3.2.1. Uniform acceleration and self-similar interface evolution at $\lambda =1$

In this section, we study the motion of the inner droplet at the early stage of drop impact and interface evolution at the intermediate stage for compound drops with equal density ($\lambda =1$). The interface of an impacting drop can be represented by the vertical component of its position vector $z=h(r, t)$. To examine the motion of the inner droplet, we track the vertical positions of its centroid (denoted by $h_C$), as well as the top and bottom of the inner droplet at the symmetry axis (denoted by $h_T$ and $h_B$, respectively; see the first image in figure 3c). The temporal evolutions of $h_T$, $h_B$ and $h_C$ are shown in figure 4 for an impacting compound drop at $We=500$, $\alpha =0.064$, $\lambda =1$ and various $\alpha$ ($=0.027$, 0.064 and 0.125). We find that the inner droplet experiences a uniform acceleration $a$ at the early stage, in the sense that $h_C$ can be approximated by

(3.1)\begin{equation} h(0, t)=h(0, 0)-t+\tfrac{1}{2}a_0 t^2,\end{equation}

where $h(0, 0)$ is the initial position and $a_0=aD/V^2$ denotes the dimensionless acceleration. Moreover, the trajectories of $h_C$ are the same for three different $\alpha$ (so as to give a constant $a_{0, C}=1.5$ for $0\le t\le 0.4$ by data fitting), which suggests that the motion of the inner droplet does not depend on $\alpha$ at the early stage of drop impact.

Figure 4.

Motion of the inner droplet in terms of time variation of $h_C$, $h_T$ and $h_B$, with $We=500$, $\lambda =1$ and various $\alpha$: (a) $\alpha =0.064$, (b) $\alpha =0.027$ and (c) $\alpha =0.125$. The dashed lines represent the prediction of (3.1), and the solid lines correspond to the prediction of (3.5). (d) Evolution of the apex of the compound droplets, where the dashed line represents the prediction of (3.1) with zero acceleration.

Interestingly, the time variations of $h_T$ and $h_B$ in figure 4(a–c) follow a manner very similar to $h_C$, and can also be described by (3.1). For example, in the case of $\alpha =0.064$, we can obtain $h_T|_{t=0}=0.7$ and $h_B|_{t=0}=0.3$ from the initial set-up, and the acceleration at the top of the inner droplet $a_{0, T}=0.83$ and that at the bottom $a_{0, B}=3.5$ by data fitting; here, the duration of acceleration depends on the vertical location, and $0\le t\le 0.5$ at the top and $0\le t \le 0.2$ at the bottom of the inner droplet. That is, the acceleration at the bottom of the inner droplet is much higher than its top, which accounts for its gradual flattening with time. The difference between $a_{0, T}$ and $a_{0, B}$ is found to increase with $\alpha$, e.g. $a_{0, B}=4.8$ and $a_{0, T}=0.75$ for $\alpha =0.125$ while $a_{0, B}=2.8$ and $a_{0, T}=0.95$ for $\alpha =0.027$. Such an acceleration process is not contradictory with zero acceleration at the drop apex at the early stage of impact of pure drops (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010). Figure 4(d) shows the trajectories of drop apex of the impacting compound drops for $\alpha =0.027$, 0.064 and 0.125. We can see that all the trajectories generally overlap and, moreover, have nearly zero acceleration until $t\approx 0.2$, at which time the pressure gradient arising from drop impact starts to affect the drop apex; see e.g. figure 5.

Figure 5.

Time variation of pressure contours (black thin lines) of an impacting compound drop (blue thick lines) with $We = 500$, $\alpha = 0.064$ and $\lambda = 1.0$ at times $t=0.05$, $0.10$, $0.15$, $0.20$, $0.25$, $0.30$.

