3. Discussion

While different in nature, D–D and D–J collisions clearly show similarities, encouraging the build of some analogy between the two collision processes, see figure 2(a–d). Before doing so, let us focus on D–D collisions. Our model is based on two ingredients: (i) a fragmentation criterion, which corresponds to a critical value of $\varPsi _d=\max [H_d(t)/D_d]$, the maximum dimensionless drop extension (see figure 2a,b) and (ii) a function, which describes the variations of $\varPsi _d$ with the liquid properties and collision parameters.

Figure 2. (a–d) Drop and jet trajectories go from left to right. The D–D collision in (a) coalescence ($D_d=341\,{\mathrm {\mu }{\rm m}}$, $X=0.41$, $We_d=31.5$, $Oh_d=0.033$) and (b) stretching separation ($D_d=340\,{\mathrm {\mu }{\rm m}}$, $X=0.61$, $We_d=31.3$, $Oh_d=0.033$). The D–J collision in (c) drops-in-jet ($D_d=275\,{\mathrm {\mu }{\rm m}}$, $\tilde {X}=1.64$, $We_{d}=30$, $Oh_j=0.246$) and (d) capsules ($D_d=292\,{\mathrm {\mu }{\rm m}}$, $\tilde {X}=1.86$, $We_{d}=48$, $Oh_j=0.246$). (e) Measured temporal evolution of $H_d/D_d$ (symbols) and its fit (dashed line) providing its maximum, $\varPsi _d$. Collision eccentricity for (f) D–D collisions, $X=x/D_d$, and (g) D–J collisions, $\tilde {X}=2\tilde {x}/D_j$.

First, $H_d(t)/D_d$ is measured for several instants after the collision and fitted with a third-order polynomial to obtain $\varPsi _d$, its maximum value with a typical measurement uncertainty of less than $3\,\%$, see figure 2(e). Note that the deformation undergone by drops colliding off-centre is not an axisymmetric lamella, as seen for head-on collisions. Thus, $H_d$, the maximal extension is the end-to-end length of a stretched entity and not a maximal diameter. We verify that the separation threshold corresponds to a constant critical value of $\varPsi _d$ of 3.25 (dashed line in figure 3a), in agreement with the numerical results of Saroka & Ashgriz (Reference Saroka and Ashgriz2015) and the recent experimental findings of Al-Dirawi et al. (Reference Al-Dirawi, Al-Ghaithi, Sykes, Castrejón-Pita and Bayly2021). This provides the fragmentation criterion (i). Note that this value remarkably close to ${\rm \pi}$, the theoretical one (Rayleigh Reference Rayleigh1892), is obtained by normalizing the critical extension with the initial drop diameter, as done by Saroka & Ashgriz (Reference Saroka and Ashgriz2015); Al-Dirawi et al. (Reference Al-Dirawi, Al-Ghaithi, Sykes, Castrejón-Pita and Bayly2021). Assuming a cylindrical shape and using volume conservation leads to a critical value of more than 5, well above the classical result of Plateau–Rayleigh.

Figure 3. D–D collisions: (a) $\varPsi _d$ as a function of $We_d$ for $Oh_d=0.033$ and different $X$. Coalescence (full symbols) and separation (empty symbols). (b) $s_d= \partial \varPsi _d/\partial We_d$ from (a). Inset: ${\rm log}(s_d/X)$ against ${\rm log}(Oh_d)$ for all experiments ($Oh_d=0.008$, 0.033, 0.325). (c) Experiments vs model – (3.2). Main graph: our data ($Oh_d=0.008$, 0.033, 0.325); inset: data of Al-Dirawi et al. (Reference Al-Dirawi, Al-Ghaithi, Sykes, Castrejón-Pita and Bayly2021) ($0.021\leq Oh_d\leq 0.214$, $0.24< X<0.55$ and $30\leq We_d \leq 130$).

