3.1.1 Time scale of compression

Following the analogy between drops and springs, the time period needed for the liquid complex to reach its maximal diameter, $t_{Dmax}$ , is expected to be a fraction of one drop oscillation period. According to Rayleigh’s theory (Rayleigh Reference Rayleigh1879), the oscillation period of a drop of mass $\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}D^{3}/6$ and surface tension $\unicode[STIX]{x1D70E}$ is proportional to $t_{oscill}=\sqrt{\unicode[STIX]{x1D70C}D^{3}/\unicode[STIX]{x1D70E}}$ where $\unicode[STIX]{x1D70E}$ is equivalent to a spring constant. This scaling differs from $t_{Dmax}\approx D/U$ which has been commonly used in most of the studies (Jiang et al. Reference Jiang, Umemura and Law1992; Qian & Law Reference Qian and Law1997). Using the collision photographs, we have recorded the temporal evolution of the disk diameter and deduced $t_{Dmax}$ as described in Roisman et al. (Reference Roisman, Planchette, Lorenceau and Brenn2011). The measurements were performed for the three kinds of collisions using a wide range of liquid properties ( $1.2<\unicode[STIX]{x1D707}<19.3$  mPa s and $17.8<\unicode[STIX]{x1D70E}<68.9~\text{mN}~\text{m}^{-1}$ ), of drop sizes ( $130<D<870~\unicode[STIX]{x03BC}\text{m}$ ) and considering velocities below and above the threshold of fragmentation ( $1<U<7~\text{m}~\text{s}^{-1}$ ). The data of Willis & Orme (Reference Willis and Orme2003) provide $t_{Dmax}$ for other conditions ( $\unicode[STIX]{x1D707}=10$ and 30 mPa s, $\unicode[STIX]{x1D70E}=20~\text{mN}~\text{m}^{-1}$ , $400<D<700~\unicode[STIX]{x03BC}\text{m}$ , $3<U<14~\text{m}~\text{s}^{-1}$ ), sometimes far above the threshold of fragmentation. We have plotted in figure 4 the ratio of the measured $t_{Dmax}$ to $t_{oscill}$ as a function of the Weber number, defined by $We=\unicode[STIX]{x1D70C}DU^{2}/\unicode[STIX]{x1D70E}$ for both binary and ternary collisions with one single liquid. For collisions with immiscible liquids, $t_{oscill}$ and $We$ are defined using the properties of the aqueous drop.

Figure 4.

$t_{Dmax}/t_{oscill}$ as a function of $We$ for binary collisions with one liquid: our data ( $\times$ ), the data of Willis & Orme (Reference Willis and Orme2003) ( $+$ ), for binary collisions with immiscible liquids (glycerol at 50 % and silicon oil M3 (△) or silicon oil M15 (○), and for ternary collisions (

). Solid lines correspond to the semi empirical time scale given by (3.1).

We observe two main regimes corresponding to low and large $We$ , with a transition around $We=50$ . For large $We$ ( $50<We$ ), the proportionality between $t_{Dmax}$ and $t_{oscill}$ is excellent. We selected the data corresponding to $50<We$ and plotted $t_{Dmax}$ as a function of $t_{oscill}$ in figure 5(a). For binary collisions, the slope between $t_{Dmax}$ and $t_{oscill}$ is found to be 0.32. The data of Willis & Orme (Reference Willis and Orme2003), which correspond to larger $Oh$ , show small deviations. The deviations, decreasing with increasing $U$ may be attributed to a change toward a viscous-dominated regime. Using immiscible liquids (silicon oils M3 and M15 with a 50 % glycerol solution), $t_{Dmax}$ of the aqueous drop (glycerol at 50 %) is recovered. The proportionality between $t_{Dmax}$ and $t_{oscill}$ is also observed for ternary drop collisions. In this case, the slope is found to be 0.49, indicating that for ternary drop collisions the disk takes 1.5 times longer to reach its maximal extension than for binary collisions. This ratio can be explained considering the association of identical springs in a series. Each of these identical springs has a mass $\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}D^{3}/6$ and a spring constant $\unicode[STIX]{x1D70E}$ . In the case of binary collisions, two such identical springs are associated in a series, forming an equivalent spring whose mass is $2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}D^{3}/6$ and spring constant is $\unicode[STIX]{x1D70E}/2$ (Symon Reference Symon1971). $t_{Dmax}$ is thus expected to be proportional to $2\sqrt{\unicode[STIX]{x1D70C}D^{3}/\unicode[STIX]{x1D70E}}$ . Similarly, for ternary collisions, the equivalent spring corresponding to the association of three identical springs in a series is expected to give $t_{Dmax}$ proportional to $3\sqrt{\unicode[STIX]{x1D70C}D^{3}/\unicode[STIX]{x1D70E}}$ . The theoretical ratio between the time period obtained with three drops and two drops is thus $3/2$ , as found experimentally.

For low $We$ ( $We<50$ ), the dynamic pressure cannot overcome the Laplace pressure. The drops being too ‘hard’ to be ‘compressed’, they coalesce with their centre of mass moving at approximately $U/2$ . $t_{Dmax}$ is of the order of $D/U$ as shown in the figure 5(b) where $t_{Dmax}$ is plotted as a function of $D/U$ for $We<20$ and $We<40$ for binary and ternary collisions, respectively. The ratio of the slopes, 1.9, is close to the theoretically expected value of 2.

Figure 5.

$t_{Dmax}$ as a function of (a) $t_{oscill}$ for $We>50$ and (b) $D/U$ for $We<20$ and $We<40$ for binary and ternary collisions, respectively. Binary collisions, our data ( $\times$ ), data from Willis & Orme (Reference Willis and Orme2003) ( $+$ ) and ternary collisions (

).

From this analysis we deduce that, for binary and ternary collisions, $t_{Dmax}$ scales with $t_{oscill}$ for high $We$ and with $D/U$ for low $We$ . At intermediate $We$ , both time scales are of the same order of magnitude. As suggested by Bazilevskii et al. (Reference Bazilevskii, Voronkov, Entov and Rozhkov1981), summing up the two time scales provides an overall good approximation of $t_{Dmax}$ which thus writes

(3.1) $$\begin{eqnarray}\displaystyle t_{Dmax}=\unicode[STIX]{x1D6FC}(t_{oscill}+\unicode[STIX]{x1D6FD}D/U). & & \displaystyle\end{eqnarray}$$

Linear least squares fitting provides values of $\unicode[STIX]{x1D6FC}$ equal to 0.27 and 0.39 and $\unicode[STIX]{x1D6FD}$ equal to 1.72 and 2.05, as plotted in figure 4, for single liquid binary and ternary collisions, respectively. For binary collisions with immiscible liquids, the instant of maximal extension is similar to the one expected using two drops of the aqueous solution.

