that $$\label{ISEOF}
I_S^{{\rm EOF}}=\frac{16 w \kappa E_S }{\mu}\left(\frac{\e k_B
T}{e}\right)^2 \left(\frac{t_w^2}{1-t_w^2}\right) \left[1+ {\cal
O}\left(\frac{1}{\kappa h}\right)\right]\cdot$$

#### Conduction current

Finally, in response to the electric field, a conduction current $I_C$ is set up in the bulk of the solution; if we denote by $\sigma$ the ionic conductivity of the electrolyte (assumed to be constant), the conduction current is given by $$\label{}
  I_C=2hw\sigma E_S.$$

#### Induced electric field

If we investigate the steady-state motion of the electrolyte solution, we require that there be no net electric current $$\label{}
  I_{S}^{\rm PD}+I_S^{{\rm EOF}}+I_{C}=0,$$ which leads to the formula for the flow-induced streaming electric field $$\label{ES}
E_s=-\frac{\d p}{\d
  x}\left(\frac{\e\zeta_w}{\sigma\mu}\right)\left[1+\frac{8\kappa}{\sigma\mu
  h}\left(\frac{\e k_B
T}{e}\right)^2 \left(\frac{t_w^2}{1-t_w^2}\right)\right]^{-1}
+{\cal O}\left(\frac{1}{\kappa h}\right)\cdot$$ As expected, the induced field $E_S$ is proportional to the applied pressure gradient[^1].

Note that within the Debye-Hückel approximation , the induced electric field can be calculated exactly for all values of $\kappa h$ [@saville; @rice; @hunter; @probstein] and we find $$\label{ESdebye}
  E_S=\frac{\d p}{\d x}
  \left(  \frac{\tanh\kappa h}{\kappa h} -1\right)
    \left[
  \frac{\sigma\mu}{\e \zeta_w}+\frac{\e\zeta_w\kappa}{4h}\left(\frac{\sinh 2\kappa h-2\kappa h}{(\cosh\kappa h)^2}
  \right)\right]^{-1}\cdot$$ In the limits where $e|\zeta_w|/k_BT\ll 1$ ([*i.e.*]{} $t_w\ll 1$) and $\kappa h \gg 1$, the expressions given by and agree and are given by $$\label{}
  E_S= -\frac{\d p}{\d
  x}\left(\frac{\e\zeta_w}{\sigma\mu}\right)\left[1+\frac{(\e\zeta_w)^2\kappa}{2\sigma\mu
  h}\right]^{-1}\cdot$$

Velocity 