2.2.1. Initial shape of the sessile drop

By choosing the length scale

(2.1)\begin{equation} R_{eq}=\left(\frac{3V}{4{\rm \pi}}\right)^{1/3}, \end{equation}

that gives the radius of an equivalent sphere of the same volume $V$, the initial shape of the drop can be characterized by the dimensionless Young–Laplace equation (Lubarda & Talke Reference Lubarda and Talke2011)

(2.2)\begin{equation} \frac{1}{h_0(1+h_0'^2)^{1/2}}-\frac{h_0''}{(1+h_0'^2)^{3/2}}={-}2h_0''(L_0)+G(L_0-z), \end{equation}

where $h_0(z)\equiv h(z,t=0^-)$ is the initial profile of the drop radius, $h_0'$ and $h_0''$ are its first and second derivatives with respect to $z$, $L_0$ is the initial height with $h_0(L_0)=0$ and

(2.3)\begin{equation} G = \frac{\rho g R_{eq}^2}{\sigma} \end{equation}

is the Bond number based on gravitational acceleration $g$. The associated boundary condition

(2.4)\begin{equation} h_0'(0)={-}\cot\theta_c, \end{equation}

involves the equilibrium contact angle $\theta _c$ that is a property of the substrate (table 1). Note that, in the $G\to 0$ limit, solution to (2.2) for $h_0(z)$ yields a spherical cap, as expected.

2.2.2. Drop evolution upon substrate acceleration

For $t>0$, in the plate's reference frame, the drop is subject to both gravity and an inertial force due to the plate's acceleration, giving a net body force $\rho \alpha \hat {\boldsymbol {z}}$ where

(2.5)\begin{equation} \alpha = a-g , \end{equation}

is the effective acceleration. We subsequently default to the inertial velocity and time references

(2.6a,b)\begin{equation} U_\alpha=\sqrt{\alpha R_{eq} },\quad T_\alpha=\sqrt{\frac{R_{eq}}{\alpha}}, \end{equation}

so the drop's flow through quiescent ambient air of negligible density and viscosity is governed by the non-dimensional Navier–Stokes equations, with pressure non-dimensionalized by $\sigma /R_{eq}$, the capillary pressure

(2.7a 2.7b)

where $\boldsymbol {u}=u\hat {\boldsymbol {r}}+w\hat {\boldsymbol {z}}$ is the velocity vector, $p$ is pressure, $\kappa$ is surface curvature, $\hat {\boldsymbol {n}}$ is the unit normal, $\delta _s$ is the surface Dirac $\delta$-distribution concentrated on the interface (Popinet Reference Popinet2018) and

(2.8a,b)\begin{equation} {\textit{Bo}} \equiv \frac{\rho \alpha R_{eq}^2}{\sigma} = \frac{\alpha}{g}G,\quad {\textit{Re}} = \frac{\rho \alpha^{1/2}R_{eq}^{3/2}}{\mu} , \end{equation}

are the acceleration-based Bond number and Reynolds number, respectively. It is also convenient here to define the Ohnesorge number ${\textit {Oh}}\equiv \mu /\sqrt {\rho \sigma R_{eq}}=\sqrt {{\textit {Bo}}}/{\textit {Re}}$ that directly compares viscous and surface tension effects. Finally, evolution of the profile $h(z,t)$ of the drop radius follows the kinematic condition

(2.9)\begin{equation} \left[\frac{\partial{h}}{\partial t}+w \frac{\partial h}{\partial z}\right]_{r=h}=u|_{r=h}. \end{equation}

2.2.3. Characteristic scales: inertial vs capillary scales

With an appropriate model for the contact dynamics prescribed at $z=0$ (discussed in § 2.3), we have obtained a closed description for the drop under inertial acceleration for all $t\geq 0$. This system is governed by four dimensionless parameters: the contact angle $\theta _c$, gravitational Bond number $G$ (or equivalently, the acceleration ratio $\alpha /g$), the inertial Bond number ${\textit {Bo}}$ and the Reynolds number ${\textit {Re}}$. Ranges of ${\textit {Bo}}$ used in the experiments are given in table 1, whereas typical values of ${\textit {Re}}$ are of $O(10^{2-4})$. Therefore, the inviscid flow assumption is reasonable, until ligament pinch-off is approached (Lister & Stone Reference Lister and Stone1998). We will also focus on results obtained for small $G$ where gravitational effects on the initial drop shape and thus early-stage motion are small.

