2 Potential flow simulations

Following the notation in RG14, lower case variables will be used in what follows to refer to dimensionless variables, which are constructed here using as scales for velocity, length and pressure $V$ , $R$ and $\unicode[STIX]{x1D70C}V^{2}$ , respectively. Since the Froude number $\mathit{Fr}=V^{2}/gR\gg 1$ and the shear stresses acting on the liquid are negligible, the only relevant dimensionless parameter characterizing the splashing of droplets in the Leidenfrost regime is the Weber number, defined here as $\mathit{We}=\unicode[STIX]{x1D70C}V^{2}R/\unicode[STIX]{x1D70E}$ .

Figure 3.

(a) The figure shows the numerical results for both $a(t)$ and the radial position of the edge lamella $r_{t}(t)$ in the case of $\mathit{We}=100$ . The vertical line indicates the ejection time predicted by (2.2) and the dashed line represents the predicted position of the rim, ejected from $r=\sqrt{3t_{e}}$ with an initial velocity $v_{t}(t_{e})=1/2\sqrt{3/t_{e}}$ , with $t_{e}$ given in (2.2). (b) The ejection times predicted by the BIMs are such that $t_{e}\propto \mathit{We}^{-2/3}$ , in agreement with (2.2).

The most relevant event taking place after the impact of a drop in the dynamic Leidenfrost regime is that at $t_{e}$ , with $t_{e}$ the ejection time – see figure 2 – an extremely thin liquid sheet of initial thickness $h_{t}(t_{e})$ is expelled tangentially to the substrate with an initial velocity $v_{t}(t_{e})$ . In RG14, an algebraic equation for $t_{e}$ was deduced based on the following facts: (i) prior to the ejection of the lamella, the time evolution of the radius of the wetted area is $a(t)=\sqrt{3t}$ (this result was derived for the very first time by Riboux and Gordillo in RG14 using the linearization of the boundary conditions for the potential flow (Wagner Reference Wagner1932)); (ii) at $t_{e}$ , the velocity of the tip of the lamella is equal to the velocity of the wetted radius, i.e. $v_{t}(t_{e})={\dot{a}}(t_{e})$ with dots denoting time derivatives; and (iii) the lamella can only be ejected if its tip advances faster than the radius of the wetted area. We concluded in RG14 that $t_{e}$ can be calculated solving the algebraic equation

(2.1) $$\begin{eqnarray}\displaystyle \frac{\sqrt{3}}{2}\,Re^{-1}\,t_{e}^{-1/2}+\mathit{We}^{-1}=\ddot{a}\,h_{t}^{2}=c^{2}t_{e}^{3/2}\quad \text{with }c=1.1, & & \displaystyle\end{eqnarray}$$

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, $\ddot{a}$ . For wall temperatures above $T_{L}$ , the edge of the lamella is not decelerated by the action of viscous shear stresses and, therefore, setting to infinity the Reynolds number $Re$ in (2.1) yields,

(2.2) $$\begin{eqnarray}\displaystyle t_{e}=(c^{2}\,\mathit{We})^{-2/3}. & & \displaystyle\end{eqnarray}$$

Using the analytical expressions for both $v_{t}(t_{e})$ and $h_{t}(t_{e})$ derived in RG14, as well as the result in (2.2), it can be concluded that $v_{t}(t_{e})=1/2\sqrt{3/t_{e}}\propto \mathit{We}^{1/3}$ and $h_{t}(t_{e})=\sqrt{12}\,t_{e}^{3/2}/\unicode[STIX]{x03C0}\propto \mathit{We}^{-1}$ . Figure 3 confirms the result in (2.2), $t_{e}\propto \mathit{We}^{-2/3}$ .

Figure 4.

