We study a fluid-dynamical model based on a coupled Navier–Stokes–Nernst–Planck–Poisson system. Of special interest are the fluid velocity, concentrations of charged particles ranging from colloidal to nano size and the induced quasi-electrostatic potential, which all depend on an externally applied electrical field. For d ≤ 3, we prove existence and in some cases uniqueness of weak solutions. Moreover, we characterize solutions via energy laws, mass conservation, non-negativity and pointwise bounds. Furthermore, the system enjoys an entropy law. Existence of locally strong solutions is verified.

Abstract. Effective Poisson-Nernst-Planck (PNP) equations are derived for macroscopic ion transport in charged porous media under periodic fluid flow by an asymptotic multi-scale expansion with drift. The microscopic setting is a two-component periodic composite consisting of a dilute electrolyte continuum (described by standard PNP equations) and a continuous dielectric matrix, which is impermeable to the ions and carries a given surface charge. Four new features arise in the upscaled equations: (i) the effective ionic diffusivities and mobilities become tensors, related to the microstructure; (ii) the effective permittivity is also a tensor, depending on the electrolyte/matrix permittivity ratio and the ratio of the Debye screening length to the macroscopic length of the porous medium; (iii) the microscopic fluidic convection is replaced by a diffusion-dispersion correction in the effective diffusion tensor; and (iv) the surface charge per volume appears as a continuous "background charge density", as in classical membrane models. The coefficient tensors in the upscaled PNP equations can be calculated from periodic reference cell problems. For an insulating solid matrix, all gradients are corrected by the same tensor, and the Einstein relation holds at the macroscopic scale, which is not generally the case for a polarizable matrix, unless the permittivity and electric field are suitably defined. In the limit of thin double layers, Poisson's equation is replaced by macroscopic electroneutrality (balancing ionic and surface charges). The general form of the macroscopic PNP equations may also hold for concentrated solution theories, based on the local-density and meanfield approximations. These results have broad applicability to ion transport in porous electrodes, separators, membranes, ion-exchange resins, soils, porous rocks, and biological tissues.

We propose and compare two classes of convergent finite element based approximations of the nonstationary Nernst-Planck-Poisson equations, whose constructions are motivated from energy versus entropy decay properties for the limiting system. Solutions of both schemes converge to weak solutions of the limiting problem for discretization parameters tending to zero. Our main focus is to study qualitative properties for the different approaches at finite discretization scales, like conservation of mass, non-negativity, discrete maximum principle, decay of discrete energies, and entropies to study long-time asymptotics.

We formulate the basic equations modeling solid-electrolyte composites without surface reactions. From these equations we achieve by the two-scale convergence method homogenized Nernst-Planck-Poisson equations. Moreover, we extend the system by including Stokes flow. Again, the two-scale convergence allows the rigorous justification of the resulting homogenized and nonlinearly coupled overall system. So called "material tensors" naturally arise by the upscaling and replace the commonly used porosity parameter from engineering. The upscaled equations derived here capture more accurately porous structures by including the microscopic geometry in a systematic way. To the author's best knowledge, this seems to be the first approach which derives the Stokes-Poisson-Nernst-Planck system being governed by porous materials and hence serves as a basis for additional specifications in the future.

We derive a new, effective macroscopic Cahn-Hilliard equation whose homogeneous free energy is represented by fourth-order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly heterogeneous domains. To the best knowledge of the authors, this seems to be the first attempt of upscaling the Cahn-Hilliard equation in such domains. The new homogenized equation should have a broad range of applicability owing to the well-known versatility of phase-field models. The additionally introduced feature of systematically and reliably accounting for confined geometries by homogenization allows for new modelling and numerical perspectives in both science and engineering. Our results are applied to wetting dynamics in porous media and to a single channel with strongly heterogeneous walls.