This work presents a finite element method for a modified Poisson-Nernst-Planck/Navier-Stokes (PNP/NS) model under the mechanical equilibrium, developed for compressible electrolytes. The modification is based on the new model proposed by Dreyer, Guhlke and Muller [39], where the diffusion flux in the classical PNP system is replaced with an implicitly involved new diffusion flux, leading to fractional nonlinearity. He and Sun [42] previously developed a numerical approach for another type of modification, where the Poisson equation in the PNP system was substituted with a fourth-order elliptic equation. Another key contribution of this work is the reduction of the equilibrium system to a modified Poisson-Boltzmann system. The proposed numerical scheme is capable of handling both compressible and incompressible regimes by employing a bulk modulus parameter, which governs the fluid's compressibility and enables seamless transition between these regimes. To emphasize practical relevance, we discuss the implications of compressible electrolytes in the context of double-layer capacitance behavior. We also conduct numerical simulations over various domains to demonstrate its applicability under various operating conditions, including temperature fluctuations and variations in the bulk modulus. The numerical results validate the accuracy and robustness of our computational scheme and demonstrate that the observed limiting behavior for the incompressible regime aligns with the theoretical trends anticipated by Dreyer et al. [39].

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