Theoretical analysis of injection-driven bubble pinch-off in capillary flows

A key challenge in the development of microfluidic devices is the control of the size and frequency of bubble formation. In a coflow geometry, the continuous phase flows through the channel and the gas phase is injected in parallel to the direction of the flow. Injection of both the gas and liquid phases at low flow rates results in bubbles that almost entirely fill the channel, with only a thin liquid film separating the bubble from the wall. These elongated (Taylor) bubbles have a large gas-liquid surface area which can increase the efficiency of chemical reactors involving heat or mass transfer.

The bubble formation process in the Taylor flow regime is controlled by the dynamics in the vicinity of the input channel, where the outer fluid squeezes the gaseous thread to produce a thinning neck. In this work, we give the first theoretical explanation of experimentally observed scaling laws for bubble length in microfluidic devices. We begin with an asymptotic analysis of Taylor bubble pinch-off in a planar geometry based on lubrication approximations, and explore the oscillatory pinch-off dynamics associated with the production of Taylor bubbles at a regular frequency. Numerical solutions are presented over a range of flow conditions, demonstrating an assortment of bubble characteristics, and compared to existing results where possible. We then progress to the analysis of Taylor bubble pinch-off in an axisymmetric capillary, which differs fundamentally from the planar analogue due to azimuthal curvature contributions. From our mathematical analysis, we provide the first theoretical explanation for the well-established experimentally observed scaling law, which states that the bubble pinch off time is inversely related to the input flux of the liquid phase.