(a) Comparison between the theoretical expression $v_{a}=2{\dot{a}}=\sqrt{3/t}$ deduced in RG14 and the numerical result. The good agreement between theory and experiments depicted in this figure is independent of the Weber number whenever $\mathit{We}\geqslant 100$ . Two of the simulations shown are carried out using $N(T=0)=501$ whereas in the case marked with an asterisk [ $^{\ast }$ ],  $N(T=0)=1001$ . (b) Comparison of the height of the lamella at $r=\sqrt{3t}$ predicted by the models in Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) [1] and Roisman (Reference Roisman2009) [2] with the numerical result. (c) Comparison between the theoretical expression $h_{a}=h(a(t),t)=(\sqrt{12}/3\unicode[STIX]{x03C0})\,t^{3/2}$ deduced in RG14, valid for $t\ll 1$ and the numerical result. The vertical line indicates the instant at which we have set the transition between the result predicted by the potential flow theory and the fitting polynomial $P(t)$ in (2.4).

Once the lamella is ejected, as also deduced in RG14, for $t>t_{e}$ and $t\ll 1$ , both the liquid velocity and the height of the liquid layer at $r=a(t)=\sqrt{3t}$ , which is the radial position where the drop meets the lamella, are respectively given by $v_{a}=2{\dot{a}}=\sqrt{3/t}$ and $h_{a}=\sqrt{12}\,t^{3/2}/(3\unicode[STIX]{x03C0})$ . Figure 4 confirm these theoretical predictions for $\mathit{We}\geqslant 100$ . More precisely, figure 4 shows that, while the analytical expression for $v_{a}$ is valid for arbitrary times, the corresponding analytical expression for $h_{a}$ departs from the numerical results for $t\geqslant 0.1$ (see figure 4 c). Therefore, the equations describing the liquid velocity in the lamella $u(r,t)$ and its height $h(r,t)$ at $r=\sqrt{3t}$ i.e. at the radial position from which the liquid sheet is ejected for times $t>t_{e}$ , are respectively given by

(2.3) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle u(r=a,t)=v_{a}=\sqrt{3/t},\quad \text{and }\\ \displaystyle h(r=a,t)=h_{a}=\sqrt{12}t^{3/2}/(3\unicode[STIX]{x03C0})\quad \text{for }t<0.1\\ \displaystyle h(r=a,t)=P(t)\quad \text{for }t\geqslant 0.1,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with

(2.4) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle P(t)=\mathop{\sum }_{i=0}^{5}(0.1p_{i})t^{i}\quad \text{and}\quad p_{0}=-2.453\times 10^{-3},\quad p_{1}=1.321,\quad p_{2}=-1.176,\\ \displaystyle p_{3}=0.4943,\quad p_{4}=-0.1047,\quad p_{5}=8.89\times 10^{-3}\end{array}\right\}\quad & & \displaystyle\end{eqnarray}$$

a polynomial which is fitted to the numerical results.

The ad hoc radial velocity field within the drop proposed by both Roisman (Reference Roisman2009) and Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) is $u(r,t)=r/(t+\unicode[STIX]{x1D70F})$ , with $\unicode[STIX]{x1D70F}$ an adjustable order-unity constant. While the stagnation point type of flow within the impacting drop hypothesized by Roisman (Reference Roisman2009) and Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) is a good approximation to the real flow field for $t\sim O(1)$ , for times such that $t>t_{e}$ , $t\ll 1$ , the velocity field at $r\simeq \sqrt{3t}$ does not correspond to a stagnation point flow (see RG14 for details). Interestingly, our theory predicts a radial velocity at $r=a(t)=\sqrt{3t}$ , $u(r=a(t),t)=a(t)/t=\sqrt{3/t}$ for $t\ll 1$ which is, by coincidence, the radial velocity corresponding to a stagnation point of flow of the type $u=r/t$ particularized at $r=\sqrt{3t}$ . This is why figure 4(a) shows a fairly good between the analytical expression $v_{a}=\sqrt{3/t}$ and the numerical results for arbitrary values of $t$ . Figure 4(b) also shows the values of $h_{a}(t)=h(r=\sqrt{3t},t)$ predicted by the model in Roisman (Reference Roisman2009),

(2.5) $$\begin{eqnarray}\displaystyle h(r,t)=8\frac{\unicode[STIX]{x1D702}}{(t+\unicode[STIX]{x1D70F})^{2}}\exp \left[-\frac{6\unicode[STIX]{x1D702}\,r^{2}}{(t+\unicode[STIX]{x1D70F})^{2}}\right], & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D702}=0.39$ , $\unicode[STIX]{x1D70F}=2\times 0.25$ and with a constant equal to $8$ because here we define dimensionless lengths using the drop radius instead of its diameter, as well as the results predicted by the model in Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010),

