2. Experimental study

Experiments are performed using the same set-up as in R&G, where spherical millimetric drops of radii $R$ fall under the action of gravity onto a dry glass slide which is partially wetted by the liquids whose physical properties are shown in table 1; the contact angle in all cases considered is close to $20^{\circ }$ , and from now on ${\it\nu}={\it\mu}/{\it\rho}$ will denote the liquid kinematic viscosity. The velocity $V$ of the drops at the instant of impact is controlled by varying the vertical distance between the exit of the needles and the substrate. The impact process is recorded with two high-speed cameras, placed perpendicular to each other and operated using two different optical magnifications and acquisition speeds (see R&G and appendix A for details). Water is not included as one of the working fluids in our experimental study, because the limitations in the spatial and temporal resolutions of our high-speed cameras impede accurate measurement of the tiny sizes (just a few microns) and large velocities ( ${\sim}20~\text{m}~\text{s}^{-1}$ ) of the fragments ejected. Nonetheless, in appendix B we will make use of the experimental results on splashing water droplets provided in Thoroddsen, Takehara & Etoh (Reference Thoroddsen, Takehara and Etoh2012) to give further support to our theory.

Figure 2. Effect of increasing viscosity on the splashing of a droplet for $V=2.21\pm 0.01~\text{m}~\text{s}^{-1}$ and $R=0.88\pm 0.023~\text{mm}$ ; from left to right, the viscosity ${\it\mu}$ is $1.3$  cP (first column), 4.6 cP (second column) and 10.0 cP (third column). From top to bottom, each sequence of images corresponds to the instants when $T$ is (a) $-0.261~\text{ms}$ , (b) 0.609 ms, (c) 0.899 ms, (d) 1.189 ms, (e) 2.349 ms and ( f) 2.929 ms. The temporal resolution is $58~{\rm\mu}\text{s}$ .

Figure 1 shows the time sequence of events occurring right after a drop of a liquid with viscosity ${\it\mu}=1~\text{cP}$ falls onto the solid at increasing impact velocities. For the smallest value of $V$ (figure 1, first column), a liquid sheet of thickness $H_{t}\ll R$ is expelled radially outwards and tangentially to the wall with a velocity $V_{t}\gg V$ for $T\geqslant T_{e}$ , where $T_{e}$ is the ejection time in R&G and $T=0$ is fixed at the instant when the drop first touches the substrate (see figure 1). When the impact speed is increased slightly (figure 1, second column), the lamella first dewets the substrate (figure 1 f, second column) and then contacts it again (figure 1 g, second column) as a consequence of the radial growth of the rim caused by capillary retraction. For velocities above the critical impact speed (image sequence in figure 1 a–k, third column), the edge of the liquid sheet dewets the substrate and then breaks into droplets of characteristic radius $R_{d}$ . These droplets are ejected outwards at a velocity substantially greater than that of the impact (figure 1 i–k, third and fourth columns).

Figure 2 shows the effect of varying the liquid viscosity while keeping the impact velocity and the drop radius at nearly constant values. The three cases depicted correspond to impact velocities above the splash threshold, i.e.  $V>V^{\ast }$ . Figure 2 shows that an increase in liquid viscosity delays the instant at which droplets are first ejected and also decreases the small angle that the droplets form with the substrate, suggesting that the vertical velocity component of the droplets is far smaller than the radial component. Figures 1 and 2 also reveal that the fragmentation process starts from the ejection of small droplets which depart right from the rim bordering the radially expanding lamella. Thus, the disintegration of the edge of the liquid sheet proceeds in a manner similar to that described by Villermaux & Bossa (Reference Villermaux and Bossa2011) and Peters, van der Meer & Gordillo (Reference Peters, van der Meer and Gordillo2013).

