The behaviour of clusters of spheres falling in a viscous fluid Part 1. Experiment

Abstract

The sedimentation of small clusters of uniform spheres, falling freely through a viscous liquid, has been studied with Reynolds numbers (based on diameter of the sphere and its velocity of free fall in the unbounded fluid) of individual spheres ranging from 10−4 to 10. The fall velocity of a cluster is, in all cases, greater than that of individual spheres, the more so when the spheres are closer together. Two spheres falling side-by-side rotate inwards and separate as they fall if Re > 0·05, but no rotation nor separation is observed for Re < 0·03. When equal-sized spheres of Re > 1 fall vertically one behind the other, the rear sphere is accelerated into the wake of the leader, rotates, round it and separates from it when the line of centres is horizontal. If two spheres of unequal size but the same individual terminal velocity fall together, the smaller always travels faster than the larger. When three similar equally spaced spheres are dropped in a horizontal line, they interchange positions but do not separate when 0·06 < Re < 0·16. But, if 0·16 < Re < 3, one sphere is always left behind; which sphere depends critically upon the initial spacing. If three to six equal spheres, of 0·06 < Re < 7, start falling as a compact cluster, they eventually draw level and arrange themselves in the same horizontal plane at the vertices of a regular polygon. The polygon expands at a decreasing rate during fall. When three spheres are arranged initially in a horizontal isosceles triangle, the spheres oscillate about their equilibrium positions but eventually the spheres form a stable equil triagnle. If Re > 7, or the cluster contains 7 or more equal spheres, it shows no tendency to form a regular polygon but breaks up into two or more groups. A regular heptagon, and a hexagon with an additional sphere at its centre, are also unstable.