Modelling bubble propagation in elasto-rigid Hele-Shaw channels

We study a model of pulmonary airway reopening where air is driven at constant volume flux into a liquid-filled Hele-Shaw channel, with an upper compliant boundary. An equivalent rigid channel supports a stable, steadily propagating air finger and a variety of unstable solutions. In the compliant channel, however, initial collapse of the channel introduces additional cross-sectional depth gradients. The induced normal and transverse depth variations alter the finger morphology and promote a variety of instabilities from tip-splitting to small-scale fingering on the curved front. A depth-averaged model for the system is simulated numerically using the open-source, finite-element library, oomph-lib, in order to explore underlying mechanisms and the relative importance of the elastic, capillary and viscous effects. The model exhibits a complex solution structure and qualitatively similar instabilities to those observed experimentally. The solution structure is related to that found in a rigid Hele-Shaw channel but here the solutions interact due to the fluid-structure interaction introduced by the compliant boundary.