Air finger propagation in elasto-rigid channels

We investigate the steady propagation of the air finger created by injecting air at a constant flow rate in an oil-filled Hele-Shaw channel topped with an elastic membrane.
We first focus on the reopening modes of this single idealised airway in terms of the injection flow rate and the initial shape (inflated or collapsed) of the membrane. We show that as the membrane is collapsed, the finger shape is increasingly dominated by the cross-sectional profile imposed prior to air injection. For very high levels of collapse, a region of near opposite wall contact develops in the middle of the channel cross-section, and a discontinuous transition in reopening modes is observed as the bubble speed increases, which is analogous to a transition observed in elastic tubes (Heap et al., 2008).
For very collapsed membranes and high finger speed, the tip of the finger becomes flat, creating a leading edge which is subject to a smaller scale viscous fingering instability: constant-depth fingers, very similar to the stubby fingers observed in radial compliant Hele-Shaw cells (Pihler-Puzovic et al., 2012) and reminiscent of the printer’s instability (Rabaud et al., 1991) travel with the front. We investigate the steady shape of those fingers in terms of the speed of the front and the geometry of the reopening region.