2.1 Collision of toroidal rims

The impact configuration with two infinite liquid cylinders, which will be analysed in § 4, is of course an idealized view. Instead of cylinders, the experiment consists in colliding two liquid tori, which constitute the edges of two circular holes opening in the static film at the same Taylor–Culick velocity (figure 2a). The latter are nucleated simultaneously, initially separated by a distance $b$. They thus travel for a time $b/2V$ from the film rupture to the rims impact. The volume of each torus $2\unicode[STIX]{x03C0}^{2}a^{2}b/2$ is equal to the liquid volume of the film filling the hole $\unicode[STIX]{x03C0}(b/2)^{2}h$, so that, at leading order, the Weber number can be written as

(2.3)$$\begin{eqnarray}We=\frac{8}{\sqrt{\unicode[STIX]{x03C0}}}\sqrt{\frac{b}{h}}.\end{eqnarray}$$

The intensity of the collision is therefore computed from directly accessible experimental control parameters $b$ and $h$, the latter being measured via the film receding velocity. The higher-order contributions from the expanding, toroidal geometry are given in § A.1.

The impact geometry exhibits at least two planes of symmetry, which are independently monitored in two experimental configurations described in the following. In the plane of the interstitial film, we denote by $x$ the direction joining the film puncture points, and $z$ the orthogonal direction: invariant in the infinite cylinders problem (see figure 2b for a three-dimensional representation). All liquid reorganization during impact occurs in the direction perpendicular to that plane, referred to as $y$. The transverse, or mid-plane $Oyz$, where $O$ is the symmetry mid-point between the two puncture points, is the second symmetry plane in that binary symmetric configuration.

2.2 Controlled rim production

The first set-up observes the collision in the transverse plane $Oyz$, orthogonal to the punctured interstitial film, with a slight inclination angle above the latter. By symmetry, it embraces all the dynamics following the impact, from the initial extension of a transverse lamella to the fragmentation later stages.

Figure 3.

Controlled rim production set-up. The impact transverse plane $Oyz$ is monitored, with a slight angle above the interstitial film plane. (a) Global view of the experiment: a Savart sheet (see text) surrounded by two couples of electrodes (framed detail). (b) Time lapse zoomed in downstream of the electrodes, from the simultaneous sparks to the encounter of the holes and rim impact. Consecutive frames are separated by 0.57 ms.

The film is a horizontal smooth Savart sheet, a radially expanding film resulting from the impact and deflection of a laminar circular jet with diameter $d_{jet}$, on a slightly larger target (Savart Reference Savart1833). The velocity field in the suspended film is strictly radial, with a constant speed $U$, uniform across the film thickness. It is essentially equal to the jet velocity as soon as viscous dissipation on impact is negligible, which is the case with water (Villermaux et al. Reference Villermaux, Pistre and Lhuissier2013). At steady state and with water considered incompressible, the volume flux $2\unicode[STIX]{x03C0}Urh(r)$, with $h(r)$ the film thickness and $r$ the radial coordinate centred on the impinging jet axis, is constant throughout the jet and the sheet so that the sheet thickness decreases with the distance $r$ to the jet as

(2.4)$$\begin{eqnarray}h(r)=\frac{d_{jet}^{2}}{8r}.\end{eqnarray}$$

Holes in the sheet are punctured by two simultaneous electrical sparks transpiercing the sheet by vaporizing the liquid locally. They are triggered on demand, by means of two couples of electrodes connected to large capacitors which on discharge produce the two simultaneous sparks. The electrodes are located on both sides of the film, separated by approximately a millimetre (the film is thinner than $100~\unicode[STIX]{x03BC}\text{m}$ there), at the same radial location $r_{0}$ from the jet, and separated by an azimuthal angle $2\unicode[STIX]{x1D703}_{0}=2\arcsin b/2r_{0}$ ($b$ is the Euclidean distance between the puncture points, figure 3a). As the holes grow and feed the rims, they are advected outwards, on diverging radial trajectories (figure 3b). This way, rims travel a distance larger than $b/2$ before they collide (see § 2.1), and hence have collected a priori more liquid when they come into contact. On the other hand, due to the thickness decrease (2.4), they experience an increasing opening velocity $V$, which now depends on the current radial coordinate: $V(r)=\sqrt{16r\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}d_{jet}^{2}}$. The impact Weber number is eventually increased, when compared with the static and uniform film case. However, for small angles $2\unicode[STIX]{x1D703}_{0}\sim b/r_{0}$, it simplifies back to the static film case (2.3). The exact expression is derived in § A.2, along with the experimental deviations to the static film approximation. Although the exact expression in (A 11) is preferred when presenting the experimental results below, we use the approximate expression (2.3) when comparing this set-up and the second one described in § 2.3.

The advected impact is observed in the laboratory fixed frame, with a high-speed camera and backlighting. Within the range of Weber numbers investigated (from 40 to 200), the set-up is time and space resolved, thanks to frame rates up to 50 kHz and spatial resolutions around $30~\unicode[STIX]{x03BC}\text{m}$ per pixel.

2.3 Random rim production

The second set-up allows us to observe the rims’ collision and subsequent fragmentation in the plane $Oxz$, the liquid film being a Savart sheet as before. Now, film punctures are nucleated by internal defects, bubbles incorporated into the liquid prior to the formation of the film (see Lhuissier & Villermaux (Reference Lhuissier and Villermaux2013) for a complete description of the set-up and method). Bubbles embedded in the sheet pop up at some distance along their radial trajectory, making the film rupture location a random variable. As a result, the location of the subsequent impact of neighbouring rims, following the merging of two growing holes, is also a distributed random variable (a single collision is illustrated in figure 4).