We then derive the evolution of $\varPsi _d$ with the liquid properties and collision parameters, i.e the second ingredient of our approach. We consider a purely geometric contribution $\varPsi _d|_{We_d\approx 0}$ and an inertial one $\varPsi _d|_{We_d \neq 0}$. The relative velocity $U$ is projected parallel ($\boldsymbol {U}_{ho}$) and normally ($\boldsymbol {U}_s$) to the line connecting the two drop centres at contact (see figure 2f). It shows how the merged drop is elongated along $\boldsymbol {U}_s$ in the absence of inertia. This elongation contributes to $\varPsi _d$ in the form of a purely geometric term $\varPsi _d|_{We_d\approx 0}=H_d|_{We_d\approx 0}/D_d$, the normalized length of the red segment in figure 2(f). The latter is a function of the dimensionless impact parameter, $X=x/D_d$, with $x$, the projection normal to $\boldsymbol {U}$ of the drop-centre-to-drop-centre segment at contact. Geometric considerations give $\varPsi _d|_{We_d\approx 0}=(X+1)/\sqrt {1-X^2}$, which can be linearized for $0.3< X<0.8$ into $\varPsi _d|_{We_d\approx 0} \approx 2.7X+0.5$. The inertial contribution is obtained by considering that some of the initial kinetic energy (${\rm \pi} \rho _d {D_d}^3 U^2 /24$) is converted into surface energy of the stretched drop ($\approx \sigma _d {\rm \pi}H_d {D_d}$) providing at first order $\varPsi _d|_{We_d \neq 0} \approx f(X, Oh_d) We_d /24$ with $f(X, Oh_d)$ a function that accounts for the ‘relevant inertia’. Here the relevant inertia causes the merged drop to stretch. Obviously, it corresponds to the inertia of the almost unaffected drop portions that continue on their initial trajectories, see not hatched portions in figure 2(f). Compared with the drops' inertia, that of the almost unaffected portions must be reduced to account for their actual mass (or volume). Neglecting strong distortion, each portion volume is given by $V=(3X^2-2X^3)V_d$ with $V_d$ the volume of one drop. The first inertia correction therefore corresponds to a factor $V/V_d=(3X^2-2X^3)$, which gives after linearization $V/V_d \approx (1.4X-0.2)$ and thus a linear variation of $f(X, Oh_d)$ with $X$. This scaling ($\varPsi _d|_{We_d \neq 0} \propto X We_d$) is specific to off-centre collisions and cannot be extrapolated to head-on impacts. For $X\lesssim 0.3$, an axisymmetric lamella forms, whose maximal diameter scales as $\sqrt {We_d}$ (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016; Planchette et al. Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017b). To go further, the viscous losses, i.e. the dependency with $Oh_d$, must be estimated.

Analytically establishing the quantitative expression of this second correction is rather complex and we decide here to use the numerical findings of Finotello et al. (Reference Finotello, Padding, Deen, Jongsma, Innings and Kuipers2017). The computed normalized remaining energy is replotted as a function of $X$ and $Oh_d$, see Appendix. It provides $f(X, Oh_d) \approx 0.31 X {Oh}^{-0.18}$, in agreement with the expected linear variation of $f$ with $X$. Consequently, we obtain $\varPsi _d|_{We_d \neq 0} \approx (0.31/24) Oh_d^{-0.18} We_d$ and thus the overall theoretical deformation:

(3.1)\begin{equation} \varPsi_{d, th} = \alpha_{d, th} Oh_d^{m_{th}} X We_d + \beta_{d, th} X + \gamma_{d, th}. \end{equation}

