The Graduate School AN ENERGETIC VARIATIONAL APPROACH TO MATHEMATICAL MODELING OF CHARGED FLUIDS : CHARGE PHASES , SIMULATION AND WELL POSEDNESS

In thesis we propose a mathematical model of electrolyte fluid and interface systems. The model is based on a coupling between the Navier-Stokes equations of an incompressible fluid, the Nernst-Plank-Poisson equations of a diffuse, binary electrolyte, and the phase field Allen-Cahn equation. The coupling is derived in the energetic variational framework and guarantees the consistent exchange of the kinetic energy of the fluid, entropic and electric energy of the charge carriers and the surface area of the interface. Using the phase field as a topological labeling of the interface, we introduce a “short range” barrier potential which selectively blocks charge migration across the interface. The model is able to capture the dynamics of both charge induced flow and selection by the interface. This is demonstrated by simulation of the coalesence of two charge selective vesicles by charge induced motion. We also develope the existence theory for global classical solutions of the NPP equations with smooth data in space dimension d ≤ 3, global weak solutions to the NPP equations coupled with the NS equations for d ≤ 3 and global weak solutions for small initial data with the additional phase field Allen Cahn equation in space dimension d ≤ 2. The NPP equations are a system of second order, divergence form, semilinear, nonlocal parabolic equations. We elucidate many of the special features of the NPP equations which are nonstandard in complex fluid systems.