General theory for the creeping motion of a finite sphere along the axis of a circular orifice

Abstract

This paper presents the first infinite-series solutions to the creeping-flow equations for the axisymmetric motion of a sphere of arbitrary size towards an orifice whose diameter is either larger or smaller than the sphere. To obtain the solution the flow field is partitioned in the plane of the opening, and for the flow to the left and right of the fluid interface separate solutions are developed that satisfy the viscous-flow boundary conditions in each region and unknown functions for the axial and radial velocity components in the plane of the opening. The continuity of the fluid stress tensor at the matching interface leads to a set of dual integral equations which are solved analytically to determine the unknown functions for the velocity components in the matching plane. A boundary collocation technique is used to satisfy the no-slip boundary conditions on the surface of the sphere.The accuracy and convergence of the present solution is tested by detailed numerical comparison with the exact bipolar co-ordinate solutions of Brenner (1961) for the drag on a sphere translating perpendicular to an infinite plane wall up to a distance of 0·1 sphere radii and is found to be in agreement to five significant digits. The converged-series collocation solutions are presented for the sphere in motion in quiescent fluid or for flow past a rigidly held sphere positioned axisymmetrically near a fixed orifice. Solutions are also presented for the zero-drag velocity of a neutrally buoyant sphere in a flow through an orifice, and the pressure–volume flow relation for a ball-valve geometry.