Numerical Methods for Fluid Interface Problems

Abstract: This thesis concerns numerical techniques for two phase flowsimulations; the two phases are immiscible and incompressible fluids. Strategies for accurate simulations are suggested. In particular, accurate approximations of the weakly discontinuousvelocity field, the discontinuous pressure, and the surface tension force and a new model for simulations of contact line dynamics are proposed.In two phase flow problems discontinuities arise in the pressure and the gradient of the velocity field due to surface tension forces and differences in the fluids' viscosity. In this thesis, a new finite element method which allows for discontinuities along an interface that can be arbitrarily located with respect to the mesh is presented. Using standard linear finite elements, the method is for an elliptic PDE proven to have optimal convergence order and a system matrix with condition number bounded independently of the position of the interface.The new finite element method is extended to the incompressible Stokes equations for two fluid systemsand enables accurate approximations of the weakly discontinuous velocity field and the discontinuous pressure.An alternative way to handle discontinuities is regularization. In this thesis, consistent regularizations of Dirac delta functions with support on interfaces are proposed. These regularized delta functions make it easy to approximate surface tension forces in level set methods.A new model for simulating contact line dynamics is also proposed. Capillary dominated flows are considered and it is assumed that contact line movement is driven by the deviation of the contact angle from its static value. This idea is used together with the conservative level set method. The need for fluid slip at the boundary is eliminated by providing a diffusive mechanism for contact line movement. Numerical experiments in two space dimensions show that the method is able to qualitatively correctly capture contact line dynamics.