that the screening lengths $\kappa^{-1}$ for the charged particle and the charged channel surface are the same, as given by equation .

The particle velocity ${\bf U}_{\rm P}(z)$ will in general be $$\label{Up}
{\bf U}_{\rm P}(z)= {\bf U}_{\rm hydro}(z)+{\bf U}_{\rm
elec}(z)+{\bf U}_{\rm k_B T},$$ which includes three contributions.

#### Hydrodynamic contribution

The first component is the hydrodynamic contribution $$\label{hydro}
{\bf U}_{\rm hydro}(z)=\left[1-{\cal O}\left(\frac{a}{d}\right)
\right] U_{\rm PD}(z){\bf e}_x,$$ where $U_{\rm PD}$ is the local pressure-driven fluid velocity. It is modified by the presence of solid walls which slow down the motion of the suspended particle. Although the analysis is in general difficult [@Happel], walls lead to a leading-order correction to the particle velocity of order of the ratio of the particle size to the distance to the walls ${\cal O} (a/d)$; this is true as long as the particle does not come too close to the wall, in which case a different contribution arises from lubrication forces. We will assume in this paper that the particle is located sufficiently far from the walls ($a\ll d=h-|z|$) so that the influence of the walls can be neglected. Such a requirement would also have to be verified in an experiment, otherwise the presence of 