3.1.2 Energy balance

The general formulation of the energy balance for the compression phase reads:

(3.2) $$\begin{eqnarray}\displaystyle E_{K,ini}+E_{\unicode[STIX]{x1D70E},ini}=E_{K,disk}+E_{\unicode[STIX]{x1D70E},disk}+E_{\unicode[STIX]{x1D707},comp}, & & \displaystyle\end{eqnarray}$$

where $E_{K,ini}$ is the initial kinetic energy, $E_{\unicode[STIX]{x1D70E},ini}$ the initial surface energy, $E_{K,disk}$ the kinetic energy stored within the disk at its maximal extension, $E_{\unicode[STIX]{x1D70E},disk}$ the surface energy of the disk and $E_{\unicode[STIX]{x1D707},comp}$ the viscous losses experienced during compression.

For the three kinds of collisions considered in this work, the initial energies are given by the equations listed in table 1. Similarly to the Weber number, we define for binary and ternary collisions with one liquid, the Reynolds and Ohnesorge numbers as $Re=\unicode[STIX]{x1D70C}DU/\unicode[STIX]{x1D707}$ and $Oh=\unicode[STIX]{x1D707}/\sqrt{\unicode[STIX]{x1D70C}D\unicode[STIX]{x1D70E}}$ . For binary collisions with immiscible liquids, these numbers will be defined in the discussion.

Table 1.

Initial energies for the three types of drop collisions considered in this work. For immiscible liquids, $\unicode[STIX]{x1D70E}_{o}$ and $\unicode[STIX]{x1D70E}_{w}$ correspond to the surface tensions of the oil and of the aqueous solution, respectively.

3.1.3 Neglecting $E_{K,disk}$

In most of the modelling approaches as from Jiang et al. (Reference Jiang, Umemura and Law1992), Qian & Law (Reference Qian and Law1997) and Planchette et al. (Reference Planchette, Lorenceau and Brenn2012), it is common to assume that $E_{K,disk}$ is negligible. To our knowledge, no experimental observations have been published to confirm this hypothesis. Numerical studies, however, seem to support this assumption, as for example Nobari, Jan & Tryggvason (Reference Nobari, Jan and Tryggvason1996), Dai & Schmidt (Reference Dai and Schmidt2005) and Nikolopoulos et al. (Reference Nikolopoulos, Nikas and Bergeles2009).

In the present work, further evaluation of $E_{K,disk}$ has been preformed using numerical simulations. For simplicity and since the numerical simulations of binary collisions have already been successfully implemented and validated (Focke & Bothe Reference Focke and Bothe2011, Reference Focke and Bothe2012; Focke et al. Reference Focke, Kuschel, Sommerfeld and Bothe2013; Liu & Bothe Reference Liu and Bothe2016), we focus here on this type of collision. The numerical simulations were performed with the in-house code Free Surface 3D (FS3D) that is based on the Volume-of-Fluid (VOF) method (Hirt & Nichols Reference Hirt and Nichols1981). In the simulations, the further improved version of the lamella stabilization algorithm from Liu & Bothe (Reference Liu and Bothe2016) is employed in order to prevent the unphysical lamella rupture at higher Weber numbers as described in Focke & Bothe (Reference Focke and Bothe2011), Focke & Bothe (Reference Focke and Bothe2012) and Liu & Bothe (Reference Liu and Bothe2016). The surface tension force is incorporated by means of the continuum surface stress (CSS) model (Lafaurie et al. Reference Lafaurie, Nardone, Scardovelli, Zaleski and Zanetti1994). All simulations are conducted using a computational domain of $500~\unicode[STIX]{x03BC}\text{m}\times 500~\unicode[STIX]{x03BC}\text{m}\times 1000~\unicode[STIX]{x03BC}\text{m}$ and 128 $\times$ 128 $\times$ 256 grid cells with three symmetry planes, resulting in a resolution of at least 87 cells per drop diameter.

Four cases are selected that correspond to experimentally investigated conditions. The drop diameter is fixed to $340~\unicode[STIX]{x03BC}\text{m}$ , glycerol solution at 50 % and silicon oil M5 are considered, and for each of these two liquids two relative drop velocities are considered: one leading to coalescence and one resulting in fragmentation. Expressed in terms of $Oh$ and $We$ , the four cases correspond to $Oh=0.03$ (glycerol 50 %) with $We=25$ and 56 and $Oh=0.06$ (silicon oil) with $We=35$ and 69. The shape of the drops obtained from the simulation are very similar to the ones observed in the experiments, as shown in figure 6 for the case $Oh=0.03$ .

Figure 6.

Temporal evolution of the drop shapes after collision as observed in experiments and obtained by numerical simulations. $Oh=0.03$ . (a) $We=25$ . (b) $We=56$ .

The temporal evolution of the different energetic contributions, as found in the numerical simulations, are presented in figure 7. At the instant of maximal disk extension – marked by black vertical lines – the internal kinetic energy represents less than 2 % of the total energy, and so for $Oh$ equal to 0.03 and 0.06, below and above the critical velocity of fragmentation. Together with the results of Nobari et al. (Reference Nobari, Jan and Tryggvason1996), Dai & Schmidt (Reference Dai and Schmidt2005) and Nikolopoulos et al. (Reference Nikolopoulos, Nikas and Bergeles2009), these findings validate the assumption that the internal kinetic energy at the end of the compression is negligible for a wide range of $Oh$ (0.02–0.38). Generalizing this result to other kinds of drop collisions, (3.2) can be approximated by

(3.3) $$\begin{eqnarray}\displaystyle E_{K,ini}+E_{\unicode[STIX]{x1D70E},ini}=E_{\unicode[STIX]{x1D70E},disk}+E_{\unicode[STIX]{x1D707},comp}. & & \displaystyle\end{eqnarray}$$

Figure 7.

Temporal evolution of the surface and kinetic energies, and of the accumulated viscous losses during the compression phase of binary drop collisions ( $E_{total}$ is the sum of $E_{K}$ , $E_{\unicode[STIX]{x1D70E}}$ and $E_{\unicode[STIX]{x1D707}}$ ). From left to right: (a) $Oh=0.03$ , $We=25$ and $We=56$ ; (b) $Oh=0.06$ , $We=35$ and $We=69$ . The black vertical lines (solid) mark the maximal extension of the disk. The kinetic energy in that state is below 2 % of the initial energy for all cases. The dashed vertical lines mark the break up of the liquid cylinder formed during relaxation.