Figure 3. Sketch of the lamella for $T>T_{e}$ and for $V>V^{\ast }$ , i.e. impact velocities above which the lamella dewets the substrate. This figure also illustrates the definitions of the different variables needed to describe the position of the rim and the region at the root of the lamella where the liquid is decelerated by viscosity. Note that since $a\propto t^{1/2}$ and $h_{a}\propto t^{3/2}$ , $h_{a}\ll a$ for $t\ll 1$ . The discontinuous line, located at a distance of $d\propto t^{3/2}\propto h_{a}$ from the jet root, limits the region from which fluid particles feed the lamella.

3. Modelling the drop ejection process

The first steps in describing the disintegration of the drop that occurs during the initial period of time after impact are to determine the values of (i) the ejection time $T_{e}$ , (ii) the initial tangential velocity of the edge of the lamella, $V_{t}(T_{e})$ , and (iii) its initial thickness, $H_{t}(T_{e})$ . For this purpose, we make use of some previous results from R&G; in that work, upon choosing $R$ , $V$ , $R/V$ and ${\it\rho}V^{2}$ as characteristic scales of length, velocity, time and pressure, respectively, with ${\it\rho}$ being the liquid density and lowercase letters denoting dimensionless variables, the following properties are shown.

1. (i) The radius of the circular area shown in figures 1 and 3 evolves in time as $a=\sqrt{3t}$ . For $T<T_{e}$ , i.e. before the lamella is ejected, $a(t)$ represents the radius of the wetted area (figure 1). For $T>T_{e}$ , $a$ is the distance between the axis of symmetry of the drop and the radial position where the root of the lamella is located; therefore $a<r_{t}$ , with $r_{t}$ being the radial position of the rim (see figure 3).

2. (ii) The tangential velocity at which the lamella is initially ejected is $v_{t}(t_{e})\simeq {\dot{a}}(t_{e})=1/2\sqrt{3/t_{e}}$ .

3. (iii) The thickness of the edge of the lamella at the instant of ejection is $h_{t}\simeq \sqrt{12}\,t_{e}^{3/2}/{\rm\pi}$ .

4. (iv) Since $v_{t}={\dot{a}}$ at $t=t_{e}$ , i.e. the wetted area and the tip of the lamella advance at the same speed, and because the lamella can only be ejected if its tip advances faster than the radius of the wetted area, $t_{e}$ is calculated from

   (3.1) $$\begin{eqnarray}c_{1}\mathit{Re}^{-1}t_{e}^{-1/2}+\mathit{Re}^{-2}\mathit{Oh}^{-2}=\ddot{a}h_{t}^{2}=c^{2}t_{e}^{3/2},\end{eqnarray}$$
   which expresses the fact that the ejection time is the instant at which the deceleration of the edge of the lamella coincides with the deceleration of the wetted area, $\ddot{a}$ . In (3.1), $\mathit{Re}={\it\rho}VR/{\it\mu}$ and $\mathit{Oh}={\it\mu}/\sqrt{{\it\rho}R{\it\sigma}}$ denote, respectively, the impact Reynolds number and the Ohnesorge number (where ${\it\sigma}$ is the interfacial tension coefficient), and $c_{1}\simeq \sqrt{3}/2$ and $c^{2}=1.2$ are constants which are adjusted experimentally. In R&G it was also shown that the ejection time given by (3.1) has high- $\mathit{Oh}$ and low- $\mathit{Oh}$ limits $t_{e}=2\mathit{Re}^{-1/2}$ and $t_{e}\propto \mathit{Re}^{-4/3}\mathit{Oh}^{-4/3}$ , respectively.

Moreover, in R&G it is shown that, once the sheet is ejected, its edge experiences a vertical lift force per unit length

which results from the addition of the lubrication force exerted by the gas in the wedge region between the advancing lamella and the substrate, $K_{l}{\it\mu}_{g}V_{t}$ , and the suction force exerted by the gas at the top part of the lamella, $K_{u}{\it\rho}_{g}V_{t}^{2}H_{t}$ . Here the subscript $g$ represents gas quantities, $K_{u}\simeq 0.3$ is a constant determined numerically, and the expression

where ${\it\lambda}$ denotes the mean free path of gas molecules, is deduced using lubrication theory if one assumes that the front part of the lamella can be approximated by a wedge of constant angle ${\it\alpha}\sim 60^{\circ }$ while it is in contact with the substrate. Let us point out here that ${\it\alpha}$ refers to the angle that the advancing front forms with the substrate at the instant of ejection $t_{e}$ (see R&G for details), so ${\it\alpha}$ is not related to the angle at which drops are ejected. Indeed, the ejected droplets form an angle with the substrate which can be approximated by the ratio between their vertical and horizontal velocity components.

