On the numerics of diffuse-interface models for two-phase flow with species transport
We have analyzed and implemented stable schemes for two-phase flow with species transport. In the first model, the species taken into account is to be interpreted as a density of charges. It interacts with external electrostatic potentials and, which is much more exciting, with the fluids. The second model presented here is similar in its form but takes non-matched mass densities into account (instead of species transport). Both models cover a set of fluid dynamic effects. For both models, we give a definition of the quantities and formulate assumptions for the physical or phenomenological parameters, such as viscosity, permeability and the interface thickness. With the intention of a rigorous numerical analysis for model 1, a specific discretization scheme in space and time is suggested. Based on this scheme, we show the discrete version of the energy estimate for both models and prove the existence of discrete solutions for the electrowetting model. Together with sufficient regularity in time, this enables us to show weak convergence for subsequences. The limits of these subsequences can be identified as solutions of the continuous weak formulation. Basic numerical analysis for the case of non-matched densities is presented. We formulate a fully discrete scheme for model 2 and prove its stability by showing a discrete energy estimate. Some selected simulations based on both models are presented. Both models have their numerical pitfalls, so we allow for basic tests to assure the validity of the code. The inhouse code EconDrop is described. The software package allows for a large variety of numerical problems, e.g. Navier-Stokes equations, generalized Poisson's equation and advection-diffusion equations. It offers adaptivity in both time and space.