Bubble racing in a Hele-Shaw cell

We study theoretically and experimentally the propagation of two bubbles in a Hele-Shaw cell under a uniform background flow. We consider the regime where the bubbles are large enough to be flattened by the cell walls into a pancake-like shape, but small enough such that each bubble remains approximately circular when viewed from above. In a system of two bubbles of different radii, if the smaller bubble is in front, it will be overtaken by the larger bubble. Under certain circumstances, the bubbles may avoid collision by rolling over one another while passing. We find that, for a given ratio of the bubble radii, there exists a critical value of a dimensionless parameter (the Bretherton parameter) above which the two bubbles will never collide, regardless of their relative size and initial transverse offset, provided they are initially well separated in the direction of the background flow. Additionally, we determine the corrections to the bubble shape from circular for two bubbles aligned with the flow direction. We find that the front bubble flattens in the flow direction, while the rear bubble elongates. These shape changes are associated with changes in velocity, which allow the rear bubble to catch the bubble in front even when they are of the same size.