Figure 4. Experimental values of the critical impact velocity $V^{\ast }$ for different values of the Ohnesorge number. The legend indicates the different fluids listed in table 1. Continuous lines represent results obtained from the model presented in R&G.

Thus, the force balance projected in the vertical direction, ${\it\rho}H_{t}^{2}\dot{V}_{v}\propto \ell$ with $\ell$ given by (3.2), provides the vertical velocity at which the lamella is initially expelled, namely,

As shown in R&G, the good agreement of the model with their own experimental data as well as with the data of Mundo et al. (Reference Mundo, Sommerfeld and Tropea1995), Xu et al. (Reference Xu, Zhang and Nagel2005), Palacios et al. (Reference Palacios, Hernandez, Gomez, Zanzi and Lopez2013) and Stevens (Reference Stevens2014) indicates that the critical impact velocity $V^{\ast }$ , above which the drop disintegrates into droplets, results from the condition that the vertical velocity (3.4) is such that $V_{v}/V_{r}\sim O(1)$ , where $V_{r}=\sqrt{2{\it\sigma}/{\it\rho}H_{t}}$ is the capillary retraction velocity (Taylor Reference Taylor1959; Culick Reference Culick1960), resulting in $\sqrt{\ell /(2{\it\sigma})}\simeq 0.14$ . Figure 4, which compares the splash threshold velocity $V^{\ast }$ predicted by $\sqrt{\ell /(2{\it\sigma})}\simeq 0.14$ with $\ell$ given by (3.2) and that measured experimentally for each of the four fluids listed in table 1, gives further support to our results in R&G. From now on, we will focus on the description of the drop fragmentation process for values of the velocity ratio $V/V^{\ast }>1$ .

So far, the theory in R&G has been tested for impact velocities below or equal to the critical speed, so first it is necessary to check whether the theory is equally applicable to calculating the initial values of $V_{v}(T_{e})$ and $V_{t}(T_{e})$ for $V>V^{\ast }$ . We will show below that there is very good agreement between experimental data and the theoretical values, and so the next step in determining the velocities and sizes of the fragments ejected from the rim is to describe the liquid flow within the lamella formed for $t>t_{e}$ and extending from $r=a(t)$ to $r=r_{t}>a$ (see figure 3).

In R&G it is shown that when viscous effects are neglected, the thickness of the lamella and the liquid velocity at $a=\sqrt{3t}$ , i.e. at the radial position where the root of the lamella is located, are given by

respectively, with these equations being valid for $a\lesssim 1$ , i.e.  $t\lesssim 1/3$ . However, in a real fluid, a boundary layer of dimensional thickness ${\it\Delta}\sim \sqrt{{\it\nu}T_{r}}=\sqrt{{\it\nu}R/V}(h_{a}/v_{a})^{1/2}$ , where $T_{r}\sim R/Vh_{a}/v_{a}$ is the characteristic residence time of fluid particles entering the lamella, develops in the region $r\simeq \sqrt{3t}$ . Indeed, from equation (6) in the supplementary data of R&G, it can be deduced that in the frame of reference moving at velocity $V{\dot{a}}$ , there exists a stagnation point in the flow at dimensionless distance $d$ from the root of the lamella, such that $\sqrt{a}\,d^{-1/2}\sim {\dot{a}}\Rightarrow d\propto t^{3/2}\propto h_{a}\ll \sqrt{3t}$ (see figures 1 and 3). Note that ${\it\Delta}$ is the width of a boundary layer that develops in a region of dimensionless width similar to the only relevant length scale characterizing this spatial region, namely the height of the root of the lamella. The effect of the viscous shear force per unit length exerted in this region, i.e.  $F_{{\it\tau}}={\it\rho}V^{2}Rf_{{\it\tau}}\sim {\it\mu}VR/{\it\Delta}v_{a}h_{a}$ , is to decrease the liquid velocity from $v_{a}$ to $v_{a}^{+}$ (see figure 3) and to increase the height of the lamella from $h_{a}$ to $h_{a}^{+}$ , with the expressions for $v_{a}$ and $h_{a}$ given in (3.5). To relate the downstream quantities $v_{a}^{+}$ and $h_{a}^{+}$ to their corresponding upstream values, we make use of integral balances of mass and momentum applied to the shaded region of figure 3, yielding $h_{a}v_{a}=h_{a}^{+}v_{a}^{+}$ and $h_{a}v_{a}^{2}-f_{{\it\tau}}=h_{a}^{+}(v_{a}^{+})^{2}$ . Therefore, the equations

