Abstract
A fully discrete Galerkin scheme for a thermodynamically consistent transient Maxwell–Stefan system for the mass particle densities, coupled to the Poisson equation for the electric potential, is investigated. The system models the diffusive dynamics of an isothermal ionized fluid mixture with vanishing barycentric velocity. The equations are studied in a bounded domain, and different molar masses are allowed. The Galerkin scheme preserves the total mass, the nonnegativity of the particle densities, their boundedness and satisfies the second law of thermodynamics in the sense that the discrete entropy production is nonnegative. The existence of solutions to the Galerkin scheme and the convergence of a subsequence to a solution to the continuous system is proved. Compared to previous works, the novelty consists in the treatment of the drift terms involving the electric field. Numerical experiments show the sensitive dependence of the particle densities and the equilibration rate on the molar masses.
Highlights
The Maxwell–Stefan equations describe the dynamics of a fluid mixture in the diffusive regime
While Maxwell–Stefan models have been investigated since several decades from a modeling and simulation viewpoint in the engineering literature (e.g., [17]), the mathematical and numerical analysis started more recently [1, 20]
We prove the existence of a weak solution to a structure-preserving fully discrete Galerkin scheme and its convergence to the continuous problem
Summary
The Maxwell–Stefan equations describe the dynamics of a fluid mixture in the diffusive regime. They have numerous applications, for instance, in sedimentation, dialysis, electrolysis, and ion exchange. The global existence of weak solutions under natural conditions was proved in [8, 26] for neutral mixtures. We prove the existence of a weak solution to a structure-preserving fully discrete Galerkin scheme and its convergence to the continuous problem. This provides, for the first time, a global existence result for Poisson–Maxwell–Stefan systems
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