On the jets ejected after the inertial collapse of cavities

Motivated by the results in Gordillo and Blanco-Rodriguez, 'Bubble bursting jets are driven by the purely inertial collapse of gas cavities', \emph{Phys. Rev. Lett., Submitted} (2023) \cite{PRL2023}, where it is found that bubble bursting jets are driven by a purely inertial mechanism, here we present a study on the dynamics of the jets produced by the collapse of gas cavities of generic shape when the implosion is forced by a far field boundary condition expressing that the flow rate per unit length, $q_\infty$, remains constant in time. Making use of theory and of numerical simulations, we first analyze the case of a conical bubble with a half-opening angle $\beta$ when the value of $q_\infty$ is fixed to a constant, finding that this type of jets converge towards a purely inertial $\beta$-dependent self-similar solution of the equations in which the jet width and velocity are respectively given, in the limit $\beta\ll 1$, by $r_{jet}\approx 2.25\tan\beta\sqrt{q_\infty\tau}$ and $v_{jet}\approx 3 q_\infty/(2\tan\beta\sqrt{q_\infty\tau})$ respectively, with $\tau$ 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 $r_c$ our theory predicts that $r_{jet}\propto \left(2 r_c\right)^{-1/2}\left(q_\infty \tau\right)^{3/4}$ and $v_{jet}\propto q_\infty \left(2r_c\right)^{1/2} \left(q_\infty\tau\right)^{-3/4}$, a result which is also in good agreement with numerical simulations. The present results might 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.