The competition between inertial acceleration and capillarity due to surface tension is a key mechanism that drives the drop deformation, therefore it is instructive to also introduce the dimensional capillary wavelength $\varLambda$, time $T_\sigma$ and velocity $U_\sigma$ units

(2.10a–c)\begin{equation} \varLambda = \sqrt{\frac{\sigma}{\rho \alpha}},\quad T_\sigma = \sqrt{\frac{\rho R_{eq}^3}{\sigma}},\quad U_\sigma = \alpha\sqrt{\frac{\rho R_{eq}^3}{\sigma}}. \end{equation}

Particularly, $U_\sigma$ measures the change of speed generated by the imposed acceleration $\alpha$ over the capillary time scale $T_\sigma$. Under the inertial references (2.1) and (2.6a,b), these become

(2.11a–c)\begin{equation} \lambda \equiv \frac{\varLambda}{R_{eq}}= \frac{1}{\sqrt{{\textit{Bo}}}},\quad t_\sigma \equiv \frac{T_\sigma}{T_\alpha} = \sqrt{{\textit{Bo}}},\quad u_\sigma \equiv \frac{U_\sigma}{U_\alpha} = \sqrt{{\textit{Bo}}}. \end{equation}

We henceforth denote $\tau =t/\sqrt {{\textit {Bo}}}$ for rescaled physical dimensional time in capillary units, while $t$ is our default time non-dimensionalized by the inertial time scale $T_{\alpha }$ from (2.6a,b).

2.3.1. Numerical framework and additional non-dimensional parameters

According to the framework given in § 2.2, the drop detachment experiments are also simulated numerically to further our understanding of the fragmentation mechanism that is critical for disease transmission applications. The key numerical methods are explained next.

We adopt Basilisk (Popinet Reference Popinet2009, Reference Popinet2015, Reference Popinet2018) and solve the axisymmetric incompressible two-phase Navier–Stokes equations with surface tension for the air–liquid system. Simulations of high density ratios like air–water and non-negligible inertia are numerically challenging, hence we employ the momentum-conserving volume-of-fluid (VoF) scheme (Arrufat et al. Reference Arrufat, Crialesi-Esposito, Fuster, Ling, Malan, Pal, Scardovelli, Tryggvason and Zaleski2021). This is particularly necessary for high ${\textit {Bo}}$. Therein, the equations of motion for air, a counterpart of (2.7), are coupled and introduce ratios for density and dynamic viscosity: $\rho _a/\rho$ and $\mu _a/\mu$, between air (subscript $_a$) and water. These add to existing list of control parameters of the numerical system: $\theta _c$, ${\textit {Bo}}$, $G$ and ${\textit {Oh}}$.

2.3.2. Numerical model with constant contact-angle model

One challenge on the numerical modelling is the appropriate capture of the liquid–solid contact line dynamics. This turns out to be important, as discussed hereafter, to capture the tip and foot breakups. We start with a no-slip boundary condition at the wall along with a constant grid-scale contact angle as a first-order approximation. We subsequently develop a different numerical model in Appendix G where the constant angle approximation is relaxed. Unless otherwise stated, DNS results presented in the rest of this manuscript are obtained using the constant contact-angle model.

Regarding the first no-slip model, it has a force singularity at the contact line (Huh & Scriven Reference Huh and Scriven1971), which is handled in the VoF formulation using an implicit numerical slip (Afkhami, Zaleski & Bussmann Reference Afkhami, Zaleski and Bussmann2009). Specifically, we consider the standard water and air conditions, and simulate sessile drops of fixed initial volume $56.5\,\mathrm {\mu }{\rm l}$ and therefore constant $G=0.755$ and $Oh=2.18\times 10^{-3}$ (comparable to the experimental values reported in table 1). The value of ${\textit {Bo}}$ is subsequently varied by changing $\alpha /g$. The case corresponding to $\theta _c = 60^\circ$ is extensively studied. The range of ${\textit {Bo}}$ in simulation is varied from 1.5 to 12. For $\theta _c = 60^\circ$, below the value of ${\textit {Bo}}=1.5$, fragmentation is suppressed.

2.4.1. The unstable breakup regime

Upon plate release, the initial sessile drop elongates vertically forming a ligament while its foot remains in contact with the substrate. For sufficiently large drop volume $V$ and effective acceleration $\alpha$, necking occurs near the ligament tip as well as its foot, which ultimately leads to breakup and partial ejection of the sessile drop volume. A typical sequence of drop deformation is shown in figure 3(a i–a v), where (a i) gives the initial shape; (a ii) shows ligament formation; (a iii) and (a iv) respectively depict tip and foot pinch-off when the ligament breaks, ejecting secondary drops/ligaments from the original sessile drop; and finally (a v) illustrates that, after the foot pinch-off, the detached ligament further fragments into multiple satellite droplets. We will show in Appendix A that the order of tip and foot breakup depends on the Bond number ${\textit {Bo}}$.

Figure 3.

(a) Successive temporal sequence of a drop undergoing inertial detachment and fragmentation. Experiment conducted with a PMMA substrate ($\theta _c=60^\circ$), and with acceleration ratio $\alpha /g=4.2$ and drop initial volume $V=80\,\mathrm {\mu }\text {l}$. A dashed line is drawn for each frame to separate the drop and its optical reflection on the substrate. A scale bar is given in (a v). (b) Stability diagram in $({\textit {Bo}}, \theta _c)$-space that distinguishes unstable drops that partially detach from the substrate. The equilibrium theory of Boucher, Evans & Kent (Reference Boucher, Evans and Kent1976) for pendant drops is compared with our experiments where only unstable cases are plotted. The red square corresponds to the example sequence given in (a) where $\theta _c=60^\circ$, ${\textit {Bo}}=4$.

