# Electroosmosis in a finite cylindrical pore: simple models of end effects

## TL;DR

This paper develops simple models to analyze electroosmotic flow in finite cylindrical pores, accounting for end effects and charge cloud spilling, with validation against numerical computations across different regimes.

## Contribution

It introduces an approximate analysis that combines limiting cases to effectively model electroosmosis in finite pores, including end effects and charge spill-out.

## Key findings

- Full numerical results agree with theory for h>>a and h=0 cases.
- Approximate analysis captures main features for small Debye length.
- Charge spill-out reduces electroosmotic flow when Debye length is large.

## Abstract

A theoretical model of electroosmosis through a circular pore of radius $a$ that traverses a membrane of thickness $h$ is investigated. Both the cylindrical surface of the pore and the outer surfaces of the membrane are charged. When $h\gg a$ end effects are negligible: results of full numerical computations of electroosmosis in an infinite pore agree with theory. When $h=0$, end effects dominate, and computations again agree with analysis. For intermediate values of $h/a$, an approximate analysis that combines these two limiting cases captures the main features of computational results when the Debye length $\kappa^{-1}$ is small compared with the pore radius $a$. However, the approximate analysis fails when $\kappa^{-1}\gg a$, when the charge cloud due to the charged cylindrical walls of the pore spills out of the ends of the pore, and the electroosmotic flow is reduced. When this spilling out is included in the analysis, agreement with computation is restored.