We propose and analyze unfitted finite element approximations for the two-phase incompressible Navier--Stokes flow in an axisymmetric setting. The discretized schemes are based on an Eulerian weak formulation for the Navier--Stokes equation in the 2d-meridian halfplane, together with a parametric formulation for the generating curve of the evolving interface. We use the lowest order Taylor--Hood and piecewise linear elements for discretizing the Navier--Stokes formulation in the bulk and the moving interface, respectively. We discuss a variety of schemes, amongst which is a linear scheme that enjoys an equidistribution property on the discrete interface and good volume conservation. An alternative scheme can be shown to be unconditionally stable and to conserve the volume of the two phases exactly. Numerical results are presented to show the robustness and accuracy of the introduced methods for simulating both rising bubble and oscillating droplet experiments.

翻译：我们提出并分析了轴对称情形下两相不可压缩Navier-Stokes流动的非拟合有限元逼近。离散格式基于二维子午半平面上Navier-Stokes方程的欧拉弱形式，以及演化界面生成曲线的参数化形式。我们分别采用最低阶的Taylor-Hood元和分片线性元对体区域Navier-Stokes方程和移动界面进行离散。我们讨论了多种格式，其中一种线性格式在离散界面上具有等分布性质并保持良好的体积守恒。另一种替代格式可证明无条件稳定且精确守恒两相体积。数值结果表明，所引入的方法在模拟上升气泡和振荡液滴实验时具有鲁棒性和精确性。