Figure 4.

Random rim production set-up. Time lapse of a selected impact event, observed in the interstitial film plane. (a) Holes opening and rim growth, impact, destabilization and late fragmentation. Consecutive frames are separated by 1 ms. Grey dashed lines are the radial trajectories of the rims centre, advected at the constant speed $U$. (b) Fragment detection and area-based size measurement.

The monitored field of view is large ($4\times 4~\text{cm}^{2}$, with a resolution of $40~\unicode[STIX]{x03BC}\text{m}$ per pixel) and focuses on a restricted angular sector of the Savart sheet, around the radius where most of the punctures take place. Together with a low acquisition frame rate (1 kHz), it allows recording of a large number of impact events. The careful selection and a posteriori characterization of some 560 hole openings constitutes a batch of 280 rim impacts. The frame rate is high enough to enable the individual tracking of the rims, from the moment they appear to the collision and, later, their destabilization into drops (figure 4). However, most of hole openings and rim impacts occur between two recorded frames. The respective time instants of the punctures and of the collision are determined with a greater precision by a linear interpolation between those two frames, on the basis of the following approximations. The film thickness is considered uniform along each rim trajectory, so that the latter are assumed to open circularly, at the constant Taylor–Culick velocity. Meanwhile, they are advected outwards with the constant radial flow of the Savart sheet, so that there is, for each couple of rims, a unique impact point. The computation of the Weber from (2.3) is made under these approximations. As explained in § 2.1, it underestimates the actual value of $We$, for which there is here no straightforward expression, but enables the comparison with the first set-up when needed (§ 2.2). For two slightly asynchronous hole punctures, resulting in an asymmetric impact, the Weber number is based on the radius $a$ of the smallest rim.

2.4 Fragments

Drops are individually detected, and calibrated. Sizes correspond to drops which have relaxed to a spherical shape and are in focus only, in both set-ups (figures 4b and 5). Their diameter $d$ is based on the measurement of their projected area $\unicode[STIX]{x03C0}d^{2}/4$, the quality of the pictures allowing for a basic intensity thresholding. Discrepancies between both set-ups are described in appendix B. Furthermore, the time-resolved monitoring of the transverse plane in the controlled rim production set-up (§ 2.2) provides extensive data about the ejection velocity of the fragments (figure 5).

Figure 5.

Snapshot of the transverse plane after a collision at $We=193$, along with droplet size and velocity detection. The uniform advection velocity $U$ is substracted during the data processing. The solid line is the axial centre line of the merged rim, the dashed lines its initial upper and lower extents prior to the collision.

3.1 Unstable transverse lamella

The collision of two rims is inelastic. Even for an impact at low $We$ leading ultimately to a unique cylinder with section $2\unicode[STIX]{x03C0}a^{2}$ (mass conservation), most of the incident kinetic energy is lost in internal irregular motions. We will come back to this point in § 5.1. However, since $Oh$ is low in the present case, the dissipation scale (${\sim}h\times Oh$, see Culick (Reference Culick1960)) is much smaller than $a$, and the collision gives rise to an – essentially inviscid – dynamics involving large deformations of the merging rims. Notably, the emergence of a thin lamella, in the direction orthogonal to the impact direction (figure 6) is systematically observed for $We$ greater than 50. This expanding lamella is itself bordered by a rim at its extremity in $y=\ell (t)$, which is uniform along the rims’ axis in the $z$-direction. This secondary rim is pulled back by surface tension conferring to $\ell (t)$ an ever decelerated motion. The phenomenon is particularly clear from the angle of view offered by the controlled rim production set-up (see also figure 10).

Figure 6.

Lamella emergence and growth, in the transverse plane, for a $We=83$ collision. Consecutive frames are separated by $22~\unicode[STIX]{x03BC}\text{s}$ (from top to bottom, then left to right).

Figure 7.

Longitudinal destabilization of a transverse growing lamella, for a $We=102$ collision. Consecutive frames are separated by $80~\unicode[STIX]{x03BC}\text{s}$.

The larger $We$, the larger the lamella maximal extension. As a consequence, the overall extension–retraction time of the lamella, i.e. its oscillation period or lifetime, increases when $We$ is increased, as will be shown in § 4.1. Regularly spaced indentations of the lamella rim (figure 7) are markedly apparent as $We$ is increased. This longitudinal destabilization ($z$-direction) features a wavelength $\unicode[STIX]{x1D706}$. It is a signature of the deceleration undergone by the lamella in the course of its development as will be analysed in § 4.4.

3.2 Transverse ligaments and the production of fines

Above a critical Weber number of the order of $We_{c}\approx 66$, the longitudinal indentations of the lamella rim give rise to regularly spaced, transverse ligaments (figures 8 and 14). The breakup of these secondary ligaments eventually forms small droplets, the fines, which are expelled, in continuation of the lamella transverse growth, in the direction perpendicular to the main liquid sheet (figure 8). These objects substantially alter the overall drop size distribution of the resulting spray, as will be seen in § 5.2.

Through this process of lamella expansion, indentation growth, ligament formations and breakup, increasingly many, and smaller, droplets (relative to $a$) are formed as $We$ is increased. They are, since the colliding rims are more finely divided, not only obviously all the more numerous (figure 9d–f) but are also more distributed in size relative to their mean, as will be seen in § 5.2.

Figure 8.

