  Motivated by recent experiments, we present a theoretical investigation of
how the electro-osmotic flow occurring in a capillary is modified when its
charged surfaces are coated by charged polymers. The theoretical treatment is
based on a three dimensional model consisting of a ternary fluid-mixture,
representing the solvent and two species for the ions, confined between two
parallel charged plates decorated by a fixed array of scatterers representing
the polymer coating. The electro-osmotic flow, generated by a constant electric
field applied in a direction parallel to the plates, is studied numerically by
means of Lattice Boltzmann simulations. In order to gain further understanding
we performed a simple theoretical analysis by extending the Stokes-Smoluchowski
equation to take into account the porosity induced by the polymers in the
region adjacent the walls. We discuss the nature of the velocity profiles by
focusing on the competing effects of the polymer charges and the frictional
forces they exert. We show evidence of the flow reduction and of the flow
inversion phenomenon when the polymer charge is opposite to the surface charge.
By using the density of polymers and the surface charge as control variables,
we propose a phase diagram that discriminates the direct and the reversed flow
regimes and determine its dependence on the ionic concentration.
