= Solution
Write $\varepsilon=\varepsilon_0\varepsilon_r$ and take $n_0$ to be the reservoir <density> of \b[each] monovalent ion species. Equal electrochemical potentials in equilibrium give the <Boltzmann ion densities in a symmetric electrolyte>,
$$
n_+=n_0e^{-e\psi/(k_BT)},\qquad n_-=n_0e^{e\psi/(k_BT)},\qquad\rho_e=e(n_+-n_-)=-2en_0\sinh\frac{e\psi}{k_BT}.
$$
Combining this with the electrostatic <Poisson equation> gives the planar <Poisson-Boltzmann equation>,
$$
\psi''=\frac{2en_0}{\varepsilon}\sinh\frac{e\psi}{k_BT},\qquad\boxed{\kappa^2=\frac{2e^2n_0}{\varepsilon k_BT}.}
$$
The <Debye length> is $\kappa^{-1}$. If the symbol $n_0$ instead denotes the combined <density> of both species, the factor two is absent; the per-species convention above makes the normalization explicit. At $|e\psi|\ll k_BT$, the <linearized planar screening solution> obeys $\psi''=\kappa^2\psi$. With $\psi(0)=\zeta<0$ and $\psi(\infty)=0$,
$$
\boxed{\psi(x)=\zeta e^{-\kappa x}.}
$$
The positive <ions> are <counterions> and the negative <ions> are <co-ions>. At weak potential,
$$
n_+\simeq n_0\left(1-\frac{e\zeta}{k_BT}e^{-\kappa x}\right),\qquad n_-\simeq n_0\left(1+\frac{e\zeta}{k_BT}e^{-\kappa x}\right).
$$
They approach $n_0$ from above and below respectively, with equal first-order excess and deficit. For stronger negative potential, interpret the source's strong-potential comparison as a statement about \b[$|e\psi|$], since $\psi$ itself is negative. <Counterions> become strongly concentrated near the wall, while <co-ions> are exponentially depleted and remain nonnegative. Linearization misses this asymmetry and would eventually predict unphysical negative <co-ion> <density>. Point-ion <mean-field theory> itself eventually needs finite-size and <correlation> corrections; it provides no steric cap at arbitrarily large <counterion> concentration.
For steady tangential <electro-osmosis>, with no imposed <pressure> gradient and negligible inertia, the <Stokes equation> is $\eta v''+\rho_eE=0$. Since $\rho_e=-\varepsilon\psi''$,
$$
\eta v''=\varepsilon E\psi''.
$$
The conditions $v(0)=0$ and $v'(\infty)=0$ give
$$
v(x)=\frac{\varepsilon E}{\eta}[\psi(x)-\zeta]=U(1-e^{-\kappa x}),\qquad\boxed{U=-\frac{\varepsilon\zeta E}{\eta}.}
$$
\b[The printed constant speed is the bulk <velocity> outside the double layer, not the <velocity> at the no-slip wall.] With negative $\zeta$ the positive <counterion> excess drives fluid along the applied field. The far-field value is the <Helmholtz-Smoluchowski electro-osmotic velocity>.<Induced-Charge Electro-osmosis and Electrophoresis by Martin Z. Bazant>[Section1,page1: diffuse charge, slip <velocity> and plug-flow limit]{ref}
For the finite channel, first model the added wall as an ideally uncharged insulating boundary with no normal field on its outer side: $\psi'(d)=0$, while $\psi(0)=\zeta$. The <one-charged-wall electro-osmotic channel> then has
$$
\psi(x)=\zeta\frac{\cosh[\kappa(d-x)]}{\cosh(\kappa d)}.
$$
Integrating the same <force> balance and enforcing $v(0)=v(d)=0$ gives
$$
\boxed{v(x)=U\left[1-\left(1-\operatorname{sech}(\kappa d)\right)\frac xd-\frac{\cosh[\kappa(d-x)]}{\cosh(\kappa d)}\right].}
$$
For $U>0$, the maximum occurs at
$$
\boxed{x_{\max}=d-\frac1\kappa\operatorname{arsinh}\left[\frac{\cosh(\kappa d)-1}{\kappa d}\right]\simeq\frac{\log(\kappa d)}\kappa\quad(\kappa d\gg1).}
$$
In that limit $v/U\simeq1-x/d-e^{-\kappa x}$: the <velocity> rises rapidly from zero at the charged wall, reaches a maximum a few screening lengths away, then falls approximately linearly to zero at the neutral wall. \b[One charged wall does not give a symmetric plug across a channel with no-slip at the other wall.]
An alternative electrode condition $\psi(d)=0$ means a wall held at reservoir potential; it is not the same as zero surface charge. Under that convention,
$$
v(x)=U\left[1-\frac xd-\frac{\sinh[\kappa(d-x)]}{\sinh(\kappa d)}\right],
$$
with $\cosh[\kappa(d-x_{\max})]=\sinh(\kappa d)/(\kappa d)$. Both boundary conventions have the same leading thin-double-layer profile above. An externally imposed <pressure> gradient would add a <pressure>-driven profile; none is specified here.
For dilute nonadsorbing uncharged <polymers> in equilibrium or sufficiently weak flow, the <entropic polymer depletion near nonadsorbing walls> suppresses chain configurations near each wall. A depletion layer of order the coil <radius of gyration> is expected at both walls, with a bulk-like central region when $d$ is much larger than the coil size. Strong confinement rounds this into a central enrichment. Neutrality removes a direct electrophoretic <force>, not excluded-volume <entropy>. Adsorption, concentration and strong-flow migration were not specified, so this is a conditional equilibrium sketch rather than a universal flowing-<polymer> profile.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2015/paper-1-ion-distributions.png]
{title=Weak screening gives symmetric first-order ion excess and deficit, while stronger potentials separate co-ion depletion from counterion accumulation}
{source=paper-1-ion-distributions.py}
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2015/paper-1-channel-and-polymers.png]
{title=A channel driven at only one wall has an asymmetric velocity maximum; nonadsorbing polymer coils are depleted near either wall}
{source=paper-1-channel-and-polymers.py}
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