Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part I: Derivation of the model and survey of the results
- Dreyer, Wolfgang
- Druet, Pierre-Étienne
- Gajewski, Paul
- Guhlke, Clemens
2010 Mathematics Subject Classification
- 35Q35 76T30 78A57, 35Q30, 76N10, 35M33, 35D30, 35B45
2008 Physics and Astronomy Classification Scheme
- 82.45Gj, 82.45.Mp, 82.60Lf
- electrolyte, electrochemical interface, chemical reaction, compressible fluid, Navier-Stokes equations, advection-diffusion-reaction equations, PDE system of mixed-type, a-priori estimates, weak solution
We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a global--in--time weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials.
- ZAMP Z. Angew. Math. Phys., 71 (2020), pp. 119/1--68, DOI 10.1007/s00033-020-01341-5 .