The constants $\beta _{d, th}=2.7$ and $\gamma _{d, th}=0.5$ come from $\varPsi _d|_{We_d \approx 0}$, and $\alpha _{d, th}= 0.31/24\approx 0.013$ and $m_{th}=-0.18$ originate from $\varPsi _d|_{We_d \neq 0}$ and thus from the estimation of the relevant inertia. Due to the uncertainty of the computed findings, the approximation caused by linearization and the crude estimation of the stretched drop surface energy, we do not expect $\alpha _{d, th}$ and $m_{th}$ to be quantitatively well predicted. We nevertheless anticipate a qualitative agreement of (3.1), which we probe by plotting the experimental data $\varPsi _{d}$ as a function of $We_d$ for different $X$. While these curves are obtained for all experiments ($0.008< Oh_d<0.325$), only those corresponding to $Oh_d=0.033$ are shown in figure 3(a). Note that the line corresponding to $\varPsi _d=3.25$ separates well the coalescence (full symbols) from the separation (empty symbols). In agreement with (3.1), $\varPsi _d$ increases linearly with $We_d$. Saroka & Ashgriz (Reference Saroka and Ashgriz2015), who numerically studied water drop collisions, reported similar variations. Further, for a given $Oh_d$ and fixed $X$, the curve slopes $s_d$ are linear in $X$ (figure 3b) as expected by (3.1) (first term). Repeating the experiments with three liquids and thus three $Oh_d$, we find that $a_d=s_d/X$ is equal to 0.078 for W ($Oh_d=0.008$), 0.066 for G ($Oh_d=0.033$) and 0.047 for SO ($Oh_d=0.325$), and thus decreases with increasing $Oh_d$. The scaling $a_d \propto {Oh_d}^{-0.14}$ (see dashed line in the insert of figure 3b) is reasonably well captured by (3.1), which predicts $a_d \propto {Oh_d}^{-0.18}$.

To obtain a quantitative agreement, the experimental results $\varPsi _{d}$ are fitted by the model equation $\varPsi _{d, mod} = \alpha _{d, mod} Oh_d^{m_{mod}} X We_d + \beta _{d, mod} X + \gamma _{d, mod}$. There, $m_{mod}$ and $\alpha _{d, mod}$ are only adjusted once, while $\beta _{d, mod}$ and $\gamma _{d, mod}$ are taken equal to their theoretical values, $\beta _{d, th}$ and $\gamma _{d, th}$, see figure 3(c). We obtain an excellent prediction with

(3.2)\begin{equation} \varPsi_{d, mod} = 0.041 Oh_d^{{-}0.128} X We_d + 2.7 X + 0.5. \end{equation}

The discrepancy between $m_{th}$ and $m_{mod}$ ($28\,\%$) could originate from the integration by Finotello et al. (Reference Finotello, Padding, Deen, Jongsma, Innings and Kuipers2017) of the losses over the whole process instead of the first instants, see Appendix for details. The fit also provides $\alpha _{d, mod}=0.041$, while the theory gives $\alpha _{d, th}=0.013$. The difference (factor 3) can be explained by the crude estimation of the stretched drop surface. All constants being the same, at least for $0.008 < Oh_d < 0.325$, (3.2) constitutes a model which is valid over a very wide domain without the need for any adjustment. The agreement is also excellent while using the data of Al-Dirawi et al. (Reference Al-Dirawi, Al-Ghaithi, Sykes, Castrejón-Pita and Bayly2021) (inset of figure 3c). This indeed indicates that the stretching separation is not purely inertial as previously reported by Al-Dirawi et al. (Reference Al-Dirawi, Al-Ghaithi, Sykes, Castrejón-Pita and Bayly2021). In fact, the authors probed fewer values of $We_d$ over a smaller range of $Oh_d$, which did not allow to identify the variations of $\partial \varPsi _d/ \partial X$ with $Oh_d$.

We then predict the separation threshold in the form of an ($X$, $We_d$) relation by fixing in (3.2) $\varPsi _{d, mod}$ to its critical value of 3.25. The results are compared to those of the literature, see figure 5(a–c). First of all, for all considered $Oh_d$, the predicted thresholds (continuous lines) perfectly match the experimental ones (symbols). We further observed a very good agreement to previously proposed models, which involved adjusted parameters while being limited to given values or narrow ranges of $Oh_d$ (Ashgriz & Poo Reference Ashgriz and Poo1990; Jiang et al. Reference Jiang, Umemura and Law1992; Gotaas et al. Reference Gotaas, Havelka, Jakobsen, Svendsen, Hase, Roth and Weigand2007; Finotello et al. Reference Finotello, Padding, Deen, Jongsma, Innings and Kuipers2017). We recall that with our approach no parameter is adjusted to cover collisions with $0.008< Oh_d<0.325$.

