The motion of a bubble in a non-uniform Hele-Shaw flow

We consider the propagation of a bubble in a non-uniform Hele-Shaw flow. We focus on a distinguished limit in which the bubble is approximately circular in plan view and its velocity is determined by a net force balance incorporating the Hele-Shaw viscous pressure and drag due to the thin films separating the bubble from the cell walls. We find that the instantaneous bubble velocity is the same as that for a bubble in a uniform flow given by the background fluid velocity evaluated at the bubble centre. We apply the model to common microfluidic components, such as a T-junction and injection sources and sinks. Using the same methodology, we derive approximate solutions for the motion of a bubble in more complicated domains including walls or obstacles. We validate the approximate approach with two test cases in which we can find the full solution analytically. First, we consider the effect of an impermeable wall on a bubble in a stagnation-point flow. Second, we consider the motion of a bubble around a circular obstacle. In both cases, the bubble deviates notably from the streamlines of the background flow only when it is very close to the wall or the obstacle.