The motion of the inner droplet is closely related to the pressure distribution inside the compound drop. Figure 5 shows the pressure contours at the early stage of drop impact (from $t=0.05$ to $0.30$), with $\lambda =1$, $We=500$ and $\alpha =0.064$. Firstly, once the drop comes into contact with the substrate, we can see that a high pressure zone is instantaneously established in the vicinity of the impact region, and the inner droplet is immediately surrounded by vertical pressure gradients. After this, the high pressure zone expands with the increase of the wetted area on the substrate, while the magnitude of the highest pressure gradually decreases with time. In principle, at the early stage of drop impact, the process of an impacting compound drop with high impact speed and equal density and viscosity (i.e. $\lambda =1$, $\mu _2=\mu _1$, $We\gg 1$ and $Re\gg 1$), is similar to that of an impacting pure drop, owing to the dominance of drop inertia over the surface tension of the inner droplet. Consequently, the interface of the inner drop can be viewed as a flow tracer, until the effect of its surface tension cannot be neglected anymore (e.g. when the curvatures at the sides of the flattened inner droplet become high). This explains why the trajectories of $h_C$ and the drop apex are independent of $\alpha$. Secondly, the flow inside the compound drop is decelerated vertically by the pressure gradients, and is then redirected near the substrate to the spreading (see also figure 3). We note that the pressure gradient varies with the vertical position. More precisely, the closer the position relative to the substrate, the stronger the pressure gradient and, therefore, the larger the acceleration of the flow. In addition, because the maximum pressure decreases with time, the pressure distribution around the inner droplet does not significantly change with the droplet descending, leading to the more or less uniform acceleration of the inner droplet. Thirdly, as the drop apex is away from the substrate, it is hardly affected by the pressure gradient at the very beginning of drop impact.

In the intermediate stage, the flow inside the drop is no more driven by the high pressure induced by the impact, and can be approximated by an axisymmetric stagnation point flow (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010). As a result, the shape evolution of an impacting pure drop would enter a self-similar inertial regime thereafter (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010; Lagubeau et al. Reference Lagubeau, Fontelos, Josserand, Maurel, Pagneux and Petitjeans2012; Gordillo, Sun & Cheng Reference Gordillo, Sun and Cheng2018; Mitchell et al. Reference Mitchell, Klewicki, Korkolis and Kinsey2019). Because the inner droplet is enclosed by the outer liquid of the compound drop, the self-similar evolution of the inner interface is supposed to start at the same time as the establishment of the stagnation point flow inside the impacting compound drop.

If the flow in the compound drop evolves into stagnation point flow, it can be expressed in the axisymmetric framework by

(3.2a,b)\begin{equation} v_r = \frac{r}{\tau} \quad \text{and}\quad v_z ={-} \frac{2z}{\tau}, \end{equation}

where $\tau$ denotes the time of the stagnation point flow, $v_r$ and $v_z$ are the radial and vertical velocity components, respectively. The subsequent interface evolution follows $Dh/D\tau =0$, or

(3.3)\begin{equation} \frac{\partial h}{\partial \tau}+ v_r\frac{\partial h}{\partial r} = v_z.\end{equation}

Equation (3.3) has the similarity solution

(3.4)\begin{equation} h(r,\tau) = \frac{1}{\tau^2} H\left(\frac{r}{\tau}\right),\end{equation}

which is valid for any function $H$ (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010). In other words, if the deformed interface has a scale of $l$ in the $z$ direction, (3.4) suggests that its scale in the $r$ direction should be of the order of $l^{-1/2}$.

Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) reported that the stagnation point flow might start from $t\approx 0.5$ for an impacting pure drop. We check whether this is also true for the impacting compound drop by examining the time variation of $h_T$ and $h_B$. Note that $\tau$ is related to the physical time $t$ by $\tau =t+t_0$, in which $t_0$ is a case-dependent constant. To determine $t_0$ in a specific flow problem, we need to take account of the condition of $v_z$ in (3.2a,b). From the numerical results at $t=0.5$, we get $h_T=0.307$ and $v_z=-0.52$ at the top of the inner interface, which suggests $t_0=0.680$. In addition, we can also obtain $h_B=0.092$ and $v_z=-0.16$ at the bottom of the inner interface at $t=0.5$, which gives $t_0=0.644$. Because the two $t_0$ only have a difference of 5 % in magnitude, it implies that the flow in the compound drop at $t=0.5$ is very close to stagnation point flow.

