A Mathematical Study of Electroconvection

Abstract

We study electroconvective models mathematically described by the Nernst-Planck-Navier-Stokes (NPNS) or Nernst-Planck-Stokes (NPS) systems. These nonlocal, semilinear parabolic systems model the time evolution of ionic concentrations in a fluid in the presence of boundaries and an applied electrical potential on the boundaries. Ions diffuse under their own concentration gradients, are convected by the fluid, and are transported by the underlyingelectrical field. In turn, the electrical field is determined nonlocally by the distribution of ions and the applied electrical potential on the boundaries; the fluid is also forced by the electrical field. We consider these systems on three dimensional bounded domains, imposed with various equilibrium and nonequilibrium boundary conditions and address four main questions: 1) global existence of strong (smooth) solutions 2) existence, regularity, and boundedness of steady state solutions 3) long time dynamics of solutions, and 4) electroneutrality in the singular limit of zero Debye length ϵ → 0. One of the main features of this thesis is the contrast of results between equilibrium and nonequilibrium boundary conditions. A primary difference between equilibrium and nonequilibrium boundary conditions is the existence and absence, respectively, of a natural dissipative structure for the corresponding NPNS/NPS system. In the case of equilibrium boundary conditions, the dissipative structure gives a natural starting point for further study of the dynamics of solutions and also gives a precise description of the asymptotic behavior of solutions in the limit of time t → ∞. On the other hand, for nonequilibrium boundary conditions, the lack of a dissipative structure manifests itself physically through more complex fluid patterns, which have been have been the subject of many experimental and numerical research efforts. In this thesis, we study this complex behavior from a rigorous mathematical viewpoint, considering both time independent and time dependent solutions, and in the latter case, considering their long time and long time averaged behavior.