Motion and deformation of a bubble in a Hele-Shaw cell

We theoretically and experimentally study the propagation of a bubble in a Hele-Shaw cell under a uniform background flow at low Reynolds number. We consider situations where both the capillary number Ca and the ratio ϵ of the cell height to the bubble diameter are small. The bubble is then flattened into a pancake-like shape, with an approximately circular profile when viewed from above, and thin liquid films lie between the bubble and the cell walls. Bubble motion and deformation are determined by an interplay between the Hele-Shaw viscous pressure, the pressure drop due to the thin films, and the capillary pressure due to the in-plane curvature of the apparent bubble boundary. Numerical and asymptotic results indicate that, with all other parameters held constant, the in-plane aspect ratio of the bubble varies nonmonotonically with its size, with smaller bubbles being flattened in the flow direction and larger bubbles being elongated. These theoretical predictions are validated experimentally, as well as the expected loss of fore-aft symmetry of the bubble shape due to differences between the advancing and retreating menisci. Measurements of the bubble velocity are also shown to agree well with theoretical predictions. We extend our results to describe a bubble moving in an inclined cell due to buoyancy. Experimental results for the bubble velocity, as well as results found in the literature, are shown to collapse under a scaling motivated by the theory. As in the case of a horizontal Hele-Shaw cell, we find that, with increasing bubble size, the bubble shape can switch from being flattened to elongated in the direction of motion.