Axisymmetric Stokes flows due to a rotlet or stokeslet near a hole in a plane wall: filtration flows

Abstract

The axially symmetric Stokes-flow problems occurring when a point source, rotlet or stokeslet is situated along the axis through the centre of a circular hole in a solid plane wall are examined. Exact solutions of the governing equations are obtained in terms of toroidal coordinates and their use in modelling the flows caused by a small particle translating and rotating near to a filter pore is considered. First-order expressions are derived for the effects of the wall and hole upon the hydrodynamic force and torque on the particle for situations in which the particle dimensions are small in comparison with its distance from the solid portion of the plane wall. The resulting expressions apply to any centrally symmetric particle, not necessarily axisymmetric. Finally, expressions are derived for the motion of a neutrally buoyant sphere suspended in a flow through a hole. It is demonstrated that such a particle will generally migrate across the streamlines of the undisturbed flow - away from or towards the symmetry axis of the flow, according as the particle is approaching or receding from the hole. Such migratory motion may be of importance in the flow of suspensions through orifices and stenoses.