Formation of secondary transverse ligaments, and ejection of fine, fast droplets from their tips, for a $We=193$ collision. Consecutive frames are separated by $38~\unicode[STIX]{x03BC}\text{s}$.

4.1 Global collision dynamics, the long time and length scales

Let two parallel cylinders aligned along $z$ and travelling at relative speed $2V$ along $x$ collide, thus adding up their mass and deflecting the momentum they carry along the transverse $y$-axis (equally split in both positive and negative directions). The fused cylinders expand along $y$ until the unbalanced surface tension force at the surface of the ensemble limits its growth. The phenomenon is reminiscent of the oscillatory dynamics of jets issuing from a non-circular orifice undergoing peristaltic pulsations studied by Rayleigh (Reference Rayleigh1879) and others (see § 6 in Eggers & Villermaux (Reference Eggers and Villermaux2008) for a historical perspective).

In all inertial fluid mechanics problems, the displacement of the fluid particles reflects the structure of the pressure field. The pressure might be viewed as the source of the motion (Cooker & Peregrine Reference Cooker and Peregrine1995; Antkowiak et al. Reference Antkowiak, Bremond, Dizès and Villermaux2007), or a consequence of it (Birkhoff et al. Reference Birkhoff, MacDougall, Pugh and Taylor1948; Riboux & Gordillo Reference Riboux and Gordillo2014), but in each case both are linked.

Ignoring first the fine details of the collision at short times, it is easy to anticipate the time scale (oscillation period $T$), and length scale (maximal extension of the ensemble $L$) of the collision coarse-grained motion after the fusion of the cylinders (figures 6 and 10a). We sketch the deforming fused ensemble as a lamella expanding in the $y$-direction with length $\ell (t)$ and width $w$ so that $2\unicode[STIX]{x03C0}a^{2}=\ell \times w$. The liquid velocity in the $y$-direction is $v(y,t)=y\dot{\ell }/\ell$. Integration of the Euler equation

(4.1)$$\begin{eqnarray}\unicode[STIX]{x1D70C}(\unicode[STIX]{x2202}_{t}v+v\unicode[STIX]{x2202}_{y}v)=-\unicode[STIX]{x2202}_{y}p\end{eqnarray}$$

between $y=0$ and $y=\ell (t)$ with $p$ the liquid pressure (see Villermaux & Bossa (Reference Villermaux and Bossa2009) for the axisymmetric version of the problem) provides

(4.2)$$\begin{eqnarray}{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}\ell \ddot{\ell }=p(0)-p(\ell ).\end{eqnarray}$$

The pressure at the expanding extremity is $p(\ell )\approx \unicode[STIX]{x1D70E}/w$ while the pressure at the contracting location is, in this coarse-grained description,

(4.3)

The Dirac delta contribution  stands for the pressure impulse caused by the impact of the cylinders at the collision location $y=0$ and time $t=0$. The momentum transfer giving rise to this pressure surge lasts in fact for the crushing time $a/V$, which is safely taken as zero provided the resulting motion of the fused ensemble lasts for a time much larger than $a/V$, as will be checked a posteriori. With $\ell (0)=a$, equation (4.2) amounts to

(4.4)

(4.5)

The initial velocity of the lamella is $V$ (see also (4.12)). The period $T$ (obtained for $\ell (T)=a$) and maximal extension $L\equiv \ell (T/2)$ of the motion are

(4.6)$$\begin{eqnarray}\displaystyle & \displaystyle T\sim \frac{a}{V}We, & \displaystyle\end{eqnarray}$$

(4.7)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{L-a}{a}\sim We. & \displaystyle\end{eqnarray}$$

The relative maximal extension of the expanded fused cylinders $(L-a)/a$ is a measure of the ratio of the motion period $T$ to the crushing time $a/V$. It is equal to $We$, a quantity larger than unity, justifying a posteriori our impulse treatment of the pressure surge. Figures 10(b) and 10(c) demonstrate the validity of the above scaling relations, up to numerical factors. The expanding lamella fragments for large $We$, that we describe next, explaining why the measurement of $T$ is meaningless above $We\approx 100$.

Figure 10.

Lamella dynamics. (a) Space–time diagram of a slice in the transverse$y$-direction. The lamella extension $\ell (t)$ is underlined with ($+$) markers and (4.5) is shown as a solid line. (b) Lamella global expansion period $T$ and (c) maximal extension $L$ as a function of the collision parameters in (4.6) and (4.7). Solid lines are best linear fit with numerical factors: (a) 0.17 and (b) 0.027.

4.2 Early time lamella formation

We have explained in § 3.1 that because viscous dissipation is made at a very small scale in the rims when $Oh\ll 1$, the momentum transfer at collision leads to the formation of tiny objects, which may be formed before the rim fusion is completed (i.e. for $t<a/V$), hence the early emergence of a thin transverse lamella. We analyse this fine-grained aspect of the phenomenon here, exploring what happens ‘inside’ the Dirac Delta of the previous section. Two regimes are distinguished.

