On the oscillatory dynamics of a Saffman–Taylor finger with a bubble at its tip

The complex behaviour of air–liquid interfaces driven into Hele-Shaw channels at high speeds could arise from oscillatory dynamics; yet both the physical and dynamical mechanisms that lead to interfacial oscillations remain unclear. We extend the experiments by Couder et al. (1986, Phys. Rev. A, vol. 34, 5175) to present a systematic investigation of the dynamics that results when a small air bubble is placed at the tip of a steadily propagating air finger in a horizontal Hele-Shaw channel. The system can exhibit steady and oscillatory behaviours, and we show that these different behaviours each occur in well-defined regions of the phase space defined by flow rate and bubble size. For sufficiently large flow rates, periodic finger oscillations give way to disordered dynamics characterised by an irregular meandering of the finger’s tip. At fixed flow rate, the oscillations commence when the bubble size is increased sufficiently that the decreased curvature of the bubble tip in the horizontal plane matches that of the finger tip. This causes the axial pressure gradient along the bubble to vanish, thus rendering the bubble susceptible to lateral perturbations. Differing time scales for finger and bubble restoral allow sustained oscillations to develop in the finger–bubble system. The oscillations cease when the bubble is sufficiently large that it can act as the tip of a single finger. The disordered dynamics at high flow rates are consistent with the transient exploration of unstable periodic states, which suggests that similar dynamics may underlie disorder in viscous fingering.