Propagation of bubbles in collapsed elasto-rigid Hele-Shaw channels

We conduct a theoretical study of a two-phase-fluid-structure interaction problem in which air is driven at constant volume flux into a liquid-filled Hele-Shaw channel whose upper boundary is an elastic sheet. A depth-averaged model in the frame of reference of the advancing air-liquid interface is used to investigate the steady and unsteady interface propagation modes via numerical simulation. In slightly collapsed channels at small propagation speeds, the steadily-propagating interface adopts a shape that is similar to the classic Saffman-Taylor finger in rigid Hele-Shaw cells. As the level of initial collapse or the propagation speed increases, the morphology of the finger varies in increasing complexity. The complex solution structure contains steady solutions present in the rigid channel, such as fingers with a curved front or asymmetric, as well as solutions that only emerge with the inclusion of the elastic membrane, such as fingers with high curvature at the tip. The unsteady calculations reveal: noise-driven tip-splitting instabilities, highly ordered small-scale fingering, and periodic oscillatory fingers. We report good quantitative agreement between our model and the experimental data from Ducloué et al. (J. Fluid Mech. vol. 819, 2017, p 121) and C. Cuttle (presented in chapter 4), provided that corrections to account for the presence of liquid films on the upper and lower walls of the channel are included in the model. Finally, we investigate the relative importance of elastic and viscous-capillary stresses as the driving force of the finger propagation.