Jetting to dripping in compound liquid jets falling under gravity

In recent years, there has been a substantial growth in technologies, which exploits the disintegration of a compound thread of fluid to produce compound droplets or capsules. In many cases, careful control of the relevant operating and material parameters can determine a range of features, including capsule sizes, production rates, and wastage. In this paper, we investigate the transition between jetting and dripping of a compound inviscid liquid jet falling under gravity in a surrounding gas. We derive an analytical expression for the dispersion relation, which takes into account the non-uniform nature of the jet, which we then solve numerically utilizing the cusp map method and its significant reduction in computational effort required in identifying saddle points of the dispersion relation. Particular attention is paid to investigating the effects of the inner-to-outer surface tension ratio σ and initial jet radii, χ, as well as the influence of gravity on critical Weber numbers, We (which mark the transition between jetting and dripping). Our results provide the convective to absolute instability boundary for a number of different parameter values.