Let us now apply these results to D–J collisions. In former studies (Planchette et al. Reference Planchette, Petit, Hinterbichler and Brenn2018; Baumgartner et al. Reference Baumgartner, Bernard, Weigand, Lamanna, Brenn and Planchette2020a,Reference Baumgartner, Brenn and Planchetteb), the spatial period of the jet $L_j$ was normalized by $D_j$ and used to build a pseudo-Rayleigh criterion. A critical value of 2 was found to roughly describe the jet fragmentation threshold in the limit of moderate jet viscosity. Here, the analogy with D–D collisions requires the introduction of a new parameter to quantify the eccentricity of the successive collisions. As sketched in figure 2(g), these collisions involve a drop and the jet portions found before and after this drop (lighter grey). Note that the jet portion directly impacted by the drop (hatched) is associated with the drop to form a compound drop. Thus, the distance between the centre of mass of the compound drop and that of the jet portions found before or after is given by $\tilde {x}=(L_j+D_d)/4$. This distance is counted twice since each compound drop interacts with two such jet sections. Using $D_j$ for normalization, the equivalent impact parameter reads $\tilde {X}=(L_j+D_d)/(2D_j)$.

We record several D–J collisions and define, similarly to $H_d(t)/D_d$, the dimensionless extension of the jet $H_j(t)/D_j$, which is measured perpendicularly to the final D–J compound trajectory (figure 2c,d). Its temporal evolution is fitted by a third-order polynomial providing its maximum value $\varPsi _j=\max (H_j(t)/D_j)$. The procedure and accuracy is similar to that of D–D collisions, see figure 2(e).

We first confirm that the jet fragmentation corresponds to a critical value of $\varPsi _j$, see figure 4(a). Interestingly, this critical value is 3.0, thus remarkably close to that found for the D–D stretching separation and underlines the relevance of our analogy. We then plot for different $\tilde {X}$, the evolution of $\varPsi _j$ with $We_{d}$ and evidence a linear dependency similar to $\varPsi _d$ for D–D collisions. The curve slopes, $s_j$, are again linear in $\tilde {X}$ (figure 4b) and increase with $Oh_j$. For D–J collisions, the relevant Ohnesorge number is that of the jet liquid, since the viscous losses mainly take place in the interstitial jet portions, which are the most stretched. We therefore propose to describe the jet stretching as

(3.3)\begin{equation} \varPsi_{j, mod} = \alpha_{j} Oh_j^n \tilde{X} We_{d} + \beta_j Oh_j^n \tilde{X} + \gamma_j. \end{equation}

Here again, $\alpha _j$, $\beta _j$ and $\gamma _j$ are constants. The first term accounts for the drop inertia reduced by viscous losses taking place in the jet liquid only. The last two terms correspond to geometrical effects. As shown in figure 4(c), the agreement is again very good. The fit provides $-0.10$ for the exponent $n$, close to $-0.128$ found for $m_{mod}$ and therefore supports the assumption that the viscosity (of drop and jet) plays, despite different geometries, a similar role in both processes (D–D and D–J collisions, respectively). Here $\alpha _j$, $\beta _j$ and $\gamma _j$ are found to be +0.0066, +3.98 and $-5.85$, respectively. The slight deviation observed for D–J could have several origins. First of all, the system centre of mass is approximated by that of the jet, which slightly affects the measurement of the collision inertia. Second, immiscible liquids are used, which modify the flow field and thus the viscous losses. Due to the lack of existing data, (3.3) could not yet be tested against results obtained with miscible liquids. It should definitively be done in future investigations. Finally and despite its similarities, the process itself is different. For D–J collisions, the system is continuous, and mainly shear occurs between the colliding elements. For D–D, the drop pairs constitute a close system, which can rotate around their centre of mass, consuming part of the available inertia.

