Front-propagating systems provide some of the most fundamental physical examples of interfacial instability and pattern formation. However, their nonlinear dynamics are rarely addressed. Here, we present an experimental study of air displacing a viscous fluid within a collapsed, compliant channel -a model system for pulmonary airway reopening. We show that compliance induces fingering instabilities absent in the rigid channel and we present the first experimental observations of the counter-intuitive 'pushing' behaviour previously predicted numerically, for which a reduction in air pressure results in faster flow. We find that pushing modes are unstable and moreover, that the dynamics of the air-fluid front involves a host of transient finger shapes over a significant range of experimental parameters.

Abstract

We examine the interaction between springlike compression of gas and viscous displacement of liquid by studying the steady compression of an air reservoir connected to a liquid-filled capillary tube.Our experiments and modelling reveal complex displacement dynamics that, for large air reservoirs, depend on a single dimensionless compressibility number. We identify two distinct displacement regimes, separated by a critical value of the compressibility number. The high-compressibility regime exhibits burst-like expulsion with applications to fluid, mechanical, or electrical systems.

No abstract

The displacement of a viscous liquid by a gas within a Hele-Shaw cell is a classical problem.The gas-liquid interface is hydrodynamically unstable, forming striking finger-like patterns that have been the subject of intense focus. Generally, both the gas and liquid phases are taken to be incompressible, with the capillary number being the key parameter that determines the severity of the instability. Here, we consider a radially outward displacement driven by the steady compression of a gas reservoir. The associated gas injection rate is then unsteady due to the compressibility of the gas. We identify a second nondimensional parameter, the compressibility number, that plays a strong role in the development of the fingering pattern. We use an axisymmetric model to study the impact of compressibility number on the unsteady evolution of injection rate and gas pressure. We use linear stability analysis to show that increasing the compressibility number delays the onset of finger development relative to the corresponding incompressible case. Finally, we present and compare a series of experiments and fully nonlinear simulations over a broad range of capillary and compressibility numbers. These results show that increasing the compressibility number systematically decreases the severity of the fingering pattern at high capillary number. Our results provide an unprecedented comparison of experiments with simulations for viscous fingering, a comprehensive understanding of the role of compressibility in unstable gas-liquid displacement flows, and insight into a new mechanism for controlling the development of fingering patterns.