3.1 General dynamics

The motion of a merged drop during a typical impact process is illustrated through a series of fluorescence images, shown in figure 2(a), which demonstrate several notable characteristics. Specifically, at the very early stage of the impact, when the drop starts penetrating into the stagnant liquid pool, a vortex ring is generated around the outer periphery of the drop due to the large shear rate (figure 2 a, $t=5~\text{ms}$ ), and becomes larger as the drop penetrates into the pool. As the rear end of the drop falls below the liquid surface, a crater is developed on the surface of the pool, which pushes the drop downwards until the crater on the liquid surface reaches a maximum depth and subsequently starts relaxing (figure 2 a, $t=20~\text{ms}$ ) to regain its original unperturbed state. The spread of the vortex ring then becomes smaller as the vortices approach each other and continue the penetration.

Figure 2.

(a) Sequence of high-speed LIF images depicting the penetration process. (b) Penetration history of one experiment ( $We=10.2$ , $D=1.9~\text{mm}$ ). (c) Penetration history for a series of experiments, $D\in [2,3,3.5~\text{mm}]$ , $We\in [8:37]$ .

Among the various dynamic and characteristic analyses that can be performed on this process, here we are mostly interested in the overall downward motion and penetration of the vortex ring. Specifically, the primary quantity of interest is the penetration depth, $h$ , which is the distance between the bottom-most point of the vortex ring, marked by the dye, and the unperturbed liquid surface, as defined in figure 2(a). The temporal evolution of $h$ is plotted for a typical experiment in figure 2(b), from which we can clearly identify three distinct stages of the penetration process. Here, $t=0$ is the instant when the drop touches the unperturbed liquid pool. At the early stage immediately after the impact, a steady increase of $h(t)$ with almost a constant slope is observed. The motion in this stage I, termed ‘Rapid penetration’, is primarily controlled by the impact inertia and as such the slope is almost constant. On the other hand, far from impact in stage III of ‘Deceleration’, we observe a monotonic penetration process with steady deceleration, marked by a continuous decay in the slope, caused by the viscous effect. Interestingly, at the intermediate or transitional period between stages I and III, the drop motion shows a non-monotonic penetration process, in that $h(t)$ first increases and then slightly decreases with time, exhibiting an oscillatory motion. Such a behaviour in this stage II, termed as ‘Non-monotonic propagation’, is caused by the crater on the pool surface created by the impact. Evolution of the penetration depth is compared for various $D$ and $We$ in figure 2(c), which does not show any similarity between different stages in different experiments, suggesting that the three stages are characterized by different length and time scales. We now explore the varying extent of influence of these three major effects, i.e. impact inertia, viscous drag and capillarity, in the drop motion in these stages and show how their influence can be scaled.

Figure 3.

(a) Inertial stage: normalized penetration history at early stage of penetration. (b) Schematic of the radial flow in liquid pool induced by the penetrating drop.

