Effect of higher-order and lubrication terms on the stability of polygonal arrangements of sedimenting spheres

Studies of the sedimentation of n identical spheres placed at the vertices of a horizontal regular polygon have shown that the arrangement is stable for n ≤ 6 and unstable for n > 6 when the spheres are widely separated. At such separations, only the Stokeslet is significant. At very small separations, higher-order terms and lubrication effects must be taken into account. The hexagon is stable at all separations. For 6 < n ≤ 12, there is a gradual increase in instability as the spheres are placed closer together. This reflects the increasing strength of the interaction terms. This is followed by a sudden and enormous decrease in instability as the distance between nearest neighbors decreases further. Both higher-order and lubrication terms contribute to this decrease. The instability increases as n increases. Irregular hexagons, in which close pairs approximate single spheres in isosceles triangles, settle with periodic motions. The same is true for irregular octagons approximating rhombi, though these octagons often break up. The orbit of a given sphere relative to the center of mass varies from nearly elliptical to egg-shaped. The orbit of each sphere in a “kite” formation is very complicated, but the boundary of the trajectory is well defined.