We show that a reformulation of the governing equations for incompressible multi-phase flow in the volume of fluid setting leads to a well defined energy rate. Weak nonlinear inflow-outflow and solid wall boundary conditions complement the development and lead to an energy estimate in terms of external data. The new formulation combined with summation-by-parts operators lead to provably nonlinear energy stability.

翻译：我们证明了在流体体积框架下对不可压缩多相流控制方程的重构可导出明确定义的能量变化率。弱非线性流入-流出边界条件及固体壁面边界条件的引入完善了理论体系，并导出了基于外部数据的能量估计。新公式结合分部求和算子，可证明实现非线性能量稳定性。