Dynamics of compressible displacement in a capillary tube

The displacement of viscous liquid in a confined geometry due to injected gas occurs in CO₂ sequestration, battery operation, and the reopening of pulmonary airways. Expelling liquid from a capillary tube by injecting air is perhaps the simplest example of such a flow, if not the simplest example of any two-phase flow. Here, however, we demonstrate that complex dynamical regimes arise in this system due to the compressibility of the injected air. In experiments, we drive silicone oil from the tube by compressing a connected reservoir of air at a fixed volumetric rate. We observe unsteady flow dynamics due to the interaction between spring-like air compressibility and diminishing viscous resistance in the draining oil slug. A simple mathematical model predicts these dynamics over a broad range of compression rates and air reservoir volumes. In the limit of large air reservoirs, the model predicts two distinct dynamical regimes, separated by a critical value of a dimensionless ‘compressibility’ number. The low- and high-compressibility regimes are associated, respectively, with quasi-steady and burst-like expulsion. Remarkably, we also observe these regimes in our experiments.