As aforementioned, unstable inertial detachment of the drop (figure 3a) occurs only if ${\textit {Bo}}$ exceeds a critical value with a very weak dependence of this value on the substrate equilibrium contact angle $\theta _c$. The analogous stability analysis for a pendant drop was performed by Boucher et al. (Reference Boucher, Evans and Kent1976), and the corresponding stability diagram, adapted to our $({\textit {Bo}},\theta _c)$ phase space, is shown in figure 3(b). Here, the ‘stable’ region according to Boucher et al. (Reference Boucher, Evans and Kent1976) was determined by quasi-static increase of the volume of a pendent drop until its detachment from a substrate of a given wetting, showing again the very weak dependence on contact angle. We highlight the $({\textit {Bo}},\theta _c)$ parameters that yield detachment in our experiments in figure 3(b). The critical ${\textit {Bo}}$ required for our sessile drops to become unstable matches well with the analysis of Boucher et al. (Reference Boucher, Evans and Kent1976).

2.4.2. Evolution simulation

Analogous to the image sequence shown in figure 3(a), the simulated evolution of the drop for ${\textit {Bo}} = 4$ at various stages is shown in figure 4. Following the constant angle model discussed in § 2.3.2, we capture the experimental features well: tip pinch-off, foot pinch-off, secondary fragmentation of the elongated middle fluid mass. Zooming in on the secondary fragmentation of the elongated structure reveals that the numerics do capture secondary daughter droplet formation as well. We further discuss the evaluation of convergence for the tip pinch-off time and volume of the ejected primary tip daughter droplet in Appendix E.

Figure 4.

Direct numerical simulation rendering of evolution of the drop for $Bo = 4.0$ and $\theta _c = 60^\circ$. The numerical values indicate time with capillary time unit $T_\sigma = 13.6$ ms (see (2.10a–c)). A total time span up to 61.2 ms is shown, roughly covering the same range as the experimental results of figure 3 and hence depicting tip pinch-off, foot pinch-off and formation of satellite drops from secondary fragmentation. The axis ratio in each image is unity and the scale is the same for all images.

2.4.3. Primary drop (tip) ejection

We focus primarily here on ejection of a tip drop from the ligament tip undergoing inertial acceleration, as shown in figure 3(a iii–a v). We characterize the tip drop with its equivalent radius, $R_{tip} = [3V_{tip}/(4{\rm \pi} )]^{1/3}$, where $V_{tip}$ is the ejected volume and the time of its neck pinch-off, $\tau _{tip}$, namely its tip breaking time, is non-dimensionalized using the capillary time scale. Figure 5(a,b) shows first that both $R_{tip}$ and $\tau _{tip}$ converge rapidly to values independent of ${\textit {Bo}}$ with sufficiently high ${\textit {Bo}}$, and also close to independent from $\theta _c$ and acceleration ratio $\alpha /g$. Second, the figures show that the dependence of $R_{tip}$ and $\tau _{tip}$ on ${\textit {Bo}}$ collapse on a master curve for different $\alpha /g$ (or equivalently gravitational Bond number $G$).

Figure 5.

(a) Dimensionless radius $\bar {R}_{tip}$ of ejected tip drop as a function of Bond number $Bo$, contact angle $\theta _c$ and acceleration ratio $\alpha /g$. The dashed lines label the critical ${\textit {Bo}}$ over which detachment occurs for the two given $\theta _c$. (b) The corresponding tip breakup time $\tau _{tip}$ is non-dimensionalized using the capillary time scale. Both $\bar {R}_{tip}$ and $\tau _{tip}$ converge rapidly to a constant as ${\textit {Bo}}$ increases for all parameters used, suggesting a universal end-pinching process controlled by capillary effects and negligible contact-line dynamics effects on the tip dynamics. Note that in (b) at ${\textit {Bo}} = 2.7$ two open circles overlap, within measurement error (see table 1).

More specifically, as ${\textit {Bo}}$ increases, the dimensionless $\bar {R}_{tip}={R}_{tip}/{R}_{eq}$ rapidly approaches ${R}_{tip}\approx 0.7$ (with the over-bar henceforth dropped) for our range of $\alpha /g$ and $\theta _c$ values explored herein. Further, in figure 5(b) $\tau _{tip}$ is also nearly independent of $Bo$ as it converges to $\tau _{tip}\approx 2.5$. Note that it does so less rapidly with $Bo$ than ${R}_{tip}$. Indeed, contrary to ${R}_{tip}$, where substrate effects are negligible, $\tau _{tip}$ converges to $\tau _{tip}\approx 2.5$ slightly slower for smaller $\theta _c$ than it does for larger $\theta _c$. This suggests that the substrate local contact-line dynamics can affect both ends of the breakup, not just the foot pinch-off. However, this local contact dynamics effect is rapidly negligible as ${\textit {Bo}}$ increases, as ${R}_{tip}$ and $\tau _{tip}$ become blind to substrate local wetting.