3.2 Inertial effect

Immediately after impact, penetration is dominated by the impact inertia of the drop. At this stage the drop kinetic energy is dominant, and the flow induced in the pool is mostly radial in nature with minimal dissipation. Recognizing that the Reynolds number of the drop is high and that the capillary disturbance of the pool surface is minimal, we expect viscous and capillary effects to be weak during this stage. Also, the Froude number ( $U/\sqrt{gD/2}$ ) for our experiments is greater than unity, showing a negligible effect of gravity. Thus, the penetration at this stage is controlled by the impact inertia. This is also evident from the linear nature of the penetration history in stage I, shown in a non-dimensional plot (figure 3 a), where $t$ is normalized by the inertial time scale, $\unicode[STIX]{x1D70F}_{i}$ , and $h(t)$ is normalized by the drop diameter, $D$ . In figure 3(a), we notice that for a wide range of $D$ and $We$ , the normalized penetration histories collapse to a linear fit until $t/\unicode[STIX]{x1D70F}_{i}\approx 1$ , beyond which they diverge, confirming the dominant role of impact inertia in stage I. Moreover, the slope of the normalized data is close to $1/2$ , which suggests that the penetration velocity, $U_{p}(=\text{d}h/\text{d}t)$ , defined as the velocity at which the coalesced drop front is moving inside the pool, is half of the impact velocity, $U_{p}\approx U/2$ . This loss in kinetic energy can be attributed to the motion induced in the pool, due to the drop impact. Using energy budget analyses for the unmerged drop, Tran et al. (Reference Tran, de Maleprade, Sun and Lohse2013) and Tang et al. (Reference Tang, Saha, Law and Sun2016) showed that the kinetic energy of the induced flow ( $E_{k,l}$ ) in the pool is approximately $3/4$ of the initial drop kinetic energy, $E_{k,0}=(1/12)\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}D^{3}U^{2}\sim U^{2}$ . Considering the drop kinetic energy after the impact is $E_{k,p}\sim U_{p}^{2}$ , the energy balance before and after the impact, i.e. $E_{k,0}=E_{k,p}+E_{k,l}$ , results in $E_{k,p}\approx (1/4)E_{k,0}$ or $U_{p}\approx (1/2)U$ . Note that this result can alternatively be obtained by following Wagner’s theory in the context of a drop impacting a shallow pool, as shown in Purvis & Smith (Reference Purvis and Smith2005). Finally, from our knowledge of the penetration speed for the inertial controlled stage I, the equation of motion in non-dimensional form can be expressed as

(3.1) $$\begin{eqnarray}\frac{h(t)}{D}=\frac{1}{2}\frac{t}{\unicode[STIX]{x1D70F}_{i}},\quad \text{for }0\leqslant \frac{t}{\unicode[STIX]{x1D70F}_{i}}\leqslant 1.\end{eqnarray}$$

3.3 Viscous drag

Figure 4.

(a) Normalized penetration velocity $\text{d}h_{p}/\text{d}t$ as a function of normalized time, $t/\unicode[STIX]{x1D70F}_{i}$ . Inset: zoomed view of stage III showing the deceleration. (b) Drop far from the pool surface: assuming spherical geometry and defined diameter ( $D_{s}$ ). (c) Normalized $D_{s}$ of the penetrating drop as a function of normalized time. $D_{s}$ becomes constant at the late stage of penetration. (d) $(h-h_{0})/h_{v}$ as a function of $\ln [(t-t_{0})/\unicode[STIX]{x1D70F}_{v}+1]$ for a wide range of $We$ and $D$ .

During the late stage of the penetration, when the vortex ring is sufficiently far away from the liquid surface, it continues to move down with a steady deceleration, as seen in stage-III of figure 2(b). This deceleration in the late stage is also clearly visible if we plot the instantaneous normalized penetration velocity for two typical experiments (figure 4 a). The penetration process in this stage almost resembles the motion of a sphere in a stagnant liquid, with the notable exception of the presence of a toroidal vortex ring (figure 4 b). However, it is interesting to note that $D_{s}$ , the virtual diameter of the sphere encompassing the vortex ring, remains almost constant at this stage, as shown for two different experiments in figure 4(c). Moreover, mixing of the dye with the surrounding liquid via the vortex is still not significant, and thus entrainment of the dye-free water from the pool is neglected here. The motion of the vortex ring, then, is modelled as a spherical mass moving through a stagnant liquid (figure 4 b), and we can write $m(\text{d}U_{p}/\text{d}t)=-F_{D}=-C_{D}A_{cs}(\unicode[STIX]{x1D70C}U_{p}^{2}/2)$ , where $m=(\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}D_{s}^{3}/6)$ is the mass of the fictitious sphere encompassing the vortex ring, $U_{p}=\text{d}h/\text{d}t$ the penetration velocity, $F_{D}$ the drag force, $C_{D}$ the drag coefficient and $A_{cs}=\unicode[STIX]{x03C0}D_{s}^{2}/4$ the cross-sectional area. By integrating the expression twice with the boundary conditions, $h(t=t_{0})=h_{0}$ and $U_{p}(t=t_{0})=U_{p,0}$ , we find

