The axially symmetric, swirl-free gas dynamics and interlinked motion of a cork stopper provoked by the opening of a champagne bottle are modelled rigorously and studied numerically. The experimental study by Liger-Belair et al. (Science Advances, 5(9), 2019) animated the present investigation. Inspection analysis justifies the inviscid treatment of the expanding jet of air enriched with dissolved carbonic acid gas initially pressurised in the bottle. Solving of the resulting Euler equations is facilitated by the open-source software Clawpack. Specific enhancements allow for resolving of the emerging supersonic pockets, associated with surprisingly complex shock structures, as well as the gas–stopper interaction with due accuracy. Our experimental effort provided modelling of the frictional behaviour, constitutive law and reversible (de-)compression of the cork material. Initially, the gas expands inside the bottleneck yet sealed by the stopper, and is hence accelerated by the gas but decelerated by dry sliding friction. Once the stopper has passed the bottle opening, the jet rapidly assumes locally supersonic speed, where a complex shock pattern is detected. Special attention is paid to the formation and dissolution of one or even two Mach discs between the opening and the released stopper. This simulated dynamics is found to be in fairly good agreement with recent experimental findings. It also provides a first insight into the generation of the typical popping sound.

The capillary breakup of the high-speed Worthington jets ejected after a cavity collapse in water occurs due to the high-Reynolds-number version of the capillary end-pinching mechanism first described, in the creeping flow limit, by Stone & Leal (J. Fluid Mech., vol. 198, 1989, p. 399). Using potential flow numerical simulations and theory, we find that the resulting drop ejection process does not depend on external noise and can be described as a function of a single dimensionless parameter, WeS = ρ R30S20/σ, which expresses the ratio of the capillary time to the inverse of the local strain rate, S0. Here, ρ and σ indicate the liquid density and the interfacial tension coefficient, respectively, and R0 is the initial radius of the jet. Our physical arguments predict the dimensionless size of the drops to scale as Ddrop/R0 ~ We−1/7S and the dimensionless time to break up as TS0 ~ We2/7S. These theoretical predictions are in good agreement with the numerical results.

At the beginning of the last century Worthington and Cole discovered that the high-speed jets ejected after the impact of an axisymmetric solid on a liquid surface are intimately related to the formation and collapse of an air cavity created in the wake of the impactor. In this paper, we combine detailed boundary-integral simulations with analytical modelling to describe the formation of such Worthington jets after the impact of a circular disk on water. We extend our earlier model in Gekle et al. (Phys. Rev. Lett., vol. 102, 2009a, 034502), valid for describing only the jet base dynamics, to describe the whole jet. We find that the flow structure inside the jet may be divided into three different regions: the axial acceleration region, where the radial momentum of the incoming liquid is converted to 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. From our modelling of the ballistic region we conclude that, contrary to the case of other physical situations where high-speed jets are also ejected, the types of Worthington jets studied here cannot be described using the theory of hyperbolic jets of Longuet-Higgins (J. Fluid Mech., vol. 127, 1983, p. 103). Most importantly, we find that the velocity and the shape of the ejected jets can be well predicted at any instant in time with the only knowledge of quantities obtained before pinch-off occurs. This fact allows us to provide closed expressions for the jet velocity and the sizes of the ejected droplets as a function of the velocity and the size of the impactor. We show that our results are also applicable to Worthington jets emerging after the collapse of a bubble growing from an underwater nozzle, although this system creates thicker jets than the disk impact.