Mass, momentum and energy partitioning in unsteady fragmentation

2.1. Experimental approach

We used two high-speed cameras to simultaneously capture both side and top views of the impact. The frame rates of the top-view and side-view cameras were 20 000 and 8000 frames per second (fps), respectively. An impacting drop was released from a needle at different heights, selecting a range of impacting speeds, $u_0$. We measured the impact speed directly from the side high-speed recordings. The drops impacted stainless-steel cylindrical rods of comparable size to that of the impacting drops to ensure formation of a horizontal inviscid expanding sheet in the air (Wang & Bourouiba Reference Wang and Bourouiba2017).

For most of this paper, we present results of a de-ionized water drop with nigrosin dye of concentration 1.2 g l$^{-1}$, with density $\rho =1.0\times 10^3\ \mathrm {kg\ m}^{-3}$, surface tension $\sigma =72\times 10^{-3}\ \text {N m}^{-1}$ and kinematic viscosity $\nu =1.0\times 10^{-6}\ \text {m}^2\ \text {s}^{-1}$. Fluids with different properties were used to assess the robustness and regime of validity of the predictions. These included glycerol–water to vary viscosity and dimethyl sulfoxide (DMSO) to vary surface tension. The ranges of the impact Weber and Reynolds numbers were selected to examine the regimes of validity of predictions. These details are discussed in § 9. Each group of experiments was repeated at least 15 times (table 1) with detailed experimental conditions summarized in tables 1 and 2.

Table 1. Experimental conditions used, including the impact drop diameter $d_0$, the impacting speed, $u_0$, and associated $We = \rho u_0^2 d_0/\sigma$ and $Re = u_0d_0/\nu$, where $\rho = 1.0\times 10^3$ kg m$^{-3}$, $\nu = 1.0\times 10^{-6}$ m$^2$ s$^{-1}$ and $\sigma = 72$ mN m$^{-1}$ are the density, kinematic viscosity and surface tension of the water drop, respectively. Here, $N_{exp}$ is the number of experiments per condition.

Table 2. Properties of fluids used in this study, with corresponding initial conditions and associated Weber, $We$, and Reynolds, $Re$, numbers. Glycerol–water mixtures were used to vary viscosity, with the percentage indicating mass fraction. DMSO was used to vary surface tension. The impact target diameters used were chosen to ensure drop-to-target size ratios, $\eta$, between 1.4 and 1.9, which ensures horizontal sheet expansion and negligible surface stresses.

2.2. Advanced image processing (AIP) algorithms

As described by Wang & Bourouiba (Reference Wang and Bourouiba2022), we developed multi-step AIP algorithms to capture all key physical quantities of the sheet, rim, ligaments and secondary droplets listed in figure 1($b$). These were well captured by our AIP algorithms despite their complex change in morphology and topology over time. For the expanding sheet, we detect the boundary between the sheet and the rim as the contour of the sheet. The radius of the sheet, $r_s$, can then be captured by measuring the total area, $A_s$, within the sheet contour and calculating, $r_s =\sqrt {A_s/{\rm \pi} }$ (figure 3$a$). The sheet spatio-temporal thickness profile, $h(r,t)$, was derived and validated via light absorption, and the sheet velocity profile, $u(r,t)$, was derived and validated by particle tracking velocimetry (Wang & Bourouiba Reference Wang and Bourouiba2017). With these quantities determined, the fluid volume, momentum and energy in the sheet are quantified experimentally.

Figure 3. ($a$) Sheet contour detection and the light absorption method using nigrosin-dyed liquid to measure the sheet thickness. ($b$) Calibration curves mapping liquid sheet thickness to transmitted light intensity ratio $I/I_0$ for different fluids laden with nigrosin dye of concentration 1.2 g l$^{-1}$. Here, $I$ is the intensity of the light transmitted through the fluid sheet, while $I_0$ is the background light intensity. WB2017 indicates data from Wang & Bourouiba (Reference Wang and Bourouiba2017), while WB2021 indicates data of the present study.

We note that, beside water, nigrosin can also be dissolved in other liquids. Table 2 includes some of these: glycerol–water mixture and DMSO. Nigrosin dye at concentration 1.2 g l$^{-1}$ was added to these liquids. Figure 3 shows that the absorption of the different nigrosin-dyed liquids follow the Beer–Lambert law of absorption well, with

(2.1)\begin{equation} \log(I/I_0) = ch \Longrightarrow h ={-}\frac{1}{c}\log(I/I_0), \end{equation}

where $I_0$ is the background light intensity and $I$ is the intensity of the light transmitted through the dyed liquid of a given thickness $h$. Here, $c$ is the liquid absorptivity. The measured fluid absorptivity of dyed liquid water $c = \frac {1}{185}\ \mathrm {\mu }$m$^{-1}$ matches that measured by Wang & Bourouiba (Reference Wang and Bourouiba2017), which further supports the robustness of this light absorption methodology for sheet thickness measurements. Glycerol–water mixtures of the same nigrosin concentration also have similar fluid absorptivity $c$ to that of water, while DMSO with the same nigrosin concentration has a slightly larger fluid absorptivity $c = \frac {1}{160}\ \mathrm {\mu }$m$^{-1}$.

For the rim and ligaments, we first detect the inner and outer contours of the rim–ligament connection. By setting the extruded parts to be the ligaments, our AIP algorithms can systematically and precisely separate the ligaments from the rim (figure 4$a$). Upon separation, the thickness of the rim, $b$, is measured locally at each angular position and averaged along the rim. The length of each ligament, $\ell$, is measured as the curvilinear distance from its root to its tip. The width of each ligament, $w$, is measured locally at each cross-section and averaged along the centreline of the ligament (inset of figure 4$a$). The volumes of the rim and ligaments are measured by assuming axisymmetric cross-sections. The tips of the ligaments detected at each frame are linked together by our ligament-tracking algorithms (Wang & Bourouiba Reference Wang and Bourouiba2018b) enabling to track the evolution of each ligament throughout the sheet evolution – expansion and retraction – and measure their tip velocity. The rim's velocity is measured by taking the derivative of the sheet radius. With the geometry and velocity of the rim and ligaments determined, the fluid volume, momentum and energy in the rim and the ligaments are quantified experimentally.

Figure 4. Key steps of our AIP algorithms, including ($a$) contour detection of the rim and ligaments and their separation; and secondary droplet ($b$) detection and ($c$) tracking.

For the secondary droplets, their sizes and positions are detected at each frame (figure 4$b$) and linked into trajectories (figure 4$c$) by our droplet-tracking algorithms (Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018). The initial time of formation of each trajectory is the time at which the droplet is shed. The ejection speed of a droplet is its speed at time of shedding. With the size, speed and shedding time of each droplet, the cumulative volume shed, $\varOmega _{d}$, as well as the associated momentum and energy of the droplets shed throughout the fragmentation, are quantified experimentally. Finally, note that the accuracy of the AIP algorithms was established by Wang & Bourouiba (Reference Wang and Bourouiba2022), who showed that the fluid volume in all sub-parts of the fragmentation system sum up, at all times, to the initial volume of the impacting drop $\varOmega _0$. The AIP algorithms, used here as well, capture both cumulative and instantaneous volume conservation.

3.1. Dynamics of rim destabilization and thickness

A number of prior studies have focused on rim destabilization via linear stability analysis (Roisman & Tropea Reference Roisman and Tropea2002; Zhang et al. Reference Zhang, Brunet, Eggers and Deegan2010; Villermaux & Bossa Reference Villermaux and Bossa2011; Peters et al. Reference Peters, van der Meer and Gordillo2013). It was shown that the rim initially destabilizes into small corrugations due to a local interplay of interfacial (Rayleigh–Plateau) and inertial (Rayleigh–Taylor) instabilities accounting for the continuous deceleration of the rim (Krechetnikov & Homsy Reference Krechetnikov and Homsy2009; Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018). However, upon fragmentation, the fluid shed from the rim is in the form of ligaments (rather than small-amplitude corrugations). Here, nonlinear effects dominate (Roisman, Berberovi & Tropea Reference Roisman, Berberovi and Tropea2009; Agbaglah et al. Reference Agbaglah, Josserand and Zaleski2013; Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018) and linear instability predictions consequently no longer capture the observations. Wang et al. (Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018) showed that, in fact, the non-Galilean frame of the rim with time-varying deceleration imposes an instantaneous self-adjustment of the rim thickness, $b$, which remains equal to the local and instantaneous capillary length as defined by the instantaneous sheet acceleration, $\ddot {r_s}$. This can be expressed as

(3.1)\begin{equation} {Bo} = \frac{\rho b^2 (-\ddot{r_s})}{\sigma} = 1 \Longrightarrow b = \sqrt{\frac{\sigma}{\rho (-\ddot{r_s})} }, \end{equation}

where $Bo$ is a local and instantaneous Bond number. Such self-adjustment of the rim thickness governed by the $Bo =1$ criterion indirectly determines the volume of fluid continuously shed from the rim during unsteady fragmentation. Here, we note that fluid shedding from the rim of a stationary sheet, such as a Savart sheet upon impact of a continuous liquid jet (Savart Reference Savart1833; Clanet & Villermaux Reference Clanet and Villermaux2002), is clearly not governed by the (3.1) constraint. Gordillo, Lhuissier & Villermaux (Reference Gordillo, Lhuissier and Villermaux2014) reported another fluid shedding mechanism for such stationary sheets, determined by the Taylor–Culick speed recession and cusps forming along the sheet, leading to a shedding velocity $v_\ell \approx u_0/5$, where $u_0$ is the impact velocity of the liquid jet. Figures 1($b$) and 2 show that the contour of the liquid sheet in unsteady fragmentation is nearly circular in contrast to the cusp-containing shape of stationary sheets (Gordillo et al. Reference Gordillo, Lhuissier and Villermaux2014). In the interest of concision, details on how rim deceleration induces fluid shedding during cusp-free unsteady sheet fragmentation discussed by Wang & Bourouiba (Reference Wang and Bourouiba2021) are not repeated here.

