\ell$ where $\ell$ is the effective ion size. Using equation , we see that the conductivity and the screening length are related by $$\label{cond2}
\sigma \approx \frac{\e k_B T}{6 \pi \mu \ell} \kappa^2.$$

Furthermore, since the conductivity $\sigma$ and the viscosity $\mu$ only appear in as their product, the estimate shows that the apparent slip length is in fact independent of the fluid viscosity. Moreover, since $\kappa\sim n_0^{1/2}$ and $\sigma\sim n_0$, and since $f(\kappa
a)$ varies only weakly with $\kappa$, we see from that the $\lambda$ is a decreasing function of the ionic strength. Also, it is clear from that the slip length always decreases with the channel size.

Finally, we note the apparent slip length vanishes for two values of the wall zeta potential: $\zeta_w=0$ and $\zeta_w=\zeta_p/f(\kappa a)$. Consequently, in between these two values, the slip length reaches a maximum value $\l$ when the wall zeta potential is equal to $\zeta_w=\zeta_m^*$, [*i.e.*]{} $\d
\lambda / \d\zeta_w(\zeta_m^*)=0$. This is illustrated in Figure \[maximum\] (left).

![Left: variation of the apparent slip length $\lambda$ for pure water as a function of the wall zeta potential $\zeta_w$ for $\zeta_p=50$ mV, $n_0=10^{-6}$ mol l$^{-1}$ (pure water), $\kappa h=10$ and $\kappa
a \ll 1$; the 