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

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 characterized 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 discretized 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 extended finite element method function. We show the applicability of our method with some numerical results.