We describe the sudden collision at velocity $2V$ of two liquid cylinders of radius $a$. The cylinders are first considered independent, moving in a dynamically inert ambient medium, contacting along their generatrix (the same reasoning holds for two spheres impacting at a point); their geometrical interpenetration radius is initially $r\sim \sqrt{Vat}$ (figure 2c). The induced flow $\boldsymbol{v}(\boldsymbol{x},t)$ ruled by $\unicode[STIX]{x2202}_{t}\boldsymbol{v}+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}=-\unicode[STIX]{x1D735}p/\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0$ obeys initially, when the velocity amplitude $|\boldsymbol{v}|$ is small enough to neglect the convective term $\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}$ (see Lamb (Reference Lamb1932), Art. 11 and Cooker & Peregrine (Reference Cooker and Peregrine1995))

(4.8)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{v}=-\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}p, & \displaystyle\end{eqnarray}$$

(4.9)$$\begin{eqnarray}\displaystyle & \displaystyle \text{and thus }\unicode[STIX]{x1D6FB}^{2}p=0. & \displaystyle\end{eqnarray}$$

In other words, if the liquid has moved over a distance $r$ in the $y$-direction, then it has moved over the same distance in the $x$-direction. The consequence of the Laplacian character of the pressure is that, since $r$ is the only initial length scale of the problem (besides $a\gg r$), the net volume of liquid whose motion is slowed down (per unit cylinder length) is of order $r^{2}$, setting its mass $m\sim \unicode[STIX]{x1D70C}\,r^{2}$ (not to be confused with the deflected volume $Vt\,r$, feeding the ejected lamellae). The cancellation of the corresponding momentum initially carried in the $x$-direction gives rise to a force $f=V{\dot{m}}$, and therefore to an isotropic pressure at the impact point given by

(4.10)$$\begin{eqnarray}\displaystyle p(0) & {\sim} & \displaystyle \frac{f}{r}\end{eqnarray}$$

(4.11)$$\begin{eqnarray}\displaystyle & {\sim} & \displaystyle \unicode[STIX]{x1D70C}V^{2}\sqrt{\frac{a}{Vt}},\end{eqnarray}$$

an early time divergence familiar in impact problems (Wagner Reference Wagner1932; Cointe & Armand Reference Cointe and Armand1987; Philippi, Lagrée & Antkowiak Reference Philippi, Lagrée and Antkowiak2016), holding both in one and two dimensions (for the impact of a spherical drop on a solid, for instance).

The pressure gradient $\unicode[STIX]{x2202}_{y}p$ in the symmetry plane of the impact (figure 2c) is of order $p(0)/r$ so that, from the dynamics in (4.8), $v/t\sim V/t$, giving simply

(4.12)$$\begin{eqnarray}v\sim V.\end{eqnarray}$$

The mass in motion $m$ increases proportionally to time as the driving force $p(0)\times r$ is constant, so that the velocity is constant. With this estimate for $v$, the amplitude of the discarded nonlinear term $|\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}|$ in the Euler equation above is of order $V^{2}/r\sim 1/\sqrt{t}$, indeed smaller than $1/t$ as $t\rightarrow 0$. The intensity of this induced flow is smaller than the geometrical expansion velocity of the interpenetration region ${\dot{r}}\sim \sqrt{Va/t}$ as long as $t<a/V$, consistent with the empirical observation that a lamella is seen to emerge from the impact region when the interpenetration distance is a fraction of $a$, and that the ejection velocity of the resulting lamella (and detached droplets), is of order $V$ when complications with liquid viscosity, ambient medium and substrate roughness are negligible (see Xu, Barcos & Nagel (Reference Xu, Barcos and Nagel2007), Riboux & Gordillo (Reference Riboux and Gordillo2015) for drops impacts).

4.3 Very early dynamics: the tiniest ejecta

The linear pressure impulse dynamics in (4.8) does not, however, apply everywhere in the interpenetration region. Close to the contact line between the cylinders (the same remark applies to a drop impacting a solid), the velocity $v$ in the symmetry plane of the impact is itself of order ${\dot{r}}$, making the nonlinear term $|\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}|$ of order $v^{2}/\unicode[STIX]{x1D6FF}(t)$ where $\unicode[STIX]{x1D6FF}(t)$ is a length scale setting the width of the pressure and velocity gradients close to the contact line. If $\unicode[STIX]{x1D6FF}(t)$ is itself initially zero and an increasing function of time, the nonlinear term is more singular than $V/t$, suggesting that the early time dynamics balances inertia with pressure, at least in a small region of size $\unicode[STIX]{x1D6FF}$. Writing $p\sim f/\unicode[STIX]{x1D6FF}$ so that $\unicode[STIX]{x2202}_{y}p\sim f/\unicode[STIX]{x1D6FF}^{2}$, the balance between $|\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}|$ and $|\unicode[STIX]{x1D735}p|/\unicode[STIX]{x1D70C}$ at the contact line can be written

(4.13a,b)$$\begin{eqnarray}\frac{v^{2}}{\unicode[STIX]{x1D6FF}}\sim \frac{1}{\unicode[STIX]{x1D70C}}\frac{f}{\unicode[STIX]{x1D6FF}^{2}},\quad \text{or}\quad v\sim V\sqrt{\frac{a}{\unicode[STIX]{x1D6FF}}},\end{eqnarray}$$

(4.14)$$\begin{eqnarray}\text{providing }\unicode[STIX]{x1D6FF}\sim Vt,\quad (\text{since }v\sim {\dot{r}})\end{eqnarray}$$

so that the local pressure in the contact line region (Mandre, Mani & Brenner Reference Mandre, Mani and Brenner2009) is now of order $f/\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D70C}Va/t$, indeed more singular than $p(0)$ in (4.11). Detailed calculations (Birkhoff et al. Reference Birkhoff, MacDougall, Pugh and Taylor1948; Riboux & Gordillo Reference Riboux and Gordillo2014) show that the contact line velocity is closer to $v=2{\dot{r}}$ (see also Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) for a fully self-similar description), albeit affected by viscous corrections when a no-slip condition applies. Expressing that the deflected mass $\unicode[STIX]{x1D70C}Vt\,r$ all enters the ejected lamella which carries its momentum provides the lamella thickness $w$ as $V\unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}Vtr)\sim \unicode[STIX]{x1D70C}v^{2}w$, that is $w\sim (V/a)r^{2}/{\dot{r}}\sim t^{3/2}$ (and consistently $w<\unicode[STIX]{x1D6FF}$ as $t\rightarrow 0$).

