A generalised free-boundary lubrication framework for dynamic capillary flows and theoretical analysis of bubble production in the coflow system

Two-phase capillary flows in channels and pipes form a fundamental problem in fluid mechanics with widespread applications in physical, biological and microfluidic systems. We show that dynamic time-dependent phenomena in such systems can be described within a theoretical framework based on free-boundary lubrication theory combined with nonlinear representation of the interfacial curvature. The framework allows complex time-dependent capillary phenomena to be analysed using a relatively simple nonlinear hyperdiffusion equation with a moving boundary. We propose the time-dependent framework and demonstrate its application to coflow, where two fluids (one viscous, the other inviscid) are injected concurrently into a channel.

The model predicts an oscillatory growth and pinch-off of the inviscid phase, producing a train of bubbles at a frequency, size and spatial pattern controlled by the capillary number and flux ratio. We map a regime diagram, partitioned based on the formation of long (Taylor) bubbles versus smaller, approximately circular (non-Taylor) bubbles. An asymptotic theory of the former regime is developed, yielding an explicit theoretical prediction for the pinch-off frequency. The result is based on identifying a quasi-static regime of the necking region that is coupled via matching to an advancing dynamic contact line along the bubble film. The analysis reveals a link between the necking dynamics controlling confined bubble pinch-off and the time-dependent spreading dynamics of classical droplets. We propose the time-dependent nonlinear-curvature framework as a means to study dynamic capillary phenomena, providing a complement to experimental and full-Stokes simulation.