Given the inviscid nature of this system, the tip detachment mode is closely related to end-pinching, which can occur for long ligaments as well as unsteady short protuberances (Stone Reference Stone1994; Wang & Bourouiba Reference Wang and Bourouiba2021). It was observed both in multiple systems, for example in numerical studies of contracting liquid filaments (Schulkes Reference Schulkes1996), and experimental studies of unsteady short rim emerging ligaments (Wang & Bourouiba Reference Wang and Bourouiba2021) that end-pinching releases a tip drop of radius proportional to the average radius of the filament, in fixed capillary time units. We further discuss the end-pinching dynamics in § 3.2.

2.4.4. Evolving contact angle and de-pinning

The apparent contact angle formed between the ligament foot and the substrate does not maintain its constant equilibrium value over time as elongation occurs. Figure 6 shows this variation in contact angle over time: figure 6(a) first shows two series of ligament diameter profiles at foot breakup time $t=t_{foot}$ but different initial static contact angles $\theta _c=60^\circ$ and $19^\circ$, and for a range of ${\textit {Bo}}$. The two series converge to a conical shape of comparable final receding angles at the substrate. To characterize these conical feet, we estimate a dynamic contact angle at the foot $\theta (t)$, obtained from $h(z,t)$ measurements in the range $z\in [0,0.2]$ for each $t$ up to $t=t_{foot}$. This yields again two time series of $\theta (t)$ plotted in figure 6(b) for the two different initial $\theta _c$. We clearly see that both series for the estimated $\theta$ approach a mean value $\bar {\theta }$ around $40^\circ$ as $t\to t_{tip}$ rather quickly, before tip pinch-off occurs, which occurs, roughly, around $t \approx 0.2 t_{tip}$ for $\theta _c=60^\circ$ and $t \approx 0.4 t_{tip}$ for $\theta _c=19^\circ$. We show in Appendix C that a constant foot angle is indeed expected if the flow is dominated by inertial acceleration. Note that we observe mild $\theta (t)$ oscillations after the convergence to the final receding angle with approximately $0.5t_{tip}$ period around the mean $\bar {\theta }$ that occurs in all experiments.

Figure 6.

(a) Drop foot profiles, $h(z)$, at foot breakup times $t=t_{foot}$ experimentally measured on the substrates of distinct wetting Results from ${\textit {Bo}}=4.2, 4.5, 5.4, 6.9, 7.8, 9.4$ are shown for $\theta _c = 60^\circ$ (solid), and ${\textit {Bo}}=5, 6, 7.9, 10$ for $\theta _c = 19^\circ$ (dashed). (b) Temporal evolution of the foot angle, $\theta (t)$, approximated by a fitted cone located in the range $0\leq z \leq 0.2$ as shown in (a) for $0\leq t/t_{tip} \leq t_{foot}/t_{tip}$. Clearly, $\theta$ from different ${\textit {Bo}}$ and $\theta _c$ converge to a common mean receding angle value around $40^\circ$ rapidly before tip breakup. This value is, at first order, independent of the initial static contact angle. (c) Evolution of the normalized contact radius $R(t)/R_0$ that corresponds to the same drops measured in (b). It shows that, for the more wetting surface, i.e. smaller initial static contact angle, the Bond number has little effect on the evolution of the contact line. In other words, the de-pinning dynamics is more independent of ${\textit {Bo}}$ than for the less wetting surface.

In parallel, figure 6(c) gives direct measurements of the receding contact radius $R(t)/R_0$, with $R(t)=h(0,t)$ and $R_0=R(0)$, as a function of time. First, for both wettings, we see a decrease of $R(t)/R_0$ with a contact line at time of foot breakup, $t=t_{foot}$, reaching essentially less than half of the initial wetting extent. Second, for smaller initial static contact angle, the Bond number has little effect on the evolution of the contact line and its de-pinning. In other words the de-pinning dynamics is less dependent on ${\textit {Bo}}$ for more wetting substrates. In fact, for the largest ${\textit {Bo}}$ examined, both surfaces show close to identical and superposable receding contact radius dynamics. Note, that we see more variation of the evolution of $R(t)/R_0$ with ${\textit {Bo}}$ for the larger initial static contact angle, with faster de-pinning with higher ${\textit {Bo}}$. Third, when the contact-angle value reaches its final receding angle, further displacement of the contact point becomes negligible. Finally, recall that fluctuations of the contact angle were seen to be more important for the more wetting surface. We also see more fluctuation of $R(t)/R_0$ for the more wetting surface. However, these fluctuations decrease in amplitude with increasing ${\textit {Bo}}$.

