We present and analyze a variational front-tracking method for a sharp-interface model of multiphase flow. The fluid interfaces between different phases are represented by curve networks in two space dimensions (2d) or surface clusters in three space dimensions (3d) with triple junctions where three interfaces meet, and boundary points/lines where an interface meets a fixed planar boundary. The model is described by the incompressible Navier--Stokes equations in the bulk domains, with classical interface conditions on the fluid interfaces, and appropriate boundary conditions at the triple junctions and boundary points/lines. We propose a weak formulation for the model, which combines a parametric formulation for the evolving interfaces and an Eulerian formulation for the bulk equations. We employ an unfitted discretization of the coupled formulation to obtain a fully discrete finite element method, where the existence and uniqueness of solutions can be shown under weak assumptions. The constructed method admits an unconditional stability result in terms of the discrete energy. Furthermore, we adapt the introduced method so that an exact volume preservation for each phase can be achieved for the discrete solutions. Numerical examples for three-phase flow and four-phase flow are presented to show the robustness and accuracy of the introduced methods.

翻译：本文提出并分析了一种用于多相流锐界面模型的变分界面追踪方法。不同相之间的流体界面由二维空间中的曲线网络或三维空间中的曲面簇表示，这些界面在三条界面相交处形成三相结，并在界面与固定平面边界相交处形成边界点/线。该模型通过体区域内的不可压缩Navier-Stokes方程描述，并包含流体界面上的经典界面条件、三相结处及边界点/线上的适当边界条件。我们提出了该模型的弱形式，该形式结合了演化界面的参数化表述与体方程的欧拉表述。通过对耦合形式采用非拟合离散化，我们获得了全离散有限元方法，该方法可在弱假设条件下证明解的存在唯一性。所构建的方法在离散能量意义下具有无条件稳定性结果。此外，我们对所提方法进行改进，使得离散解能实现各相的精确体积守恒。文中给出了三相流与四相流的数值算例，以展示所提方法的鲁棒性与精确性。