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Predicting the size of droplets that pinch off from a liquid jet is important to applications ranging from bubble-initiated atmospheric aerosols to inkjet printing. These predictions are complicated by smaller satellite drops that form when a thin liquid thread develops and breaks up faster than its ends fully retract. Typically this process is modeled by perturbing a cylindrical liquid thread with an amplitude that is small relative to the cylinder diameter. Yet early on in the pinch-off process, the ends of the liquid thread are conical and lack a characteristic length scale from which to normalize a finite perturbation. Here we numerically simulate the retraction of nearly inviscid conical filaments and introduce self-similar perturbations to drive the system into a discretely self-similar retraction that can enable breakup without biasing a particular length scale. We find that for most cone angles, the perturbation amplitude must exceed a threshold for satellite drops to form. We show that this critical perturbation amplitude depends on the cone angle and can be accurately predicted by an argument based on the static stability of the initial perturbed cone.

Threshold for discretely self-similar satellite drop formation from a retracting liquid cone