A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids

Abstract

This paper is the first in a series of investigations having the overall objective of developing a new technique for treating the slow viscous motion past finite assemblages of particles of arbitrary shape. The new method, termed the multi-pole representation technique, is based on the theory that any object conforming to a natural co-ordinate system in a particle assemblage can be approximated by a truncated series of multi-lobular disturbances in which the accuracy of the representation is systematically improved by the addition of higher order multipoles. The essential elements of this theory are illustrated by examining the flows past finite line arrays of axisymmetric bodies such as spheres and spheroids which conform to special natural co-ordinate systems. It is demonstrated that this new procedure converges more rapidly and is simpler to use than the method of reflexions and represents the desired boundaries more precisely than the point-force approximation even when the objects are touching one another. Comparison of these solutions with the exact solutions of Stimson & Jeffery (1926) for the two sphere problem demonstrates the rapidity of convergence of this multipole procedure even when the spheres are touching. Drag results are also presented for flows past chains containing up to 101 spheres as well as for chains containing up to 15 prolate or oblate spheroids. The potential value of the technique is suggested by the rapidity with which the drag calculations were made, the 101 sphere problem requiring about 10 seconds on an IBM 360–65 computer to determine both the fluid flow and the drag coefficient.

A method for determining Stokes flow around particles near a wall or in a thin film bounded by a wall and a gas-liquid interface

“…[22][23][24][25][26][27][28][29][30][31][32][33] We first consider the case of a single spherical particle of radius a held fixed in a parabolic flow given by…”

Suspensions of prolate spheroids in Stokes flow. Part 1. Dynamics of a finite number of particles in an unbounded fluid

“…Incidentally, the multipole expansion solution presented in this work may guide the selection of the appropriate range completer for the completed double-layer method, and thereby increase the performance of the technique for many-body problems involving spheroidal particles. The multipole collocation method has mostly been applied to the motion of two or three spheres (Hassonjee, Ganatos & Pfeffer 1988) and to axisymmetric flows past chains of ellipsoids (Gluckman, Pfeffer & Weinbaum 1971 ;Liao & Krueger 1980). It is best suited for problems involving a finite number of identical particles positioned in a very symmetric arrangement.…”

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

“…We emphasize that our method can be reduced to a 60-line routine that, with the help of subprograms for the special functions (Legendre functions) calculates the entire collection of functions. Others have solved various subsets of this problem using bispherical coordinates (Stimson and Jeffery 1926;Goldman, Cox and Brenner 1966;Lin, Lee and Sather 1970), method of reflections (Happel and Brenner 1965) and lubrication theory (O'Neill and Majumdar 1970), as reviewed by Jeffrey and Onishi (1984). These authors also present a comprehensive solution of the important subset involving the force and torque in an ambient *, field composed of a uniform stream and vorticity field, using the twin-multipole variation of the method of reflections.…”

“…spheres, the points are distributed in equal numbers between the two, at equidistant spacings (Gluckman, Weinbaum and Pfeffer 1971). Furthermore, we decompose each resistance problem into subproblems that exploit the fore-aft mirror symmetry with respect to the XY plane.…”