Stable bubble formations in a Hele-Shaw channel

Abstract

Deformable bubbles propagated by the flow of a viscous liquid in a planar Hele-Shaw channel of uniform depth tend to travel steadily along the channel's streamwise axis and pairs of neighbouring bubbles will either separate or coalesce because an individual bubble's propagation speed increases monotonically with its size. Thus, any group of bubbles will eventually rearrange itself in order of decreasing size and all of the bubbles will separate. We show that, by introducing a small geometric perturbation to the channel in the form of an axially-uniform depth reduction along its centreline, the system supports a multitude of stable bubble formations and this can disrupt the usual reordering by bubble size. The constituent bubbles of a stable formation lie in alternation on opposite sides of the depth-perturbation, retain fixed shapes and propagate steadily at the same speed. A stable formation is always led by the smallest of its constituent bubbles, which would be the slowest bubble in isolation. The leading bubble propagates as if it were isolated and the trailing bubbles reduce their speeds by adjusting their shapes and overlaps of the depth-perturbation in the perturbation fields of their preceding nearest-neighbours in order to match that of the leading bubble. The trailing bubbles can be arranged in any order and, hence, the number of stable formations increases factorially as the number of bubbles is increased.