by the flow.

#### Pressure-driven current

First, the pressure-driven motion of the screening cloud of counter-ions near the charged surface leads to an advection-of-charge electric current $I_{S}^{PD}$ given by $$\label{ISPDbl}
I_{S}^{\rm PD}=\int_{-h}^hw\rho_e(z)U_{{\rm PD}}(z)\d z=\frac{2\e
wh k_B T}{\mu e} \left(\frac{\d p}{\d x} \right)I_1,$$ where we have used the electrostatic equation to relate the net charge density in the liquid to the electrostatic potential, ${\rho_e}= -\e \nabla^2 \psi$ and where $I_1$ is given by $$\label{I1}
I_1=\phi_w-\int_0^1\phi(\z)\d \z,$$ with the same dimensionless notations as in section \[surfacecharge\]. In the limit where $\kappa h \gg 1$, plugging in the solution into leads to $$\label{I1limit}
I_1=\phi_w - \frac{2}{\kappa h} \int_0^{\kappa h}\ln
\left(\frac{1+t_we^{-x} }{1-t_we^{-x} }\right)\d x,$$ so that $$\label{ISPD}
I_{S}^{\rm PD}=\frac{2\e wh\zeta_w}{\mu }\left(\frac{\d p}{\d
x}\right)\left[1+{\cal O}\left(\frac{1}{\kappa h}\right)\right].$$

#### Electroosmotic current

If an electric field is induced by the flow, the streaming current has a second component $I_S^{{\rm EOF}}$, given by the advection of counter-ions by the induced electroosmotic flow $$\label{ISEOFbl}
I_S^{{\rm EOF}}=\int_{-h}^hw\rho_e(z)U_{\rm EOF}(z)\d z
=\frac{2wE_S}{h\mu}\left(\frac{\e k_BT}{e}\right)^2I_2,$$ where $I_2$ is given by $$\label{I2}
I_2=\int_0^1\left(\frac{\d\phi}{\d\z}\right)^2\d\z.$$ In the limit where $\kappa h \gg 1$, the boundary layer solution leads to the leading order expression for $I_2$ in powers of $1/\kappa h$, $$\label{I2limit}
I_2=\frac{8\kappa h t_w^2 (1-e^{-2\kappa
h})}{(1-t_w^2)(1-t_w^2e^{-2\kappa h})},$$ so 