Theory of the jets ejected after the inertial collapse of cavities with applications to bubble bursting jets

Abstract

The dynamics of the axisymmetric jets originated from the bursting of bubbles in a liquid of density ρ, viscosity μ, and interfacial tension coefficient σ can be rationalized as a two-stage process in which, initially, the pressure jump ∼σ/Rb accelerates the liquid towards the axis of symmetry, inducing a far-field flow rate per unit length Q∞VcRb, with Rb and Vc=σ/(ρRb) indicating the radius of the bubble and the capillary velocity, respectively. The second stage, during which a fast jet of radius Rjet(T) Rb and velocity Vjet(T)≫Vc is ejected, is driven by the far-field radial velocity field established initially, which forces the collapse of the cavity walls while keeping Q∞ practically constant in time because liquid inertia and mass conservation prevent appreciable changes of this quantity during the very short timescale characterizing the ejection of the jet. Our theoretical predictions for Rjet(T) and Vjet(T) reproduce fairly well the time evolution of the jet width and of the jet velocity for over three decades in time, obtaining good agreement with numerical simulations from the instant of jet inception until Rjet∼Rb. The analytical expressions for the jet width and for the jet velocity provided here constitute the initial conditions for the explicit solution of the ballistic equations deduced in Gekle and Gordillo [J. Fluid Mech. 663, 293 (2010)0022-112010.1017/S0022112010003526], which, hence, can be straightforwardly used in order to quantify the size and velocity of the first drop ejected and the fluxes of mass, momentum, and energy transferred from the ocean into the atmosphere. In addition, motivated by the results obtained for the particular case of bubble bursting jets, we also present here a unified theoretical framework aimed at quantifying the dynamics of the type of generic jets produced by the collapse of axisymmetric gas cavities of arbitrary shape when their implosion is forced by the radial velocity induced by a far-field boundary condition expressing that the dimensionless liquid flow rate per unit length directed towards the axis of symmetry, q∞, remains constant in time. Making use of theory and of full numerical simulations, we first analyze the case of the collapse of a conical bubble with a half-opening angle β finding that, when the value of q∞ is fixed to a constant, this type of axisymmetric jets converge towards a purely inertial β-dependent self-similar solution of the inviscid Navier-Stokes equations, described here for the first time, which is characterized by the fact that the jet width and velocity are respectively given, in the limit β 1, by rjet≈2.25tanβq∞τ and vjet≈3q∞/(2tanβq∞τ), with τ indicating the dimensionless time after the jet is ejected. For the case of parabolic cavities with a dimensionless radius of curvature at the plane of symmetry rc, our theory predicts that rjet(2rc)-1/2(q∞τ)3/4 and vjetq∞(2rc)1/2(q∞τ)-3/4, a result which is also in good agreement with full numerical simulations. The present results might also find applications in the description of the very fast jets, with velocities reaching up to 1000 m s-1, produced after a bubble cavitates very close to a wall and in the quantification of the so-called bazooka effect.