(3.2) $$\begin{eqnarray}\frac{h(t)-h_{0}}{h_{v}}=\ln \left[\frac{t-t_{0}}{\unicode[STIX]{x1D70F}_{v}}+1\right],\end{eqnarray}$$

where, $h_{v}=(4D_{s})/(3C_{D})$ and $\unicode[STIX]{x1D70F}_{v}=(4D_{s})/(3C_{D}U_{p,0})$ . It can be seen that $h_{v}({\sim}D_{s})$ and $\unicode[STIX]{x1D70F}_{v}({\sim}D_{s}/U_{p,0})$ are the effective length and time scales for this stage. In the current experiments, $\unicode[STIX]{x1D70F}_{v}$ ranges from 46.1 to 120.5 ms. We identify $t_{0}$ as the time when the penetration velocity $\text{d}h/\text{d}t$ attains a local maximum and the subsequent penetration process displays a monotonic decrease in $\text{d}h/\text{d}t$ , as shown in figure 4(a). Subsequently, the $t_{0}$ , $h_{0}$ and $U_{p,0}$ values are extracted from individual experiments. The effective Reynolds number ( $Re=\unicode[STIX]{x1D70C}U_{p}D_{s}/\unicode[STIX]{x1D707}$ ) at this stage is approximately $10^{3}{-}10^{4}$ , and for which the drag coefficient $C_{D}$ , for flow over a smooth sphere, has a value close to 0.5 (NASA 1999; Kundu & Cohen Reference Kundu and Cohen2008). Alternatively, a ‘best-fit’ value of $C_{D}$ for each condition can also be obtained by comparing the experimental data with the theoretical prediction in (3.2). These individual ‘best-fit’ $C_{D}$ values are also found to have values close to 0.5 (details in supplementary material available online at https://doi.org/10.1017/jfm.2019.503). Consequently, we use $C_{D}=0.5$ for all the experimental conditions and plot $(h-h_{0})/h_{v}$ as a function of $\ln [(t-t_{0})/\unicode[STIX]{x1D70F}_{v}+1]$ in figure 4(d), which shows very close resemblance with the scaling. It is noted that (3.2) represents an exact solution, which does not require any proportionality constant, as supported by the near-unity slope in figure 4(d). In passing, we further note that for the current experiments $(D_{s}g)/(2C_{D}U_{p}^{2})$ is less than unity, suggesting that the gravity effect is negligible compared to that of viscous drag.

3.4 Capillary effect

Capillary effect of the pool surface plays a critical role in the intermediate stage, which is marked by the non-monotonic changes in $h$ versus $t$ (figure 2 b). This oscillation is caused by the formation (and relaxation) of the crater on the pool surface (figure 2 a, $t>5~\text{ms}$ ) due to the impact. The crater eventually reaches a maximum depth due to the capillary effect, and then the highly distorted pool surface retracts to attain its original shape (figure 5 a). The formation and retraction of the crater induce an additional downward and upward motion respectively, in the pool, which periodically accelerates and decelerates the penetration of the vortex ring visible through oscillations in stage II, as seen in figure 2(b). The primary two parameters, which characterize the formation and relaxation of the pool surface, are the maximum crater depth $H_{md}$ and the time $t_{md}$ taken (after the initial contact between the drop and the pool surface) to create this maximum crater depth. In figure 5(b), we plot $t_{md}$ normalized by the capillary time scale of the pool surface, $\unicode[STIX]{x1D70F}_{c}$ , and it is found to be almost constant (figure 5 b) for all the experiments across different drop sizes and impact velocities. This confirms that the process is controlled by capillarity and the effects of inertia are minimal. On the other hand, if plotted with $We$ in figure 5(c), the non-dimensional $H_{md}$ , normalized by $D$ , displays a power law dependence with an exponent close to 0.5, i.e. $H_{md}/D\sim We^{1/2}$ (figure 5 c). Such a power law dependence can be explained by an energy balance assuming that the kinetic energy of the penetrating drop, $E_{k,p}=\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}D^{3}U_{p}^{2}/12$ , has been converted to the surface energy of the crater, $E_{s,H}$ , and we have already shown that $U_{p}=1/2U$ . Furthermore, we assume that the crater has a hemispherical shape, shown in figure 5(d), whose surface area is $A_{Hd}\approx 2\unicode[STIX]{x03C0}H_{md}^{2}$ . This leads to $E_{s,H}=\unicode[STIX]{x1D70E}A_{Hd}\approx 2\unicode[STIX]{x1D70E}\unicode[STIX]{x03C0}H_{md}^{2}$ , or $H_{md}/D\sim We^{1/2}$ , which is closely supported by the experimental data shown in figure 5(c). The scalings of $H_{md}$ and $t_{md}$ confirm that the crater formation process is capillarity driven. It is also noted that for the present experiments, the Bond number, $Bo=[(\unicode[STIX]{x1D70C}g)/(\unicode[STIX]{x1D70E}k^{2})]$ is less than unity, suggesting that the capillary wave is stronger than the gravity wave and as such the latter has been neglected.