constitute the initial conditions to be used next to predict the spatiotemporal evolution of the lamella; here $\sqrt{2}$ is a fixed proportionality constant and $v_{a}$ and $h_{a}$ are as given in (3.5). In (3.6), the viscous deceleration term is of order ${\sim}O(0.1)$ for the liquid with the smallest viscosity and ${\sim}0.5$ for the liquid with ${\it\mu}=10~\text{cP}$ .

To describe the dynamics of the portion of the liquid sheet that extends from $r\simeq a(t)$ up to the radial position where the rim is located, i.e.  $r=r_{t}$ , we make use of the fact that the viscous shear stresses can be neglected in this region because the liquid is no longer in contact with the substrate; see the sketch in figure 3 and the images in figure 1(d,e), third and last columns. Additionally, since the local Weber number $\mathit{We}=\mathit{Re}^{2}\mathit{Oh}^{2}$ characterizing the flow within the sheet is such that $\mathit{We}(v_{a}^{+})^{2}h_{a}\gg 1$ , the gradients of capillary pressure are negligible, and so the momentum equation within the lamella reduces to

with $u(r,t)$ denoting the liquid velocity inside the lamella and $\text{D}/\text{D}t\equiv \partial /\partial t+u\partial /\partial r$ the material derivative. Equation (3.7) states that fluid particles conserve their velocities for $\sqrt{3t}\lesssim r<r_{t}(t)$ , i.e. up to the radial position $r_{t}(t)$ where the rim is located. The equation for the thickness of the liquid sheet $h(r,\,t)$ in the region $\sqrt{3t}\lesssim r<r_{t}(t)$ is deduced by making use of the mass balance, yielding

Equations (3.7) and (3.8), which represent the ballistic motion of fluid particles along the lamella, are solved subject to the initial conditions $u(r=\sqrt{3t},t)=v_{a}^{+}$ and $h(r=\sqrt{3t},t)=h_{a}^{+}$ , with $v_{a}^{+}$ and $h_{a}^{+}$ given in (3.6). Notice that, since $h_{a}\ll \sqrt{3t}$ , we impose the boundary condition at $r=a(t)$ and neglect the width of the region ${\sim}O(h_{a})$ where the boundary layer develops. To find the solution of the system (3.7) and (3.8), we used a Lagrangian numerical method which is quite similar to that described in Gekle & Gordillo (Reference Gekle and Gordillo2010).

The radial and vertical positions of the edge of the lamella, $r_{t}(t)$ and $z_{t}(t)$ , as well as its thickness $h_{t}(t)$ , are deduced by applying the integral balances of mass and momentum at the rim (Taylor Reference Taylor1959):