(2.6) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle h(r,t)=\frac{1}{(t+\unicode[STIX]{x1D70F})^{2}}\,H(x)\\ \displaystyle H(x)=\frac{3.19}{(1+C\,x^{2})^{6}}\quad \text{with }\,\,x(r,t)=\frac{r}{t+\unicode[STIX]{x1D70F}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with $C=0.604$ and $\unicode[STIX]{x1D70F}=1$ (Lastakowski et al. Reference Lastakowski, Boyer, Biance, Pirat and Ybert2014). Clearly, neither the model by Roisman (Reference Roisman2009) not that by Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) are in agreement with the numerical results.

3 Prediction of the critical Weber number for splashing and of the maximum spreading radius for droplets impacting in the Leidenfrost regime

For a given instant of time $t>t_{e}$ , the lamella extends from the position where the drop meets the liquid sheet i.e. $r=\sqrt{3t}$ , down to $r=r_{t}(t)$ , with $r_{t}(t)$ indicating the radial position of the rim. In RG15, we showed that both the liquid velocity $u(r,t)$ and the height of the liquid layer within the lamella, $h(r,t)$ , can be calculated using the pair of equations (Gordillo & Gekle Reference Gordillo and Gekle2010; Villermaux & Bossa Reference Villermaux and Bossa2011),

(3.1a,b ) $$\begin{eqnarray}\displaystyle \frac{\text{D}u}{\text{D}t}=0\quad \text{and}\quad \frac{\text{D}\ln (rh)}{\text{D}\,t}=-\frac{\unicode[STIX]{x2202}\,u}{\unicode[STIX]{x2202}\,r}, & & \displaystyle\end{eqnarray}$$

with $\text{D}/\text{D}t\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\,t+u\,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r$ the material derivative. The former of the two equations in (3.1a,b ) expresses that fluid particles conserve their velocities within the lamella from $r=\sqrt{3t}$ down to the radial position $r_{t}(t)$ where the rim is located; the latter is the continuity equation. The pair of equations in (3.1a,b ) representing the ballistic motion of fluid particles along the lamella are solved subject to the initial conditions given in equation (2.3) by means of the Lagrangian numerical method already used in RG15. The results of the model for $h(r,t)$ , shown in figure 5, are in excellent agreement with the numerical simulations.

Figure 5.

Comparison, for $\mathit{We}=100$ , between the height of the lamella predicted by the model – ballistic equations (3.1a,b ) subjected to the boundary conditions given in (2.3) – and the numerical result. The vertical line illustrates the radial position $r=a(t)=\sqrt{3t}$ for several instants of time as well as the height of the lamella at $r=\sqrt{3t}$ , $h_{a}(t)=h(r=\sqrt{3t},t)$ ; $v_{a}$ is the liquid velocity at $r=a(t)=\sqrt{3t}$ , namely $v_{a}=u(r=\sqrt{3t},t)$ . (a) $t=0.19$ , (b) $0.25$ , (c) $0.32$ , (d) $0.45$ , (e) $0.55$ , (f) $0.70$ , (g) $0.90$ and (h) $t=1.10$ .

The radial position of the edge of the lamella, $r_{t}(t)$ , 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; Culick Reference Culick1960),

(3.2) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \frac{\text{d}r_{t}}{\text{d}t}=v_{t},\\ \displaystyle \frac{\unicode[STIX]{x03C0}}{4}\frac{\text{d}h_{t}^{2}}{\text{d}t}=[u(r_{t},t)-v_{t}]h(r_{t},t),\\ \displaystyle \frac{\unicode[STIX]{x03C0}\,h_{t}^{2}}{4}\frac{\text{d}v_{t}}{\text{d}t}=[u(r_{t},t)-v_{t}]^{2}\,h(r_{t},t)-2\,\mathit{We}^{-1}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The system of equations (3.2) is solved subjected to the following initial conditions,

(3.3) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle r_{t}=a(t_{e})=\sqrt{3\,t_{e}},\\ \displaystyle v_{t}=v_{t}(t_{e})=1/2\sqrt{3/t_{e}},\\ \displaystyle h_{t}=h_{t}(t_{e})=\sqrt{12}\,t_{e}^{3/2}/\unicode[STIX]{x03C0},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with $t_{e}$ given by (2.2).