Figure 5.

(a) Shadowgraph image: formation and relaxation of the crater. The dark region is the crater. The maximum crater depth, $H_{md}$ is also identified. (b) Non-dimensional crater formation time as a function of $We$ for three different drop diameters. (c) Non-dimensional maximum crater depth as a function of $We$ in log–log scale. (d) Schematic of the crater formation.

To evaluate the effect of crater formation on the penetration of the vortex ring, we first formulate the crater formation process, which can be expressed as a capillary wave moving through the surface of the pool. The beginning of crater formation is marked by the instant when the rear end of the drop (outside the pool), moving with velocity $U$ , reaches the unperturbed liquid surface, i.e. at $t=\unicode[STIX]{x1D70F}_{i}$ , which also represents the end of stage I. Since the crater formation is a capillary driven process, its effect on the penetration velocity of the drop can be expressed as $U_{c}=U_{p}\cos [\unicode[STIX]{x1D714}_{c}(t-\unicode[STIX]{x1D70F}_{i})]$ and its effect on the penetration depth as $h_{p,c}=(U_{p}/\unicode[STIX]{x1D714}_{c})\sin [\unicode[STIX]{x1D714}_{c}(t-\unicode[STIX]{x1D70F}_{i})]$ . The influence of the capillary wave, as manifested by the formation and relaxation of the crater, will be felt by the vortex ring, as long as it is in close vicinity to the pool surface. Although one can expect that the transition to the next stage will occur between $1/4$ of the capillary period when the capillary deformation is maximum, and $1/2$ of the capillary period when the pool surface reaches its initial state during the retraction process, the exact duration of the capillary effect on penetration, i.e. duration of stage II, is difficult to ascertain simply from scaling. However, as shown in figure 4(c), from experiments we can identify $t=t_{0}$ , beyond which the penetration velocity monotonically decreases due to a pure viscous effect without any effect of capillary oscillation in the pool surface. Thus, we use $t=t_{0}$ as the end of stage II, where the capillary effect on penetration is important. Although capillarity has a strong effect on the drop motion in stage II, the viscous effect is equally important, as evident from the increasing mean position of the drop during this stage (figure 2 b). It is noted that the relative contributions from the viscous and capillary effects to propagation during this stage are comparable (shown in supplementary materials), and hence the viscous term cannot be neglected. Recognizing that stage II includes both viscous and capillary effects, the equation of motion for this stage can be expressed as

(3.3) $$\begin{eqnarray}\frac{h(t)}{D}=\frac{1}{2}+\frac{\unicode[STIX]{x1D70F}_{c}}{\unicode[STIX]{x1D70F}_{i}}\frac{1}{4\unicode[STIX]{x03C0}}\sin \left[2\unicode[STIX]{x03C0}\frac{t-\unicode[STIX]{x1D70F}_{i}}{\unicode[STIX]{x1D70F}_{c}}\right]+\frac{h_{v}}{D}\ln \left[\frac{t-\unicode[STIX]{x1D70F}_{i}}{\unicode[STIX]{x1D70F}_{v}}+1\right].\end{eqnarray}$$