An infinite-series solution for the creeping motion through an orifice of finite length

Abstract

This paper presents an infinite-series solution to the creeping viscous motion of a fluid through low- and moderate-aspect-ratio pores. The flow field is divided into two simply bounded regions: a cylindrical volume bounded by the walls of the pore and the entrance and exit planes, and an infinite half-space outside the pore. Analytic solutions are first obtained in each region for unknown functions representing arbitrary axial and radial velocity profiles at the pore entrance (exit). These unknown functions are then determined by matching the normal and tangential stress at the pore opening. The results indicate that the velocity profile approaches to within 1·5 per cent of a Poiseuille profile after a short entrance distance of half the pore radius. In the far field the solution matches exactly the streamline pattern for a flow through an orifice of zero thickness obtained by Sampson (1891). The pressure drop across the pore exhibits linear dependence on the aspect ratio and is closely approximated (less than one per cent error) by a simple algebraic expression.