2.4.5. Curvature change, necking onset and early-time selection of the volume of tip drop

Here, we link the tip-drop volume selection to the early dynamics of the deforming sessile drop. In fact, we find that the volume of the tip drop is essentially already determined, for the most part, at the very early stage of curvature inflection of the drop contour and preceding the development of a clear elongated ligament. As we shall show quantitatively next, in figure 7, plotting $h(z,t)$, that the tip-drop volume is already selected, to $\pm 15\,\%$, by the time the final receding angle of the foot, $\bar {\theta } \approx 40^\circ$, is reached at the contact line (figure 6). These times are $t/t_{tip}=0.2$ for $\theta _c=60^\circ$ and $t/t_{tip}=0.4$ for $\theta _c=19^\circ$. In fact, by these times, a sign switch of $\partial _z^2 h$ and $\kappa$ along $z$-axis occurs, i.e. a curvature inflection of $h$ develops. At the time of clear curvature sign change, the volume enclosed between the point of curvature inflection and the top of the deforming contour can be converted into an equivalent spherical drop radius of

(2.12)\begin{equation} R_{equi}(\xi,t) = \left(\frac{3 \int_\xi^{L(t)} h(z,t)^2 \,{\rm d}z}{4} \right)^{1/3}, \end{equation}

where $0\leq \xi \leq L(t)$ specifies the volume domain and $L(t)$ is the ligament height at time $t$. In figure 7, the locations $\xi$ that respectively yield $R_{equi}/R_{tip}=0.85, 1, 1.15$ are labelled for all instants. When a clear curvature inflection of the sessile drop contour appears, $R_{equi}$ at $z=\xi$, near such an inflection point, recovers the size of the ejected tip drop, $R_{tip}$ (shown in figure 5), within $\pm 15\,\%$ error of the actual a posteriori post-pinch-off direct $R_{tip}$ measurement. In sum, for all ${\textit {Bo}}$ and wetting substrates, the convergence of the contact angle to its final value $\bar {\theta }$, at the contact line, associated with little variation of apparent contact-line radius or contact angle subsequently, also coincides with the clear appearance of a curvature inflection of the drop contour profile. Such conditions impose the geometric constraint that selects already, at this early time, for the most part, the tip-drop volume; that is even prior to development of a clear ligament. Moreover, we can also conclude that the end-pinching ultimately leading to the ejection of the primary tip drop is only weakly influenced by the substrate dynamics/wetting. In § 3 we will further examine this insight analytically and numerically.

Figure 7.

Evolution of the drop radius contour profile $h(z,t)$ measured for different wettings and for increasing ${\textit {Bo}}$. Here, the axial $z$-axis of symmetry is in the horizontal direction. In (a–c), $\theta _c = 60^\circ$ and ${\textit {Bo}}=3.8, 5.9, 8.7$. Results are shown at three instants $t/t_{tip}=0.05,0.13,0.2$ for each ${\textit {Bo}}$. In (d–f), $\theta _c = 19^\circ$, ${\textit {Bo}}=7.2, 7.9, 10$, and for each panel $t/t_{tip}=0.2,0.3,0.4$. Along each contour, the portion of the deformed drop whose volume generates an equivalent radius of $R_{equi}/R_{tip}=0.85, 1, 1.15$ is marked by solving (2.12). We show that the first time of appearance of a clear inflection point of the curvature coincides, at first order, with the selection of the final receding contact angle at the foot (figure 6), i.e. ${\approx }0.2 t_{tip}$ and ${\approx }0.4 t_{tip}$, for $\theta _c = 60^\circ$ and $\theta _c = 19^\circ$, respectively. And that at this early time of appearance of the inflection the volume between the $z$-plane of this first inflection region and the tip of the deforming sessile drop, the final size of the tip drop ejected, $R_{tip}$, is already selected for the most part to within $15\,\%$ error.

3.2.1. Estimating the tip-drop pinch-off time

Building on the insights of the preceding section where the drop detachment process is two staged with a shift in regime from inertia to capillary dominated, we discuss here an estimation of the tip breakup time $\tau _{tip}$. As seen in figures 8 and 24 already and discussed, the duration of the inertial regime, $\tau _{inertia}$, has a weak dependence on the wetting of the substrate, a function of equilibrium contact angle $\theta _c$, i.e. $\tau _{inertia}=\tau _{inertia}(\theta _c)$. For example, from figure 8(a), $\tau _{inertia}(60^\circ )\approx 0.6$ and $\tau _{inertia}(19^\circ )\approx 0.9$. Once the system transitions into the second regime, surface tension dominates the local dynamics of necking, ultimately leading to the pinch off of the tip drop. This pinch-off dynamics is analogous to end-pinching for inviscid filaments (e.g. Schulkes Reference Schulkes1996), shown numerically to have a dimensional tip-drop pinch-off time $5T_\sigma$, where $T_\sigma$ is defined as in (2.10a–c) taking dimensional $\tilde {R}_{eq}$ to be a uniform filament radius $\tilde {R}_f$ proportional to the dimensional tip-drop size, with $\tilde {R}_f\approx \tilde {R}_{tip}/1.55$. Therefore, combining the insights gained from the two regimes and using the capillary time scale (2.10a–c) for non-dimensionalization, we obtain

(3.1)\begin{equation} \tau_{tip}(\theta_c)\approx\tau_{inertia}(\theta_c)+5\left(\frac{{R}_{tip}}{1.55}\right)^{3/2}. \end{equation}

Approximating $R_{tip} = \tilde {R}_{tip}/R_{eq}=0.8$, (3.1) evaluates to $\tau _{tip}(60^\circ )\approx 2.5$ and $\tau _{tip}(19^\circ )\approx 2.8$. These estimations capture the experimental measurements shown in figure 5(b) very well. Notably, the drop radius-to-ligament width ratio of ${\approx }1.5$ was also observed experimentally to be robust and independent of the impact Weber number for the continuous end-pinching of ligaments along the rim of canonical unsteady sheet fragmentation upon drop impact on a substrate of comparable size to that of the drop (Wang & Bourouiba Reference Wang and Bourouiba2018), reflecting, also there, a robust local capillary-dominated dynamics.

