Pressure drop due to the motion of a sphere near the wall bounding a Poiseuille flow

An expression is derived for the (low Reynolds number) additional pressure drop created by a relatively small sphere moving near the wall of a circular tube through which there is a Poiseuille flow. Two specific applications are examined: (i) the sedimentation of a homogeneous non-neutrally buoyant sphere in a quiescent fluid; and (ii) the motion of a neutrally buoyant sphere. In the latter case a pronounced increase in the additional pressure drop is predicted when the separation between the sphere and the tube wall is reduced to zero. This analysis, which includes the behaviour for a sphere in contact with the tube wall, supplements previous ‘method of reflexions’ treatments valid only when the distance from the sphere centre to the wall is large compared with the sphere radius.