Suppose that the flow enters the regime of stagnation point flow at $\tau =\tau _0$, it is easy to obtain from (3.4) that the subsequent time variation of $h_T$ (and $h_B$) should obey

(3.5)\begin{equation} \frac{h(0, \tau)}{h(0, \tau_0)}=\left(\frac{\tau_0}{\tau}\right)^ 2. \end{equation}

We plot in figure 4 the prediction of $h_C$, $h_T$ and $h_B$ by (3.5), with $\tau _0=0.5+t_0$ and $t_0=0.662$ (i.e. the averaged value of the two $t_0$). We can see that the theoretical prediction agrees well with the numerical results for $t\in [0.5, 1]$. Furthermore, it is also interesting to see in figure 4(a) that the self-similar solution of $h_C$ can be extended to as early as $t=0.24$, while it is $t=0.16$ for $h_B$ and $t=0.44$ for $h_T$. This observation suggests that the lower part of flow enters the self-similar inertial regime at an earlier time.

Figure 6(a) shows the snapshots of the inner interface in the stagnation point flow at $We=500$, $\lambda =1$ and $\alpha =0.064$ (specifically from time $t=0.5$ to $0.9$, with a time interval of $0.1$). To check whether the interface evolution is self-similar, we firstly choose $l=h_T-h_B$ as the characteristic length of the deformed inner droplet in the $z$ direction. If the inner interface undergoes self-similar evolution, it would have $1/\sqrt {h_T-h_B}$ as its characteristic length in the $r$ direction. Then, we rescale the corresponding droplet shapes by the self-similar scales, i.e. ($r\sqrt {h_T-h_B}$, $(z-h_B)/(h_T-h_B)$), and the results are shown in figure 6(b). We can see that the inner interface at different times appears to overlap in the self-similar coordinate, except the region near its spreading front. The deviation from the self-similar solution at the spreading front is reasonably expected at the late times ($t=0.8$ and $0.9$). Because the local curvature progressively increases with time (also see figure 6a), the effect of surface tension on the drop shape cannot be neglected at the late times.

Figure 6.

(a) The superimposed shapes of the inner droplet at different times at $We=500$, $\lambda =1$ and $\alpha =0.064$. (b) Illustration of self-similar evolution of the inner droplet, in which the droplet shapes are rescaled by the self-similar length scales.

3.2.2. Density effect on the motion of inner droplet

In order to assess the effect of density ratio on the dynamics of the inner droplet, we plot the time evolution of $h_C$ at $We=500$, $\alpha =0.064$ and various $\lambda (=0.25, 0.5, 1, 2, 4)$ in figure 7. We can see that, the lower the density of the inner droplet, the higher the position of $h_C$. This indicates that the inner droplets encounter different accelerations, of which the magnitude increases monotonically with $\lambda$. The density difference also affect the pressure distribution inside the compound drop. Figure 8 shows the pressure contours of impacting compound drops at the early stage (more precisely, $t=0.15$) with $We = 500$, $\alpha = 0.064$ and various $\lambda$ ($=0.1$, $1.0$, $4.0$). We can see that the maximum pressure increases with $\lambda$, because initial momentum of the inner droplet also increases with $\lambda$ and a denser droplet is harder to stop. Similar to the cases with $\lambda =1$ in § 3.2.1, a uniform acceleration process is expected for the inner droplet with $\lambda \ne 1$ at the early stage of drop impact. If so, the trajectory of the inner droplet can also be described by (3.1), by replacing $a_0$ by a $\lambda$-dependent acceleration $a_\lambda$.

Figure 7.

Evolution of $h_C$ at $We = 500$, $\alpha =0.064$ and various $\lambda$. The dashed lines correspond to the prediction of (3.1) using the acceleration $a_\lambda$ calculated by (3.7).

Figure 8.

Pressure contours (lines with numbers) of impacting compound drops (thick lines) with $We = 500$, $\alpha = 0.064$ and various $\lambda =0.1$, $1.0$, $4.0$ at time $t=0.15$.