3.2. Non-Galilean Taylor–Culick's law for the sheet radius, $r_s(r, t)$

Wang & Bourouiba (Reference Wang and Bourouiba2018b) showed that droplets are shed continuously during the sheet evolution – expansion and retraction – with most shedding, in fact, occurring prior to the sheet's maximum extension. Using our AIP algorithms, we (Wang & Bourouiba Reference Wang and Bourouiba2022) showed that the fluid shed from the rim has a leading-order effect on the sheet evolution, namely, the unsteady sheet evolution is coupled with the fluid shedding and resulting droplet fragmentation during the entire process. Wang & Bourouiba (Reference Wang and Bourouiba2022) derived a modified theory of unsteady sheet evolution which incorporated the coupled inertial and fluid shedding effects, leading to the following governing equation for the sheet radius, $r_s(r,t)$:

(3.2) \begin{equation} \left. \begin{array}{c@{}} \dot{(ar_s)} = q_{in}-q_{out}, \\ (\rho ar_s)\ddot{r_s} - \rho q_{in}(u(r_s,t) - \dot{r_s}) + \rho q_{out}v_{\ell} + 2\sigma r_s = 0, \\ \text{with } a = {\rm \pi}b^2/4 \text{ and } q_{in}(t) = h(r_s,t)r_s(t)(u(r_s,t)-\dot{r_s}), \end{array} \right\} \end{equation}

where $\langle \cdot \rangle$ is the derivative with respect to time, the sheet velocity profile is $u(r_s,t) = r_s/t$ (Wang & Bourouiba Reference Wang and Bourouiba2017), $a$ is the cross-sectional area of the rim, $q_{in}$ is the volume rate – volume per unit of time and unit radian – entering the rim, $q_{out}$ is the volume rate of fluid shed from the rim per unit of time and radian, and $v_\ell$ is the speed of fluid leaving from the rim into the ligaments in the reference frame of the rim. Based on the self-adjustment of the rim thickness governed by the $Bo = 1$ criterion, the volume rate $q_{out}$ is directly determined by mass conservation in (3.2). In addition, Wang & Bourouiba (Reference Wang and Bourouiba2022) showed that the fluid outward speed $v_\ell$ is determined by the local dynamics of the ligament–rim junction. With $Bo = 1$ and $v_\ell$ determined, the governing equation of unsteady sheet radius is shown to reduce to

(3.3)\begin{equation} -\rho h(r_s,t)\left(\frac{r_s}{t} - \dot{r_s}\right)^2 + \left(2-\frac{\rm \pi}{7}\right)\sigma = 0, \end{equation}

referred to as a non-Galilean Taylor–Culick law, due to its similarity with the classical Taylor–Culick relation, but with an additional term $-{\rm \pi} \sigma /7$ shown to incorporate the unsteadiness, inertial and shedding effects.

By choosing the characteristic length scale to be the impacting drop diameter, $d_0$, and the characteristic time scale to be the capillary time, $\tau _{cap} = \sqrt {\rho \varOmega _0/{\rm \pi} \sigma } =\sqrt {\rho d_0^3/6\sigma }$, the non-dimensional time $T$, sheet radius $R_s$ and thickness $H$ are

(3.4a–c)\begin{equation} T = \frac{t}{\tau_{cap}}, \quad R_s = \frac{r_s}{d_0} \quad \text{and} \quad H = \frac{h}{d_0}. \end{equation}

Thus, the non-dimensional form of (3.3) reads

(3.5)\begin{equation} -6H(R_s,T)\left(\frac{R_s}{T}-\dot{R_s}\right)^2 + \left(2-\frac{\rm \pi}{7}\right) = 0, \end{equation}

where $H(R_s,T)$, the sheet thickness at the rim, is

(3.6)\begin{equation} H(R_s,T) = \frac{T\sqrt{{6We}}}{6a_3R_s(T)^3 +a_2R_s(T)^2T\sqrt{{6We}} + a_1R_s(T)T^2{We}}, \end{equation}

where $a_1$, $a_2$ and $a_3$ are the constant coefficients of the sheet thickness profile that was derived and validated by Wang & Bourouiba (Reference Wang and Bourouiba2017).

3.3. Time-to-peak invariance

Armed with (3.5), a nonlinear equation, Wang & Bourouiba (Reference Wang and Bourouiba2022) revealed and validated that the unsteady sheet dynamics has a peculiar property of invariance of time-to-peak: the evolution of the sheet radius, $R_s(T)$, over dimensionless time $T$ non-dimensionalized by the capillary time scale $\tau _{cap}$, is independent of impact $We$ or energy. Namely, the sheet radius, $R_s(T)$, is expressed as

(3.7)\begin{equation} R_s(T) = \sqrt{{We}} Y_0(T), \end{equation}

reaches its maximum value at a time $T_m = 0.43$ independent of the impact $We$ or energy, with an approximate analytical expression of $Y_0(T)$ derived to be

(3.8) \begin{equation} \left. \begin{array}{c@{}} Y_0(T) = 0.15(T-T_m)^3-0.4(T-T_m)^2 + \mathcal{R}_m, \\ \text{with } T_m = 0.43 \text{ and } \mathcal{R}_m = R_m/\sqrt{{We}} = 0.12. \end{array} \right\} \end{equation}

In the remainder of this paper, we show that such a property of time invariance has important implications for the temporal evolution of volume/mass, momentum and energy of the fluid system and their partition in its sub-parts: from sheet, to rim to fluid shed.

5.1. Volume/mass fraction in the rim

Taking the rim to be cylindrical, its dimensional volume, $\mathcal {\varOmega }_{rim}$, is

(5.1)\begin{equation} \varOmega_{rim}(t) = 2{\rm \pi} r_s(t) \frac{\rm \pi}{4}b^2(t), \end{equation}

which in dimensionless form is

(5.2)\begin{equation} \mathcal{V}_{rim}(T) = \frac{\varOmega_{rim}/2{\rm \pi}}{d_0^3} = \frac{\rm \pi}{4}R_s(T) B(T)^2= \frac{\rm \pi}{4}Y_0(T)\varPsi^2(T), \end{equation}

where the expressions of sheet radius (3.7) and rim thickness (4.4) were used. In addition, the impacting drop volume $\varOmega _0 = {\rm \pi}d_0^3/6$ is non-dimensionalized as

(5.3)\begin{equation} \mathcal{V}_0 = \frac{\varOmega_0/2{\rm \pi}}{d_0^3} = \frac{1}{12}. \end{equation}

Thus, the volume fraction in the rim becomes

(5.4)\begin{equation} \frac{\mathcal{V}_{rim}(T)}{\mathcal{V}_0} = 3{\rm \pi} Y_0(T)\varPsi^2(T), \end{equation}

so the evolution of the fluid volume fraction in the rim governed by the $Bo = 1$ criterion is clearly independent of $We$. This is indeed verified by our data (figure 7$a$).

Figure 7. ($a$) Measured time evolution of the volume fraction in the rim, ${\mathcal {V}_{rim}(T)}/{\mathcal {V}_0}$, for different impact $We$. All data collapse on a single curve, showing independence from $We$. The solid line shows its prediction (5.4). The inset shows the ratio of measured data and prediction, which remains close to one, indicating good agreement between predictions and data. ($b$) Measured time evolution of the rate of change of the rim volume $\dot {\mathcal {V}}_{rim}$ for different $We$, which also collapse on a single curve and is well captured by our prediction (5.7). The inset shows the difference between measurement and prediction, which is close to zero, indicating good agreement between prediction and data. Error bars indicate the standard deviation across all repeated experiments in each group.

5.2. Volume/mass fraction in the fluid shed from the rim: ligaments and droplets

In terms of fluid shed from the rim, in the form of ligaments and droplets, the key quantity is the fluid volume shed from the rim per unit of time, $Q_{out}$, which from the rim's mass conservation, (3.2), is

(5.5)\begin{equation} Q_{out}(T) = Q_{in}(T) - \dot{\mathcal{V}}_{rim}(T). \end{equation}

To derive an explicit expression, we next determine the volume entering the rim per unit of time and radian, $Q_{in}(T)$, and the rate of change of the rim's volume, $\dot {\mathcal {V}}_{rim}(T)$, separately.

