Periodic dynamics in viscous fingering

Abstract

The displacement of a viscous liquid by air inside a Hele-Shaw channel is an archetype for nonlinear pattern-forming systems. Typically, bubbles or fingers of air propagate steadily at low values of the capillary number Ca = μ U * / σ , where μ is the viscosity of the liquid, U * is the speed of the bubble or finger and σ is the surface tension at the air–liquid interface. However, as the capillary number increases, the advancing interface can exhibit disordered pattern-forming dynamics. We demonstrate experimentally that a sustained, remote perturbation of the bubble tip can drive time-periodic dynamics. We exploit the propensity of a group of bubbles to propagate in steady spatial arrangements inside a Hele-Shaw channel with a centralized depth-reduction to apply a sustained pressure perturbation ahead of the bubble as it propagates. The oscillation cycle is initiated by the splitting of the bubble tip, via the Saffman–Taylor instability, which is promoted by the local flow-field conditions. The restoral of the bubble 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. The splitting and restoral mechanisms do not rely on the specific perturbation used in this study, suggesting a generic mechanism for periodic dynamics of propagating curved fronts subject to steady flow-field perturbations.