Eliminating Spurious Velocities with a Stable Approximation of Viscous Incompressible Two-Phase Stokes Flow

TL;DR

This paper introduces a stable finite element method for two-phase Stokes flow that effectively eliminates spurious velocities and ensures volume conservation, improving accuracy and stability in simulating free boundary problems.

Contribution

The authors develop a novel variational formulation and a mesh-quality preserving scheme with volume conservation and stability guarantees for two-phase flow simulations.

Findings

Spurious velocities are eliminated in classical test cases.

The method maintains mesh quality over time.

Exact volume conservation is achieved with pressure space enrichment.

Abstract

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. In addition, the mesh quality of the parametric approximation of the interface does not deteriorate, in general, over time; and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, on using a simple XFEM pressure space enrichment, we obtain exact volume conservation for the two phase regions. Furthermore, our fully discrete finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results which, in particular, demonstrate that spurious velocities can be avoided in the classical test cases.