3.3.1. The slender-jet model governing equations

Formally derived as the $(L/R)^{-1}\to 0$ asymptotic limit of the system (2.7) and (2.9), the inviscid slender-jet equations (Papageorgiou & Orellana Reference Papageorgiou and Orellana1998) under our default inertial scaling (2.10a–c) read

(3.2a)$$\begin{gather} \partial_t h ={-}w\partial_z h -\frac{h}{2}\partial_z w , \end{gather}$$

(3.2b)$$\begin{gather}\partial_t w ={-}w \partial_z w-\frac{1}{Bo}\partial_z\kappa +1, \end{gather}$$

where $\partial _t$ and $\partial _z$ denote partial derivatives with respect to $t$ and $z$, respectively, and the curvature $\kappa$ is

(3.3)\begin{equation} \kappa=\kappa(z,t) = \frac{1}{h\left(1+(\partial_zh)^2\right)^{1/2}}-\frac{\partial^2_zh}{\left(1+(\partial_zh)^2\right)^{3/2}}. \end{equation}

Equivalently, the slender-jet system (3.2) with boundary condition $h(L,t)=0$ ensures conservation of volume, momentum and energy, respectively, as follows (Eggers & Dupont Reference Eggers and Dupont1994; Contò et al. Reference Contò, Marín, Antkowiak, Castrejón-Pita and Gordillo2019):

(3.4a)$$\begin{gather} \frac{{\rm d}}{{\rm d}t}\int_{z_b}^{L(t)}h^2\,{\rm d}z=\frac{{\rm d}}{{\rm d}t}\left(\frac{V_{jet}}{\rm \pi}\right)=\left[h^2w\right]_{z=z_b}, \end{gather}$$

(3.4b)$$\begin{gather}\frac{{\rm d}}{{\rm d}t}\int_{z_b}^{L(t)}wh^2\,{\rm d}z=\frac{V_{jet}}{\rm \pi}+\left[h^2w^2-\frac{1}{{\textit{Bo}}}K_1(z,t)\right]_{z=z_b}, \end{gather}$$

(3.4c)$$\begin{gather}\frac{{\rm d}}{{\rm d}t}\int_{z_b}^{L(t)}\!\left\{\!\left(\frac{w^2}{2}-z\right)h^2+\frac{2h}{{\textit{Bo}}}\!\left(1+(\partial_zh)^2\right)^{1/2}\right\}{\rm d}z=\left[\frac{w^3h^2}{2}-wh^2z+\frac{1}{{\textit{Bo}}}K_2(z,t)\right]_{z=z_b}\!, \end{gather}$$

where

(3.5)\begin{equation} \left. \begin{gathered} K_1(z,t)=\frac{h}{\left(1+(\partial_zh)^2\right)^{1/2}}+\frac{h^2\partial^2_zh}{\left(1+(\partial_zh)^2\right)^{3/2}},\\ K_2(z,t)=\kappa h^2 w-\frac{2 (\partial_z h) (\partial_t h) h}{\left(1+(\partial_zh)^2\right)^{1/2}}. \end{gathered} \right\} \end{equation}

Here, the integral terminal $z_b\in (0,L)$ is a constant parameter that defines a jet's lower boundary. Accordingly, $V_{jet}$ is the jet volume. Note that $z_b\neq 0$ because (3.2) does not account for any solid–liquid stress at the contact plane $z=0$. If we choose $z_b\ll 1$, a thin capillary boundary layer is then expected for $z\in [0,z_b)$, where surface tension and boundary stresses would govern the drop foot region.

Notably, it is possible to identify the kinetic, potential and surface energies of the jet domain as $E_k=\int _{z_b}^Lw^2h^2/2\,\textrm {d}z$, $E_p=\int _{z_b}^L-zh^2\,\textrm {d}z$ and $E_s=\int _{z_b}^L2h\sqrt {1+h_z^2}/{\textit {Bo}} \,\textrm {d}z$, respectively, in the energy conservation law (3.4c). In general, this energy equation is auxiliary to the already closed system (3.2) for an incompressible jet. However, as we will see, we utilize this additional equation to enable closure for the jet's breaking topology without having to solve the evolution equations over time.

