Abstract
We consider an improved Nernst–Planck–Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non-equilibrium. The elastic deformation of the medium, that induces an inherent coupling of mass and momentum transport, is taken into account. 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 non-trivial kernel. In this paper, we establish the existence of a global-in-time weak solution, allowing for a general structure of the mobility tensor and for chemical reactions with fast nonlinear 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.
Highlights
Increasing the efficiency of actual high-performance energy storage systems requires an exact understanding of their fundamental physical principles
In the neighbourhood of interfaces, the classical description using the Nernst–Planck theory is failing for various reasons: first of all, the Nernst–Planck model neglects the high pressures induced by the Lorentz force that affect the charge transport
Let us here mention the papers [31] and [34] where models of compressible mixtures with energy balance, but without electric field, were studied. These models are not derived from exactly the same thermodynamic principles that are used in our study: the constitutive equations for the pressure, for the diffusion fluxes and for the reaction terms, are different in [31] and in [12]
Summary
Increasing the efficiency of actual high-performance energy storage systems requires an exact understanding of their fundamental physical principles. Let us here mention the papers [31] and [34] where models of compressible mixtures with energy balance, but without electric field, were studied These models are not derived from exactly the same thermodynamic principles that are used in our study: the constitutive equations for the pressure, for the diffusion fluxes and for the reaction terms, are different in [31] and in [12]. The compactness question occurs there like in our analysis but is solved assuming a special structure of the viscosity tensor, called Bresch–Desjardins condition The latter allows to obtain estimates on the density gradient, a device which is not at our disposal here. The third part (Sects. 12, 13, 14) is concerned with the investigation of compactness properties, and with the proof of convergence of the approximation scheme
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