5.2.1. Rate of change of volume fraction in the rim: $\dot {\mathcal {V}}_{rim}$

From (5.1), the rate of change of the rim volume over time is

(5.6)\begin{equation} \dot{\varOmega}_{rim}(t) = \frac{\textrm{d}}{\textrm{d} t} \left[2{\rm \pi} r_s(t) \left(\frac{\rm \pi}{4}b^2(t)\right)\right], \end{equation}

and in non-dimensional form is

(5.7)\begin{equation} \dot{\mathcal{V}}_{rim}(T) = \frac{\dot{\varOmega}_{rim}/2{\rm \pi} }{d_0^3/\tau_{cap}} = \frac{\textrm{d}}{\textrm{d} T} \left[\frac{\rm \pi}{4} Y_0(T)\varPsi^2(T) \right], \end{equation}

which is also independent of $We$ as verified by our data, that is well captured by our prediction (5.7) (figure 7$b$).

5.2.2. Volume fraction entering the rim

We start with the rate of volume injection into the rim from the sheet per radian, $q_{in}(t)$, which was expressed in (3.2). In non-dimensional form, it reads

(5.8)\begin{equation} Q_{in}(T) = \frac{q_{in}}{d_0^3/\tau_{cap}} = H(R_s,T)R_s(T)\left(\frac{R_s(T)}{T}-\dot{R_s}(T)\right). \end{equation}

Using the sheet radius expression (3.7) and the sheet thickness at the rim, (4.3) reads

(5.9)\begin{align} Q_{in}(T) = \frac{1}{{We}}\varPhi(T) {We} Y_0(T)\left(\frac{Y_0(T)}{T}-\dot{Y_0}(T)\right) = \varPhi(T)Y_0(T) \left(\frac{Y_0(T)}{T}-\dot{Y_0}(T)\right), \end{align}

which is interestingly also independent of impact $We$ . This expression, (5.9), is an explicit form of the volume entering the rim per unit of time and radian, $Q_{in}(T)$. It can be further simplified using (3.5) to

(5.10)\begin{equation} \frac{R_s(T)}{T}-\dot{R_s}(T) = \frac{\alpha}{\sqrt{H(R_s,T)}} \quad \text{with } \alpha = \sqrt{\frac{2-{\rm \pi}/7}{6}}. \end{equation}

Substituting (5.10) into (5.9) gives

(5.11)\begin{equation} Q_{in}(T) = \alpha\sqrt{H(R_s,T)}R_s(T) = \alpha\sqrt{\varPhi(T)}{Y_0(T)}. \end{equation}

The solid line in figure 8($a$) shows that our prediction (5.11) captures the data well.

Figure 8. ($a$) Measured time evolution of the rate of volume entering the rim per unit radian, $Q_{in}$, for different impact $We$, which all collapse on a single curve, indicating independence from $We$. The solid line shows that our prediction (5.11) captures the data well. ($b$) Measured time evolution of the cumulative volume fraction entering the rim, $\mathcal {V}_{in} = \varOmega _{in}(T)/\varOmega _0$, is also independent of $We$ and is well captured by our prediction (5.13) (solid line). Both insets show that the ratios of measurement to prediction remain close to one, which indicates good agreement between prediction and data. Error bars indicate the standard deviation across all repeated experiments in each group.

The non-dimensionalized cumulative volume entering the rim from the sheet is thus

(5.12)\begin{equation} \mathcal{V}_{in}(T) = \int_{0}^{T} Q_{in}(T')\, \mathrm{d}T' = \int_{0}^{T} \alpha\sqrt{\varPhi(T')}{Y_0(T')}\, \mathrm{d}T'. \end{equation}

Normalized by the impacting drop volume $\mathcal {V}_0$, the fraction of cumulative volume entering the rim is

(5.13)\begin{equation} \frac{\mathcal{V}_{in}(T)}{\mathcal{V}_0} = 12\int_{0}^{T} \alpha\sqrt{\varPhi(T')}{Y_0(T')}\, \mathrm{d}T', \end{equation}

which is also independent of the impact $We$ as verified by the data in figure 8($b$). Thus, at each instant of dimensionless time, the cumulative volume fraction emanating from the sheet into the rim, with respect to the impacting drop volume is, in fact, independent of $We$.

Having determined the volume entering the rim per unit of time and radian, $Q_{in}$, with (5.11), and the rate of change of rim volume, $\dot {\mathcal {V}}_{rim}$, with (5.7),

(5.14)\begin{equation} Q_{out}(T) = \alpha\sqrt{\varPhi(T)}Y_0(T) - \frac{\rm \pi}{4}\frac{\textrm{d}}{\textrm{d} T}[Y_0(T)\varPsi^2(T)], \end{equation}

where $\alpha$ is the constant given in (5.10). Since both $Q_{in}$ and $\dot {\mathcal {V}}_{rim}$ are independent of $We$, the volume shed per unit of time and radian from the rim, $Q_{out}$, is also independent of $We$. This is verified by our data in figure 9$a$, which are very well captured by our prediction (5.14) (solid line). In addition, the volume rate shed, $Q_{out}$, appears to monotonically increase with time at a nearly constant rate. Taking this functional form (see § A.4) would read

(5.15)\begin{equation} Q_{out} = p_0 + p_1(T-T_m), \end{equation}

where $T_m = 0.43$ is the time of maximum sheet radius (Wang & Bourouiba Reference Wang and Bourouiba2022), $p_0 = 0.09$ is the volume shed per unit of time and radian at $T_m$, and $p_1 = 0.15$ is the approximate increase rate of the volume rate shed throughout the sheet evolution. Note that $p_1$ and $p_2$ are not fitted but both are derived theoretically in § A.4.

Figure 9. ($a$) Measured time evolution of the volume rate shed per radian from the sheet into the rim for different impact $We$. All data collapse on a single curve, which indicates independence from $We$. The solid and dashed lines show the full prediction (5.14) and the approximate explicit expression (5.15) of $Q_{out}$, respectively. The inset shows that the ratio of the measured data and prediction remains close to one, which indicates good agreement between prediction and data. ($b$) Measured time evolution of the cumulative volume fraction $\mathcal {V}_{out}(T)/\mathcal {V}_0$ shed from the rim into ligaments–droplets, for different $We$, is also independent of $We$ and is well captured by its full prediction (5.17), as well as the approximate expression (5.18). The inset shows the difference between measured data and prediction, which remains close to zero, indicating good agreement between prediction and data. Error bars indicate the standard deviation across all repeated experiments in each group.

From the prediction of volume rate shed, $Q_{out}$, we can directly determine the time evolution of the non-dimensionalized volume in the ligaments and droplets shed from the rim,

(5.16)\begin{equation} \mathcal{V}_{out} = \int_{0}^{T} Q_{out}(T')\, \mathrm{d}T' = \int_{0}^{T} \alpha\sqrt{\varPhi(T')}Y_0(T')\, \mathrm{d}T' - \frac{\rm \pi}{4} Y_0(T)\varPsi^2(T). \end{equation}

Normalized by the impacting drop volume $\mathcal {V}_0$, the fraction of volume shed from the rim gives

(5.17)\begin{equation} \frac{\mathcal{V}_{out}}{\mathcal{V}_{0}} = 12\int_{0}^{T} \alpha\sqrt{\varPhi(T')}Y_0(T')\, \mathrm{d}T' - 3{\rm \pi} Y_0(T)\varPsi^2(T), \end{equation}

which can be shown (see § A.4) to reduce to

(5.18)\begin{equation} \frac{\mathcal{V}_{out}}{\mathcal{V}_0} = 12p_0(T-T_m) + 6p_1(T-T_m)^2. \end{equation}

Figure 9($b$) shows that both the full prediction (5.17) of the time evolution of measured cumulative volume shed from the rim normalized by $\varOmega _0$ and its approximate explicit form (5.18) capture the data very well for all $We$. Interestingly, the resulting cumulative volume fraction shed amounts to 70 % of the original drop volume, regardless of impact energy.

Recalling our first question in § 1 : How does the volume fraction of fluid shed throughout the fragmentation vary with impact $We$ or impact energy? We have shown that

1. (i) the temporal evolution of the dimensionless rate of volume shed from the rim, $Q_{out}(T)$, to form ligaments and droplets, is approximately linearly increasing over time; and

2. (ii) that the volume fraction of fluid shed, $Q_{out}$, from the rim at each instant and cumulatively, $\mathcal {V}_{out}$, are both also independent of the impact $We$ or impact energy.

In sum, the fractions of impacting drop volume shed at any given time and cumulatively, are, in fact, both independent of the impact $We$ or energy. A resulting 70 % of the impacting drop volume/mass is shed, regardless of impact energy, by the dimensionless time $T = 0.7$, up to which point the sheet conserves its integrity free of inner punctures. Beyond that time, the sheet can punctures around the solid surface of the rod, depending on its surface properties, and particularly for higher impact $We$, since the sheet thickness monotonically decreases with the impact $We$ (Wang & Bourouiba Reference Wang and Bourouiba2017). For example, the measured sheet thickness around the edge of the rod at $T = 0.7$ for $We = 963$ is less than 20 $\mathrm {\mu }$m. This rupture of sheets around the edge of the solid surface from which they emanate along with collision of rims were reported and studied in prior work (Villermaux & Bossa Reference Villermaux and Bossa2011; Lejeune, Gilet & Bourouiba Reference Lejeune, Gilet and Bourouiba2018; Néel, Lhuissier & Villermaux Reference Néel, Lhuissier and Villermaux2020).