3.1.4 Estimating $E_{\unicode[STIX]{x1D707},comp}$ via scaling argument

In this section we propose to estimate $E_{\unicode[STIX]{x1D707},comp}$ based on scaling arguments. The results, established for binary collisions, are later compared to numerical and experimental data.

Figure 8.

Schematic representation of liquid complex during compression (cut in the centre) showing the two main locations of viscous losses. The blue crown corresponds to the volume in the lamella while the orange torus is the rim volume.

Despite the similarity between binary drop collisions and impact of drops on solid walls, scaling relations cannot be interchanged due to the development of a viscous boundary layer at the vicinity of the wall (Roisman Reference Roisman2009b ). It is therefore crucial to figure out where the viscous losses take place for binary drop collisions. Nikolopoulos et al. (Reference Nikolopoulos, Nikas and Bergeles2009) numerically calculated the viscous losses taking place both in the liquid and gaseous phases and showed that losses in the surrounding gaseous phase are a small fraction of the total losses. Thus, thereafter, only the liquid phase is considered. The numerical simulations by Nobari et al. (Reference Nobari, Jan and Tryggvason1996) indicate that, for bouncing drops, i.e. small $We$ , the highest dissipation per unit volume is found at the outer edges of the drops and not in a thin layer near the stagnation point as proposed by Jiang et al. (Reference Jiang, Umemura and Law1992). Roisman (Reference Roisman2004), Roisman et al. (Reference Roisman, Planchette, Lorenceau and Brenn2011) who focused on the inertia-dominated regime (high $We$ and $Re$ ), localized the losses in the lamella just upstream from the rim and, to a smaller extend, in the rim itself. Accounting for these findings, we propose different possible locations of the losses. We then estimate the losses and discuss the regime where they may dominate.

(i) Losses in the lamella – just upstream from the rim: $E_{\unicode[STIX]{x1D707},comp,1}$ . For the inertia-dominated regime, the liquid flows from the centre of the disk towards its outer edge, where it is slowed down by surface tension forcing the liquid to accumulate in a rim. The biggest velocity gradients are thus found in the lamella just upstream from the rim, see figure 8. The difference of velocities between the flow in the rim and in the lamella can be approximated by $U$ , while the length scale of deceleration is the one of the viscous boundary layer developing just in front of the rim. As a consequence:

(3.4) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},comp,1}\approx \unicode[STIX]{x1D707}\left(\frac{U}{l_{\unicode[STIX]{x1D707}}}\right)^{2}\unicode[STIX]{x1D6FA}_{1}T_{1}, & & \displaystyle\end{eqnarray}$$

with $l_{\unicode[STIX]{x1D707}}\approx \unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}U$ . The volume $\unicode[STIX]{x1D6FA}_{1}$ on which this dissipation occurs is the volume of a crown of width $l_{\unicode[STIX]{x1D707}}$ and thickness $h$ , the thickness of the lamella close to the rim. The diameter of the crown can be approximated by $D_{max}$ , the maximum diameter of the disk, which scales at first order like $D\sqrt{We}$ (Jiang et al. Reference Jiang, Umemura and Law1992; Willis & Orme Reference Willis and Orme2000, experimental results of § 3.1.6). As in Roisman (Reference Roisman2004) and Roisman et al. (Reference Roisman, Planchette, Lorenceau and Brenn2011), we use the result of Taylor (Reference Taylor1959) to estimate $h\approx \unicode[STIX]{x1D70E}/2\unicode[STIX]{x1D70C}U^{2}$ .

The time period $T_{1}$ on which these losses develop could at first be estimated by $t_{Dmax}$ . In the first instants of the collision, however, the lamella is not yet formed, and approximating $T_{1}$ by $t_{oscill}$ seems more appropriate here.

We obtain

(3.5) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},comp,1}\propto \unicode[STIX]{x1D70C}D^{3}U^{2}. & & \displaystyle\end{eqnarray}$$

Note that taking $T_{1}\approx t_{Dmax}$ would influence the proposed scaling only for small $We$ which does not correspond to the inertia-dominated regime.

(ii) Losses in the rim: $E_{\unicode[STIX]{x1D707},comp,2}$ . In the work of Roisman et al. (Reference Roisman, Planchette, Lorenceau and Brenn2011) focusing on the inertia-dominated regime, losses in the rim are added to the dominating ones taking place in the lamella. The velocity in the rim is approximated by the interface velocity, itself derived from volume conservation between the lamella and the rim. The length scale of the velocity gradient is estimated by $l_{\unicode[STIX]{x1D707}}$ . With these assumptions, the losses in the rim scale as those of the lamella but are found to be much smaller.

We propose here an alternative scaling that accounts for smaller $Re$ , i.e. going beyond the limit of inertia-dominated regime. In the rim, capillary effects dominate over inertia and the liquid velocity can be estimated by $V_{rim}\approx \sqrt{\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}r_{rim}}$ where $r_{rim}$ designates the rim radius. This velocity is expected to be much smaller than the one in the lamella, and for high viscosity, the viscous boundary layer in the rim becomes probably as large as the radius of the rim itself. Thus, the losses in the rim are:

(3.6) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},comp,2}\approx \unicode[STIX]{x1D707}\left(\frac{V_{rim}}{r_{rim}}\right)^{2}\unicode[STIX]{x1D6FA}_{2}T_{2}. & & \displaystyle\end{eqnarray}$$

Noting that the lamella is very thin and does not represent a significant volume of liquid, volume conservation allows writing $r_{rim}\approx DWe^{-1/4}$ . The volume $\unicode[STIX]{x1D6FA}_{2}$ on which these losses occur corresponds to the rim volume, i.e. to $r_{rim}^{2}D_{max}\propto D^{3}$ , in orange in figure 8. The time period $T_{2}$ can be taken as $t_{Dmax}$ , since the rim starts forming immediately after impact. In this case, we obtain:

(3.7) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},comp,2}\propto \unicode[STIX]{x1D70C}D^{3}U^{2}\frac{We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4}}{Re}. & & \displaystyle\end{eqnarray}$$

This expression is valid if the viscous boundary layer thickness becomes effectively comparable to the rim radius, i.e. in the limit of small $Re$ . Note that approximating $T_{2}$ by $t_{oscill}$ would result in $E_{\unicode[STIX]{x1D707},comp,2}\propto \unicode[STIX]{x1D70C}D^{3}U^{2}We^{1/4}Re^{-1}$ . Differences would be mainly found for low $We$ , where the drops hardly deform into a lamella.