Note on dimensionality: the reasoning above also applies to a drop of radius $a$ impacting a solid or another identical drop, expanding in two dimensions with now $m\sim \unicode[STIX]{x1D70C}r^{3}$ leading, from $f\sim V{\dot{m}}$, to $v\sim V(a/\unicode[STIX]{x1D6FF})$ and $p\sim f/(r\unicode[STIX]{x1D6FF})\sim \unicode[STIX]{x1D70C}V^{2}(a/\unicode[STIX]{x1D6FF})$, identically to the one-dimensional case (see e.g. Riboux & Gordillo Reference Riboux and Gordillo2014).

We now consider the case of cylinders initially linked by a quiescent film of thickness $h<a$ relevant to the present situation (the thickness $h$ is in practice related to the velocity $V=\sqrt{2\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}h}$ by the Taylor–Culick relation). The presence of the film de-singularizes the pressure at contact since it is both uniform and finite over a region of order $h$. The above reasoning thus starts to apply as soon as $\unicode[STIX]{x1D6FF}=h$, suggesting, given $v\sim V\sqrt{a/\unicode[STIX]{x1D6FF}}$ in (4.13), that the ejection speed of the lamella at very short times, possibly fragmented into very fine droplets above a critical Weber number $We_{c}$ to be determined (next § 4.4), will be of order

(4.15)$$\begin{eqnarray}\displaystyle V_{e} & {\sim} & \displaystyle V\sqrt{\frac{a}{h}}\end{eqnarray}$$

(4.16)$$\begin{eqnarray}\displaystyle & \propto & \displaystyle V\sqrt{We-We_{c}},\end{eqnarray}$$

a dependence to which the observations reported in § 5.4 below give some support.

This very early dynamics holds as long as $\unicode[STIX]{x1D6FF}<r$, that is $t<a/V$, then leaving place to the one in § 4.2.

Figure 11.

Longitudinal instability. (a) Growth time $\unicode[STIX]{x1D70F}$ and (b) wavelength $\unicode[STIX]{x1D706}$ are plotted against their expected scalings $h/V$ and $h$, respectively from (4.19), collapsing data from impacts at different $We$. Solid lines are best linear fits with factors: (a) 47 and (b) 3.9.

4.4 Lamella deceleration and instability

Once ejected from the interpenetration region with a thickness of order $h$, the lamella expands ballistically, only arrested at its border by capillary retraction, thus giving rise to an ever decelerated regime. The corresponding dynamics for $\ell (t)$ then simply reads

(4.17)$$\begin{eqnarray}\ddot{\ell }\sim -\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}h^{2}},\end{eqnarray}$$

featuring a negative and constant acceleration. The lamella border is a density interface which separates a dense liquid from a dynamically inert medium. Being decelerated, this interface is unstable in the sense of Rayleigh–Taylor, a fact which is commonplace for slowing (Villermaux & Bossa Reference Villermaux and Bossa2011), retracting (Lhuissier & Villermaux Reference Lhuissier and Villermaux2011) or spinning sheets (Fraser, Dombrowski & Routley Reference Fraser, Dombrowski and Routley1963; Eisenklam Reference Eisenklam1964), explaining the origin of the developing longitudinal indentations observed in the $z$-direction (figure 7). The characteristic time of growth $\unicode[STIX]{x1D70F}$, and wavelength $\unicode[STIX]{x1D706}$ of this instability are

(4.18a,b)$$\begin{eqnarray}\unicode[STIX]{x1D70F}\sim \sqrt{\frac{\unicode[STIX]{x1D706}}{|\ddot{\ell }|}};\quad \unicode[STIX]{x1D706}\sim \sqrt{\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}|\ddot{\ell }|}},\end{eqnarray}$$

which translate, given $\ddot{\ell }$ in (4.17), to

(4.19a,b)$$\begin{eqnarray}\unicode[STIX]{x1D70F}\sim \frac{h}{V};\quad \unicode[STIX]{x1D706}\sim h.\end{eqnarray}$$

These relationships are relatively convincing at the scaling level (they offer a good collapse for different $We$, in particular), with however large pre-factors, as seen from figure 11.

Figure 12.

Onset of the instability. Lamella oscillation period $T$ (hollow circles) and instability growth rate $\unicode[STIX]{x1D70F}$ (plain triangles), in units of the crushing time scale $a/V$ versus $We$. Dashed and solid lines are the predicted scalings from (4.20), along with best fit amplitudes. They cross at the instability threshold and define $We_{c}\simeq 66$.

4.5 Transition to splashing

The global oscillation period of the fused cylinders $T$ given in (4.6), and the lamella instability time scale $\unicode[STIX]{x1D70F}$ in (4.19) both written in units of $a/V$ depend on the Weber number as

(4.20a,b)$$\begin{eqnarray}T\sim \frac{a}{V}We;\quad \unicode[STIX]{x1D70F}\sim \frac{a}{V}We^{-1}.\end{eqnarray}$$

One is longer (namely $T$) and the other shorter (namely $\unicode[STIX]{x1D70F}$) when $We$ increases. Obviously, a cross-over occurs whose meaning is the following: at moderate $We$, the expansion–recoil period of the fused rims is too short for the expanded lamella to destabilize given its slow pace; destabilization is faster and has also more time to develop at larger $We$, and is thus now favoured. Both trends are visible in figure 12. The instability growth time cannot, de facto be measured below the cross-over threshold at the intersection of the two curves, which occurs for $We_{c}\approx 66$, a value which was qualitatively anticipated by Lhuissier & Villermaux (Reference Lhuissier and Villermaux2013).