5.3. Volume/mass fraction in the sheet and summary of volume/mass partitioning

With the fraction of cumulative volume having entered the rim $\mathcal {V}_{in}(T)$ at time $T$ determined in (5.13), the fraction of impacting drop volume remaining in the sheet at that time $T$ is

(5.19)\begin{equation} \frac{\mathcal{V}_{s}(T)}{\mathcal{V}_0} = 1 - \frac{\mathcal{V}_{in}(T)}{{\mathcal{V}_0}} = 1 - 12\int_{0}^{T} \alpha\sqrt{\varPhi(T')}{Y_0(T')}\, \mathrm{d}T', \end{equation}

which is clearly independent of $We$ (figure 10$a$). In sum, we have discovered that the volume fractions in the sheet, $\mathcal {V}_{s}$, in the rim, $\mathcal {V}_{rim}$, and in the fluid shed from the rim in the form of ligaments and droplets, $\mathcal {V}_{out}$, are in fact all independent of impact $We$. This is verified by the summary of all measured volume fractions in the fluid system (figure 10$b$). When examining the effect of volume shed from the rim on the sheet evolution, the cumulative volume shed should be compared with the instantaneous rim volume rather than the total impacting drop volume. This is because the dynamics of the rim is derived from control volume theory, with the rim being the control volume. Hence, although the cumulative fluid shedding amounts to approximately 20–30 % (figure 10$b$) of the impacting drop volume, the cumulative volume shed from the rim is in fact of the same order of magnitude as that of the rim at any given time during most of the sheet expansion (figure 10$b$). We therefore cannot neglect shedding. Doing so would amount to an overestimation of the rim's volume by 100 %.

Figure 10. ($a$) Measured time evolution of the volume fraction in the sheet, $\mathcal {V}_s = \varOmega _s(T)/\varOmega _0$, including the sheet in the air and on the rod, for different $We$. All data collapse on a single curve, which indicates independence of the the mass fraction in the sheet (air and rod) from $We$. The solid line shows that the prediction (5.19) matches the data well. The inset shows that the ratio of measured data to prediction is close to one, which indicates a good agreement between prediction and data. ($b$) Measured time evolution of the volume fractions in different sub-parts of the system. For each sub-part, the data for different $We$ collapse on a single curve, which indicates independence from $We$. The solid lines show the predictions of the volume fractions in each sub-part are in good agreement with the data and recovering the total impacting drop volume, $\varOmega _0$. This confirms that the cumulative volume shed, at a given time, from the rim remains of the same order of magnitude as the instantaneous rim volume.

Finally, given incompressibility, the mass fractions in each part of the fluid system – sheet, rim and ligaments–drops shed – are immediately deduced from the volume fractions derived above, by simply multiplying volumes by the fluid density $\rho$. The mass fractions in each sub-part are thus also all independent of impact $We$.

In the next sections, we turn to the momentum and energy partitioning in each sub-part of the system.

6.1. Fraction of momentum in the rim

We now examine the evolution of the momentum in each sub-part of the system. Starting with the rim, its momentum is

(6.1)\begin{equation} p_{rim}(t) = \rho \varOmega_{rim}(t) \dot{r_s}(t), \end{equation}

and in dimensionless form reads

(6.2)\begin{equation} P_{rim}(T) = \frac{p_{rim}/2{\rm \pi}}{\rho d_0^4/\tau_{cap}} = \frac{1}{12}\mathcal{V}_{rim}(T) \dot{R_s}(T) = \sqrt{{We}} \frac{\rm \pi}{4} Y_0(T)\varPsi^2(T)\dot{Y_0}(T), \end{equation}

where (5.4) and (3.7) were used. Equation (6.2) explicitly depends on $\sqrt {{We}}$. The non-dimensional impacting drop momentum, $p_0 = \varOmega _0 u_0$ is

(6.3)\begin{equation} P_0 = \frac{\rho\varOmega_0 u_0/2{\rm \pi}}{\rho d_0^4/\tau_{cap}} = \frac{1}{12\sqrt{6}}\sqrt{{We}}. \end{equation}

Thus, the momentum fraction in the rim is

(6.4)\begin{equation} \frac{P_{rim}(T)}{P_0} = 3\sqrt{6}{\rm \pi} Y_0(T)\varPsi^2(T)\dot{Y_0}(T), \end{equation}

which is independent of $We$ as verified by our data (figure 11$a$). The solid line in figure 11($a$) shows that our prediction of fraction of momentum in the rim, (6.4), captures the data very well and is independent of $We$.

Figure 11. Measured time evolution of ($a$) the fraction of momentum in the rim and ($b$) the cumulative momentum fraction shed from the rim for different impact $We$. All data collapse on a single curve, which indicates that both the fraction of momentum in the rim and the fraction of momentum shed from the rim are independent of $We$. The solid lines show that our predictions (6.4) and (6.11) capture the data very well. Both insets show that the difference between measured data and prediction remains zero, which indicates good agreement between prediction and data. Error bars indicate the standard deviation across all repeated experiments in each group.

6.2. Fraction of momentum in the fluid shed: ligaments and droplets

Turning to the momentum in the ligaments–droplets sub-parts of the system, consider the rate of volume shedding from the rim, $q_{out}$, per radian. The associated rate of momentum shedding from the rim into the ligaments–droplets, $\dot {p}_{out}$, per radian, is then

(6.5)\begin{equation} \dot{p}_{out}(t) = \rho q_{out}(t)[\dot{r_s}(t) + v_\ell(t)], \end{equation}

where $v_\ell (t)$ is the fluid speed shed from the rim in the reference frame of the rim. Based on the local dynamics of the ligament growth, Wang & Bourouiba (Reference Wang and Bourouiba2022) derived and verified that the rate of momentum shed from the fluid ejected at speed, $v_\ell$, can be approximated to be

(6.6)\begin{equation} \rho q_{out}(t)v_{\ell}(t) = \frac{1.34{\rm \pi}}{{We}^{3/8}}\sigma r_s(t). \end{equation}

Using the scale of momentum change rate per radian as $\rho d_0^4/\tau _{cap}^2$, the dimensionless momentum shed from the rim per unit of time is

(6.7)\begin{equation} \dot{P}_{out}(T) = \frac{\dot{p}_{out}}{\rho d_0^4/\tau_{cap}^2 } = Q_{out}(T)\dot{R_s}(T) + \frac{1.34{\rm \pi}}{6{We}^{3/8}}R_s(T). \end{equation}

Using (3.7), (6.7) becomes

(6.8)\begin{equation} \dot{P}_{out}(T) = \sqrt{{We}}\left[ Q_{out}(T)\dot{Y_0}(T) + \frac{1.34{\rm \pi}}{6{We}^{3/8}}Y_0(T) \right]. \end{equation}

Wang & Bourouiba (Reference Wang and Bourouiba2022) showed that the fluid speed $v_\ell$ shed from the rim in the reference frame of the rim is much smaller than the rim velocity $\dot {r_s}$. Thus, the contribution of the second term in the bracket of (6.8) is much smaller than the first term. For the intermediate-$We$ range, $250<{We}<10^4$, over which common fragmentation processes occur (Wang & Bourouiba Reference Wang and Bourouiba2022), the variation caused by $We$ on the value of the bracket is within 10 %, a weak $We$-dependence. Without loss of generality, the Weber-dependent coefficient of the second term in the bracket can be approximated to take the average value in the intermediate-$We$ range, ${1.34{\rm \pi} }/{(6{We}^{3/8})} \approx {\rm \pi}/56$. Thus, the momentum partition of the fluid shed from the rim is approximately

(6.9)\begin{equation} \dot{P}_{out}(T) \approx \sqrt{{We}}\left[ Q_{out}(T)\dot{Y_0}(T) + \frac{\rm \pi}{56}Y_0(T) \right]. \end{equation}

Taking the integral of $\dot {P}_{out}$ with time, the cumulative momentum shed from the rim, which is transferred into the combination of ligaments and droplets, is thus

(6.10)\begin{equation} P_{out}(T) = \sqrt{{We}}\int_0^{T}\left[ Q_{out}(T')\dot{Y_0}(T') + \frac{\rm \pi}{56}Y_0(T') \right]\, \mathrm{d}T', \end{equation}

which, similar to $P_{rim}$, is also proportional to $\sqrt {{We}}$. Thus, the fraction of cumulative momentum shed from the rim at a given time $T$ is

(6.11)\begin{equation} \frac{P_{out}(T)}{P_0} = \int_0^{T}\left[ Q_{out}(T')\dot{Y_0}(T') + \frac{\rm \pi}{56}Y_0(T') \right]\, \mathrm{d}T', \end{equation}

which is independent of the impact $We$.