(iii) Low $Oh$ : $E_{\unicode[STIX]{x1D707},comp,3}$ . We observe for collisions at low $Oh$ that the lamella is less pronounced and its diameter remains small in comparison to $D_{max}$ , while it is surrounded by a large rim. We attribute this difference to the strong capillarity (small $We$ ), which limits the extension of the disk and possibly to the low viscosity (high $Re$ ) which slows down the flow development. It is worth noting that $Oh=t_{oscill}/t_{\unicode[STIX]{x1D707}}$ and consequently $Oh\approx t_{Dmax}/t_{\unicode[STIX]{x1D707}}$ , with the viscous time scale $t_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D70C}D^{2}/\unicode[STIX]{x1D707}$ . In the case of low $Oh$ , and hence low $t_{Dmax}/t_{\unicode[STIX]{x1D707}}$ , the viscous boundary layer may not have enough time to fully develop before the disk recoils. Let us thus consider a viscous layer of thickness $l_{\unicode[STIX]{x1D707}}^{\ast }=\sqrt{(\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C})t_{Dmax}}$ ( $\ll l_{\unicode[STIX]{x1D707}}$ ). The losses can be obtained by integrating $\unicode[STIX]{x1D707}U^{2}/{l_{\unicode[STIX]{x1D707}}^{\ast }}^{2}$ over $t_{Dmax}$ in a volume proportional to $D^{2}l_{\unicode[STIX]{x1D707}}^{\ast }$ . The losses are expected to scale as $\unicode[STIX]{x1D70C}D^{3}U^{2}\sqrt{t_{Dmax}/t_{\unicode[STIX]{x1D707}}}$ . Approximating $t_{Dmax}$ by $t_{oscill}$ is equivalent to neglecting $\unicode[STIX]{x1D6FD}$ by comparison to $\sqrt{We}$ and seems practically acceptable for all fragmentation conditions ( $\unicode[STIX]{x1D6FD}<\sqrt{We}$ for $We>3$ ). Thus, we obtain:

(3.8) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},comp,3}\propto \unicode[STIX]{x1D70C}D^{3}U^{2}\sqrt{Oh}. & & \displaystyle\end{eqnarray}$$

This analysis offers three kinds of losses. For small $Oh$ , the lamella does not totally develop and (3.8) is expected to apply. When the lamella forms, purely inertial losses are expected that scale as predicted by (3.5). In the limit of low $Re$ , additional losses in the rim must be added which follow the scaling of (3.7).

3.1.5 Surface at maximal deformation

To compare the predictions of (3.5), (3.7) and (3.8) with the experimental results, we first evaluate $E_{\unicode[STIX]{x1D70E},disk}$ from collision photographs. $E_{\unicode[STIX]{x1D707},comp}$ is obtained in a second step via (3.3). Yet, the exact shape of the liquid complex at maximal extension is difficult to describe, and the radius of the rim is difficult to measure. In the case of binary collisions, Willis & Orme (Reference Willis and Orme2003) showed that this surface can be well approximated by the one of a pancake of diameter $D_{max}$ and thickness $e\approx 4/3\,D^{3}D_{max}^{-2}$ (volume conservation). The difference from the surface of a torus of the same maximal diameter was found to be of the order of typical experimental uncertainty. Thus, we estimate the surface energy of the liquid complex at its maximal extension by:

(3.9) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D70E},disk}=\frac{\unicode[STIX]{x03C0}}{2}\unicode[STIX]{x1D70E}{D_{max}}^{2}\left(1+\frac{8}{3}\left(\frac{D}{D_{max}}\right)^{3}\right). & & \displaystyle\end{eqnarray}$$

This estimation is in good agreement with the numerical simulations. For $Oh=0.03$ and $D_{max}/D=1.80$ (or $D_{max}/D=2.10$ ), we obtain $E_{\unicode[STIX]{x1D70E},disk}/E_{\unicode[STIX]{x1D70E},ini}$ equal to 1.18 (or 1.42) using (3.9), and to 1.11 (or 1.44) from the simulations. For $Oh=0.06$ , with $D_{max}/D=1.85$ (or $D_{max}/D=2.13$ ), $E_{\unicode[STIX]{x1D70E},disk}/E_{\unicode[STIX]{x1D70E},ini}$ equals 1.20 (or 1.43) with (3.9), and 1.13 (or 1.42) from the simulations. The differences decrease with increasing $D_{max}/D$ and remain below 7 % for $D_{max}/D>1.8$ validating the use of (3.9).

For binary drop collisions with immiscible liquids, we use by analogy

(3.10) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D70E},disk}=\frac{\unicode[STIX]{x03C0}}{2}{D_{max}}^{2}\left((\unicode[STIX]{x1D70E}_{o}+\unicode[STIX]{x1D70E}_{ow})+\frac{8}{3}\left(\frac{D}{D_{max}}\right)^{3}\left(\unicode[STIX]{x1D70E}_{o}+\frac{\unicode[STIX]{x1D70E}_{ow}}{2}\right)\right) & & \displaystyle\end{eqnarray}$$

and in the ternary case

(3.11) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D70E},disk}=\frac{\unicode[STIX]{x03C0}}{2}\unicode[STIX]{x1D70E}{D_{max}}^{2}\left(1+4\left(\frac{D}{D_{max}}\right)^{3}\right). & & \displaystyle\end{eqnarray}$$

3.1.6 Experimental estimation of $E_{\unicode[STIX]{x1D707},comp}$

For each liquid and droplet size we vary $U$ , measure $D_{max}$ on the photographs, obtain the experimental $E_{\unicode[STIX]{x1D70E},disk}$ and further compute $E_{\unicode[STIX]{x1D707},comp}$ which we plot as a function of $E_{K,ini}$ . This approach is similar to the one followed by Jiang et al. (Reference Jiang, Umemura and Law1992) and Willis & Orme (Reference Willis and Orme2003), where the evolution of ${\hat{S}}=E_{\unicode[STIX]{x1D70E},disk}/E_{\unicode[STIX]{x1D70E},ini}$ with $We^{\ast }=E_{K,ini}/E_{\unicode[STIX]{x1D70E},ini}$ was studied. Indeed, noting  $E_{\unicode[STIX]{x1D707},comp}=f(E_{K,ini})$ gives ${\hat{S}}=1+We^{\ast }-f(We^{\ast })$ . Since it has been reported that ${\hat{S}}$ linearly increases with $We^{\ast }$ (Jiang et al. Reference Jiang, Umemura and Law1992; Willis & Orme Reference Willis and Orme2003), $E_{\unicode[STIX]{x1D707},comp}$ is expected to linearly increase with $E_{K,ini}$ . Typical results are shown in figure 9 for each kind of collision.