Figure 6(a) shows the solution of the system of equations (3.1a,b )–(3.3) for a particular value of the Weber number, whereas figure 6(b) illustrates the comparison between the predicted maximum spreading radius and the experimental data in Tran et al. (Reference Tran, Staat, Prosperetti, Sun and Lohse2012). As was pointed out in Wilderman et al. (Reference Wilderman, Visser, Sun and Lohse2016), there are two clearly differentiated regions in the plot $r_{max}$ versus $\mathit{We}$ of figure 6(b): the impact is elastic for $\mathit{We}\lesssim 10$ since the initial kinetic energy is transformed into surface energy, as it apparent from the capillary waves developing at the interface of the impacting drop whereas, for $\mathit{We}\gtrsim 10$ , the drop spreading process is dominated by inertia. It is discussed in Wilderman et al. (Reference Wilderman, Visser, Sun and Lohse2016) that approximately one-half of the initial kinetic energy is dissipated in the sudden expansion connecting the liquid sheet with the rim for $\mathit{We}\geqslant 10$ , a fact which is implicitly taken into account in the mass and momentum conservation equations (3.2). In spite of the fact that the system of equations (3.2) is analogous to that used in Roisman (Reference Roisman2009), Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010), Villermaux & Bossa (Reference Villermaux and Bossa2011) and Lastakowski et al. (Reference Lastakowski, Boyer, Biance, Pirat and Ybert2014), the crucial difference between the present study and previous ones is that, in our case, we do not impose a specific form for the velocity field within the lamella. Instead, since fluid particles conserve their velocities within the liquid sheet, our system of equations (3.2) subjected to the real initial conditions given in (2.3), with $t_{e}$ calculated self-consistently through equation (2.2), provides us with the correct expressions for both the liquid velocity $u(r_{t})$ and the height of the lamella $h(r_{t})$ upstream of the toroidal rim, as can be inferred from the good agreement existing between experiments and predictions depicted in figure 6(b).

Figure 6.

(a) Radial position $r_{t}$ , velocity $v_{t}$ and height $h_{t}$ of the edge of the lamella as function of the dimensionless time $t$ for $\mathit{We}=150$ . The maximum spreading radius is reached when $r_{t\,max}=R_{max}/R\;\Longleftrightarrow \;v_{t}\approx 0$ . (b) The predicted maximum spreading radius is of the form $R_{max}/R\propto \mathit{We}^{1/2}$ for $\mathit{We}\gtrsim 20$ . The experiments in Tran et al. (Reference Tran, Staat, Prosperetti, Sun and Lohse2012) are plotted for comparison. Two different regimes are observed: for Weber numbers $\mathit{We}\lesssim 10$ , capillary waves develop, distorting the initial shape of the drop, whereas for $\mathit{We}>10$ , the impact is dominated by inertia. The black line represents the maximum spreading radius resulting from the self-consistent theory presented here.

Figure 7.

(a) Ratio $t_{c}/t_{h}$ as function of $t$ for several values of the Weber number. The theory in RG15 predicts the disintegration of the toroidal rim when $0.075\leqslant t_{c}/t_{h}\leqslant 0.095$ (area coloured in blue). Therefore, the predicted critical Weber number is $\mathit{We}^{\ast }\simeq 50$ . (b) For $\mathit{We}>\mathit{We}^{\ast }$ , the rim breaks at $t_{b}\propto \mathit{We}^{-4/3}$ . (c) The diameters and velocities of the droplets ejected for $\mathit{We}>\mathit{We}^{\ast }$ calculated numerically are approximated here as $d_{d}=h_{t}(t_{b})\propto \mathit{We}^{-4/3}$ and $v_{d}=v_{t}(t_{b})\propto \mathit{We}^{2/3}$ respectively.

