An Adaptive Finite Element Method for Multiphase Flows with Surface Tension

An adaptive finite element method for solving incompressible steady-state free surfaces flow problems, driven by surface tension, is presented. Eulerian free surface capturing, the pseudo-concentration technique, is used to locate interfaces. This approach adds a transport equation to the Navier-Stokes equations. This system of equations is solved in a coupled manner using stabilized finite element methods. Adaptivity of the computational mesh is done, using local least-squares projection a posteriori error estimation on the dependent variables, to help capture interfacial phenomena. Imposition of conditions at interfaces is known to be easier using a Lagrangian surface tracking strategy. In our case, the continuum surface force (CSF) model, which trades the zero thickness interface for a smooth transition zone, has been included in our adaptive finite element methodology. This makes the inclusion of surface tension in the interface boundary conditions a simple matter. It is shown that the developed adaptive finite element method is easy to implement, and permits the modeling of free surfaces phenomena, with surface tension, at a precision never attained before, and at a reasonable cost. The finite element formalism helps the modeling of flows with interfaces in complex geometries. Steady-state free surface phenomena, such as bubbles dynamics, are modeled to show the capability of the methodology.