Figure 11($b$) shows the measured time evolution of the cumulative momentum fraction shed from the rim into the ligaments and droplets. At each time, the cumulative momentum is measured by accounting for the momentum of all ligaments along the rim and the cumulative momentum of secondary droplet shed up to that time. In addition, during the shedding, the growth of ligaments is restricted by surface tension. Thus, we also account for the momentum lost due to surface tension as part of the cumulative momentum shed from the rim. The experimental measurement of the cumulative momentum shed from the rim is thus

(6.12)\begin{equation} p_{out}^{(exp)}(t) = \sum_{n=1}^{N(t)}\rho\varOmega_{ln}(t) u_{ln}(t) + \sum_{m=1}^{M(t)}\rho\varOmega_{dm}(t) u_{dm}(t) + \int_0^{t} \sum_{n=1}^{N(t')}\frac{\rm \pi}{2}\sigma w_n(t')\, \mathrm{d}t', \end{equation}

where $N(t)$ is the number of ligaments at time $t$, and $\varOmega _{ln}(t)$ and $u_{ln}(t)$ are the volume and average velocity of the $n$th ligament at time $t$, respectively. Here, $N(t)$ is the total number of droplets shed at time $t$, while $M(t)$ is the total number of droplets shed by time $t$. The $\varOmega _{dm}(t)$ and $u_{dm}(t)$ are the volume and velocity, respectively, of the $m$th droplet shed before time $t$. The $w_n(t)$ is the width of the $n$th ligament and ${\rm \pi} \sigma w_n(t)/2$ is the surface tension force acting on that ligament at time $t$. All the quantities in (6.12) are measured with high precision by the AIP algorithms, as described in § 2.2.

Figure 11($b$) shows that the measured time evolution of the cumulative momentum fraction in the fluid shed from the rim, in the form of ligaments and drops, (6.12), for different impact $We$. All data collapse on a single curve, which shows that the fraction of momentum in the fluid shed from the rim is indeed independent of $We$ and is very well captured by our prediction (6.11) (solid line).

6.3. Fraction of momentum in the sheet

By analogy to the mass analysis in § 5, the momentum of the sheet, which includes both the part in the air and on the rod, is

(6.13)\begin{equation} p_{s}(t) = p_0 - p_{in}(t) \quad \text{with } p_{in}(t) = \int_0^{t} 2{\rm \pi} \rho q_{in}(t') u(r_s,t')\, \mathrm{d}t', \end{equation}

which is the difference between the impacting drop momentum and the momentum entering the rim. Non-dimensionalized by the momentum scale per radian $\rho d_0^4/\tau ^2$,

(6.14)\begin{equation} P_{s}(T) = \frac{p_s/2{\rm \pi}}{\rho d_0^4/\tau_{cap}^2} = P_0 - P_{in}(T) \quad \text{with} \ P_{in}(T) = \int_0^{T} Q_{in}(T') U(R_s,T')\, \mathrm{d}T'. \end{equation}

Using the expression of $Q_{in}(T)$ (5.11), the sheet velocity profile $U(R_s,T) = R_s/T$ and the sheet radius of $R_s$ (3.7), the cumulative momentum entering the rim $P_{in}(T)$ becomes

(6.15)\begin{equation} P_{in}(T) = \sqrt{{We}}\int_{0}^{T} \alpha\frac{\sqrt{\varPhi(T')}}{T'}Y_0^2(T')\, \mathrm{d}T'. \end{equation}

The fraction of cumulative momentum entering the rim at time $T$ is

(6.16)\begin{equation} \frac{P_{in}(T)}{P_0} = 12\sqrt{6}\int_{0}^{T} \alpha\frac{\sqrt{\varPhi(T')}}{T'}Y_0^2(T')\, \mathrm{d}T', \end{equation}

which is independent of the impact $We$ as verified by the data (figure 12$a$). It shows that the measured cumulative momentum entering the rim approaches the impacting drop momentum $P_0$, which shows that no momentum is lost in the sheet. The solid line in figure 12($a$) shows that the prediction (6.16) captures the data at first order. However, a small but systematic offset from the data is observed. To understand the origin of the gap, we examine the rate of momentum injection into the rim per unit of time,

(6.17)\begin{equation} \dot{P}_{in}(T) = \alpha \sqrt{{We}} \frac{\sqrt{\varPhi(T)}}{T}Y_0^2(T). \end{equation}

Figure 12($b$) shows the momentum rate entering the rim for different $We$. Normalized by $\sqrt {{We}}$, all data collapse on a single curve and is well captured by our prediction (6.17) at late time. However, the prediction deviates slightly from the data at early time. This indicates that a second-order effect, not captured by our first-order sheet dynamics theory (Wang & Bourouiba Reference Wang and Bourouiba2022), takes place at early time. We leave to future work the deciphering of the physics underlying the early-time, second-order effect of the crushing of the drop.

Figure 12. ($a$) Measured time evolution of the cumulative momentum fraction entering the rim from the sheet $P_{in}(T)/P_0$ for different impact $We$. All data collapse on a single curve, which shows the independence from $We$. The measured cumulative momentum entering the rim approaches the impacting drop momentum, which shows that no momentum is lost in the sheet. The solid line shows that the prediction (6.16) matches the data at first order, but with a slight underestimation. The inset shows the difference between the measured data and the prediction. ($b$) Measured time evolution of the rate of momentum fraction $\dot {P}_{in}$ entering the rim, which is proportional to $\sqrt {{We}}$. The solid line shows that the prediction (6.17) captures the data at first order, but with small underestimation at early time, which leads to the deviation of the prediction from the data in panel ($a$). The inset shows the ratio of measured data with prediction. Error bars indicate standard deviation across repeated experiments in each group.

Thus, the fraction of the momentum in the sheet with respect to the impacting drop momentum can be expressed as

(6.18)\begin{equation} \frac{P_{s}(T)}{P_0} = 1 - \frac{P_{in}(T)}{P_0} = 1-12\sqrt{6}\int_{0}^{T} \alpha\frac{\sqrt{\varPhi(T')}}{T'}Y_0^2(T')\, \mathrm{d}T', \end{equation}

which is also independent of $We$ as verified by the data (figure 13$a$), which is captured, at first order by our prediction (6.18). Note that given that the prediction (6.16) slightly underestimates the cumulative momentum, $P_{in}$, entering the rim from the sheet, the prediction (6.18) is expected to slightly overestimate the momentum in the sheet. These slight deviations arise from second-order effects at early crushing time, not captured by our sheet dynamics theory.

Figure 13. ($a$) Measured time evolution of the momentum fraction in the sheet $P_s/P_0$ for different $We$. All data collapse on a single curve showing independence from $We$. The prediction (6.18) (solid line) captures the data at first order, but with a slight overestimation, consistent with the underestimation of $P_{in}(T)/P_0$ in figure 12($a$). ($b$) Measured time evolution of the cumulative surface tension force $P_\sigma$ acting on the rim for different $We$, normalized by the impacting drop momentum $P_0$. All data collapse on a single curve, which shows independence from $We$, and is well captured by our prediction (6.21). Both insets show the difference between measured data and prediction. Error bars indicate the standard deviation across all repeated experiments in each group.

6.4. Momentum balance across the sub-systems: sheet, rim and fluid shed

We have now verified that the fractions of momentum in each sub-part of the fragmenting fluid system are independent of $We$. However, distinct from volume/mass partitioning, the sum of the momentum in each sub-part does not recover the impacting drop momentum, $P_0$. Instead, the total momentum summed over all sub-parts decreases over time due to surface tension forces acting on the rim, which reduces the momentum of the rim and the fluid shed from the rim. The cumulative surface tension force acting on the rim at a given time $t$ (impulse) is

(6.19)\begin{equation} p_\sigma(t) = \int_0^t 4{\rm \pi} \sigma r_s(t')\, \mathrm{d}t', \end{equation}

and in non-dimensional form reads

(6.20)\begin{equation} P_\sigma(T) = \frac{p_\sigma/2{\rm \pi}}{\rho d_0^4/\tau_{cap}} = \int_0^T \frac{1}{3} R_s(T')\, \mathrm{d}T' = \sqrt{{We}}\int_0^T \frac{1}{3} Y_0(T')\, \mathrm{d}T', \end{equation}

where (3.7) was used. The fraction of initial drop momentum, $P_0$, taken up by this force by time $T$ is

(6.21)\begin{equation} \frac{P_\sigma(T)}{P_0} = \int_0^T 4\sqrt{6} Y_0(T')\, \mathrm{d}T', \end{equation}

which is also independent of $We$ as verified by the data (figure 13$b$), and is very well captured by our prediction (6.21) (solid line in figure 13$b$).

Finally, figure 14 shows that, incorporating the impulse of the surface tension force, the total momentum recovers most of the initial momentum of the impacting drop at first order. In addition, the figure shows that the fractions of impacting drop momentum in each sub-part of the fluid system, as well as the cumulative impulse of the surface tension force, all follow universal time functions, which are independent of the impact $We$ or energy.

Figure 14. The partition of momentum fractions in each sub-part of the fluid system is independent of $We$. The small derivation of total momentum from unity originates from a second-order effect on the sheet dynamics at early time (as shown in figure 12$b$).

