Theory of Taylor bubble pinch-off in planar coflow

Two-fluid capillary flows in channels and pipes form a fundamental problem in fluid mechanics with widespread applications in physical, biological and microfluidic systems. A key phenomenon in such flows is the generation of long capsular bubbles, known as Taylor bubbles. A simple configuration that forms Taylor bubbles is coflow, where two fluids (one viscous, the other inviscid) are injected concurrently into a pipe or channel, producing a periodic pinch-off of inviscid bubbles at a regular frequency. By considering a lubrication theory for the necking region between the bubble injection nozzle and the developing Taylor bubble, we conduct a mathematical study of the formation of two-dimensional Taylor bubbles, yielding analytical insight into injection-driven pinch-off dynamics and the parametric control of its frequency. A key result is to demonstrate a quasi-static limiting theory of the necking dynamics for small capillary numbers. The theory is based on coupling an outer near-static regime of the pinching film with an apparent dynamic contact line that matches the necking zone to the film surrounding the Taylor bubble. This asymptotic structure, which becomes more refined in the limit of small capillary number, reveals a fundamental link between capillary pinch-off dynamics and the quasi-static regimes underlying spreading droplets. The theory yields a prediction for the dimensionless pinch-off time, t Subscript asterisk Baseline proportional to d Superscript 7 divided by 3 Baseline divided by q Subscript upper F Superscript 4 divided by 3 t ∗ ∝ d 7 / 3 / q F 4 / 3 t_* ∝ d7/3 / qF4/3 , where q Subscript upper F q F qF represents the flux of viscous fluid and d d d the size of the capillary. The study thereby yields new mathematical understanding of flow regimes that underlie bubble generation in capillary systems from first principles, and the first step towards application of quasi-static analysis to more complex axisymmetric and three-dimensional examples of capillary pinch-off.