Generalized local Darcys model of Teorells oscillations PDEs

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized 

Here / is the electric current density in the filter—the time-independent control parameter in the galvanostatic regime. 

The flow velocity in the filter v is again related to the pressure and the electric potential gradients via a generalized Darcy s law of the form 

Below we shall need some order of magnitude estimates for the hydraulic permeability i and the electro-osmotic coefficient u . Such estimates are provided by the expressions 

Equation (6.3.3b) results from assuming a Poiseille flow in a filter s pore of typical radius r, i is the dynamic viscosity of the fluid. Equation (6.3.3c) is a common expression for the electro-osmotic coefficient [13], with d and , respectively, the dielectric constant of the fluid and the -potential of the pore wall. For the time being, we shall assume il constant (independent of C(x,t)). 

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