7.1. Fraction of energy in the rim

The energy of the rim can be separated into two parts, the kinetic energy and surface energy. Knowing the rim mass, $m_{rim}$, and velocity, $\dot {r_s}$, the rim kinetic energy is

(7.3)\begin{equation} e_{rim}^k(t) = \tfrac{1}{2}\rho \varOmega_{rim} \dot{r_s}^2. \end{equation}

Thus, the non-dimensional form of this energy per unit radian reads

(7.4)\begin{equation} E_{rim}^k(T) = \frac{e_{rim}^k/2{\rm \pi}}{\rho d_0^5/\tau_{cap}^2} = \frac{1}{2}{We}\left[\frac{\rm \pi}{4} Y_0(T)\varPsi^2(T) \right] \dot{Y_0}^2, \end{equation}

where (5.1), (3.7) and (4.4) were used. Equation (7.4) has an explicit dependence on impact $We$. When normalized by the impacting drop energy, $E_0$, the fraction of total energy taken up by the rim's kinetic energy is

(7.5)\begin{equation} \frac{E_{rim}^k}{E_0} = 18{\rm \pi} [ Y_0(T)\varPsi^2(T) ] \dot{Y_0}^2, \end{equation}

which is independent of impact $We$ as verified by the data (figure 15$a$), and is very well captured by our prediction (7.5) (solid line).

Figure 15. ($a$) Measured time evolution of the fraction of total energy taken up by the rim's kinetic energy $E_{rim}^{(k)}(T)/E_0$ for different impact $We$. All data collapse on a single curve showing independence from $We$. The solid line shows that our prediction (7.5) captures the data well. ($b$) Measured time evolution of the fraction of energy taken up by the rim's surface energy $E_{rim}^{(k)}(T)/E_0$, for different impact $We$. Contrary to the kinetic energy, the fraction of the rim's surface energy depends on the impact $We$, which is also captured well by our prediction (7.8). Both insets show that the difference between measured data and prediction remains zero, which indicates good agreement between the prediction and data. Error bars indicate the standard deviation across all repeated experiments in each group.

Turning to the surface energy of the rim, consider the rim to have a cylindrical shape. Its surface energy is

(7.6)\begin{equation} e_{rim}^s(t) = 2{\rm \pi} r_s({\rm \pi} b)\sigma. \end{equation}

In non-dimensional form, its surface energy per radian thus reads

(7.7)\begin{equation} E_{rim}^s(T) = \frac{e_{rim}^s/2{\rm \pi}}{\rho d_0^5/\tau_{cap}^2} = \frac{\rm \pi}{6} R_s B = \frac{\rm \pi}{6} {We}^{1/4}[{-}6\ddot{Y}_0(T)]^{{-}1/2}Y_0(T), \end{equation}

where the expressions for the rim thickness $B$ (4.4) and the sheet radius $R_s$ (3.7) were used. Equation (7.7) has an explicit dependence on $We^{1/4}$.

When normalized by the total impact energy, $E_0$, the fraction of total energy stored in the rim's surface energy is

(7.8)\begin{equation} \frac{E_{rim}^s(T)}{E_0} = 24{\rm \pi} {We}^{{-}3/4}[{-}6\ddot{Y}_0(T)]^{{-}1/2}Y_0(T), \end{equation}

which has an explicit dependence on $We^{-3/4}$ distinct from the energy fraction in the form of the rim's kinetic energy, which was shown to be independent of the impact $We$. Figure 15$b$ shows the measured temporal evolution of the fraction of total energy in the rim's surface energy and shows that it is captured very well by our prediction (7.8). Comparing figures 15($a$) and 15($b$), the magnitude of the surface energy of the rim is approximately half that of the kinetic energy and accounts for less than 5 % of the total impacting drop energy. This confirms that the surface tension force and energy corresponding to the rim can be considered as negligible in the unsteady fragmentation.

Note that the effect of the surface energy of the rim was not reflected in the governing equation of the sheet radius (3.5). In fact, the surface tension force acting on the rim per radian, $2r_s \sigma$, given in the momentum balance equation (3.2) only corresponds to the surface energy of the evolving sheet. Analogous to the surface tension force of the capillary hydraulic jump problem (Bush & Aristoff Reference Bush and Aristoff2003), the surface tension force corresponding to the surface energy of the rim, associated with the curvature of the rim, would scale as $\sigma b$. It would thus have a distinct $We$-dependence from that of the dominant sheet surface tension force $\sigma r_s$, as well as other terms in the governing equation (3.5) as confirmed in (7.8). Finally, given that for most impacts of interest $b \ll r_s$, this surface tension force corresponding to the rim's contribution is expected to be negligible.

7.2. Fraction of energy in the fluid shed: ligaments and droplets

Based on the rate of fluid volume shed from the rim, $q_{out}$, the rate of energy feeding the fluid shed, per radian, is

(7.9)\begin{equation} \dot{e}_{out}(t) = \tfrac{1}{2}\rho q_{out}(t)[\dot{r_s}(t) + v_\ell(t)]^2, \end{equation}

where $\dot {r_s}$ is the rim velocity and $v_\ell$ is the outflux speed shed from the rim. As mentioned in § 6, $\dot {r_s}$ is much larger than $v_\ell$. Thus, the rate of kinetic energy passed on the fluid shed is approximately

(7.10)\begin{equation} \dot{e}_{out}(t) \approx \tfrac{1}{2}\rho q_{out}(t) \dot{r_s}^2(t) + \rho q_{out}(t)v_\ell(t)\dot{r_s}(t). \end{equation}

Using the approximate expression for $\rho q_{out} v_\ell$, (6.6), gives

(7.11)\begin{equation} \dot{e}_{out}(t) \approx \frac{1}{2}\rho q_{out}(t) \dot{r_s}^2(t) + \frac{1.34{\rm \pi}}{{We}^{3/8}}\sigma r_s(t)\dot{r_s}(t), \end{equation}

which in non-dimensionalized form reads

(7.12)\begin{equation} \dot{E}_{out}(T) = \frac{e_{out}}{\rho d_0^5/\tau_{cap}^3} = \frac{1}{2}Q_{out}(T)\dot{R_s}^2(T) + \frac{1.34{\rm \pi}}{6{We}^{3/8}}R_s(T)\dot{R_s}(T). \end{equation}

As discussed in § 6.2, the Weber-dependent coefficient of the second term in the bracket can be approximated to take an average value in the intermediate-$We$ range, which gives ${1.34{\rm \pi} }/{(6{We}^{3/8})} \approx {\rm \pi}/56$. Thus, the energy of the fluid shed from the rim is approximately

(7.13)\begin{equation} \dot{E}_{out}(T) = \frac{e_{out}}{\rho d_0^5/\tau_{cap}^3} = \frac{1}{2}Q_{out}(T)\dot{R_s}^2(T) + \frac{{\rm \pi} }{56}R_s(T)\dot{R_s}(T). \end{equation}

Using the expression of the sheet radius $R_s$, (3.7), gives

(7.14)\begin{equation} \dot{E}_{out}(T) = \frac{1}{2}{We}\left[ Q_{out}(T)\dot{Y_0}^2(T) + \frac{{\rm \pi} }{28}Y_0(T)\dot{Y_0}(T)\right]. \end{equation}

The cumulative fraction of total energy injected into the shed fluid at time $T$ is thus

(7.15)\begin{equation} \frac{E_{out}(T)}{E_0} = \int_0^{T} 72\left[ Q_{out}(T')\dot{Y_0}^2(T') + \frac{{\rm \pi} }{28}Y_0(T')\dot{Y_0}(T')\right]\, \mathrm{d}T', \end{equation}

which is independent of the impact $We$ as confirmed by our data (figure 16$a$). Here, experimental measurement of the cumulative kinetic energy shed from the rim is obtained from the measured volume rate, $q_{out}$, shed from the rim and the measured outflux speed $v_\ell$ using the AIP algorithms.

Figure 16. ($a$) Measured time evolution of the rate of kinetic energy of fluid shed from the rim for different impact $We$ normalized by impact energy $E_0$. All data collapse on a single curve, which shows independence from $We$. The solid line shows that the prediction (7.15) captures the data well. The inset shows the difference between measured data and prediction remains zero, which indicates good agreement between prediction and data. ($b$) Measured time evolution of the fractions of total energy of the ligaments and total energy of the cumulative droplets shed over time, for $We = 679$. Error bars indicate the standard deviation across repeated experiments in each group.

To confirm the accuracy of the measurements, the fluid shed from the rim becomes the ligaments and the cumulative secondary droplets. Thus, the total energy of the ligaments and the cumulative secondary droplets shed from the rim, including both the kinetic and surface energy, should be equal to the kinetic energy shed from the rim. Namely,

(7.16)$$\begin{gather} e_{out}(t) = \dot{e}_{drop}^k(t) + e_{drop}^s(t) + e_{liga}^k(t) + e_{liga}^s(t), \end{gather}$$

(7.17a,b)$$\begin{gather}\text{with } e_{liga}^k(t) = \sum_{n=1}^{N(t)}\frac{1}{2}\rho\varOmega_{ln}(t) u_{ln}^2(t), \quad e_{liga}^s(t) =\sum_{n=1}^{N(t)}\sigma A_{ln}(t), \end{gather}$$

(7.18a,b)$$\begin{gather}\text{and}\quad e_{drop}^k(t) = \sum_{m=1}^{M(t)}\frac{1}{2}\rho\varOmega_{dm}(t) u_{dm}^2(t), \quad e_{drop}^s(t) =\sum_{n=1}^{N(t)}\sigma A_{dm}(t), \end{gather}$$

where $A_{ln}$ is the surface area of the $n$th ligament and $A_{dm}$ is the surface area of the $m$th droplet. Other quantities were introduced in (6.12). Figure 16($b$) shows the measured temporal evolution of the fraction of total energy in the ligaments and the cumulative droplets with respect to the impacting drop energy $E_0$. It shows that the total energy of the ligaments and droplets is very well captured by our prediction (7.15). Thus, the kinetic energy shed from the rim is indeed converted to the energy required for the formation of the ligaments and droplets. Note that matching also confirms the accuracy of our AIP algorithms.

