Pattern formation due to viscous fingering exhibits a fascinating range of complex dynamics. For example, when air displaces a viscous liquid in the narrow gap between two parallel plates – a Hele-Shaw channel – the resulting steadily propagating finger of air can undergo a relatively abrupt transition to disordered front propagation. This motivates an exploration of the system’s nonlinear dynamics, which in turn requires the finite-amplitude perturbation of the finger. In the search for a suitable perturbation, we were inspired by Couder et al.’s [1] seminal work on viscous fingering, where he obtained narrow steadily propagating fingers when a small bubble was aggregated at the tip of the finger. This is because the bubble modifies the curvature of the finger’s tip and, thus, changes the finger’s shape. We have explored the dynamics of this system as a function of the driving flow rate and the size of the bubble at the tip. Fig. 1a shows that there are two regions of steady propagation of the finger (blue and green) separated by a red region where the propagation is unsteady. We explore the steady to oscillatory transitions and interpret the dynamics based on competing physical mechanisms. For modest flow rates, the finger oscillates transversely as described by Couder et al. [1], but as the flow rate increases, the dynamics become disordered (Fig. 1b). For high flow rates, the finger appears to meander randomly about the channel’s centreline for a range bubble sizes which may indicate a transient exploration of weakly unstable states.

References

[1] Y. Couder, N. Gérard and M. Rabaud, Narrow fingers in the Saffman–Taylor instability, Phys. Rev. A 34, 1986.