Stable bubble formations in a Hele-Shaw channel

Deformable air bubbles flowing in a uniform Hele-Shaw channel filled with a viscous liquid will either separate or aggregate because a bubble's speed increases monotonically with its size. Hence, there are no stable multi-bubble states. We show that the introduction of a small geometric perturbation to the channel in the form of a central axially-uniform depth reduction supports a multitude of stable bubble formations in which bubbles lie in alternation on opposite sides of the perturbation and retain constant separations as they propagate. In all cases, a formation is led by the smallest of its constitutive bubbles and the trailing bubbles can be arranged in any order. Hence, for a group of $N$ bubbles, there are $(N-1)!$ stable formations. These steady multi-bubble states arise because each of the trailing bubbles can decrease their speed to that of the leading bubble through a reduction in the local viscous dissipation, caused by an increase in the bubble's overlap of the depth-reduced region. We find that such formations exist for depth perturbations of less than $1\%$.