Three-dimensional finite element model of the motion of a viscous incompressible fluid with a free surface, taking into account the surface tension

The description of the implicit finite element method for the numerical solution of the Navier-Stokes equations in a three-dimensional formulation with allowance for surface tension is presented. The mathematical description of the process is based on the equations of motion, continuity, and natural boundary conditions on a free surface. The numerical solution of the problem is based on the finite element method with the approximation of the main variables of the problem (the vector of velocity and pressure), satisfying the condition of their compatibility. In addition, to reduce pressure oscillations in the vicinity of the free surface, increased pressure integration schemes on the element are used. The evolution of the free surface of a drop under the action of surface tension is considered as an example. The stability and computational efficiency of a model with approximate convergence and conservatism with an increase in the time step by two orders of magnitude relative to the currently used integration schemes is shown.