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.

翻译：本文提出了一种具有黏弹性的两相Navier-Stokes流的参数化有限元逼近方法。该自由边界问题由两个流体相中的黏弹性Navier-Stokes方程描述，并通过界面上的跳跃条件相互连接。流体中的弹性采用可能包含应力扩散的Oldroyd-B模型进行表征。该模型最初被引入用于逼近不可压缩牛顿流体与超弹性neo-Hookean固体之间的流固耦合问题，这类问题可作为本模型的极限情况。我们采用非拟合有限元方法对模型的变分形式进行逼近，该方法使用分段线性参数化有限元。针对体区域内的两相Navier-Stokes-Oldroyd-B系统，我们采用了一种离散化方案，确保耦合的体-界面系统具有无条件可解性和稳定性。当压力逼近空间借助XFEM函数进行增强时，可观察到两相具有良好的体积守恒特性。我们通过若干数值结果展示了该方法的适用性。