3.3.2. Slender-jet free-fall regime

Having introduced the slender-jet equations (3.2), we now develop an analytical solution starting with a steady velocity ansatz $w$ for the force balance (3.2b) that also satisfies the boundary condition $\lim _{z\to 0}w=0$. This solution would describe a free-falling ligament in the ${\textit {Bo}}\to \infty$ limit. In this limit and for these conditions, we obtain for all $t>0$

(3.6)\begin{equation} w(z,t) = \sqrt{2z}. \end{equation}

Accuracy of such a steady velocity profile is directly assessed in figure 10 with DNS of a constant contact-angle model using $\theta _c=40^\circ$ (10a,b) and $\theta _c=60^\circ$ (10c,d). For example, for ${\textit {Bo}}=7$, figure 10(a,c) gives the central axial velocity $w(r=0,z,t)$ as a function $z$ at four different times up to tip-drop pinch off. All the profiles indeed follow the steady profile (3.6) except for the region near the tip as $z\to L$. Clearly, away from the extremity of the ligament the steady solution does capture a universal velocity profile that is robust for a range of ${\textit {Bo}}$, as seen in figure 10(b,d). Indeed, these figures show an ensemble time average $\bar {w}$ taken from $w(0,z,t)$ over all simulated times $t$ for a range of ${\textit {Bo}}$. These observations support (3.6) to capture the profile of the ligament region away from its two extremities. Notably, we qualify the success of (3.6) strictly under the conditions that a) the inviscid flow limit applies, and b) the imposed inertial acceleration is constant (see (3.2b)). This is because both viscous effects and variable forcing over time can significantly alter the ligament's stretching dynamics (Villermaux Reference Villermaux2012; Vincent et al. Reference Vincent, Duchemin and Villermaux2014).

Figure 10.

(a) Central axial velocity $w(r=0,z,t)$ as a function of $z$, for ${\textit {Bo}}=7$, from DNS of constant $\theta _c=40^\circ$ at different times (solid), compared with the steady profile $w = \sqrt {2z}$ (3.6) (dashed). (b) Time-averaged DNS velocity profiles $\bar {w}=\bar {w}(z)$ for different ${\textit {Bo}}$ (solid) shown in log–log form in order to more easily show the range. The inset gives an example of $\bar {w}$ (black solid line) from various temporal profiles of $w$ (blue and dotted) for ${\textit {Bo}}=5$. Panels (c,d) give the corresponding results to (a,b) for $\theta _c=60^\circ$. This DNS benchmark validates that (3.6) gives the correct universal steady velocity profile for a ligament in its middle region sufficiently far from its two ends, i.e. $z\gtrsim O(1)$ and $L-z\gtrsim O(1)$, at all Bond numbers.

Having established support for (3.6), the continuity equation (3.2a) can be further solved with the general solution $h(z,t)={H}(t-\sqrt {2z})z^{-1/4}$, where $H({\cdot })\in C^1$ is arbitrary. To determine $H$, we first specify the initial condition at time $t=t_s$ when the SJM becomes applicable, by denoting

(3.7a–c)\begin{equation} \tilde{h}\left(z,\tilde{t} \right)\equiv h(z,t_s+\tilde{t}),\quad \tilde{h}_0(z)\equiv\tilde{h}(z,0), \quad \tilde{t}\equiv t-t_s\geq 0. \end{equation}

Further, since we are interested in the jet domain $z\in [z_b, L]$, an additional boundary condition at $z=z_b$ is required. By taking $z_b\to 0$, so that physically $w|_{z=z_b}\to 0$ at the substrate, we assume (3.2a) evaluated within the capillary layer for $z\in [0,z_b)$ follows the dominant balance $|w\partial _z h/\partial _t h|\ll 1$ and $|w\partial _z h/ (h\partial _z w)|\ll 1$. As such, we arrive at the following approximate equation:

(3.8)\begin{equation} \partial_t h +\frac{h}{2\sqrt{2z}} = 0, \end{equation}

for $h$ using (3.6), whose solution at $z=z_b$ as a boundary condition is given by

(3.9)\begin{equation} \tilde{h}(z_b,\tilde{t}) \equiv\tilde{h}_b(\tilde{t}) = \exp\left(-\frac{\tilde{t}}{2\sqrt{2z_b}}\right)\tilde{h}_0(z_b), \end{equation}

where the initial value (3.7a–c) is used. Note that the asymptotic conditions that lead to (3.9) as $z_b\to 0$ may be assessed upon direct substitution into (3.2a), yielding

(3.10)\begin{equation} \left|\frac{w\partial_z h}{\partial_t h}\right|_{z=z_b}\sim\frac{w}{z_b},\quad \left|\frac{w\partial_z h}{h\partial_z w}\right|_{z=z_b}\sim\frac{w z_b^{{-}3/2}}{\partial_zw}. \end{equation}

Therefore, an essential singularity of the form $w\sim o(\textrm {e}^{-1/z_b})$ is needed for the second term in (3.10) to vanish, while any algebraic profile, e.g. (3.6) derived in the ${\textit {Bo}}\to \infty$ limit, leads to divergence. Indeed, we observe in figure 10 that, as $z\to 0$, the simulated $w$ from DNS approaches zero much faster than the steady profile. Presently, we do not explore the capillary boundary layer for $z\in [0,z_b)$ in full detail, but assume that there is a sharp transition of the flow dominance from surface tension to inertia at the critical slice $z=z_b$, hence (3.9) is obtained as an approximation. This hypothesis enables the analytical solution to follow, and is subsequently justified by experiments.

