ON THE NONLINEAR INTERACTION OF CHARGED PARTICLES WITH FLUIDS

Abstract

We consider three different phenomena governing the fluid flow in the presence of charged particles: electroconvection in fluids, electroconvection in porous media, and electrodiffusion. Electroconvecton in fluids is mathematically represented by a nonlinear drift-diffusion partial differential equation describing the time evolution of a surface charge density in a two-dimensional incompressible fluid. The velocity of the fluid evolves according to Navier-Stokes equations forced nonlinearly by the electrical forces due to the presence of the charge density. The resulting model is reminiscent of the quasi-geostrophic equation, where the main difference resides in the dependence of the drift velocity on the charge density. When the fluid flows through a porous medium, the velocity and the electrical forces are related according to Darcy’s law, which yields a challenging doubly nonlinear and doubly nonlocal model describing electroconvection in porous media. A different type of particle-fluid interaction, called electrodiffusion, is also considered. This latter phenomenon is described by nonlinearly advected and nonlinearly forced continuity equations tracking the time evolution of the concentrations of many ionic species having different valences and diffusivities and interacting with an incompressible fluid. This work is based on [1, 2, 3] and addresses the global well-posedness, long-time dynamics, and other features associated with the aforementioned three models. REFERENCES:[1] E. Abdo, M. Ignatova, Long time dynamics of a model of electroconvection, Trans. Amer. Math. Soc. 374 (2021), 5849–5875. [2] E. Abdo, M. Ignatova, Long Time Finite Dimensionality in Charged Fluids, Nonlinearity 34 (9) (2021), 6173–6209. [3] E. Abdo, M. Ignatova, On Electroconvection in Porous Media, to appear in Indiana University Mathematics Journal (2023).