 Helmholtz-Smoluchowski electro-osmotic velocity

In a half-space with no pressure gradient, Stokes balance and Poisson electrostatics give $\eta v\prime \prime =\epsilon E\psi \prime \prime$. No slip at the wall and zero bulk shear imply $v=\epsilon E(\psi -\zeta )/\eta$. The bulk velocity is $U=-\epsilon \zeta E/\eta$; it is not the fluid speed at the no-slip wall.