Because for small $\alpha$ the inner and outer interfaces of the impacting compound drops evolve independently at the early stage of impact, the motion of the inner droplet can be modelled as a droplet subject to an impulsive acceleration $a_0$ of the surrounding fluid in an open space. Without loss of generality, we use the Atwood number $A=(\rho _1-\rho _2)/(\rho _1+\rho _2)=(1-\lambda )/(1+\lambda )$ to represent the effect of $\lambda$ on the acceleration $a_\lambda$. A linear approximation of $a_\lambda$ as a function of $A$ can be obtained by Taylor expansion around $A=0$ (or equivalently $\lambda =1$),

(3.6)\begin{equation} a_\lambda=a_0+\frac{\partial a_\lambda}{\partial A} A + O(A^2). \end{equation}

At the early stage of drop impact, the buoyancy force and the effect of added mass play a major role in the acceleration process (and both are a linear function of density), while the viscous force exerted by the surrounding fluid can be neglected. It is reasonable to assume that the quadratic term on the right-hand side of (3.6) is negligibly small.

To determine the value of $({1}/{a_0}) ({\partial a_\lambda }/{\partial A})$, we check $a_\lambda /a_0$ under two extreme conditions, i.e. $A=-1$ and 1 (corresponding to $\lambda \rightarrow \infty$ and $\lambda \rightarrow 0$, respectively). When $A \rightarrow -1$, it is straightforward to have $a_\lambda /a_0\rightarrow 0$. The case of $A \rightarrow 1$ corresponds to a gas bubble immersed in an accelerated dense liquid. We assume that the bubble is more or less spherical, so that the added mass can be approximated by $\frac {1}{2}\rho _1\varOmega$, where $\varOmega$ denotes the volume of the bubble. Taking the added mass and the buoyancy force ($\approx \rho _1\varOmega a_0$) into account, we can get $a_\lambda /a_0 \rightarrow 2$ for $A \rightarrow 1$. It is noteworthy that we obtain $({1}/{a_0}) ({\partial a_\lambda }/{\partial A})\approx 1$ from either $A \rightarrow -1$ or $A\rightarrow 1$. This fact is further confirmation that the acceleration of the inner droplet at the early stage of drop impact theoretically follows

(3.7)\begin{equation} a_\lambda=a_0(1+A).\end{equation}

Equation (3.7) can be verified by comparing the prediction of $h_C$ by (3.1) against the numerical results. Figure 9 shows the evolution of $h_C$ at $We = 500$ and various $\alpha$ ($=0.027$, 0.064 and 0.125) and $\lambda$ ($=0.25$ and 4), along with the prediction of (3.1), in which $a_{0, C}=1.5$ obtained from numerical results at $\lambda =1$ is used to calculate $a_\lambda$ by (3.7). It is interesting to see that the trajectories of $h_C$ for the three different $\alpha$ are virtually overlapped at the early stage of impact, specifically $t\le 0.25$ for $\lambda =0.25$ and $t\le 0.35$ for $\lambda =4$. That is, the time variation of $h_C$ is not sensitive to $\alpha$ for relatively small $\alpha$. Also, the predictions are in good agreement with the numerical results in the respective range of time, as expected.

Figure 9.

Evolution of the vertical position of the centroid of the inner droplet ($h_C$) at $We = 500$ and various $\alpha$ ($=0.027$, 0.064 and 0.125). In particular, $\lambda =0.25$ in (a), and $\lambda =4$ in (b). The dashed line corresponds to the prediction of (3.1), with $a_\lambda$ calculated by (3.7). The inset shows the superimposed interfaces of the inner droplets with different $\alpha$.

3.2.3. Density effect on the inner droplet deformation at the immediate stage

At the intermediate stage of drop impact, the deformation of the inner droplets with $\lambda <1$ is very different from that with $\lambda >1$ (cf. the middle column in figure 3). For $\lambda <1$, the non-alignment of pressure and density gradients at the bottom of the inner droplet gives rise to the baroclinic effect and the subsequent generation of vorticity at the interface (cf. figure 10a). This phenomenon consequently enhances the ascending movement of the bottom of the inner droplet. At a later time, an upward counter-jet is induced by the vorticity, and breaks up the inner droplet (see the two right images in figure 3b). Such a counter-jet has been experimentally observed in a wide range of $Re$ and $We$ after the impact of hollow drops, and it could be so strong as to penetrate the whole compound drop, especially with large impact velocity and $\alpha$ (Gulyaev & Solonenko Reference Gulyaev and Solonenko2013). When the inner droplet with $\lambda \ge 1$ is approaching the substrate, its deformation is characterized by the formation and subsequent evolution of a lamella at the spreading front. Figure 10 shows the snapshots of impacting drops at $\alpha =0.064$, $We=500$ and various density contrasts ($\lambda =1, 2$ and $4$). We notice that, with the increase of $\lambda$, the lamella becomes thinner and thinner, probably accompanied by curling up.

