Oscillatory Fingering in an Elasto-Rigid Hele-Shaw Channel

We study the propagation of an air finger in a Hele-Shaw channel where the top boundary is an elastic membrane. The injection of air at a constant flow rate works against elastic, viscous and capillary forces to form an approximately steadily propagating blistering finger (Juel et al. 2018), which depends on the capillary number Ca -- the ratio of viscous to surface tension forces -- and the level of initial collapse of the channel cross-section. A stable Saffman–Taylor finger is formed when the membrane is initially flush with the side walls, and in the rigid channel this mode is linearly stable for all values of Ca. By contrast, increasing the initial deflection of the membrane leads to the broadening of the finger and the formation of small-scale fingering at its tip, which is advected around the finger as it advances. This form of propagation is oscillatory in the reference frame of the moving finger. We present evidence of a simple geometric mechanism for the switching between two modes of oscillatory fingering. One resembles the tip-splitting instabilities observed to occur subcritically in rigid channels above a critical value of Ca and the other is linked to stubby finger formation in tapered channels, such as in the famous tape-peeling study of McEwan & Taylor (1961).