In (3.9), the geometry of the rim has been approximated by a torus of minor radius $h_{t}(t)/2$ . Moreover, the smallness of the angle that the spray forms with the substrate suggests approximating the projection of the capillary force per unit length in the direction tangential to the wall by $\simeq 2{\it\sigma}$ ; as a consequence of the smallness of the ratio ${\it\rho}_{g}/{\it\rho}$ , air drag is neglected with respect to the flux of momentum entering the lamella. The equations in (3.9) need to be complemented with the force balance in the vertical direction,

where the projection of the capillary force per unit length in the direction normal to the wall has been approximated by $\simeq 2{\it\sigma}z_{t}/r_{t}$ and the lift coefficient has been taken to be a constant, $C_{l}=1$ . We assume that the influx of vertical momentum into the rim is negligible and, to simplify the model as much as possible, we do not take into account the dependence of $C_{l}$ either on the angle of incidence or on the Reynolds number, because our description does not keep track of the orientation of the tip of the lamella with respect to the surrounding atmosphere. Since, as will be shown below, the theory in R&G is valid also for describing the ejection of the lamella for impact velocities above the critical velocity, (3.9) and (3.10) are solved subject to the following initial conditions:

where $t_{e}$ is calculated using (3.1) for ${\it\mu}<10~\text{cP}$ and $t_{e}=2\mathit{Re}^{-1/2}$ for ${\it\mu}=10~\text{cP}$ .

Figure 5. Calculated ratio $T_{c}/T_{h}=\sqrt{\mathit{We }/8}\,(h^{1/2})\,\text{d}h/\text{d}t$ for different impact velocities and the different liquids used in the experimental study: (a) ethanol, (b) decamethyltetrasiloxane, (c) poly(dimethylsiloxane) and (d) 10 cP silicone oil. The circles (○) indicate the instants $t_{b}=T_{b}V/R$ at which the model predicts the droplets will be ejected. Each of the curves starts at  $t_{e}$ .

The final step to complete our theory is to model the breakup time $t_{b}$ , i.e. the time at which drops are ejected from the edge of the lamella. In Lhuissier & Villermaux (Reference Lhuissier and Villermaux2011), Agbaglah, Josserand & Zaleski (Reference Agbaglah, Josserand and Zaleski2013), Peters et al. (Reference Peters, van der Meer and Gordillo2013) and references therein, it is clearly shown that the droplets making up the spray result from the amplification of Rayleigh–Taylor and capillary instabilities developing in the azimuthal direction (Thoroddsen et al. Reference Thoroddsen, Takehara and Etoh2012). The growth rates of these instabilities, however, are highly attenuated as a consequence of simultaneous growth of the rim thickness. Thus, drops will only be ejected when the time characterizing the radial growth of the rim, $T_{h}=(1/H_{t}\,\text{d}H_{t}/\text{d}T)^{-1}$ , is substantially greater than either the capillary time $T_{c}=({\it\rho}H_{t}^{3}/8{\it\sigma})^{1/2}$ or the viscous time $T_{v}={\it\mu}H_{t}/{\it\sigma}$ . In the present case, since the local Ohnesorge number is such that $T_{v}/T_{c}={\it\mu}/\sqrt{{\it\rho}H_{t}{\it\sigma}}\ll 1$ , the growth of perturbations will be controlled by the capillary rather than the viscous time. Figure 5 shows the time evolution of $T_{c}/T_{h}(t)=\sqrt{\mathit{We }/8}\,(h^{1/2})\,\text{d}h/\text{d}t$ , calculated from (3.7)–(3.10) and solved subject to the initial conditions (3.6) and (3.11) for the different experimental conditions investigated. The function $T_{c}/T_{h}(t)$ reaches a constant value ${\sim}0.075$ for times $T\approx T_{b}$ , with $T_{b}$ such that $(\text{d}/\text{d}T)[T_{c}/T_{h}](T=T_{b})=0$ . In our model, we assume that drops are ejected precisely at $t_{b}$ . This choice of $t_{b}$ is justified by taking into account the following facts (Eggers & Villermaux Reference Eggers and Villermaux2008): (i) the characteristic time of growth of a capillary instability is ${\sim}3T_{c}$ ; (ii) Rayleigh’s stability analysis of capillary perturbations reveals that the dimensionless wavenumber $k$ with fastest growth rate is $k={\rm\pi}H_{t}/{\it\lambda}\simeq 0.7$ , where ${\it\lambda}$ is the wavelength of the perturbation, which is kept constant in time; (iii) the growth of capillary instabilities is inhibited for $k\geqslant 1$ . Therefore, in order for capillary corrugations with characteristic initial wavenumbers $k=0.7$ to be amplified up to the point that drops are ejected from the rim, it is necessary that in time $3T_{c}$ , the variation of the dimensionless wavenumber is such that ${\rm\Delta}k<0.3$ . This condition is equivalent to $\text{d}H_{t}/\text{d}T\times 3T_{c}\lesssim 0.3H_{t}\Rightarrow T_{c}/T_{h}\lesssim 0.1$ , a value that is very close to and slightly larger than the minimum value of $T_{c}/T_{h}$ , which is approximately $0.075$ . Note from figure 5 that our criterion for choosing the instant of drop ejection is consistent with the previous arguments. However, the way $T_{b}$ is calculated is nothing but a plausible assumption: indeed, in a real experiment, the breakup time will depend strongly on the initial amplitude of azimuthal perturbations. In addition, the determination of $T_{b}$ from very flat curves such as those in figure 5 may result in a non-negligible uncertainty in the determination of the breakup time.

