Time-periodic viscous fingering

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. In this talk, we demonstrate that this system naturally supports a host of time-periodic states, which are a fundamental building-block of disordered dynamics. The introduction of a bubble downstream is used to provide a sustained perturbation to the finger's tip and this can lead to its splitting and the subsequent advection of the tip-disturbance. The finger's continued propagation provides an in-built restoring mechanism because it broadens in order to reattain its preferred width set by the balance of viscous and capillary forces. The interplay between these two effects leads to a time-periodic tip instability and the finger deposits a spatially periodic pattern as it propagates. We show that this mechanism generalises to a range of sustained perturbations to the finger's front.