Impact of freely falling liquid containers and subsequent jetting

Abstract

When a container, partially filled with liquid, is dropped from a certain height onto a floor, it will undergo a sudden deceleration followed by a rebound. At the moment of the container impact, the free-surface meniscus experiences large radial pressure gradients forming a high-velocity surface jet. We report experimental results and scaling analysis of the jet formation, showing that the jet initiation could also occur during the sudden deceleration phase. We show that the jet velocity scales as the geometric mean of the impact velocity and curvature-deformation velocity scale. We also report results for a second-jet originating from the tip of the evolving first-jet that resembles the tubular jets observed earlier in liquid entry to a pipe problem (Lorenceau et al. in Phys Fluids 14(6):1985–1992, 2002, Bergmann et al. in J Fluid Mech 600:19–43, 2008). We show that the second-jet follows a capillary velocity scale, unlike the tubular jet. The second-jet is caused by the collapse of an unstable cavity at the first-jet tip. The cavity radius follows an inertia-capillary scaling: ${r\sim ~(t_s-t)^{1/2}}$, where $t_s-t$ is the time to singularity.

Graphical Abstract

Appendices

Appendix

A second-jet formation in $D=20$ mm water

Fig. 20

Stages of first and second-jet formation in a container D = 20 mm, $h_l=10$ mm and h = 405 mm

The container under free fall just touches the Aluminium block in Fig. 20a. Note that the meniscus height ${\Delta h}_f$ is smaller than that in Fig. 4a which yields a smooth tapering jet (Type 1 jet, ${\Delta h}_f/R>1/2$). Formation of first-jet with a dimple at the tip (Type 2 jet, ${\Delta h}_f<1/2$), while the container still in contact with the Aluminium block, is shown as (a–e). For (f–j) the bottom part is moving away from the floor, coinciding with the formation of second-jet.

B Meniscus height just before impact ${\Delta h}_f$ data

Fig. 21

Representation of impact velocity, meniscus height pair ($U_i$, ${\Delta h}_f$)

The meniscus height ${\Delta h}_f$ values are measured independently for each impact experiment along with the impact velocity $U_i$. Each data point in Fig. 21 represents the impact velocity, meniscus height pair ($U_i$, ${\Delta h}_f$) which are the control parameters of this study for specific fluid and ambient conditions. Figure 21 provides ${\Delta h}_f$ values for Figs. 7, 9, 10.

C Comparison of jet velocity data

Fig. 22

Comparison of the jet velocity data (JSD) with previous researches. , Antkowiak et al. (2007b) (both glass and Aluminium bottom); , $D=14.2$ mm, measured from the normal jetting (non-cavitation) images of Kiyama et al. (2016); (with dot), $D=14.2$ mm, normal velocity data tabulated in Kiyama et al. (2016); , Onuki et al. (2018). See Table 3 for the remaining symbols

Figure 22 compares the first-jet velocity data from our experiments with the theoretical prediction of Gordillo et al. (2020), experimental data from Antkowiak et al. (2007b), Kiyama et al. (2016) and Onuki et al. (2018) presented in Gordillo et al. (2020). It is underlined that for all the previous researches, $V_\mathrm{rebound}$ is the independent parameter in the x-axis, wherein our experiments, the impact velocity $U_i$ is the independent parameter. It is worth noting that, of all the experimental results, only Onuki et al. (2018) data corresponds to “striker hitting the container bottom”-type experiments. The rest of them are all “container dropped from a height” experiments. The meniscus is approximated as an inverted spherical cap, with the contact angle as the control parameter in the theoretical formulation of Gordillo et al. (2020). The three parallel lines represent the theoretical prediction in three different contact angles, where $U_j$ is linearly varying with $V_\mathrm{rebound}$. The green line $--$ is the upper-velocity limit when the contact angle approaches zero, and the yellow line $-.-$ is in which the contact angle approaches $90^{\circ }$, the bottom limit of velocity. The contact angle for our smallest container ($D = 9$ mm), shortly before the impact, is around $30^{\circ }$, the same as that of Onuki et al. (2018), shown by a dotted line in Fig. 22.

