  Using experiments and a depth-averaged numerical model, we study
instabilities of two-phase flows in a Hele-Shaw channel with an elastic upper
boundary and a non-uniform cross-section prescribed by initial collapse.
Experimentally, we find increasingly complex and unsteady modes of air-finger
propagation as the dimensionless bubble speed, Ca, and level of collapse are
increased, including pointed fingers, indented fingers and the feathered modes
first identified by Cuttle et al.(J. Fluid Mech., vol. 886, 2020, A20). By
introducing a measure of the viscous contribution to finger propagation, we
identify a Ca threshold beyond which viscous forces are superseded by elastic
effects. Quantitative prediction of this transition between 'viscous' and
'elastic' reopening regimes across levels of collapse establishes the fidelity
of the numerical model. In the viscous regime, we recover the non-monotonic
dependence on Ca of the finger pressure, which is characteristic of benchtop
models of airway reopening. To explore the elastic regime numerically, we
extend the depth-averaged model introduced by Fontana et al. (J. Fluid Mech.,
vol. 916, 2021, A27) to include an artificial disjoining pressure which
prevents the unphysical self-intersection of the interface. Using time
simulations, we capture for the first time the majority of experimental finger
dynamics, including feathered modes. We show that these disordered states
continually evolve, with no evidence of convergence to steady or periodic
states. We find that the steady bifurcation structure satisfactorily predicts
the bubble pressure as a function of Ca, but that it does not provide
sufficient information to predict the transition to unsteady dynamics which
appears strongly nonlinear.
