A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion

Abstract

This paper presents the first ‘exact’ solutions to the creeping-flow equations for the transverse motion of a sphere of arbitrary size and position between two plane parallel walls. Previous solutions to this classical Stokes flow problem (Ho & Leal 1974) were limited to a sphere whose diameter is small compared with the distance of the closest approach to either boundary. The accuracy and convergence of the present method of solution are tested by detailed comparison with the exact bipolar co-ordinate solutions of Brenner (1961) for the drag on a sphere translating perpendicular to a single plane wall. The converged series collocation solutions obtained in the presence of two walls show that for the best case where the sphere is equidistant from each boundary the drag on the sphere predicted by Ho & Leal (1974), using a first-order reflexion theory, is 40 per cent below the true value when the walls are spaced two sphere diameters apart and is one order-of-magnitude lower at a spacing of 1.1 diameters.

Motion of a rigid cylinder between parallel plates in stokes flow

“…Our analysis in Part 1 for particle motion in a quiescent fluid shows that in certain cases the behavior in 2-D and 3-D cases was very similar, whereas in other cases there were qualitative differences. Here, we compare our results to those of Bungay and Brenner [9, 10] for a spherical particle in a circular tube in Poiseuille flow, and of Refs [11][12][13] for a spherical particle between two plane parallel boundaries in Poiseuille and Couette flows. The results in these papers are presented in different forms than the results of the present paper, so some additional calculations were required to rescale the variables.…”

The resistance and mobility functions of two equal spheres in low-Reynolds-number flow

“…From here on, without loss of generality, we shall restrict our attention to the "11" and "12" functions. (Ganatos, Pfeffer and Weinbaum 1978), the motion of a sphere between two parallel infinite plates (Ganatos, Pfeffer and Weinbaum 1980) and the sedimentation of a sphere in an inciined channel (Ganatos, Weinbaum and Pfeffer 1982). For our two-sphere problem, a suitable co-ordinate system exists.…”

Numerical solution of two-dimensional stokes equations for flow with particles in a channel of arbitrary shape using boundary-conforming coordinates

“…The authors reported good agreement with analytical solutions [4,5] for a train of spherical particles in axisymmetric creeping flow. Recently, collocation solutions were obtained for axisymmetric and nonaxisymmetric motion of a spherical particle between two parallel planes in Poiseuille and Couette flows, and in the case of sedimentation [6,7,8].…”

“…The derivatives of pressure can be expressed in terms of the derivatives of vorticity (6) Consider a coordinate transformation (7) defined as a solution of two elliptic quasilinear partial differential equations [9], (8) where (9) G is the Jacobian of the transformation (10) and P = P(ξ, η) and Q = Q(ξ, η) are functions chosen to control the spacing of coordinate lines in the x−y plane.…”

“…Once the stream function is known, the vorticity can be found from (5) and the pressure distribution from (6).…”