T}$ due to thermal motion, which can be significant. A solid spherical particle of radius $a$, located far from boundaries, has a diffusivity $D$ given by the Stokes-Einstein relation $D=k_B T/ 6\pi \mu a$ [@saville], corresponding to a root mean square velocity on the order of $U_{k_BT} \sim D/a \sim
k_B T/ 6\pi \mu a^2$. At 25C in water, $a= 10$ nm leads to $U_{k_BT}\sim 1$ mm/s; this value is of the same order as the fluid velocity in a circular capillary of radius $R\sim 100$ $\mu$m and flow rate $Q\sim 1$ $\mu$L/min, typical values for microfluidic devices. Consequently, we cannot assume that the Peclet number, $Pe=U/U_{k_B T}=Ua/D$, is necessarily large and thermal motion cannot in general be neglected. However, in the experiments reported to date, velocity measurements are cross correlated (as in [@Meinhart]) or averaged (as in [@Pit]) so that the random thermal motion disappears, and we will therefore not consider it in this paper.

#### Summary

Under the previous assumptions, we can write the velocity for the particle as $$\label{summary}
U_{\rm P}(z)  =U_{{\rm PD}}(z)+
 \frac{\e
E_{S}}{\mu}(f(\kappa a)\zeta_p-\zeta_w) + {\cal
O}\left(\frac{a}{d}\right),$$ where the velocity should be understood as an ensemble average over different experimental realizations.

Apparent slip length
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