As expected, $E_{\unicode[STIX]{x1D707},comp}$ increases nearly linearly with $E_{K,ini}$ , which implies that, at first order, $D_{max}$ scales like $D\sqrt{We}$ . For wide ranges of $U$ , however, we observe some deviations. To our knowledge, the deviation from linearity has not been discussed in the literature so far but was already visible on the measurements of Willis & Orme (Reference Willis and Orme2003). To overcome this difficulty we restrict ourselves to velocity ranges for which the deviations from the linear evolution are not significant. As a result, for some cases, we obtain two different slopes, one for rather low velocities and the other one for large velocities. The local slope of $E_{\unicode[STIX]{x1D707},comp}$ as a function of $E_{K,ini}$ is thus systematically recorded. We denote this local slope as $a$ and calculate $Re$ and $We$ for the median value of $U$ .

Figure 9.

Examples of $E_{\unicode[STIX]{x1D707},comp}$ as a function of $E_{K,ini}$ . (a) Binary collisions with one liquid, (b) ternary collisions, (c) binary collisions with immiscible liquids ( $D=200~\unicode[STIX]{x03BC}\text{m}$ ). For wide ranges of $U$ , deviations to the expected linear relation are observed (a) and (b) with SO M5.

This procedure is followed for binary collisions with $0.007<Oh<0.12$ , including aqueous solutions of glycerol and silicon oils, and drop diameters between 130 and $380~\unicode[STIX]{x03BC}\text{m}$ . The values corresponding to $0.1<Oh<0.33$ deduced from the data of Willis & Orme (Reference Willis and Orme2003) are also considered. For ternary collisions, the studied conditions correspond to $0.02<Oh<0.12$ with $130<D<380~\unicode[STIX]{x03BC}\text{m}$ . Even though rather moderate $D_{max}/D$ are found for low $Oh$ , most of the linear fits are made for $D_{max}/D>1.8$ , excluding very large errors in estimating $E_{\unicode[STIX]{x1D70E},disk}$ . The results are shown in figure 10. In the upper graph, we have chosen to plot $a$ as a function of $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re$ ; $a$ as a function of $Oh$ , $We$ and $Re$ are successively plotted in the lower graphs. For binary collisions with immiscible liquids, we use $We$ defined on the encapsulated drop only, since it fixes $t_{Dmax}$ . At this stage of the collision, the viscous losses are assumed to happen in both phases, thus an average Reynolds number defined as $\tilde{Re}=(\unicode[STIX]{x1D70C}_{o}+\unicode[STIX]{x1D70C}_{w})DU/(\unicode[STIX]{x1D707}_{o}+\unicode[STIX]{x1D707}_{w})$ is used.

Figure 10.

(a) $a$ (slope of $E_{\unicode[STIX]{x1D707},comp}$ as a function of $E_{K,ini}$ ) as a function $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re$ . For collisions with one liquid, binary ( $\times$ and $+$ )  $\unicode[STIX]{x1D6FC}=0.27$ , $\unicode[STIX]{x1D6FD}=1.72$ and ternary collisions (

) $\unicode[STIX]{x1D6FC}=0.39$ , $\unicode[STIX]{x1D6FD}=2.05$ . For binary collisions of immiscible liquid drops (▵ and ▫) $\unicode[STIX]{x1D6FC}=0.27$ , $\unicode[STIX]{x1D6FD}=1.72$ , $We$ is defined with the encapsulated liquid properties and $Re$ is taken as $(\unicode[STIX]{x1D70C}_{o}+\unicode[STIX]{x1D70C}_{w})DU/(\unicode[STIX]{x1D707}_{o}+\unicode[STIX]{x1D707}_{w})$ . (b–d) $a$ , (same points, one liquid) as a function of $Oh$ , $We$ and $Re$ from top to bottom. Black line, (3.8).

For every kind of collision, we observe a plateau on which $a$ is constant. For binary collisions with one liquid, this plateau is preceded and followed by different regimes. At low $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re$ , a fast increase of $a$ is observed, while for large $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re$ a rather moderate increase is visible. The value of $a$ on the plateau indicates that 65 % of the initial kinetic energy has been lost at this stage of the process. For ternary collisions, only two of the three regimes are seen. The plateau at moderate $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re$ , which corresponds to a loss of 44 % of the initial kinetic energy, and the regime for larger $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re$ , where $a$ slightly increases. The absence of experimental points for very low $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re$ can explain that the first regime of fast increase is not observed. Finally, for binary collisions of immiscible liquid drops, only the plateau is visible. Considering the wide range of parameters probed, the sensitivity of viscous dissipation to $We$ and $Re$ appears to be weak. The value of $a$ is found to be 0.97, close to 0.96 as reported in Planchette et al. (Reference Planchette, Lorenceau and Brenn2012), where the surface of the liquid complex at maximal extension was approximated by disks only. The dissipated energy during compression is thus much higher for immiscible liquid drops than for collisions with one single liquid.

3.3.1 Estimating $E_{\unicode[STIX]{x1D70E},ini}-E_{\unicode[STIX]{x1D70E},cyl}$ at the fragmentation threshold

At the fragmentation threshold, two drops of equal sizes are produced (Rayleigh criterion). For binary collisions, the volume conservation requires that these two drops have the diameter $D$ of the initial drops. Consequently, the surface energy of the drops resulting from the fragmentation, which is by definition equal to the one of the cylinder, $E_{\unicode[STIX]{x1D70E},cyl}$ , must be equal to $E_{\unicode[STIX]{x1D70E},ini}$ . We deduce $E_{\unicode[STIX]{x1D70E},cyl}=E_{\unicode[STIX]{x1D70E},ini}$ .

In the case of ternary collisions, we obtain: $E_{\unicode[STIX]{x1D70E},ini}-E_{\unicode[STIX]{x1D70E},cyl}=(1-(2/3)^{1/3})E_{\unicode[STIX]{x1D70E},ini}$ which can also be expressed as $72(1-(2/3)^{1/3})E_{K,ini}/We$ .