The jet velocity data in Fig. 22 span two orders of magnitude. The lowest rebound velocity data are obtained by Antkowiak et al. (2007b) and Kiyama et al. (2016) from their container drop experiments. The data from the fine capillary for which the size is lesser than the capillary length by Onuki et al. (2018) match with the theory ($30^{\circ }$ contact angle) as was shown in Gordillo et al. (2020). We note that our data set $D=9$ mm ($\circ$) also shows the same trend. It is because of the relatively smaller diameter of the container, where the meniscus shape could possibly be approximated as an inverted spherical cap. However, the larger diameter ones ($D=$16 mm, 20 mm, 30 mm, 40 mm) do not necessarily follow the theory. In fact, the majority of data sets deviate from the theoretical trend. It is worth noting that our entire data approximately lie in the band (the area bordered with the upper and lower bounds of the theory). Interestingly, the container dropping experimental data from Antkowiak et al. (2007b) and Kiyama et al. (2016) stays apart by about an order of magnitude for the same rebound velocity. Furthermore, Antkowiak et al. (2007b)’s data vary from the theoretical trend in smaller rebound velocities ($V_\mathrm{rebound}<1$m/s) and Kiyama et al. (2016) recorded higher magnitudes of jet velocity than predicted by the theory.

To summarise, there are two main observations on jet velocity from Fig. 22. Firstly, as seen in our data, the jet velocity from larger containers ($D\geqslant 16$ mm) varies from the theoretical trend. A possible reason is the nonspherical shape of the meniscus in larger containers, exemplified by the two images presented as insets in Fig. 3. Secondly, as it appears, the jetting from the “container dropped from height”-type experiments is subtly different to their “striker hitting the container-bottom” counterparts. This is because for the container-drop type experiments, the meniscus effectively undergoes a sudden velocity change during the impact and subsequently then rebounds (separated by a time gap $t_d$). As mentioned in Sects. 1 and 2, $t_d$ is a sensitive parameter that decides JSD and JSA in container dropping from height type experiments. The significant data variation between Antkowiak et al. (2007b) and Kiyama et al. (2016) is possibly due to this complexity mentioned above intrinsic to such experiments, as suggestive from the $U_j=2(U_i+V_\mathrm{rebound})$ scaling of Kiyama et al. (2016). Note that our data, which are gathered from a wide range of container diameters and different fluids and oscillating meniscus (Sect. 3), cover a range of meniscus shapes and hence could indicate the extent of jet velocity spread due to various meniscus shapes. Since the jet velocity mismatch between Antkowiak et al. (2007b) and Kiyama et al. (2016) is larger than the jet velocity spread in our data, it is unlikely that the difference in their meniscus shapes causes the considerable difference in jet velocity.

D Critical criteria for second-jets

For the radial cavity to begin to collapse, the kinetic energy flux of the first-jet $(1/2)\rho \pi {r_0}^2 {U_j}^3$ should overcome the potential energy at a length scale $l_{2j}$, namely $~\rho \pi {r_0}^2 U_j g l_{2j}$. In other words $(1/2)~\rho \pi {r_0}^2 {U_j}^3 > \rho \pi {r_0}^2 U_j~ g~ l_{2j}$. We recall from Fig. 14 that $l_{2j}= d_0$ and hence the inequality yields ${{U_j}}/{\sqrt{g~d_0}}~>~\sqrt{2}.$

From the inset of Fig. 15, we note that $d_0\simeq R/2$. Based on Fig. 9, $U_I/U_i$ is taken as order one constant. These two relations, in conjunction with the scaling law of (3) and the inequality, lead us to the following criterion based on the container radius R, the impact velocity $U_i$ and assigning $\sqrt{1-k_1}\simeq 1/2$ as ${U_i}/{\sqrt{g~R}}~>~2.$ This is a sufficient condition to ensure that the kinetic energy flux from the first-jet is larger than the potential energy at the lengthscale $l_{2j}$. For a freely falling container, this criterion reduces to a ratio between the drop-height (h) and the container radius, ${h}/{R}~>~2.$

A viscosity-based criterion for second-jet formation is as follows. A comparison of the viscous dissipation rate $\mu \pi {r_0}^2 l_{2j} {U_j}^2/{r_0}^2$ of evolving first-jet to its kinetic energy flux yields the following inequality $({1}/{2})\rho \pi {r_0}^2 {U_j}^3 \gg \mu \pi {r_0}^2 l_{2j} {{U_j}^2}/{{r_0}^2}$, to prevent the suppression of first-jet before reaching the length $l_{2j}$. This inequality gives local (jet) Reynolds number condition $Re_j= {\rho U_j r_0}/{\mu } ~\gg ~{2~l_{2j}}/{r_0}~=~4$. A sufficient condition for the above relation but given in terms of the Reynolds number based on impact velocity and container radius is $Re_R={\rho U_i R}/{\mu }~\gg ~32$,

where we have used the scaling relation (3) with $\sqrt{1-k_1}\simeq 1/2$, dimple radius $r_0~\simeq ~R/4$ (see inset in Fig. 15) and approximating $U_I/U_i$ as a order one constant.