Dipl.-Math. André Fischer
Evolution of fluids with charged dissolved species
A proper understanding of the movement of ions is of essential importance in industrial applications such as saline water desalination. The so-called “nanofiltation” uses a membrane with fixed electric charge for filtering biased liquids. In order to set up an appropriate model for this technique not only diffusion and convection, but also electro-migration has to be taken into account. The latter in general is driven by an external as well as an internal electrical potential caused by the distribution of the dissolved ionic species. Recent works often simplified this situation by imposing an electroneutrality condition throughout the volume. Experiments, however, showing that at least the vicinity of interfaces is not electrically neutral demand the drop of this electroneutrality condition.
We are concerned with the determination of a thermodynamically valid model for the above phenomena without electroneutrality and its well-posedness by means of the existence and uniqueness of strong solutions. A very challenging task in this respect is the accurate choice of boundary conditions for all involved quantities. The main interest lies in the question for global existence of unique solutions and their asymptotics. For this purpose we apply the mathematical theory of maximal regularity and suitable à priori estimates via the energy method. In order to obtain those à priori estimates it is crucial to understand the physical and mathematical interaction of the different kinds of energy occurring.
Based on works of W. Nernst and M. Planck in the late 19th century a common model for the ionic movement in electrolytes is given by the Nernst-Planck equations. We examined these equations coupled with the Navier-Stokes equations for the fluid movement and the Poisson equation for the electrical potential. By maximal regularity it was possible to derive the local-in-time existence of unique solutions in any space dimension d>=2. Using a suitable energy function the solutions were proved to exist for all time if d=2.
I would like to thank for financial support from the DFG within the cluster of excellence of „Center of Smart Interfaces“.