Periodic dynamics in viscous fingering

Abstract

The displacement of a viscous liquid by air in the narrow gap between two parallel plates - a Hele-Shaw channel - is an exemplar of complex pattern formation. Typically, bubbles or fingers of air propagate steadily at low values of the driving parameter. However, as the driving parameter increases, they can exhibit disordered pattern-forming dynamics. In this paper, we demonstrate experimentally that a remote perturbation of the bubble's tip can drive time-periodic bubble propagation: a fundamental building block of complex unsteady dynamics. We exploit the propensity of a group of bubbles to self-organise into a fixed spatial arrangement in a Hele-Shaw channel with a centralised depth-reduction in order to apply a sustained perturbation to a bubble's shape as it propagates. We find that the bubble with a perturbed shape begins to oscillate after the system undergoes a supercritical Hopf bifurcation upon variation of the tip perturbation and dimensionless flow rate. The oscillation cycle features the splitting of the bubble's tip and advection of the resulting finger-like protrusion along the bubble's length until it is absorbed by the bubble's advancing rear. The restoral of the bubble's tip follows naturally because the system is driven by a fixed flow rate and the perturbed bubble is attracted to the weakly unstable, steadily propagating state that is set by the ratio of imposed viscous and capillary forces. Our results suggest a generic mechanism for time-periodic dynamics of propagating curved fronts subject to a steady shape perturbation.