Electrophoresis of tightly fitting spheres along a circular cylinder of finite length

Electrophoresis of a tightly fitting sphere of radius $a$ along the centreline of a liquid-filled circular cylinder of radius $R$ is studied for a gap width $h_0=R-a\ll a$. We assume a Debye length $\kappa ^{-1}\ll h_0$, so that surface conductivity is negligible for zeta potentials typically seen in experiments, and the Smoluchowski slip velocity is imposed as a boundary condition at the solid surfaces. The pressure difference between the front and rear of the sphere is determined. If the cylinder has finite length $L$, this pressure difference causes an additional volumetric flow of liquid along the cylinder, increasing the electrophoretic velocity of the sphere, and an analytic prediction for this increase is found when $L\gg R$. If $N$ identical, well-spaced spheres are present, the electrophoretic velocity of the spheres increases with $N$, in agreement with the experiments of Misiunas & Keyser (Phys. Rev. Lett., vol. 122, 2019, 214501).