Parametric finite element approximation of two-phase Navier--Stokes flow with viscoelasticity

TL;DR

This paper introduces a stable, finite element method for simulating two-phase viscoelastic fluid flows with free boundaries, demonstrating good volume conservation and applicability through numerical experiments.

Contribution

It develops a novel unfitted finite element approach for two-phase Navier-Stokes-Oldroyd-B systems with guaranteed stability and volume conservation.

Findings

Unconditionally stable discretization for coupled bulk-interface system

Good volume conservation with enriched pressure space

Numerical results demonstrating method applicability

Abstract

In this work, we present a parametric finite element approximation of two-phase Navier-Stokes flow with viscoelasticity. The free boundary problem is given by the viscoelastic Navier-Stokes equations in the two fluid phases, connected by jump conditions across the interface. The elasticity in the fluids is characterised using the Oldroyd-B model with possible stress diffusion. The model was originally introduced to approximate fluid-structure interaction problems between an incompressible Newtonian fluid and a hyperelastic neo-Hookean solid, which are possible limit cases of the model. We approximate a variational formulation of the model with an unfitted finite element method that uses piecewise linear parametric finite elements. The two-phase Navier-Stokes-Oldroyd-B system in the bulk regions is discretised in a way that guarantees unconditional solvability and stability for the coupled bulk-interface system. Good volume conservation properties for the two phases are observed in the case where the pressure approximation space is enriched with the help of an XFEM function. We show the applicability of our method with some numerical results.