Moreover, since $T_{c}\ll T_{b}$ , drops will be ejected slightly after $T_{b}$ is reached. Hence we neglect the contribution of $T_{c}$ and assume that drops will be ejected right at $T_{b}$ , with their velocities and sizes determined from the solution of (3.7)–(3.10) subject to the initial conditions (3.6) and (3.11) at the particular instant $t_{b}=T_{b}V/R$ . Let us point out that, in our description, we assume that the mean diameters of the first drops ejected can be approximated by the thickness of the rim at $t=t_{b}$ . The reason behind this approximation is the following: the volume of liquid contained in a rim of diameter $h(t_{b})$ along a distance in the azimuthal direction equivalent to that of the most unstable capillary wavelength ( $\simeq 4.5h(t_{b})$ ) would give rise to the ejection of a droplet with diameter $\simeq 1.9h_{t}(t_{b})$ . However, only a fraction of this volume will be ejected as a droplet; in fact, the simulations in Agbaglah et al. (Reference Agbaglah, Josserand and Zaleski2013) and the experiments of Thoroddsen et al. (Reference Thoroddsen, Takehara and Etoh2012) show that the diameters of the drops ejected are quite similar to the diameter of the rim, and this is the reason that in our model we identify the diameters of the drops with that of the rim at  $t=t_{b}$ .

Figure 6, which compares the measured ejection time with the value of $t_{b}$ calculated as $(\text{d}/\text{d}t)[h^{1/2}\,\text{d}h/\text{d}t]\,(t=t_{b})=0$ (where $t_{b}$ is the value marked with a circle in figure 5) for each of the four fluids investigated and for different impact velocities, validates our approach.

Figure 6. Comparison of the predicted breakup time $t_{b}$ with the experimentally obtained value for (a) ethanol, (b) decamethyltetrasiloxane, (c) poly(dimethylsiloxane) and (d) 10 cP silicone oil.

4. Comparison between observations and the model

Figures 7 and 8 show the horizontal ( $v_{t}$ ) and vertical ( $v_{v}$ ) velocity components of the tip of the lamella at the instant of ejection $t_{e}$ and at the breakup time $t_{b}$ , for each of the four fluids investigated. The tangential and vertical velocities at $t_{e}$ are calculated, respectively, from $v_{t}(t_{e})={\dot{a}}(t_{e})=1/2\sqrt{3/t_{e}}$ and (3.4), with $C_{v}=2$ and $t_{e}$ determined either using (3.1) for ${\it\mu}<10~\text{cP}$ or by means of the high- $\mathit{Oh}$ limit $t_{e}=2\mathit{Re}^{-1/2}$ for ${\it\mu}=10~\text{cP}$ . The results shown in figures 7 and 8 indicate that the theoretical description of the ejection of the lamella in R&G can indeed be extended to describe the drop fragmentation process for impact velocities above the critical velocity, i.e. for $V>V^{\ast }$ . Moreover, observe that for a given fluid, $v_{t}(t_{e})\gg v_{v}(t_{e})$ , which implies that the trajectory of the edge of the lamella forms a small angle with the substrate, in accordance with the experimental evidence shown in figures 1 and 2. This observation is more evident in figures 9 and 10, where the measured radial ( $r_{t}$ ) and vertical ( $z_{t}$ ) positions of the edge of the lamella are compared with our model predictions. Note that the agreement is remarkable for all four types of fluid investigated and also, as anticipated above, $z_{t}\ll r_{t}$ . Furthermore, the trajectories of the tip of the lamella in figures 9 and 10, represented up to $t=0.4>1/3$ , are calculated consistently with the range of validity of (3.5). Indeed, figure 11 shows that the fluid particles entering the rim at the instant $t$ were ejected from the root of the jet at $t_{a}<1/3$ .

