Mode-switching of fingering instabilities in an elastic Hele-Shaw channel

We study the propagation of a curved front in a Hele-Shaw channel where the top boundary is an elastic membrane, and the channel is initially collapsed and filled with liquid – a benchtop model of pulmonary airway reopening. The injection of air at a constant flow rate works against elastic, viscous and capillary forces to form an approximately steadily propagating blistering finger, which depends on the level of initial collapse of the channel cross-section and the capillary number Ca – the ratio of viscous to surface tension forces. We find that the initial deflection of the membrane destabilises the finger into an oscillatory propagation mode, where small-scale fingering at the finger-tip is advected towards the rear of the finger as it advances. We present evidence of a switching transition between two oscillatory modes which are connected to the steady pushing and peeling regimes underlying airway reopening. One resembles the tip-splitting instabilities observed to occur sub-critically in rigid channels above a critical value of Ca, while the other is linked to stubby finger formation in tapered channels, e.g. tape peeling and the printer’s instability.