Mobility of a Small Sphere in a Viscous Fluid Confined in a Rigid Circular Cylinder of Finite Length

Abstract

A theoretical study is made on the effect of finiteness of the length of a circular cylinder on the translational motion of a small sphere along its centerline. Fluid occupying that region is assumed viscous and incompressible, and the analysis is based on the Stokes equations. The method of reflections is employed and first-order correction of the force experienced by the sphere is obtained. For longer cylinders, end-effects on the force are important only when the sphere is very close to the endwalls, while for cylinders whose length is the same order of magnitude as its radius, end-and sidewalls play equally significant roles. Among cylindrical boxes containing the same amount of fluid volume, additional resistance force at the center becomes minimum for a cylinder with its length/diameter ratio of about 0.56. Streamlines are also shown.