  In this paper we report the theory describing the electro-osmotic flow in
charged nanopores with constant radius and charge density driven by alternating
current. We solve the ion and solution transport in unsteady conditions as
described by the Navier-Stokes and Nernst-Planck equations considering the
electrical potential inside the charged nanopore uniform in the radial
direction (Uniform Potential model approximation). We derive the transport
equation system in the case in which the pore is connected to two boundary
diffusion layers and the cations and anions have different diffusion
coefficients. This approach allows the theoretical description of the
characteristic frequency dependence of the phase shift between the applied
current density and the electro-osmotic flow. Additionally we show how the
analysis of the dynamic response of the electro-osmotic coupling factor versus
the AC frequency allows us to quantify the apparent ion diffusion coefficient
in membranes. Notably, the frequency window where the phase-shift is predicted
is well inside the commonly used sampling rate for electro-osmotic flow
experiments $f\in\left[10^{-4},10^1\right]$ Hz, hence the proposed method can
be useful to determine the apparent diffusion coefficient of ions (such as
redox complex for flow batteries) in charged membranes.
