Capillary waves control the ejection of bubble bursting jets

Abstract

Here we provide a theoretical framework describing the generation of the fast jet ejected vertically out of a liquid when a bubble, resting on a liquid–gas interface, bursts. The self-consistent physical mechanism presented here explains the emergence of the liquid jet as a consequence of the collapse of the gas cavity driven by the low capillary pressures that appear suddenly around its base when the cap, the thin film separating the bubble from the ambient gas, pinches. The resulting pressure gradient deforms the bubble which, at the moment of jet ejection, adopts the shape of a truncated cone. The dynamics near the lower base of the cone, and thus the jet ejection process, is determined by the wavelength $\unicode[STIX]{x1D706}^{\ast }$ of the smallest capillary wave created during the coalescence of the bubble with the atmosphere which is not attenuated by viscosity. The minimum radius at the lower base of the cone decreases, and hence the capillary suction and the associated radial velocities increase, with the wavelength $\unicode[STIX]{x1D706}^{\ast }$. We show that $\unicode[STIX]{x1D706}^{\ast }$ increases with viscosity as $\unicode[STIX]{x1D706}^{\ast }\propto Oh^{1/2}$ for $Oh\lesssim O(0.01)$, with $Oh=\unicode[STIX]{x1D707}/\sqrt{\unicode[STIX]{x1D70C}R\unicode[STIX]{x1D70E}}$ the Ohnesorge number, $R$ the bubble radius and $\unicode[STIX]{x1D70C}$, $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70E}$ indicating respectively the liquid density, viscosity and interfacial tension coefficient. The velocity of the extremely fast and thin jet can be calculated as the flow generated by a continuous line of sinks extending along the axis of symmetry a distance proportional to $\unicode[STIX]{x1D706}^{\ast }$. We find that the jet velocity increases with the Ohnesorge number and reaches a maximum for $Oh=Oh_{c}$, the value for which the crest of the capillary wave reaches the vertex of the cone, and which depends on the Bond number $Bo=\unicode[STIX]{x1D70C}gR^{2}/\unicode[STIX]{x1D70E}$. For $Oh>Oh_{c}$, the jet is ejected after a bubble is pinched off; in this regime, viscosity delays the formation of the jet, which is thereafter emitted at a velocity which is inversely proportional to the liquid viscosity.

Jetting in finite-amplitude, free, capillary-gravity waves

“…(2002)). A predictive model of jetting recently proposed in Gordillo & Rodríguez-Rodríguez (2019) and validated through direct numerical simulations in Blanco-Rodríguez & Gordillo (2020) supports this conclusion, where the jet velocity increases as before reaching a peak at a critical and subsequently decreasing as . These interesting predictions need to be tested on the jets reported here using a nonlinear viscous IVP solution.…”

“…It was first shown by Duchemin et al (2002) that the jet velocity has a non-monotonic dependence on viscosity, increasing initially with increasing viscosity but showing a peak at a critical value of viscosity (see figure 12 in Duchemin et al (2002)). A predictive model of jetting recently proposed in Gordillo & Rodríguez-Rodríguez (2019) and validated through direct numerical simulations in Blanco-Rodríguez & Gordillo (2020) supports this conclusion, where the jet velocity increases as Oh 1/2 before reaching a peak at a critical Oh c and subsequently decreasing as Oh −1 . These interesting predictions need to be tested on the jets reported here using a nonlinear viscous IVP solution.…”

Multitude of dimple shapes can produce singular jets during the collapse of immiscible drop-impact craters

“…This is an order of magnitude faster than predicted for the bursting bubbles (Deike et al. 2018; Gañán-Calvo 2018; Gordillo & Rodríguez-Rodríguez 2019). The jet diameters of are also two orders of magnitude thinner than those predicted by the bursting-bubble theory (Gañán-Calvo 2018), see also the Comment of Gordillo & Rodríguez-Rodríguez (2018), who question the underlying scaling of this theory.…”

“…What is the role of capillary waves in setting up the dimple for the inertial focusing? For the singular jets the dimple dynamics has until recently been formulated in the self-similar capillary-inertial formalism (Zeff et al 2000;Duchemin et al 2002;, while the final cylindrical collapse has most recently been shown to follow pure inertial focusing (Thoroddsen et al 2018;Gordillo & Rodríguez-Rodríguez 2019). One can therefore expect a dynamical transition in the vicinity of the final jet formation.…”

Continuous formation of microbubbles during partial coalescence of bubbles from a submerged capillary nozzle

“…The formed liquid jet and the surface waves accelerates the departure of the mother bubble and the next coalescence, leading to more microbubbles being generated in one father‐bubble cycle. The formation mechanism of the fast liquid jet has been investigated and explained in previous studies of bubble bursting at a flat gas–liquid interface, represented by the coalescence of a bubble with an infinitely large bubble (flat interface) 27‐29 . The sudden change in capillary pressure inside the bubble just after coalescence or bursting deforms its base to adopt a truncated cone shape, as shown in Figure 3c at t = 0.25 ms. As the convergence of the last capillary wave, the liquid flow in the vicinity of the cone induces the fast jet after the generation of microbubble, as shown in Figure 3c at t = 0.45 ms.…”