7.3. Fraction of energy in the sheet

The energy of the sheet – both on the rod and in the air – includes two parts, the kinetic energy due to the fluid motion in the sheet and the surface energy. We will start with the kinetic energy. Similar to the mass and momentum of the sheet, the sheet kinetic energy can be evaluated as

(7.19)\begin{equation} e_s^{(k)}(t) = e_0 - e_{in}(t) \quad \text{with } e_{in}(t) = \int_0^t \frac{1}{2} 2{\rm \pi} \rho q_{in}(t')u(r_s,t')^2 \, \mathrm{d}t', \end{equation}

which is the total impacting drop energy, $e_0$ in (7.1), minus the cumulative energy that has entered the rim at time $t$, $e_{in}(t)$. In non-dimensionalized form, it reads

(7.20)\begin{align} E_s^{(k)}(T) = \frac{e_s/2{\rm \pi} }{\rho d_0^5/\tau_{cap}^2} = E_0 - E_{in}(T) \quad \text{with } E_{in}(T) = \int_0^T \frac{1}{2}Q_{in}(T')U(R_s,T')^2 \, \mathrm{d}T'. \end{align}

Using the expression of $Q_{in}$ (5.11), the sheet velocity profile $U(R_s,T)= R_s/T$ and the sheet radius $R_s$ (3.7), the cumulative energy entering the rim becomes

(7.21)\begin{equation} E_{in}(T) = {We}\int_0^T \frac{1}{2} \alpha \frac{\sqrt{\varPhi(T')}}{T'^2}Y_0^3(T')\, \mathrm{d}T'. \end{equation}

Normalized by the impacting drop energy $E_0$,

(7.22)\begin{equation} \frac{E_{in}(T)}{E_0} = 72\int_0^T \alpha \frac{\sqrt{\varPhi(T')}}{T'^2}Y_0^3(T'), \mathrm{d}T', \end{equation}

which is independent of $We$ as confirmed by the data (figure 17$a$). It shows that the measured cumulative energy entering the rim approaches that of the impacting drop energy, $E_0$, which shows that no energy is lost in the sheet prior to entering the rim. However, similar to the cumulative momentum entering the rim, a small systematic gap between the data and the prediction is observed accounting for approximately 15 % of the total impacting drop energy $E_0$. To understand the origin of the gap, consider the energy entering the rim per unit of time:

(7.23)\begin{equation} \dot{E}_{in}(T) = {We} \frac{1}{2} \alpha \frac{\sqrt{\varPhi(T)}}{T^2}Y_0^3(T), \end{equation}

which deviates from the data slightly at the early time (figure 17$b$). This is consistent with the cumulative momentum entering the rim (figure 12$b$), which confirms that a second-order effect, not captured by our first-order sheet dynamics (Wang & Bourouiba Reference Wang and Bourouiba2022) theory, takes place at early impact time.

Figure 17. ($a$) Measured time evolution of the fraction of cumulative energy entering the rim from the sheet $E_{in}/E_0$ for different impact $We$. All data collapse on a single curve, which indicates independence from $We$. The measured cumulative energy entering the rim approaches to the impacting drop energy, which indicates no energy loss in the sheet. The solid line shows that the prediction (7.22) captures the data at first order, but with certain underestimation, consistent with that of the cumulative momentum $P_{in}/P_0$ entering the rim (figure 12$a$). The inset shows the difference between measured data and prediction. ($b$) Measured time evolution of the rate of energy injection into the rim $\dot {E}_{in}$, normalized by ${We}$ to which $\dot {E}_{in}$ is proportional. The solid line shows that the prediction (7.23) deviates slightly from data at early time, which leads to the offset in panel ($a$). The inset shows the ratio of measured data with prediction. Error bars indicate the standard deviation across repeated experiments in each group.

Thus, the partition of the kinetic energy in the sheet with respect to the impacting drop energy becomes

(7.24)\begin{equation} \frac{E_{s}^{(k)}(T)}{E_0} = 1 - \frac{E_{in}(T)}{E_0} = 1 - 72\int_0^T \alpha \frac{\sqrt{\varPhi(T')}}{T'^2}Y_0^3(T')\, \mathrm{d}T', \end{equation}

which is independent of $We$ as confirmed by the data (figure 18$a$). Since the fraction of cumulative energy entering the rim is underestimated by 15 % (figure 17$a$), the kinetic energy in the sheet is overestimated by 15 % (figure 18$a$).

Figure 18. Measured time evolution of the fraction of energy in the sheet with its ($a$) kinetic energy, $E_{s}^{(k)}/E_0$, and ($b$) surface energy, $E_{s}^{(s)}/E_0$, for different impact $We$. All data collapse on a single curve, which indicates independence from $We$. The solid line in panel $(a)$ shows that the prediction (7.24) captures the data at first order, but with certain overestimation, consistent with the underestimation of $E_{in}/E_0$ in figure 17($a$). The solid line in panel $(b)$ shows that the prediction (7.27) captures the data very well. Both insets show the difference between measured data and prediction. Error bars indicate the standard deviation across repeated experiments in each group.

Taking a two-dimensional (2-D) horizontal sheet, the surface energy of the sheet is

(7.25)\begin{equation} e_s^{(s)}(t) = 2\sigma {\rm \pi}(r_s)^2, \end{equation}

and in dimensionless form reads

(7.26)\begin{equation} E_{s}^{(s)} = \frac{e_s^{(s)}/2{\rm \pi} }{\rho d_0^5/\tau_{cap}^2} = \frac{\sigma (r_s)^2}{\rho d_0^5/\tau_{cap}^2} = \frac{1}{6}R_s^2 = \frac{{We}}{6} Y_0^2(T), \end{equation}

where $R_s$ (3.7) was used. Normalized by the impacting drop energy, $E_0$, (7.2),

(7.27)\begin{equation} \frac{E_{s}^{(s)}}{E_0} = 24Y_0^2(T), \end{equation}

which is also independent of $We$ as confirmed by figure 18$b$. Thus, the energy fraction of the sheet, including both the kinetic and surface energy, is independent of $We$.

In sum, except for the surface energy of the rim, the fractions of energy in each sub-part of the system from sheet to rim to fluid shed are all independent of impact $We$, which is very well captured by our predictions (figure 19$a$). The rim surface energy contribution is less than 5 % of the impacting drop energy. Thus, the sum of the contributions of the energy of the sheet, rim and fluid shed remain mostly independent of impact $We$ (figure 19$a$).

Figure 19. ($a$) The fractions of energy in each sub-part of the fluid system are all independent of $We$. ($b$) Measured time evolution of the energy loss in the system as shown in panel ($a$). The solid line shows the prediction (8.8) of energy loss in the rim, which accounts for 30 % of the original impacting drop energy. The remaining gap in predicted energy loss, of 15 %, in the system is due to a second-order mechanism occurring at early time (seen in figure 17$b$). The inset shows the difference between measured data and prediction. Error bars indicate the standard deviation across repeated experiments in each group.

9.1. Comparison with the experiments using fluids of various properties

To verify the robustness of our prediction of partition of volume/mass, momentum and energy, in unsteady fragmentation, we conducted experiments using fluids of different properties (table 2). We used glycerol–water mixtures to vary the fluid viscosity and DMSO to vary the fluid surface tension. As shown in §§ 4–7, all predictions are derived based on the $Bo = 1$ criterion that governs the rim dynamics (§ 3.1) and the non-Galilean Taylor–Culick law that governs the sheet dynamics (§ 3). These two conditions hold for impact $We$ and $Re$ that are much larger than unity so as to ensure early onset of shedding and inviscid regimes of sheet dynamics, respectively. Regarding the regime of validity of the $Bo = 1$ criterion for rim thickness self-adjustment, we showed in prior work that it is robust for local rim $Re:\ \widehat {{Re}}=\sqrt {\sigma b/\rho \nu ^2} < 6\sqrt {2}$ (Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018).

For low-impact $We$, the rim thickness $B\sim {We}^{-1/4}$ (4.4) becomes thicker. Thus, the capillary time of the rim $\tau _b = \sqrt {\rho b^3/8\sigma }$, characteristic of rim destabilization, increases, which results in the delays of fluid shedding from the rim. Riboux & Gordillo (Reference Riboux and Gordillo2015) showed that rim destabilization ceases when the ratio of the rim capillary time to the rim-thickening time $\tau _h = b/\dot {b}$ is larger than 0.1, namely $\tau _b/\tau _h > 0.1$, where the $Bo = 1$ criterion is also violated.