The onset of this instability (explaining the formation of the ‘arms’ of Worthington (Reference Worthington1876)), whose outcome is the production the fines analysed in the next section, is very similar to the so-called ‘splashing’ transition in drop impact (i.e. the production of disjointed fragments besides the recoil of the main drop, see Josserand & Thoroddsen (Reference Josserand and Thoroddsen2016)) for which we have given here a mechanistic interpretation.

5.1 Capillary breakup with corrugations ($We<We_{c}$)

For collisions with $We$ below the critical splashing Weber number $We_{c}$, drops simply form the breakup of the fused rim cylinder, whose mean radius is $\sqrt{2}a$ (see e.g. figure 1a). In that case, we know that the outcome is a droplet distribution of sizes $d$ given by

(5.1)$$\begin{eqnarray}p(\unicode[STIX]{x1D701})=\frac{m^{m}}{\unicode[STIX]{x1D6E4}(m)}\unicode[STIX]{x1D701}^{m-1}\text{e}^{-m\unicode[STIX]{x1D701}},\quad \text{with }\unicode[STIX]{x1D701}=d/\langle d\rangle\end{eqnarray}$$

in units scaled by their mean $\langle d\rangle$, a mean approximately given by the standard Plateau–Rayleigh expectation $d_{PR}\approx 1.89\times 2\sqrt{2}a$ (see figure 13a,b) modulo a weak correction on $m$, whose value reflects the initial relative corrugations of the cylinder (Eggers & Villermaux Reference Eggers and Villermaux2008). If, right after the collision, the cylinder radius is $\sqrt{2}a+\unicode[STIX]{x1D709}(z)$, with $\unicode[STIX]{x1D709}(z)$ a longitudinal modulation with zero mean and variance $\langle \unicode[STIX]{x1D709}^{2}\rangle$, then

(5.2)$$\begin{eqnarray}m\sim \frac{a^{2}}{\langle \unicode[STIX]{x1D709}^{2}\rangle }.\end{eqnarray}$$

As seen in figure 9(a–c), the corrugations are mostly reminiscent of the state of the rims prior to the collision, and the variance $\langle \unicode[STIX]{x1D709}^{2}\rangle$ may be interpreted from an energy balance (see the Appendix in Bremond, Clanet & Villermaux (Reference Bremond, Clanet and Villermaux2007)). Only half of the kinetic energy is used for the rim recess, the other half, ultimately dissipated by viscosity, being at the origin of turbulent-like motions, the cause of the rim interface corrugations (see the Appendix in Villermaux & Bossa (Reference Villermaux and Bossa2011)). Balancing the available kinetic energy with the (transient) surface energy excess in the corrugated state $2\unicode[STIX]{x03C0}a\unicode[STIX]{x1D70E}\langle \sqrt{1+\unicode[STIX]{x1D709}^{\prime 2}}-1\rangle$ where $\unicode[STIX]{x1D709}^{\prime }=\text{d}\unicode[STIX]{x1D709}/\text{d}z$, we have, at lowest order and up to (large) prefactors,

(5.3a,b)$$\begin{eqnarray}a\unicode[STIX]{x1D70E}\langle \unicode[STIX]{x1D709}^{\prime 2}\rangle \sim a^{2}\unicode[STIX]{x1D70C}V^{2},\qquad \text{or}\quad \langle \unicode[STIX]{x1D709}^{\prime 2}\rangle \sim We.\end{eqnarray}$$

Corrugations are injected at the scale of the incoming rims radius $a$ which sets the fluctuation scale of $\unicode[STIX]{x1D709}$ so that $\langle \unicode[STIX]{x1D709}^{\prime 2}\rangle \approx \langle \unicode[STIX]{x1D709}^{2}\rangle /a^{2}$ (see figure 9) and

(5.4)$$\begin{eqnarray}m\sim We^{-1},\end{eqnarray}$$

indicating that more violent collisions lead to broader size distributions [remember that], such as that shown in figure 13(a) at $We=58$ for which $m=5$.

Figure 14.

Transverse ligament mediated fragmentation, at $We=193$. The sequence shows the emergence and destabilization of the lamella (left panel, from top to bottom), then its retraction into transverse ligaments with volume $s^{3}$ (central picture) which finally fragment into droplets with individual size $d$ (right picture).

5.2 Above the splashing transition ($We>We_{c}$)

Above the splashing transition described in § 4.5, the transverse indentations produce ligaments which fragment into increasingly many smaller drops as $We$ is increased. Consequently, the mean drop size deviates from the Plateau–Rayleigh prediction (figure 13b) because the drops now come from the fused rim cylinder chopped off into smaller pieces, namely the finer transverse ligaments (figure 14). An estimation of the mean size of these fine droplets is as follows: assume the fused rims convert entirely (this is all the more true when $We$ is larger) into an assembly of transverse ligaments each with length $L\sim a\,We$ and spaced by $\unicode[STIX]{x1D706}\sim h$ (see § 4). Volume conservation gives $\unicode[STIX]{x03C0}(\sqrt{2}a)^{2}\unicode[STIX]{x1D706}\sim \unicode[STIX]{x03C0}L\langle d\rangle ^{2}$ with $\langle d\rangle$ the transverse ligament mean diameter, also setting that of the fines after capillary breakup (Plateau Reference Plateau1873; Rayleigh Reference Rayleigh1879). We thus expect, when the conversion is complete, that $\langle d\rangle$ will saturate at a value of order

(5.5)$$\begin{eqnarray}\langle d\rangle \sim h\end{eqnarray}$$

above $We_{c}$, a very different trend than the one prevailing below (for which $\langle d\rangle /h\sim We$, see figure 13b). Note that it is precisely because the set-up with random hole production misses the finest drops (see appendix B) that this transition is delayed compared with that observed with the controlled set-up, and which indeed occurs at $We_{c}$.

