Global stability of axisymmetric flow focusing

In this paper, we analyse numerically the stability of the steady jetting regime of gaseous flow focusing. The base flows are calculated by solving the full Navier–Stokes equations and boundary conditions for a wide range of liquid viscosities and gas speeds. The axisymmetric modes characterizing the asymptotic stability of those flows are obtained from the linearized Navier–Stokes equations and boundary conditions. We determine the flow rates leading to marginally stable states, and compare them with the experimental values at the jetting-to-dripping transition. The theoretical predictions satisfactorily agree with the experimental results for large gas speeds. However, they do not capture the trend of the jetting-to-dripping transition curve for small gas velocities, and considerably underestimate the minimum flow rate in this case. To explain this discrepancy, the Navier–Stokes equations are integrated over time after introducing a small perturbation in the tapering liquid meniscus. There is a transient growth of the perturbation before the asymptotic exponential regime is reached, which leads to dripping. Our work shows that flow focusing stability can be explained in terms of the combination of asymptotic global stability and the system short-term response for large and small gas velocities, respectively.