Impulse-induced liquid jets from bubbles with arbitrary contact angles

TL;DR

This work develops an analytical pressure-impulse framework for impulsively generated jets from bubbles with arbitrary contact angles in axisymmetric containers. By solving a 3D Laplace problem in the small-bubble limit using toroidal coordinates and Legendre functions, the jet velocity is shown to decompose as $v(\theta)=v_f(\theta)+h\,v_g(\theta)$, where curvature focusing and submersion redistribution compete to determine the maximum jet speed. The authors provide a series-based semi-analytical solution for general $\theta$ and submerged configurations, and validate the theory with experiments that reveal a non-monotonic dependence of jet speed on bubble height, with the optimal geometry shifting with submersion depth. This framework informs geometry-controlled jet engineering for applications requiring high-speed, focused jets.

Abstract

This paper investigates the relationship between the contact angle of a spherical bubble attached to a tube submerged in a container and the jet speed induced by an impulsive acceleration at its base. While it has been well established that bubble geometry strongly influences the ejection speeds of liquid jets, mathematical studies of liquid jets with arbitrary bubble shapes remain limited. In this work, we derive a pressure impulse in the small-cavity limit as a tractable integral of classical Legendre functions. It is shown that the jet speed can be divided into two components: (i) the velocity induced by the hydrostatic pressure impulse distribution created by the curvature of the bubble, and (ii) the velocity induced by the distribution of the submersion of the tube in a container. This decomposition reveals that an optimal bubble curvature emerges only when the tube is submerged: the optimality is absent for non-submerged configurations, where the jet speed increases monotonically with bubble depth. Experiments confirm this non-monotonicity and quantitatively support the predicted shift of the optimal geometry with submersion depth.

Figures (12)

• Figure 1: A liquid jet created by an impact on the ground. At the initial time $t^*=0$, a falling container experiences an impact on its bottom, which produces a variation of the pressure impulse $\Pi^*$ in a cylindrical container. The bubble region on the tip of the tube is assumed to be spherical, which is consistent with some experimental results (see Onuki2018-jx). After the impact, a liquid jet is generated by the non-uniform distribution of the pressure impulse.
• Figure 2: (i) The non-dimensional boundary value problem for a harmonic pressure impulse $\Pi$ and (ii) the geometry $D_\infty$ in the small-bubble limit. When $\lambda\rightarrow 0$, the geometry is a semi-infinite half 3D space outside a spherical-shaped bubble, denoted by $D_\infty$. Note that the value $H$ is related to $\theta$ as $H = \tan(\theta/2)$. The impulse $\Pi$ is approximated well by $\Pi_\infty$ as $\lambda \rightarrow 0$.
• Figure 3: Toroidal coordinates $(\alpha,\beta)$ for solving the mixed boundary value problem for a pressure impulse $\Pi_\infty$. Cylindrical coordinates $(\xi,z)$ are mapped to the new coordinates $(\alpha,\beta)$ using the transformation in (\ref{['eq:change_xiz']}). The boundary of the bubble is expressed by $\beta=\pi+\theta$.
• Figure 4: Contour plots for the harmonic pressure impulse $\Pi_\infty(\xi,z)$ in (\ref{['eq:pressure_asymp1']}) and the velocity in (\ref{['eq:velocity1']}). (i) $H=0.4$ and (ii) $H=1.5$. The value $H$ is related to the angle $\theta$ as $H = \tan(\theta/2)$.
• Figure 5: Contour plots for the harmonic pressure impulse $\Pi_\infty(\xi,z)$ in (\ref{['eq:pressure_asymp1']}) and velocity in (\ref{['eq:velocity1']}) for $H=1.5$. Note that $\Pi_\infty(\pm 1,0)$ is discontinuous due to the boundary conditions. (i) $h=1.0$ and (ii) $h=2.5$. The cavity height is fixed at $H=1.5$.