A variational front-tracking method for multiphase flow with triple junctions

TL;DR

This paper introduces a variational front-tracking finite element method for simulating multiphase flows with complex interfaces and triple junctions, ensuring stability and volume conservation, validated by numerical examples.

Contribution

It develops a novel variational front-tracking approach combining parametric and Eulerian formulations for multiphase flow with triple junctions, with proven stability and volume preservation.

Findings

Method is unconditionally stable with discrete energy

Exact volume preservation achieved in discrete solutions

Numerical examples demonstrate robustness and accuracy

Abstract

We present and analyze a variational front-tracking method for a sharp-interface model of multiphase flow. The fluid interfaces between different phases are represented by curve networks in two space dimensions (2d) or surface clusters in three space dimensions (3d) with triple junctions where three interfaces meet, and boundary points/lines where an interface meets a fixed planar boundary. The model is described by the incompressible Navier--Stokes equations in the bulk domains, with classical interface conditions on the fluid interfaces, and appropriate boundary conditions at the triple junctions and boundary points/lines. We propose a weak formulation for the model, which combines a parametric formulation for the evolving interfaces and an Eulerian formulation for the bulk equations. We employ an unfitted discretization of the coupled formulation to obtain a fully discrete finite element method, where the existence and uniqueness of solutions can be shown under weak assumptions. The constructed method admits an unconditional stability result in terms of the discrete energy. Furthermore, we adapt the introduced method so that an exact volume preservation for each phase can be achieved for the discrete solutions. Numerical examples for three-phase flow and four-phase flow are presented to show the robustness and accuracy of the introduced methods.