Fully implicit and accurate treatment of jump conditions for two-phase incompressible Navier-Stokes equation

We present a numerical method for two-phase incompressible Navier-Stokes equation with jump discontinuity in the normal component of the stress tensor and in the material properties. Although the proposed method is only first-order accurate, it does capture discontinuity sharply, not neglecting nor omitting any component of the jump condition. Discontinuities in velocity gradient and pressure are expressed using a linear combination of singular force and tangential derivatives of velocities to handle jump conditions in a fully implicit manner. The linear system for the divergence of the stress tensor is constructed in the framework of the ghost fluid method, and the resulting saddle-point system is solved via an iterative procedure. Numerical results support the inference that the proposed method converges in $L^\infty$ norms even when velocities and pressures are not smooth across the interface and can handle a large density ratio that is likely to appear in a real-world simulation.