Nonlinear dynamics of viscous fingering

Abstract

Viscous fingering is an archetype of pattern formation underpinned by a classical hydrodynamic instability, which arises at the interface between a gas bubble displacing viscous fluid in the narrow gap between two parallel plates. Beyond onset, a radially expanding interface develops into highly-branched, continually evolving patterns of fingers. In contrast, an initially flat interface propagating in a rectilinear channel can grow into a steadily advancing symmetric finger, which may in turn destabilise beyond a threshold value of the driving parameter. We approach the complex pattern formation due to viscous fingering from a dynamical systems theory perspective, and discuss the propensity of viscous fingers to exhibit complex dynamics, where the system’s transient evolution corresponds to a meandering between weakly unstable states. We review classical results and examine recent evidence from studies of viscous fingering subjected to controlled perturbations through this lens, thereby providing an alternative perspective on this classical problem, and highlighting open questions.