Another of the advantages of not imposing an ad hoc velocity field within the lamella rests in the fact that both the critical Weber number and the diameters and the velocities of the droplets ejected for impact velocities $V>V^{\ast }$ , which strongly depend on $h(r_{t})$ , can be calculated self-consistently. Indeed, in RG15, following the ideas in Villermaux & Bossa (Reference Villermaux and Bossa2011) and Agbaglah, Josserand & Zaleski (Reference Agbaglah, Josserand and Zaleski2013), we deduced a criterion for the disintegration of the edge of the liquid sheet based on the fact that the droplets composing the spray result from the amplification in the azimuthal direction of Rayleigh–Taylor and capillary instabilities. The growth rates of the disturbances developing in the azimuthal direction of the toroidal rim are highly attenuated as a consequence of the simultaneous growth of its thickness. In consequence, we concluded that drops will only be ejected when the time characterizing the radial growth of the rim, $T_{h}=(R/V)t_{h}=(1/H_{t}\,\text{d}H_{t}/\text{d}T)^{-1}$ , is substantially larger than the capillary time $T_{c}=(R/V)t_{c}=(\unicode[STIX]{x1D70C}\,H_{t}^{3}/8\unicode[STIX]{x1D70E})^{1/2}$ . The breakup time $t_{b}$ is fixed at the instant at which $t_{c}/t_{h}\simeq 0.085$ for the reasons explained in RG15, which are reproduced here for the sake of clarity: (i) the characteristic time of growth of a capillary instability is ${\sim}3\,T_{c}$ , (ii) a Rayleigh’s stability analysis of capillary perturbations reveals that the dimensionless wavenumber $k$ with a fastest growth rate is $k=\unicode[STIX]{x03C0}H_{t}/\unicode[STIX]{x1D706}\simeq 0.7$ , where $\unicode[STIX]{x1D706}$ indicates the wavelength of the perturbation, which is kept constant in time and (iii) the growth of capillary instabilities is inhibited for $k\geqslant 1$ (Eggers & Villermaux Reference Eggers and Villermaux2008). 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 a time $3\,T_{c}$ , $\unicode[STIX]{x1D6E5}k<0.3$ or, equivalently, that $(\text{d}H_{t}/\text{d}T)\times 3T_{c}\lesssim 0.3H_{t}\Rightarrow T_{c}/T_{h}\lesssim 0.1$ , a value which is very close to $t_{c}/t_{h}\simeq 0.085$ .

It is depicted in figure 7(a) that the ratio $t_{c}/t_{h}$ is above the threshold $0.085$ for $\mathit{We}\lesssim 50$ , a fact meaning that the drop never splashes for values of the Weber number, based on $R$ , below 50. Therefore, the critical Weber number predicted by our approach is $\mathit{We}^{\ast }=50$ , a value which is rather close to the experimental one reported in Staat et al. (Reference Staat, Tran, Geerdink, Riboux, Sun, Gordillo and Lohse2015), who find that $\mathit{We}^{\ast }$ based on $R$ slightly varies with temperature within the range $\mathit{We}^{\ast }\simeq 55{-}100$ . Let us point out that $\mathit{We}^{\ast }$ is insensitive to the specific value chosen for the threshold $t_{c}/t_{h}$ , whenever this value is included within the range $0.075\leqslant t_{c}/t_{h}\leqslant 0.095$ (see the shaded region in figure 7). For values of the Weber number above the critical, figure 7(b) shows that the dimensionless breakup time decreases with $\mathit{We}$ as $t_{b}\propto \mathit{We}^{-3/4}$ and, motivated by the good agreement existing between experiments and theory reported in RG15, the diameters and the velocities of the tiny droplets ejected radially outwards from the toroidal rim are calculated respectively as $d_{d}=h_{t}(t_{b})$ and $v_{d}=v_{t}(t_{b})$ . The numerical values for both $d_{d}\propto \mathit{We}^{-4/3}$ and $v_{d}\propto \mathit{We}^{2/3}$ , which need experimental validation, are provided in figure 7(c).