Figure 20($a$) compares snapshots of unsteady sheet fragmentation upon impact of a drop of water, DMSO and glycerol (50 %)–water (50 %) mixture at the time of maximum sheet radius $T_m = 0.43\tau _{cap}$. This comparison shows no qualitative difference in fragmentation patterns. The impact conditions for all three fluids lie in the regime $\widehat {{Re}}=\sqrt {\sigma b/\rho \nu ^2} > 6\sqrt {2}$ where the $Bo = 1$ criterion holds, as confirmed in the inset of figure 20($b$). The time evolution of sheet radius for all three fluids also follows the non-Galilean Taylor–Culick law (figure 20$b$). Figure 20($b$) shows the time evolution of the volume fractions of fluid in the sheet, in the rim and shed from the rim throughout the entire fragmentation for the three different fluids. For each quantity, the data of all three fluids collapse on a single curve and are well captured by our prediction (solid lines), which confirms the robustness of our prediction for the partition of unsteady fragmentation as long as $Bo = 1$ (3.1) and the non-Galilean Taylor–Culick law of sheet expansion are both satisfied.

Figure 20. ($a$) Comparison of snapshots of unsteady fragmentation upon drop impact using three fluids (water, DMSO and glycerol (50 %)–water (50 %) mixture) at the time of maximum sheet radius $T_m=0.43\tau _{cap}$. It shows no notable difference in fragmentation phenomenology. Full details of impact conditions and fluid properties for each fluid are given in tables 1 and 2. ($b$) Measured time evolution of the sheet radius throughout the entire fragmentation for different fluids is well captured by the non-Galilean Taylor–Culick law (3.5). The inset shows that the rim thickness for different fluids follows the $Bo = 1$ criterion (3.1). Error bars indicate the standard deviation across all repeated experiments for each group. ($c$) Measured time evolution of the volume fractions in different sub-parts of the system during fragmentation for different fluids. For each sub-part, the data for different fluids collapse on a single curve, and are well captured by the prediction, which indicates robustness of the theory.

9.2. Regime map in terms of $We$ and $Re$ where the partition theory holds

When fluid viscosity is sufficiently large for the local Reynolds number of the rim to satisfy $\widehat {{Re}} < 6\sqrt {2}$, droplet shedding ceases throughout the entire sheet evolution. Figure 21($a$) compares the unsteady fragmentation phenomena upon drop impact for increasing fluid viscosity. As seen in the figure for the glycerol (70 %)–water (30 %) mixture, the cumulative volume fraction, $\varOmega _{out}/\varOmega _0$, of fluid shed from the rim becomes systematically smaller than that of the other two lower viscosity mixtures used. We note that the sheet radius normalized by $\sqrt {{We}}$ for the glycerol (70 %)–water (30 %) mixture is systematically smaller than that for the other lower viscosity cases. This is not due to suppression or reduction of shedding. The reduction of fluid shedding would in fact increase the maximum radius of the sheet (Wang & Bourouiba Reference Wang and Bourouiba2022) due to less momentum and mass losses from the rim (figure 21$b$). However, viscous surface stresses $\tau \sim \mu \frac {\textrm {d} u}{\textrm {d} z}$ on the rod surface increases with $\mu$, which result in the reduction of the momentum entering the rim in the first place. We can confirm support for this effect with the departure of the sheet thickness and velocity profiles form the inviscid limit as well. Here, $u(r,t)$ is the sheet velocity profile and $z$ is the direction along the sheet thickness.

Figure 21. ($a$) Comparison of sequential snapshots from unsteady fragmentation upon drop impact using fluids from low to high viscosity. Fragmentation in the most viscous case is clearly reduced compared with the less viscous cases. ($b$) Measured time evolution of the sheet radius for the three cases is shown in panel ($a$). The radii temporal evolution collapse for the two fluids of lower viscosity. Viscous effects clearly start affecting the sheet radius evolution in the third case. The inset shows that data for the first two cases follow the $Bo = 1$ criterion (3.1), while the data for the third case do not. ($c$) The evolution of the volume fractions of fluid shed from the rim remains essentially inviscid, matching the prediction (5.17) assuming a non-Galilean Taylor–Culick law of evolution of the sheet and the rim's $Bo = 1$ criterion, the volume fractions shed from the rim collapse on a single curve, while the impact for the fluid of higher viscosity clearly deviates and shows a smaller cumulative shedding. Error bars indicate the standard deviation across all repeated experiments for each group.

Systematic experimental results shown in figures 20 and 21 confirm that the $We$ independence of partitioning of the volume/mass, momentum and energy in each sub-part of the fragmentation system holds if and only if the $Bo = 1$ criterion holds at the rim and the sheet radius continues to be governed by the non-Galilean Taylor–Culick law, where viscous effects are negligible. Recall that the validity of the rim's $Bo = 1$ criterion is for $\widehat {{Re}} > \widehat {{Re}}_c = 6\sqrt {2}$ (Wang et al. Reference Wang, Dandekar, Bustos, Poulain and Bourouiba2018). For the rim's local $Bo = 1$ to be ensured,

(9.1)\begin{equation} \widehat{{Re}}_{min}= \frac{\sqrt{\rho \sigma b_{min}}}{\mu} > \widehat{{Re}}_c = 6\sqrt{2} \end{equation}

has to hold, where $b_{min}$ is the minimum thickness of the rim during the fragmentation. Physically, the rim thickness should grow from zero at the very beginning of the impact. The fluid emanating from the sheet rapidly thickens the rim. Thus, it is reasonable to use the asymptotic value of the rim thickness at $t=0$, derived based on the $Bo = 1$ criterion (4.4), as the minimum rim thickness during fragmentation (thickness increases over time as seen in figure 6$b$), which gives

(9.2)\begin{equation} b_{min} = 0.2{We}^{{-}1/4}d_0. \end{equation}

Substituting (9.2) into (9.1) gives

(9.3)\begin{equation} \widehat{{Re}}_{min} = \frac{\sqrt{0.2\rho \sigma d_0}}{\mu} {We}^{{-}1/8} = \sqrt{0.2}{Re}{We}^{{-}5/8} > 6\sqrt{2}. \end{equation}

Thus, the regime of validity of the local instantaneous $Bo = 1$ rim criterion reads

(9.4)\begin{equation} {Re} {We}^{{-}5/8} > \frac{6\sqrt{2}}{\sqrt{0.2}} \approx 19.0. \end{equation}

Having quantified the effect of viscosity on the partition theory of unsteady fragmentation, we shift our focus to the effect of impact $We$ on the fragmentation. Figure 22($a$) compares the sheets and fragments upon drop impact for different impact $We$. As the impact $We$ decreases, fluid continues to be shed from the rim for ${We} > 250$, where the $Bo = 1$ holds (inset of figure 22$b$) and the sheet radius follows the non-Galilean Taylor–Culick law (figure 22$b$). However, for $We$ as low as 170, both the $Bo = 1$ criterion and the non-Galilean Taylor–Culick law no longer hold (inset of figure 21$b$). Consequently, the cumulative volume fraction, $\varOmega _{out}/\varOmega _0$, of fluid shed from the rim becomes systematically smaller than that of the ${We} > 250$ regime (figure 22$b$). However, the reduction of fluid shed from the rim for low $We$ is not due to viscous effects. For $We = 170$, from table 2, we have that

(9.5)\begin{equation} {Re} {We}^{{-}5/8} \approx 297 \gg 19, \end{equation}

which satisfies the criterion (9.4) for a negligible viscous effect. The absence of shedding, which leads to a violation of the rim's $Bo=1$ criterion, is at play here. This violation is associated with a rim thickening that delays its destabilization: as $We$ decreases, the rim thickness $B\sim {We}^{-1/4}$ increases, leading to the increase of the rim capillary time, $\tau _b = \sqrt {\rho b^3/8\sigma }$, which delays rim destabilization and associated shedding into droplets. Furthermore, Riboux & Gordillo (Reference Riboux and Gordillo2015) showed that instead of merely delaying rim destabilization and droplet ejection, droplet ejection ceases when the ratio of the rim capillary time to rim-thickening time becomes larger than 0.1, i.e. $\tau _b/\tau _h > 0.1$. Choosing the characteristic length scale to be $d_0$, and the characteristic time scale $\tau _{cap}=\sqrt {\rho d_0^3/6\sigma }$, this rim stabilization criterion reads

(9.6)\begin{equation} \frac{\tau_b}{\tau_h} = \dot{b}\sqrt{\frac{\rho b}{8\sigma}} = \dot{B}\sqrt{\frac{3}{4}B} > 0.1. \end{equation}

Figure 22. ($a$) Sequential snapshots of sheet upon drop impact for different impact $We$. Droplet shedding during sheet expansion takes place as long as $We > 170$. ($b$) Measured time evolution of sheet radius for the four cases shown in panel ($a$). Data for all cases of ${We} > 250$ collapse on a single curve and are well captured by the non-Galilean Taylor–Culick law (3.5). Data for $We = 170$ deviate. The inset shows that data for all cases of ${We} > 250$ continue to follow the $Bo = 1$ criterion (3.1), while the data for