Circular bubbles in a Hele-Shaw channel: a Hele-Shaw Newton's cradle

We consider the propagation of inviscid bubbles in a Hele-Shaw cell under a uniform background flow. We focus on the distinguished limit in which the hydrodynamic pressure gradient due to the external flow balances viscous drag effects due to thin liquid films between the bubbles and the cell walls (Bretherton, J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188), with the ratio between these two effects measured by a single dimensionless parameter that we label $\delta$ . In this regime, we find that each bubble remains approximately circular, and its propagation velocity is determined by a net force balance. The analytical solution for the problem of an isolated bubble in an infinite Hele-Shaw cell is found to agree well with experimental data in the literature. In particular, we find that the bubble may move faster or slower than the background fluid speed, depending on whether $\delta >1$ or $\delta <1$ , or precisely with the background flow if $\delta =1$ . When the model is generalised to include the effects of multiple bubbles and boundaries in the Hele-Shaw cell, we still find that the sign of $\delta -1$ causes striking changes in the qualitative behaviour. For a train of three or more bubbles moving along a Hele-Shaw channel, we observe longitudinal waves that propagate forwards or backwards along the bubble train, depending on whether $\delta >1$ or $\delta <1$ , resembling a Hele-Shaw Newton's cradle.