Figure 7. Tangential velocity of the lamella for impact velocities greater than the critical velocity $V^{\ast }$ , at the ejection time $t=t_{e}$ (●) and at the breakup instant $t=t_{b}$ (○), for (a) ethanol, (b) decamethyltetrasiloxane, (c) poly(dimethylsiloxane) and (d) 10 cP silicone oil; the lines represent results obtained from the model.

Figure 8. Vertical velocity of the lamella for impact velocities greater than the critical velocity $V^{\ast }$ , at the ejection time $t=t_{e}$ (●) and at the breakup instant $t=t_{b}$ (○), for (a) ethanol, (b) decamethyltetrasiloxane, (c) poly(dimethylsiloxane) and (d) 10 cP silicone oil; the lines represent results obtained from the model.

Figure 9. Horizontal position of the edge of the lamella plotted as a function of time. The panels correspond to the different fluids listed in table 1: (a) ethanol, with $V/V^{\ast }=1.15$ ; (b) decamethyltetrasiloxane, with $V/V^{\ast }=1.20$ , (c) poly(dimethylsiloxane), with $V/V^{\ast }=1.24$ ; (d) 10 cP silicone oil, with $V/V^{\ast }=1.00$ . Continuous lines represent results obtained from the model.

Figure 10. Vertical position of the edge of the lamella plotted as a function of time. The panels correspond to the different fluids listed in table 1: (a) ethanol, with $V/V^{\ast }=1.15$ ; (b) decamethyltetrasiloxane, with $V/V^{\ast }=1.20$ , (c) poly(dimethylsiloxane), with $V/V^{\ast }=1.24$ ; (d) 10 cP silicone oil, with $V/V^{\ast }=1.00$ . Continuous lines represent results obtained from the model.

Figure 11. Plotted along the vertical axis, $t_{a}$ is the instant at which a fluid particle entering the rim at instant $t$ was ejected from the root of the lamella. The symbols indicate the breakup times $t=t_{b}$ for the following cases: (♦) ethanol, with $V/V^{\ast }=1.11$ ; (●) decamethyltetrasiloxane, with $V/V^{\ast }=1.21$ ; (▪) poly(dimethylsiloxane), with $V/V^{\ast }=1.26$ ; (◀ ) 10 cP silicone oil, with $V/V^{\ast }=1.12$ .

Figures 7 and 8 also reveal that, while $v_{t}(t_{e})$ increases slightly with $V/V^{\ast }$ , $v_{v}(t_{e})$ remains practically unchanged within the range of impact velocities investigated. Also, observe from figures 7 and 8 that both $v_{t}(t_{e})$ and $v_{v}(t_{e})$ decrease when the liquid viscosity is increased. Observe that all the experimental measurements at $t_{e}$ are captured by our theory, not only qualitatively but also quantitatively.

The main purpose of our model, however, is to predict the horizontal and vertical velocity components as well as the diameters of the droplets formed at the edge of the lamella. Figures 7 and 8 show that, as a consequence of the capillary deceleration of the rim, $v_{t}(t_{e})>v_{t}(t_{b})$ and $v_{v}(t_{e})>v_{v}(t_{b})$ , with the velocities at $t_{b}$ calculated by solving the system (3.7)–(3.10) subject to the initial conditions predicted by our theory, i.e. (3.6) and (3.11). The agreement between our theory and the experimental measurements is quantitative for all the fluids listed in table 1, except for the liquid with the largest viscosity.

