Slow viscous flow past a sphere in a cylindrical tube

A theoretical treatment is presented for the slow flow of a viscous fluid through a cylindrical container within which a small spherical particle is confined. The sphere is situated in an arbitrary position within the cylinder and moves at constant velocity parallel to the walls. Approximate expressions are derived which give the frictional drag, rotational couple, and permanent pressure drop caused by the presence of this obstacle in the original Poiseuillian field of flow. The primary parameters involved are the ratio of sphere to cylinder radius and fractional distance of the particle from the longitudinal axis of the cylinder. With appropriate modifications, the results are also applicable to a sphere settling in a quiescent fluid. This yields the necessary boundary corrections to Stokes law arising in connection with devices such as the falling ball viscometer when the sphere is eccentrically located.