Stable bubble formations in a depth-perturbed Hele-Shaw channel

The propagation of polydisperse bubbles inside a rectangular Hele-Shaw channel by the constant volumetric flux flow of an ambient viscous liquid is generally unsteady; pairs of neighbouring bubbles will either separate or coalesce because their propagation speeds increase monotonically with their sizes. Thus, any group of bubbles will eventually rearrange itself in order of decreasing size, individual bubbles will separate and, crucially, there are no stable multiple-bubble states. Remarkably, the introduction of a small axially uniform elevation to the channel’s lower boundary (on the order of a percent of the channel’s depth) leads to the creation of a multitude of stable multiple-bubble states and, thus, disrupts the usual reordering by bubble sizes. The constituent bubbles of such states lie in alternation on opposite sides of the elevation. The elevation provides a mechanism for the trailing bubbles to reduce their propagation speeds when approaching their preceding nearest-neighbours. The stable multiple-bubble states are always led by the smallest bubble whilst the trailing bubbles can be arranged in any order and, thus, their number grows factorially as the number of bubbles increases.