End loss for Stokes flow through a slippery circular pore in a barrier of finite thickness

Abstract

An analytical model based on the fluid cylinder approximation and eigenfunction expansions is developed for Stokes flow through a slippery circular pore in a barrier of finite thickness. The hydraulic resistance, which comprises the end resistance and Poiseuille resistance, is determined as a function of the pore thickness, slip length of the pore wall, and proximity of pores. The results are presented to reveal how wall slip may change, quantitatively and qualitatively, the effect of the pore thickness on the end resistance. It is shown, in particular, that the use of Sampson's formula may underestimate the end loss under the effect of wall slip. Velocity slip on the wall will cause a greater departure of the velocity profile at the inlet from that of the fully developed flow, and therefore, a longer entrance length is required for the flow to attain its final state. Empirical formulas are proposed to facilitate quick calculation of the end resistance as a function of the controlling parameters.