This paper introduces a novel unfitted finite element method for simulating two-phase flows with insoluble surfactants, ensuring mass conservation, handling non-conforming meshes, and accurately capturing discontinuities across evolving interfaces.

Contribution

The paper presents a new space-time cut finite element approach for two-phase flows with surfactants, allowing for non-conforming meshes and improved accuracy in interface problems.

Findings

Method conserves surfactant mass accurately.

Handles non-conforming meshes with evolving interfaces.

Demonstrates effectiveness in 2D and 3D simulations.

Abstract

We propose a new unfitted finite element method for simulation of two-phase flows in presence of insoluble surfactant. The key features of the method are 1) discrete conservation of surfactant mass; 2) the possibility of having meshes that do not conform to the evolving interface separating the immiscible fluids; 3) accurate approximation of quantities with weak or strong discontinuities across evolving geometries such as the velocity field and the pressure. The new discretization of the incompressible Navier--Stokes equations coupled to the convection-diffusion equation modeling the surfactant transport on evolving surfaces is based on a space-time cut finite element formulation with quadrature in time and a stabilization term in the weak formulation that provides function extension. The proposed strategy utilize the same computational mesh for the discretization of the surface Partial Differential Equation (PDE) and the bulk PDEs and can be combined with different techniques for representing and evolving the interface, here the level set method is used. Numerical simulations in both two and three space dimensions are presented including simulations showing the role of surfactant in the interaction between two drops.