The reason for the discrepancies observed with the liquid of ${\it\mu}=10~\text{cP}$ is probably related to the fact that the breakup time in this case is so large that $t_{a}>1/3$ , exceeding the limit of validity of (3.5) (see figure 11). Therefore, for $t_{a}>1/3$ , the equations in (3.5) should be replaced, for instance, by analogous expressions such as those in Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010). This step is beyond the scope of the present study, which, as shown above, captures very well the sizes and velocities of the drops ejected in the cases of low-viscosity liquids.

In spite of the fact that the model is unable to accurately predict the velocities of the fragments ejected for ${\it\mu}=10~\text{cP}$ , it is noteworthy that our approach gives good quantitative predictions of the sizes of the first drops ejected from the rim, as figure 12 shows. Finally, as a further test of our theory, in figure 16 of appendix B we compare the predictions of our model with the experimental results of Thoroddsen et al. (Reference Thoroddsen, Takehara and Etoh2012), who studied the velocities and sizes of fragments ejected after the impact of high-speed water droplets; good agreement is obtained. The evidence in appendix B indicates that the model presented here is useful not only for quantifying the ejection of the fastest fragments generated after drop splashing at $t_{b}$ , but also in describing the continuous drop disintegration process that takes place for larger times.

Figure 12. Comparison of the predicted radius of the first droplets ejected with experimentally measured values for (a) ethanol, (b) decamethyltetrasiloxane, (c) poly(dimethylsiloxane) and (d) 10 cP silicone oil.

5. Concluding remarks

In this paper we have presented a model for predicting the sizes and velocities of fragments ejected after a drop impacts a smooth wall at a velocity above the critical speed described in Riboux & Gordillo (Reference Riboux and Gordillo2014), extending and providing further support for our theory in R&G. The present description can be summarized as follows. Fluid particles conserve the tangential velocity $V_{a}^{+}$ given in (3.6) that they possessed downstream of a narrow region of typical length ${\sim}H_{a}$ located where the drop meets the substrate, $\simeq \sqrt{3RVT}$ . Downstream of the region of length ${\sim}H_{a}$ in which the fluid is decelerated by viscous friction, the sheet is no longer in contact with the substrate, and therefore the radial velocities of fluid particles are conserved up to the radial position where the rim is located. The rim itself is accelerated radially outwards as a consequence of the influx of momentum transported by fluid particles ejected from the location $\simeq \sqrt{3RVT}$ , and is decelerated due to the action of the capillary forces. The rim, of initial thickness $H_{t}(T_{e})\ll R$ , also translates in the direction perpendicular to the wall, thanks to the initial vertical velocity imparted by the lift force per unit length, $\ell$ , given in (3.2), as well as by the lift force $1/2{\it\rho}_{g}V_{t}^{2}H_{t}C_{l}$ ; capillarity contributes to decelerate the rim vertically. Finally, droplets are ejected because the development of capillary and Rayleigh–Taylor instabilities favours the generation of corrugations of increasing amplitude, which give rise to the formation and ejection of drops at the instant when the ratio $T_{h}/T_{c}$ becomes large enough. Here we have characterized this instant $T_{b}$ as the time at which $T_{c}/T_{h}$ reaches a minimum, with $T_{c}/T_{h}\simeq 0.075$ . Here $T_{h}$ and $T_{c}$ denote the characteristic times of growth of the rim thickness and of capillary instabilities, respectively. The diameters of the drops are close to the diameter of the rim at $T_{b}$ , and their velocities are approximately equal to the rim velocity at this instant.

Let us point out here that these results are valid for the case of smooth and partially wetting solids with a static contact angle of ${\sim}20^{\circ }$ . We would like to extend the theory in R&G to a wider range of static contact angles and also to rough surfaces.