  Helped by the careful analysis of their experimental data, Worthington (1897)
described roughly the mechanism underlying the formation of high-speed jets
ejected after the impact of an axisymmetric solid on a liquid-air interface. In
this work we combine detailed boundary-integral simulations with analytical
modeling to describe the formation and break-up of such Worthington jets in two
common physical systems: the impact of a circular disc on a liquid surface and
the release of air bubbles from an underwater nozzle. We first show that the
jet base dynamics can be predicted for both systems using our earlier model in
Gekle, Gordillo, van der Meer and Lohse. Phys. Rev. Lett. 102 (2009).
Nevertheless, our main point here is to present a model which allows us to
accurately predict the shape of the entire jet. Good agreement with numerics
and some experimental data is found. Moreover, we find that, contrarily to the
capillary breakup of liquid cylinders in vacuum studied by Rayleigh, the
breakup of stretched liquid jets at high values of both Weber and Reynolds
numbers is not triggered by the growth of perturbations coming from an external
source of noise. Instead, the jet breaks up due to the capillary deceleration
of the liquid at the tip which produces a corrugation to the jet shape. This
perturbation, which is self-induced by the flow, will grow in time promoted by
a capillary mechanism. We are able to predict the exact shape evolution of
Worthington jets ejected after the impact of a solid object - including the
size of small droplets ejected from the tip due to a surface-tension driven
instability - using as the single input parameters the minimum radius of the
cavity and the flow field before the jet emerges.