Figure 15.

Size distribution, at $We=193$. (a) Ligament size $s$ distribution, measured before (●) and after ($\star$) fragmentation (respectively expressions (5.6) and (5.7)), along with a gamma distribution (5.1) with order $m=40$ (solid line). (b) Drop size $d$ distribution, resulting from the transverse ligament fragmentation: (●) 322 drops, for which origin ligaments are unspecified; ($\star$) 117 drops issued from the ligaments in (a). The solid line is the distribution (5.8) with orders $m=40,n=5$. Data come from the controlled rim production set-up.

5.3 Distributions

Having understood the origin of the fines and why they indeed feature small sizes, we now turn to their distribution. Each transverse ligament fragments into a variable number $i$ of droplets with mean $\langle i\rangle \sim L/\langle d\rangle$, with $\langle d\rangle$ the mean droplet size (typically $\langle i\rangle =O(10)$ at $We=193$). The volume of a ligament can be obtained either at its greatest extension $L$, prior to its fragmentation by measuring its diameter, or by summing the droplet volumes produced after its fragmentation (sizes $d_{i}$), leading to an equivalent ligament size $s$ given by

(5.6)$$\begin{eqnarray}\displaystyle s^{3} & = & \displaystyle \frac{\unicode[STIX]{x03C0}}{4}Ld_{L}^{2}\end{eqnarray}$$

(5.7)$$\begin{eqnarray}\displaystyle & = & \displaystyle \frac{\unicode[STIX]{x03C0}}{6}\mathop{\sum }_{i}d_{i}^{3},\end{eqnarray}$$

a size which is distributed among the ligament population, with mean $\langle s\rangle$. The fluctuation of $s$ reflects the one of the fused cylinder corrugations, which are themselves described by the distribution in (5.1), so that we expect the normalized size as $\unicode[STIX]{x1D701}=s/\langle s\rangle$ averaged over several realizations, involving several ligaments at a given $We$, to be distributed according to (5.1) with the parameter $m$ possibly depending on $We$ as in (5.4).

Figure 15 reports measurements of $s$ made both before (41 ligaments over 6 impacts) and after (subset of 22 ligaments) their fragmentation, together with the resulting drop sizes $d$ obtained in the transverse instability region. The distribution of $s$ is very narrow ($m=40$), suggesting that an efficient size selection by the instability mechanism operates, which not only sets the inter-ligament distance $\unicode[STIX]{x1D706}$ deterministically, but also the volume embarked in each of them, thus screening the corrugations of the rims. In this two-stage fragmentation scenario, the transverse ligaments are corrugated, for the same reason the fused cylinder below $We_{c}$ is, and we call $n$ their corrugation index. When restricted to this relatively homogeneous ligament population (large $m$), the width of $p(d/\langle d\rangle )$ is thus essentially given by that of the intrinsic ligament breakup with $n=5$, as for $We<We_{c}$ (see figures 13a and 15b).

Figure 16.

Orders $n$ of the gamma distribution (5.1) for $We<We_{c}$, and $n$ and $m$ of the Bessel-compound distribution (5.8) for $We>We_{c}$. The expectation $n\sim We^{-1}$ from (5.4) is plotted as a solid line. Data come from the random rim production set-up.

The random hole production set-up is better suited to appreciating the different $s$ in the overall spray content. Small ligaments will produce smaller droplets than larger ligaments. The overall distribution of sizes $d$ has thus two sources of variability, the one coming from the distribution of $s$, and the one coming from the distribution of $d$ for a given $s$. Since these two factors are independent of each other, we expect the final fine droplet distribution to be given by a linear superposition of the two effects as (Villermaux & Bossa Reference Villermaux and Bossa2009)

(5.8)$$\begin{eqnarray}p\left(\unicode[STIX]{x1D701}=\frac{d}{\langle d\rangle }\right)=\frac{2(mn)^{(m+n)/2}}{\unicode[STIX]{x1D6E4}(m)\unicode[STIX]{x1D6E4}(n)}\unicode[STIX]{x1D701}^{(m+n)/2-1}K_{m-n}(2\sqrt{mn\unicode[STIX]{x1D701}}),\end{eqnarray}$$

where $K_{m-n}$ is a modified Bessel function of the second kind of order $m-n$ (Abramowitz & Stegun Reference Abramowitz and Stegun1964), and where $m$ reflects the roughness of the distribution in ligament sizes $s$, while $n$ reflects that of the ligament corrugations in this construction. In the final expression (5.8), the roles of $m$ and $n$ are interchangeable (the distribution is unchanged by the permutation $m\leftrightarrow n$), both variabilities adding up in an additive fashion, by construction: indeed, the square of the standard deviation $\langle \unicode[STIX]{x1D701}^{2}\rangle /\langle \unicode[STIX]{x1D701}\rangle ^{2}-1$ of $p(\unicode[STIX]{x1D701})$ in (5.8) is

