Equilibrium configurations of two-dimensional bubbles in a channel: N-bubble case*

Microfluidic systems often involve flow of droplets or bubbles through narrow channels. How droplets or bubbles arrange in channels influences in turn how they flow. Here the focus is upon bubbles confined in a Hele-Shaw cell. Depending on bubble size relative to channel size, a zig-zag pattern can be adopted with bubbles alternating from side-to-side of the Hele-Shaw channel. Systems with odd numbers of bubbles are of interest as they break topological symmetry, with more bubbles (and hence, in moving systems, more drag) on one side of a channel than the other. Systems with up to N=3 bubbles are dominated by edge effects, with bubbles bulging at the ends of the structure. In systems with N=5 or more bubbles, edge effects remain present, but are weaker: with N=5 or more bubbles, there is at least one bubble confined in the interior of the zig-zag which cannot bulge freely. Edge effects are then defects superposed on a large-N structure. Even weak edge effects are of interest however, since very short bubble films often appear towards edges of structures. These short films represent weak points at which topology of the bubble structure is liable to change. By locating such short films, both maximum and minimum bubble sizes can be obtained to retain a zig-zag pattern.