For collisions with immiscible liquids, this energy difference depends on the distribution of the encapsulated liquid within the two resulting drops. It can be written as: $E_{\unicode[STIX]{x1D70E},ini}-E_{\unicode[STIX]{x1D70E},cyl}=\unicode[STIX]{x03C0}D^{2}(\unicode[STIX]{x1D70E}_{w}-\unicode[STIX]{x1D70E}_{o}-2\unicode[STIX]{x1D714}\unicode[STIX]{x1D70E}_{ow})$ where $\unicode[STIX]{x1D714}$ varies from $2^{-2/3}\approx 0.63$ for equally distributed liquids to 0.5 in the limit case where all the encapsulated liquid is in one drop only. The relative variations are quite small, and for a given combination of surface and interfacial tensions $E_{\unicode[STIX]{x1D70E},ini}-E_{\unicode[STIX]{x1D70E},cyl}$ can be regarded as constant.

3.3.2 Scaling relations for $E_{\unicode[STIX]{x1D707},relax}$

The estimation of the viscous losses during the relaxation phase is quite difficult. There exist different sources of velocity gradients, such as the Laplace pressure difference between the torus and the lamella responsible for the torus to drain into the lamella, as well as the interface retraction for the minimization of surface energy and resulting in the collapse of the torus. The length scales are also very different, involving large dimensions ( $D$ , $D_{max}$ and $L_{max}$ , the maximal cylinder length) and much smaller ones, such as the thickness of the lamella $h$ or of the viscous boundary layer $l_{\unicode[STIX]{x1D707}}$ . To approach this problem, we distinguish different sub-periods or sub-cases during the relaxation, as in the work of Qian & Law (Reference Qian and Law1997) and Roisman (Reference Roisman2004). Schematically, the first one extends until the lamella disappears engulfed in the retracting torus, the second one follows until the cylinder reaches its maximal elongation and eventually breaks up. This evolution is represented in figure 13.

(i) First instants. For the first sub-period, the general expression of the viscous losses reads

(3.14) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},relax,3}=\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x0394}V}{\unicode[STIX]{x0394}X}\right)^{2}\unicode[STIX]{x1D6FA}_{3}T_{3}. & & \displaystyle\end{eqnarray}$$

This sub-period corresponds more or less to the relaxation of a distorted drop in the oblate phase. The time period $T_{3}$ scales as $t_{oscill}$ and the velocity $\unicode[STIX]{x0394}V$ as $D_{max}/t_{oscill}\approx U$ . We thus expect, as for the compression phase, the velocity variations to remain limited to the viscous boundary layer of thickness $l_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}U$ . We obtain the velocity gradient $\unicode[STIX]{x0394}V/\unicode[STIX]{x0394}X\approx \unicode[STIX]{x1D70C}U^{2}/\unicode[STIX]{x1D707}$ . The volume $\unicode[STIX]{x1D6FA}_{3}$ is difficult to estimate. We propose to estimate this volume as the overlap of the lamella and the torus. As a result, $\unicode[STIX]{x1D6FA}_{3}$ scales like the lamella thickness (again estimated by $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}U^{2}$ ) and a part of the torus immediately at the contact with the lamella, estimated by $r_{rim}^{2}$ . This leads to $\unicode[STIX]{x1D6FA}_{3}\approx (D^{2}\unicode[STIX]{x1D70E})/(\unicode[STIX]{x1D70C}We^{1/2}U^{2})$ . The losses scale as

(3.15) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},relax,3}\approx b\frac{\unicode[STIX]{x1D70C}U^{2}D^{3}}{OhWe^{1/2}}, & & \displaystyle\end{eqnarray}$$

where $b$ is a constant. This expression is valid as long as $b/(OhWe^{1/2})$ is less than $(1-a)$ .

(ii) Cylinder elongation. During the second sub-period, which can be seen as a drop oscillation in the prolate phase, the losses read

(3.16) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},relax,4}=\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x0394}V}{\unicode[STIX]{x0394}X}\right)^{2}\unicode[STIX]{x1D6FA}_{4}T_{4}. & & \displaystyle\end{eqnarray}$$

To estimate the velocity gradient, the velocity $V$ is approximated by $L_{max}/T_{4}$ , with $L_{max}$ the maximal length reached by the cylinder, and $T_{4}$ the oscillation time scale of the merged drop. By analogy with a water spring, the liquid complex relaxes over a time scale of $\sqrt{2}t_{oscill}$ and $\sqrt{3}t_{oscill}$ for the binary and ternary cases, respectively. Similarly, the variations of velocity typically take place on $2^{1/3}D$ and $3^{1/3}D$ . Finally, the volume $\unicode[STIX]{x1D6FA}_{4}$ can be estimated by $2D^{3}$ and $3D^{3}$ for binary and ternary cases. Following these scalings leads to

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle E_{\unicode[STIX]{x1D707},relax,4}=\unicode[STIX]{x1D707}\left(\frac{L_{max}}{2^{1/3}D}\right)^{2}\frac{\sqrt{2}D^{3}}{t_{oscill}}\quad \text{binary collisions}, & \displaystyle\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle E_{\unicode[STIX]{x1D707},relax,4}=\unicode[STIX]{x1D707}\left(\frac{L_{max}}{3^{1/3}D}\right)^{2}\frac{\sqrt{3}D^{3}}{t_{oscill}}\quad \text{ternary collisions}. & \displaystyle\end{eqnarray}$$

(iii) On the relative importance of these losses. The distinction between the two sub-periods is rather arbitrary, since the disk retraction and the cylinder elongation do not happen sequentially, but rather in parallel. The liquid system at the end of the compression is neither a pancake nor an ellipsoid, which makes the first instants of its retraction difficult to describe. Qualitatively, its phenomenology depends on its exact shape, and the two extreme cases represented in figure 14 can be distinguished. Denoting $D_{merged}$ as the diameter of the merged drop ( $D_{merged}=2^{1/3}D$ or $3^{1/3}D$ ), the first limit corresponds to large $D_{max}/D_{merged}$ . For a given $We$ , $D_{max}/D_{merged}$ is larger for binary than for ternary collisions, and it increases with $We$ for a given collision type. In the limit where $D_{max}/D_{merged}$ is comparable to unity, the flux from the torus to the lamella remains small since the Laplace pressure gradient is moderate. The torus collapses on itself. The velocity gradients are mainly limited to the viscous boundary layer, and the losses occur on the volume corresponding to the overlap of the lamella with the torus. The scaling proposed for $E_{\unicode[STIX]{x1D707},relax,3}$ seems relevant. After this collapse, the cylinder extends and further losses are expected in the form of $E_{\unicode[STIX]{x1D707},relax,4}$ . The total losses can be estimated by summing up $E_{\unicode[STIX]{x1D707},relax,3}$ and $E_{\unicode[STIX]{x1D707},relax,4}$ . For large $D_{max}/D_{merged}$ , the small rim radius ( $r_{rim}\propto DWe^{-1/4}$ ) induces a large pressure gradient which drains the torus into the lamella before the disk significantly retracts (retraction velocity scales as $D_{max}/t_{oscill}$ ). The scaling proposed for the losses of the first instants, based on the torus collapse, is not fully relevant any more, and the total losses are expected to scale as $E_{\unicode[STIX]{x1D707},relax,4}$ , the expression proposed for the second sub-period.