(5.9)$$\begin{eqnarray}\frac{1+m+n}{mn},\end{eqnarray}$$

meaning, for instance, that the width of the final drop size distribution relative to its mean is given by that of the ultimate ligament breakup ($1/\sqrt{n}$) when the ligaments are all alike ($m\rightarrow \infty$), either because they have been sorted as such as above, or because they are naturally produced as such (as in effervescent atomization, see Lhuissier & Villermaux (Reference Lhuissier and Villermaux2013)). Figures 15(b) and 17 show that this double-index distribution is a fair fit to the final drop size distribution, obtained here with moderate statistics (322 drops produced by 4 impacts in figure 15b); much better fits are achieved in less controlled situations offering no analytical clues about the precise origin of the drops, but which on the other hand involve many more droplets and thus offer better converged distributions (Kooij et al. Reference Kooij, Sijs, Denn, Villermaux and Bonn2018, Reference Kooij, Astefanei, Corthals and Bonn2019).

Figure 17.

Drop size $d$ distribution, for increasing Weber number, along with the predicted distributions (right column, solid lines). (a) Dimensional distributions $P(d)$ for $We=28,74,135$. Distributions $p(\unicode[STIX]{x1D701}=\text{d}/\langle \text{d}\rangle )$ for (b) $We=28$, with gamma distribution (5.1) (order $n=12$); (c) $We=74$ with Bessel-based distribution (5.8) (indices $m=10,n=12$); (d) $We=135$ with rain distribution (5.10). All data (hollow symbols) come from the random rim production set-up.

The precise value of the indices $m$ and $n$ depend, in general, on the way the ligaments providing the final drops are formed through various sequential instabilities, possibly coupled to each other, or not. In the present case, both $m$ and $n$ follow the trend anticipated in (5.4) as $We$ is varied, with $m$ (related to the source ligament size) being always larger than $n$ (related to the ligament breakup), as seen in figure 16.

The distribution in (5.8) presents two limit cases: when the fused cylinder transverse instability is not too noisy, it selects a unique transverse wavelength $\unicode[STIX]{x1D706}$, corresponding to a unique ligament size $s$ and in that case $m\rightarrow \infty$. In that limit, which suits gentle collisions with $We$ close to $We_{c}$, or even below (hence with no secondary instability), the distribution (5.8) is simply the gamma distribution in (5.1) with order $n$ and $\unicode[STIX]{x1D701}=d/\langle d\rangle$, as seen in figure 17(b), identically to figures 13 and 15(b). The other limit suits ligaments which are broadly distributed in size, and also very corrugated themselves (with, say, $m=n=4$), a case describing strong collisions at high Weber number for which

(5.10)$$\begin{eqnarray}p(\unicode[STIX]{x1D701})={\textstyle \frac{2^{15}}{9}}\unicode[STIX]{x1D701}^{3}K_{0}(8\sqrt{\unicode[STIX]{x1D701}}),\end{eqnarray}$$

as seen in figure 17(d). The general form in (5.8) interpolates smoothly between these two limits across the whole $We$ range (figure 17c).

Because it involves two successive processes, the drop generation mechanism in this rim collision scheme thus not only produces finer droplets in the mean, but it produces broader size distributions than when only one process is at play – a distinct feature of the fines, which are called so because they coexist with much bigger droplets.

5.4 Ejection speed of the early fines

Above $We_{c}$ and for times of the order of its instability onset $\unicode[STIX]{x1D70F}=h/V$ in (4.19), fines are ejected in the transverse lamella plane. Given that there are, at least, two different mechanisms responsible for the formation of these tiny drops (§§ 4.2 and 4.3), their ejection velocities are, not surprisingly, broadly distributed.

Figure 18.

Fine ejection velocity. (a) Velocity of the 10 % fastest droplets (circle symbol, dashed line), compared with the square root law (5.11) (solid line, with adjusted amplitude 0.15) and droplet mean velocity (square symbol, dotted line). (b–d) Two-dimensional histograms of droplet size and velocity, for increasing Weber number: (b) $We=62$; (c) $We=112$; (d) $We=193$.

The mean ejection speed $V_{e}$ is found (figure 18a) to be typically smaller than, or of the order of, the collision speed $V$ at large $We$, a fact already observed in Thoroddsen, Takehara & Etoh (Reference Thoroddsen, Takehara and Etoh2012) and Riboux & Gordillo (Reference Riboux and Gordillo2015) or Wang & Bourouiba (Reference Wang and Bourouiba2018), for instance, consistent with the early dynamics described in § 4.2. Of course, to the size distribution of fines corresponds a distribution of speeds $V_{e}$ and a clear correlation exists for different sizes $d$ (of order $h$) at a given $We$, and for different $We$: the smallest, because they are the earliest, are also the fastest (figure 18b–d). As opposed to the mean speed, these tiny droplets are faster (once their velocity is normalized by $V$) as $We$ is increased. The ejection velocity of the 10 % fastest drops can be approximately described by (figure 18a)

(5.11)$$\begin{eqnarray}V_{e}\sim V\sqrt{We-We_{c}},\end{eqnarray}$$

where the offset $We_{c}$ is of the order of the lamella transverse instability threshold, a trend not incompatible with the one anticipated by the very early dynamics described in (4.15) of § 4.3.

This early time detachment process involves a comparatively small fraction of the liquid to be fragmented after the crushing time $a/V$, but it does contribute to broadening of the final drop size distribution.