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.

Considerable advances have been made in the past few years in treating a variety of problems in slender-body Stokes flow (Taylor 1969; Batchelor 1970; Cox 1970, 1971; Tillett 1970). However, the problem of treating the creeping motion past bluff objects, whose boundaries do not conform to a constant co-ordinate surface of one of the special orthogonal co-ordinate systems for which the Stokes slow-flow equation is simply separable, is still largely unsolved. In the slender-body Stokes flow studies mentioned above, the viscous-flow boundary-value problem is formulated approximately as an integral equation for an unknown distribution of Stokeslets over a line enclosed by the body. The theory is valid for only very extended shapes, since the error in drag decays inversely as the logarithm of the aspect ratio of the object. By contrast, the present authors show that the boundary-value problem for the axisymmetric flow past an arbitrary convex body of revolution can be formulated exactly as an integral equation for an unknown distribution of ring-like singularities over the surface of the body. The kernel in this integral equation is closely related to the fundamental separable solutions of the Stokes slow-flow equation when written in an oblate spheroidal co-ordinate system of vanishing aspect ratio. The two lowest-order appropriate spheroidal singularities are found to provide a complete description for all surface elements, except those perpendicular to the axis. Higher-order singularities of all orders are required to describe axially perpendicular surfaces, such as the ends of a cylinder or the blunt base of an object. The newly derived integral equation is solved numerically to provide the first theoretical solutions for low aspect ratio cylinders and cones. The theoretically predicted drag results are in excellent agreement with experimentally measured values.

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