Generation and Breakup of Worthington Jets After Cavity Collapse

Helped by the careful analysis of their experimental data, Worthington & Cole (1897, 1900) described roughly the mechanism underlying the formation of high-speed jets ejected after the impact of an axisymmetric solid on a liquid-air interface. They made the fundamental observation that the intensity of these sharp jets was intimately related to the formation of an axisymmetric air cavity in the wake of the impactor. 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. In our model, the flow structure inside the jet is divided into three different regions: The axial acceleration region, where the radial momentum of the incoming liquid is converted into axial momentum, the ballistic region, where fluid particles experience no further acceleration and move constantly with the velocity obtained at the end of the acceleration region and the jet tip region where the jet eventually breaks into droplets. 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 (1878), 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. Combining these three regions for the base, the jet, and the tip 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.