Figure 10.

Snapshots of impacting compound droplets with $\alpha =0.064$, $We=500$ and various $\lambda$. In particular, (a) $\lambda =0.1$, $t=0.25$, (b) $\lambda =1$, $t=1.25$, (c) $\lambda =2$, $t=0.9$, (d) $\lambda =4$, $t=0.6$. The horizontal velocity contour (left) and vorticity (right) are shown.

Figure 11(a) shows the lamella thickness of the inner droplet ($h_l$) as a function of $We$ for $\lambda =1$, $1.5$, $2$ and $4$. Here, $h_l$ is defined as the mean thickness of the lamella (over a distance of 0.03 starting from the lamella front), which is measured at the moment when the velocity of the inner droplet centroid approaches zero. Despite of the differences between the results at various $\lambda$, the dimensionless lamella thickness $h_l/D$ is seen to have a scaling of $We^{-1/2}$ for the cases with the same $\lambda$. This observation is very similar to the deformation of a pure drop after impacting onto a substrate, for which Clanet et al. (Reference Clanet, Béguin, Richard and Quéré2004) experimentally found that the dimensionless lamella thickness follows $h_l/D\sim We^{-1/2}$. Moreover, they attributed this to the balance between the surface tension and the vertical deceleration of the drop ($\sim V^2/D$). The balance gives rise to an effective capillary length ($\sim \sqrt {\sigma D/(\rho V^2)}=D \,We^{-1/2}$), which is nothing but the scale of the lamella thickness. The idea of Clanet et al. (Reference Clanet, Béguin, Richard and Quéré2004) is adopted here to analyse the lamella thickness of the inner droplet. At the early stage of drop impact, the inner droplet experiences more or less uniform acceleration $a_0(1+A)V^2/D$, as indicated in (3.7). When the inner droplet is approaching the substrate (at $t\approx 0.5$), its vertical velocity approximately reduces to $U_i=V(1-A)/2$. Then, after a time scale of $d/U_i$, the vertical momentum of the inner droplet is supposed to completely transform to its horizontal spreading. In other words, the inner droplet encounters an acceleration of scale $U_i^2/d$ in the process of momentum transformation. Therefore, from the balance between surface tension and drop acceleration as in Clanet et al. (Reference Clanet, Béguin, Richard and Quéré2004), the dimensionless lamella thickness of the inner droplet with $\lambda \ge 1$ should follow

(3.8)\begin{equation} \frac{h_l}{D}\sim [\lambda (1-A)^2\alpha^{-{1}/{3}}We]^{-{1}/{2}}.\end{equation}

Figure 11(b) shows $h_l$ vs $\lambda (1-A)^2\alpha ^{-{1}/{3}}We$ on a log–log scale with different $\lambda$ and $\alpha$. The results roughly collapse onto a straight line, which corresponds to the theoretical predictions in (3.8).

Figure 11.

(a) Variation of lamella thickness of the inner droplet ($h_l/D$) as a function of $We$ at $\alpha =0.064$ on a log–log scale. (b) The lamella thickness as a function of $\lambda (1-A)^2\alpha ^{-{1}/{3}}We$ on a log–log scale, in which $We$ ranges from 50 to 500, $\lambda$ from 1 to 4 and $\alpha$ from 0.027 to 0.125.

The rolling up of the lamella of the inner droplet is caused by a mechanism that is the same as that of the ejected sheet in the impact of a pure drop (Riboux & Gordillo Reference Riboux and Gordillo2014). This is a combined effect of the lubrication force exerted by the surrounding fluid near the lamella front, and the suction force due to the Bernoulli pressure acting on top of the spreading lamella. Obviously, the mechanism becomes more significant for a thinner lamella.