Figure 13.

Schematic temporal evolution of the disk relaxing into a cylinder (cross-section). During the first sub-period, the lamella is engulfed in the collapsing torus. In the second one, the cylinder elongates. The volumes assumed to have the highest velocity gradients are shown in grey.

For drop collisions with immiscible liquids, this approach must be modified to take into account the structure of the merged entity which consists of an aqueous core and an encapsulating oil shell. As already reported in Planchette et al. (Reference Planchette, Lorenceau and Brenn2012), the losses are assumed to be mainly located in the oil shell, which provides the scaling

(3.19) $$\begin{eqnarray}\displaystyle E_{\unicode[STIX]{x1D707},relax}\approx \unicode[STIX]{x1D707}_{o}\frac{L_{max}}{D}\frac{D^{3}}{t_{oscill,o}}. & & \displaystyle\end{eqnarray}$$

Here $t_{oscill,o}$ is calculated using the density and surface tension of the oil.

Figure 14.

Possible evolution (qualitative) of the liquid complex during relaxation (cross-section). (a) Laplace pressure gradients drain the torus in the lamella before it collapses. (b) The torus collapses on itself, engulfing the lamella.

3.4.1 Scaling laws

With the help of the modified Rayleigh criterion (see figure 12), $L_{max}/D$ can be taken equal to $\unicode[STIX]{x1D701}_{crit}d/D$ where $d$ is the cylinder diameter, for all collisions with $Oh<0.1$ .

For binary collisions with one liquid, combining the results of § 3.1, the facts that $E_{\unicode[STIX]{x1D70E},ini}=E_{\unicode[STIX]{x1D70E},cyl}$ and $d/D\approx 0.8$ , with the scaling obtained for $E_{\unicode[STIX]{x1D707},relax}$ (3.15) and (3.17), provides the theoretical threshold velocity

(3.20) $$\begin{eqnarray}\displaystyle U_{t,2D}^{theo}=0.8\unicode[STIX]{x1D701}_{crit}\sqrt{\frac{24\sqrt{2}/\unicode[STIX]{x03C0}}{1-a-b_{2D}/(We^{1/2}Oh)}}\sqrt{\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D70C}\,t_{oscill}}}. & & \displaystyle\end{eqnarray}$$

For ternary collisions, $d/D\approx 0.9$ and we similarly obtain

(3.21) $$\begin{eqnarray}\displaystyle U_{t,3D}^{theo}=0.9\unicode[STIX]{x1D701}_{crit}\sqrt{\frac{24\sqrt{3}/\unicode[STIX]{x03C0}}{1-a-b_{3D}/(We^{1/2}Oh)+72/(We(1-(2/3)^{1/3}))}}\sqrt{\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D70C}\,t_{oscill}}}, & & \displaystyle\end{eqnarray}$$

where $b_{2D}$ and $b_{3D}$ are two constants specifying the relative importance of the losses in the first and second sub-periods of the relaxation.

For binary collisions with immiscible liquids, the scaling law accounting for surface energy variations, is

(3.22) $$\begin{eqnarray}\displaystyle U_{t,IM}^{theo}\approx \sqrt{A\unicode[STIX]{x1D701}_{crit}\frac{\unicode[STIX]{x1D707}_{o}}{\unicode[STIX]{x1D70C}_{o}+\unicode[STIX]{x1D70C}_{w}}\sqrt{\frac{\unicode[STIX]{x1D70E}_{o}}{\unicode[STIX]{x1D70C}_{o}D^{3}}}-\frac{48(\unicode[STIX]{x1D70E}_{w}-\unicode[STIX]{x1D70E}_{o}-2w\unicode[STIX]{x1D70E}_{ow})}{(1-a_{I})(\unicode[STIX]{x1D70C}_{o}+\unicode[STIX]{x1D70C}_{w})D}}, & & \displaystyle\end{eqnarray}$$

where $A$ is a constant. By treating the surface energy terms as a constant, as in Planchette et al. (Reference Planchette, Lorenceau and Brenn2012), it simplifies to:

(3.23) $$\begin{eqnarray}\displaystyle U_{t,IM}^{theo}\approx 50\sqrt{\unicode[STIX]{x1D701}_{crit}\frac{\unicode[STIX]{x1D707}_{o}}{\unicode[STIX]{x1D70C}_{o}+\unicode[STIX]{x1D70C}_{w}}\sqrt{\frac{\unicode[STIX]{x1D70E}_{o}}{\unicode[STIX]{x1D70C}_{o}D^{3}}}}. & & \displaystyle\end{eqnarray}$$

This modelling approach is a priori valid as long as the Rayleigh criterion is followed, i.e. for $Oh<0.1$ .

3.4.2 Comparison to experimental results

For each kind of collision we experimentally determined the fragmentation threshold using a wide range of liquid properties ( $887<\unicode[STIX]{x1D70C}<1935~\text{kg}~\text{m}^{-3}$ , $1.2<\unicode[STIX]{x1D707}<19.3~\text{mPa}~\text{s}$ and $17.8<\unicode[STIX]{x1D70E}<68.9~\text{mN}~\text{m}^{-1}$ ) and droplet diameters (between 130 and $870~\unicode[STIX]{x03BC}\text{m}$ ). Practically, the threshold velocity is taken as the average of the highest velocity leading to coalescence and the lowest one resulting in fragmentation, which corresponds to typical errors of $\pm 0.05~\text{m}~\text{s}^{-1}$ . These experimental thresholds are plotted in figure 15 as a function of the theoretical ones $U_{t,2D}^{theo}$ , $U_{t,3D}^{theo}$ . The scaling predictions are first evaluated with $b_{2D}$ and $b_{3D}$ equal to zero, $a$ equals to $a_{plat,2D}$ and $a_{plat,3D}$ for values of $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re$ corresponding to the plateaus and with the values of $a$ estimated in figure 10 for other cases. For collisions with immiscible liquids, the value $a_{I}=0.97$ is integrated in the prefactor of (3.23), and $\unicode[STIX]{x1D701}_{crit}=3$ for all cases. Overall, the agreement between the predictions and the experimental data is very good. Given the variety of configurations considered, it indicates that our modelling approach is very general and meaningful. The results obtained with immiscible liquids (not reproduced here) are very satisfactory, as already reported in Planchette et al. (Reference Planchette, Lorenceau and Brenn2012). For the binary collisions with one liquid, we observe good agreement for the points corresponding to $Oh<0.1$ . These points are found along a line through the origin with a slope of 2.5, indicating the right order of magnitude. The validity limit of the Rayleigh criterion explains the validity limit of the predictions for higher $Oh$ . It is interesting to note that by excluding the data points for $Oh>0.1$ , we also exclude the points corresponding to $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re>0.01$ found in the third regime of figure 10. The velocity threshold predicted for the points found in the first regime of figure 10 for which $a<a_{plat,2D}$ are underestimated.

