On the Nernst–Planck–Navier–Stokes system

Abstract

We consider ionic electrodiffusion in fluids, described by the Nernst-Planck-Navier-Stokes system in bounded domains, in two dimensions, with Dirichlet boundary conditions for the Navier-Stokes and Poisson equations, and blocking (vanishing normal flux) or selective (Dirichlet) boundary conditions for the ionic concentrations. We prove global existence and stability results for large data.where n is outer normal at the boundary of Ω, is termed "blocking boundary conditions". These boundary conditions model situations in which boundaries are impermeable: the ions are not allowed to cross them.

Charged Fluids in Porous Media

“…The same result holds in three dimensions if the fluid velocity remains regular for all time [17]. When the ionic concentrations satisfy either the blocking boundary conditions or the uniformly selective boundary conditions, strong solutions are global in two dimensions [6], and in three dimensions provided that the initial data is a small perturbation of a steady state [8]. If both the ionic concentrations and the electrical potential obey the Dirichlet boundary conditions, global strong solutions exist in three dimensions as long as the fluid velocity is regular [9].…”

“…∇ • u = 0, (1.5) where c i : T d ×[0, T ] → R + are the i-th ionic species concentrations, z i ∈ Z are corresponding valences, D i > 0 are constant diffusivities, Φ : T d × [0, T ] → R is the nondimensional electrical potential, ρ : T d × [0, T ] → R is the nondimensional charge density, u : T d × [0, T ] → T d is the fluid velocity field, p : T d × [0, T ] → R is the fluid pressure, and ε > 0 is a constant proportional to the square of the Debye length [6,20]. Here, d = 2, 3 is the space dimension and T d is a d-dimensional torus.…”

Weak sequential stability for a nonlinear model of nematic electrolytes

“…a Q-tensor in the LC terminology, instead of the vectorvalued one, n, as done for instance in [26]. The current work is related to work done in certain simpler systems that can be regarded as subsets of our equations, such as Nernst-Planck-Navier-Stokes system (see for instance [8] and the references therein) and liquid crystal equations (see for instance the review [21]). It should be noted that there exists a large body of literature concerned with the Nernst-Planck-Navier-Stokes system, concerned with the fundamental issues of existence of weak and strong solutions (see for instance [8,14,25]), as well as regularity (see the recent work [15] and the references therein).…”

“…The current work is related to work done in certain simpler systems that can be regarded as subsets of our equations, such as Nernst-Planck-Navier-Stokes system (see for instance [8] and the references therein) and liquid crystal equations (see for instance the review [21]). It should be noted that there exists a large body of literature concerned with the Nernst-Planck-Navier-Stokes system, concerned with the fundamental issues of existence of weak and strong solutions (see for instance [8,14,25]), as well as regularity (see the recent work [15] and the references therein). The additional level of complexity generated by the addition of the liquid crystal equations (1.5) makes most of the approaches in the Nernst-Planck-Navier-Stokes literature largely inapplicable in a direct manner and generates additional challenging technical difficulties.…”

Existence of solutions to the Poisson--Nernst--Planck system with singular permanent charges in $\mathbb{R}^2$

“…The Poisson-Nernst-Planck (PNP) system is a basic model for the transport of charged particles. It has been applied in many applications both in physics and biology [4,5,7,9,16,30,32,36,37,38,40,43], sometimes modified by taking into account such as the sizes of ions, the interaction between ions and thermal effects [14,18,20,22,23,28,29,41,42]. The classical PNP system can be represented as…”