$$\label{inner}
\tanh\left(\frac{\phi_{\rm\, in}(\z)}{4} \right) = \tanh
\left(\frac{\phi_w}{4} \right) e^{-\kappa h (1-|\z|)}.$$ Finally, since $\phi_{\rm \,out}=0$, the inner solution is also equal to the composite solution $\phi(\z)$, uniformly valid throughout the channel as $\kappa h \to \infty$, at leading order in $1/\kappa h$. For convenience, equation can be rewritten as $$\label{comp}
\phi(\z)=2\ln \left(\frac{1+t_w e^{-\kappa h (1-|\z|)}}{1-t_w
e^{-\kappa h (1-|\z|)}} \right),$$ where we have defined $t_w=\tanh (e\zeta_w/4k_B T)$.

Hydrodynamics and electrokinetics
---------------------------------

When a pressure-driven flow occurs in the channel, the fluid velocity is unidirectional ${\bf U}=U(z){\bf e}_x$, where ${\bf
e}_x$ is the streamwise direction. In the absence of electrical effects, the fluid velocity is simply Poiseuille’s pressure-driven formula [@batchelor], which we will denote $U_{{\rm PD}}$, and is given by $$\label{PD}
  U_{\rm PD} (z)=-\frac{h^2}{2\mu}\frac{\d p}{\d x}\left[1-\frac{z^2}{h^2}
  \right]\cdot$$

Furthermore, if an external, or induced, electric field ${\bf
E}_{S}=E_{S}{\bf e}_x$ is also applied to the channel, the presence of a net charge density near the solid surface moving in response to the field leads to an additional velocity component known as electroosmotic flow (EOF) [@saville]. It is directed in the $x$-direction, is given by $$\label{EOF}
  U_{{\rm EOF}}(z)=\frac{\e E_{S}}{\mu}\Big[\psi(z)-\zeta_w
  \Big],$$ and is valid for any value of $\zeta_w$.

Streaming potential and electroviscous effect {#stream}
---------------------------------------------

