A Hele-Shaw Newton's cradle: The motion of bubbles in Hele-Shaw cells

We present a model for the motion of approximately circular bubbles in a Hele-Shaw cell. The bubble velocity is determined by a balance between the hydrodynamic pressures from the external flow and the drag due to the thin films above and below the bubble. We find that the qualitative behaviour depends on a dimensionless parameter $\delta \propto \mathrm{Ca}^{1/3}R/h$, where $\mathrm{Ca}$ is the capillary number, $R$ is the bubble radius and $h$ is the cell height. An isolated bubble travels faster than the external fluid if $\delta>1$ or slower if $\delta<1$, and the theoretical dependence of the bubble velocity on $\delta$ is found to agree well with experimental observations. Furthermore, we show how the effects of interaction with cell boundaries and/or other bubbles also depend on the value of $\delta$. For example, in a train of three identical bubbles travelling along the centre line, the middle bubble either catches up with the one in front (if $\delta >1$) or is caught by the one behind (if $\delta <1$), forming what we term a Hele-Shaw Newton's cradle.