Figure 15.

Experimental threshold velocity $U_{exp}$ as a function of the theoretical predictions $U_{t}^{theo}$ . (a) Binary collisions $U_{t,2D}^{theo}$ is evaluated with (3.20) taking $b_{2D}=0$ . The diagonal line indicates a slope of 2.5. (b) Ternary collisions $U_{t,3D}^{theo}$ is evaluated with (3.21) taking $b_{3D}=0$ .

The agreement is also very good for ternary collisions. The slope between the experimental and theoretical points is similar to the one observed for binary collisions (1.8 instead of 2.5). In contrast to the binary collisions we note the existence of a relatively large intercept corresponding to an underestimation of the threshold and tending to decrease with increasing $Oh$ . We do not explain this intercept by the difference between $E_{\unicode[STIX]{x1D70E},ini}$ and $E_{\unicode[STIX]{x1D70E},cyl}$ , since we accounted for it in our scaling.

Including the losses of the first relaxation sub-period allows to reduce the deviations for binary collisions at low values of $a$ as well as the intercept for the ternary collisions. The results are presented in figure 16 where the scaling laws are evaluated with $b_{2D}=0.025(1-a_{2D})$ and $b_{3D}=0.08(1-a_{3D})$ . The reason for taking $b_{2D}$ and $b_{3D}$ as functions of $(1-a_{2D})$ and $(1-a_{3D})$ , respectively, comes from the dependency of the excess surface energy on $(1-a)$ . Indeed, the excess surface energy available at the end of the compression scales as $E_{\unicode[STIX]{x1D70E},disk}-E_{\unicode[STIX]{x1D70E},ini}\approx E_{K,ini}(1-a)$ . Thus, the losses caused by the collapse of the torus on itself may increase with increasing $(1-a)$ . With this correction, the agreement becomes excellent for the binary collisions. Indeed, all experimental points satisfying the Rayleigh criterion ( $Oh<0.1$ ) are represented in figure 16. This includes glycerol solutions with concentrations ranging from 10 % to 65 %, silicon oils and drop diameters between 130 and $870~\unicode[STIX]{x03BC}\text{m}$ . All these points are found well aligned, so that the experimental threshold, $U_{exp}$ , is well predicted by $2.38\,U_{t,2D}^{theo}$ . For ternary collisions, the intercept is reduced from 1.55 to 0.61, which indicates the relevance of this correction. Note that for one point corresponding to $Oh=0.06$ the correction could not be applied because $b/(OhWe^{1/2})>1-a$ . All other experimental data points obtained for ternary collisions are shown in figure 16. The liquids include glycerol solutions with concentrations varying between 30 % and 70 % and silicon oils, while the drop diameters range between 130 and $550~\unicode[STIX]{x03BC}\text{m}$ . For all these cases, $U_{exp}$ can be very well approached by $2.20\,U_{t,3D}^{theo}+0.61$ . Besides the difficulties to estimate the losses in the relaxation phase, it seems that our analysis is qualitatively correct and that, even though the sum of proposed scalings is tentative, the result captures the major contributions properly.

Figure 16.

Experimental threshold velocity $U_{exp}$ versus the theoretical prediction $U_{t}^{theo}$ . (a) Binary collisions $U_{t,2D}^{theo}$ evaluated with (3.20), taking $b_{2D}=0.025(1-a_{2D})$ . (b) Ternary collisions $U_{t,3D}^{theo}$ evaluated with (3.21), taking $b_{3D}=0.08(1-a_{3D})$ .

Finally, it is interesting to reformulate, for collisions using one liquid, the threshold velocity in terms of a critical $Re$ versus a critical $Oh$ , as represented in Planchette et al. (Reference Planchette, Lorenceau and Brenn2012) and Hinterbichler et al. (Reference Hinterbichler, Planchette and Brenn2015). The exponents deduced from fitting the critical experimental $Re$ as a power of $Oh$ for $Oh<0.1$ are $-$ 0.75 and $-$ 0.90 for binary and ternary collisions, respectively, using a subset of the data set of this paper. The change of exponent between the ternary and binary cases seems to indicate a principal difference between these configurations. Denoting a constant as $C$ , the relations (3.20) and (3.21) assume the form

(3.24) $$\begin{eqnarray}\displaystyle Re^{2}\propto \frac{b}{1-a}Re\,Oh^{-2}+\frac{C}{1-a}Oh^{-1}. & & \displaystyle\end{eqnarray}$$

Assuming $a$ and $b$ to be independent of $Oh$ and $Re$ , which is verified on a wide range of $\unicode[STIX]{x1D6FC}(We^{1/4}+\unicode[STIX]{x1D6FD}We^{-1/4})/Re>0.01$ , we can derive three exponents corresponding to three limiting cases. If $b/Oh^{2}$ is negligible against the other terms, we obtain $Re\propto Oh^{-1/2}$ . This is the case if most of the losses during relaxation occur during the second sub-period, in the entire liquid system. If $C$ is small, e.g. when most of the relaxation losses are caused by the collapse of the torus on itself at the very first instants, $Re$ scales as $Oh^{-2}$ . Finally if $Re$ is small enough to neglect the left-hand side of equation (3.24), i.e. for very large losses during the compression phase, the scaling reads $Re\propto Oh$ . These exponents are different from those obtained by fitting the experimental data. Yet, the three terms are probably of the same order, and it may be difficult to further reduce (3.24). Indeed, re-plotting the experimental critical values of $Re$ versus $Oh$ in terms of critical $Re$ versus $We$ gives quite dispersed data points (plot not shown). Although the representation given in Hinterbichler et al. (Reference Hinterbichler, Planchette and Brenn2015) appears at first promising, it may be misleading due to the fact that $Oh=\sqrt{We}/Re$ , which considerably reduces the dispersion with $We$ in contrary to the proposed scalings.