${\bf E}_S=E_S{\bf e}_x$ is aligned with the channel direction, the electrophoretic velocity ${\bf U}_{\rm
EPH}=U_{\rm EPH}\,{\bf e}_x$ is given by $$\label{phoretic}
U_{\rm EPH}=\frac{\e E_S (f(\kappa a)\zeta_p-\zeta_w)}
{\mu}\left[1-{\cal O}\left(\frac{a^3}{d^3}\right) \right]\cdot$$ This velocity first includes the “pure” electrophoretic mobility of the particle [@saville; @hunter; @AnnRev], characterized by the function $f(x)$, which satisfies $f(0)=2/3$ (Hückel’s result for thick screening length) and $f(\infty)=1$ (Smoluchowski’s result for thin screening length). Note that we can use these classical electrophoretic formulae because since $\kappa h \gg 1$, the perturbation of the ion distribution in the double layer around the particle is not modified by the local shear flow. The velocity also includes the electroosmotic back-flow resulting from the motion of excess charges near the channel walls and proportional to the wall zeta potential $\zeta_w$. Furthermore, the presence of a wall always influences the electrophoretic mobility at cubic order in the ratio of the particle size to the distance to the wall, as long as double layers do not overlap [@Ennis; @Yariv]; since we already assumed the particle to be located far from the wall, we will neglect the wall influence here as well.

#### Thermal contribution

Finally, the particle velocity has a random contribution ${\bf
U}_{\rm k_B 