Figure 4. D–J collisions: (a) $\varPsi _j$ against $We_d$ for different $\tilde {X}$, $Oh_j=0.246$ and $\varDelta =1.0$. Coalescence (full symbols) and separation (empty symbols); (b) $s_j=\partial \varPsi _j/ \partial We_d$ against $\tilde {X}$ from (a); (c) experiments vs model (3.3) with $0.021< Oh_j<0.25$ and $0.7<\varDelta <1.3$.

To explain why, in contrast to $\varPsi _d|_{We_{d}\approx 0}$, $\varPsi _j|_{We_{d}\rightarrow 0}=\beta _j Oh_j^n \tilde {X} + \gamma _j$ is a function of $Oh_j$, it is useful to recall that Ohnesorge numbers can be seen as the ratio of a bulk motion time scale, $t_{\mu }=\mu _jD_j/\sigma _j$ (Stone & Leal Reference Stone and Leal1989), and an interfacial time scale, $t_{\sigma }=\sqrt {\rho _j {D_j}^3/\sigma _j}$ (Rayleigh Reference Rayleigh1892). At intermediate time scales, when $\varPsi _j$ is measured, the morphology of the compound jet depends on their relative kinetics and therefore on the jet liquid properties via its Ohnesorge number. For high $Oh_j$, $t_{\mu }>t_{\sigma }$, the capillary effects are fast enough to significantly flatten the outer jet surface, leading to small $\varPsi _j|_{We_{d}\rightarrow 0}$. The contrary happens for small $Oh_j$. We also verify that increasing $L_j$ or $D_d$ as well as decreasing $D_j$ (thus increasing $\tilde {X}$) leads – as expected – to greater $\varPsi _j|_{We_{d}\rightarrow 0}$.

Note that given the definition of $\tilde {X}$ and the value of $\beta _j$, $\gamma _j$ must be negative to represent separated successive and not overlapping drop collisions. We verify that $\varPsi _j|_{We_{j}\rightarrow 0}>1$ in all experiments.

As for D–D collisions, fixing $\varPsi _{j, mod}$ to 3.0 in (3.3) enables the prediction of the transition between continuous (circles) and fragmented jet (diamonds), see solid lines in figure 5(d–f). The agreement between the model (solid lines) and the experiments (symbols) is very good, significantly better than with the former criterion of $L_j/D_j \approx 2.0$ (horizontal dashed line). It is valid over a wide range of $Oh_j$ ($0.021 < Oh_j < 0.246$) and for different drop and jet diameters ($0.7<\varDelta <1.3$) without adjusting any parameter.

Figure 5. (a–c) The D–D collisions: separation transition for $Oh_d=0.008$ (a), $Oh_d=0.033$ (b) and $Oh_d=0.325$ (c) with coalescence (circles) and separation (stars). Solid lines, (3.2) with $\varPsi _{d, mod}=3.25$; dashed lines, Jiang et al. (Reference Jiang, Umemura and Law1992); dash-double-dotted lines, Finotello et al. (Reference Finotello, Padding, Deen, Jongsma, Innings and Kuipers2017); dotted line, Ashgriz & Poo (Reference Ashgriz and Poo1990); dot-dashed line, Gotaas et al. (Reference Gotaas, Havelka, Jakobsen, Svendsen, Hase, Roth and Weigand2007). (d–f) The D–J collisions: transition between continuous (circles) and fragmented jet (stars) for $Oh_j=0.021$ (d), $Oh_j=0.073$ (e) and $Oh_j=0.246$ (f). Solid lines: (3.3) with $\varPsi _{j, mod}=3.0$. Dashed line (e): former criterion $L_j/D_j=2$.