With initial-boundary values specified in (3.7a–c) and (3.9), the 1-D transport equation (3.2a) can be solved for all $z,t$ following a standard method-of-characteristic procedure. As a result

(3.11)\begin{equation} \tilde{h}(z,\tilde{t})=\tilde{h}_{free\text{-}fall}\equiv \begin{cases} \sqrt{1-\dfrac{\tilde{t}}{\sqrt{2z}}}\, \tilde{h}_0\left(\dfrac{1}{2}\left(\tilde{t}-\sqrt{2z}\right)^2\right), & \zeta(z_b,\tilde{t}) \leq z \leq \zeta({\tilde{L}_0,\tilde{t}}),\\ \left(\dfrac{z_b}{z}\right)^{1/4}\exp\left(-\dfrac{\tilde{t}-\sqrt{2}\left(\sqrt{z}-\sqrt{z_b}\right)}{2\sqrt{2z_b}}\right)\tilde{h}_0(z_b), & z_b\leq z \leq \zeta(z_b,\tilde{t}), \end{cases} \end{equation}

where $\tilde {L}_0$ is the ligament length at $\tilde {t}=0$, i.e. $\tilde {h}_0(\tilde {L}_0)=0$, and

(3.12)\begin{equation} \zeta(Z,\tilde{t}) = \frac{1}{2}\left(\tilde{t}+\sqrt{2Z}\right)^2. \end{equation}

Here, $\zeta (Z,\tilde {t})$ gives the advected position of a Lagrangian slice, parameterized by its original location $z=Z$ at $\tilde {t}=0$, after time $\tilde {t}$ by the uniform axial velocity $w$. That is, $\zeta$ solves the ordinary differential equation (ODE) system

(3.13a,b)\begin{equation} \partial_{\tilde{t}}\zeta=w(\zeta(Z,\tilde{t})), \quad \zeta(Z,0)=Z\in [z_b, \tilde{L}_0], \end{equation}

with $w$ given by (3.6).

Figure 11 shows experimental evidence that the free-fall solution (3.11) captures the measurements made for both contact angles well. First, with $\theta _c=60^\circ$ and ${\textit {Bo}}=7$, figure 11(a) gives the corresponding drop radius $h$ as a function of capillary time $\tau$ for an increasing sequence of $z/\lambda =0.5,1,\ldots,12$, where $\lambda$ is the capillary wavelength defined in (2.11a–c). Using a logarithmic scale, we observe that, at each fixed $z$ that is sufficiently large ($z/\lambda \gtrapprox 2$), there always exists a transitioning time when $h$ peaks and exponential thinning occurs afterwards. However, at small distances closer to the substrate, e.g. $z/\lambda =0.5,1$, the rate of $h$ thinning appears slower than exponential, indicating the effect of the capillary boundary layer. Therefore, we can empirically identify a critical time $t_s$ and location $z_b$ from measurements such that the approximated boundary condition (3.9) is reasonable. For example, by choosing $\tau _s=t_s/\sqrt {{\textit {Bo}}}=0.2$ and $z_b=2\lambda$ (labelled in figure 11a), we compare the $h$ data directly against (3.11) using similarity variables $\xi \equiv [\tilde {t}-\sqrt {2}(\sqrt {z}-\sqrt {z_b})]/(2\sqrt {2z_b})$ and $\eta \equiv (z/z_b)^{1/4} \tilde {h}(z,\tilde {t})/\tilde {h}_0(z_b)$ in figure 11(b). It is shown that the roughly parallel curves in figure 11(a), as well as additional data for larger $z/\lambda =16,20,24$, all collapse onto the theoretical solution $\eta =\textrm {e}^{-\xi }$. The biggest deviation is found for $z/\lambda =2$ in this case as $\xi$ (or equivalently, $t$) increases, again exhibiting long term capillary influences. Similarly, figure 11(c,d) reveals the same findings using experimental data for $\theta _c=19^\circ$ and ${\textit {Bo}}=6$. Compared with the previous case, here, $\tau _s=1.7$ and $z_b=4\lambda$ are chosen for $(\xi,\eta )$ in figure 11(d), indicating a thicker boundary layer with respect to the capillary length for a more wetting surface.

Figure 11.

(a) Drop radius $h(z,\tau )$ as a function of capillary time $\tau$ measured experimentally at different heights of wavelength $z/\lambda =0.5,1,1.5,2,4,8,12$, using $\theta _c=60^\circ$ and ${\textit {Bo}}=7$ ($\alpha /g=9.9$). (b) Radius measurements at different heights $z/\lambda =2,4,8,12,16,20,24$ plotted by solid lines in term of similarity variables $\xi \equiv [\tilde {t}-\sqrt {2}(\sqrt {z}-\sqrt {z_b})]/(2\sqrt {2z_b})$ and $\eta \equiv (z/z_b)^{1/4} \tilde {h}(z,\tilde {t})/\tilde {h}_0(z_b)$, with parameters $\tau _s=0.2$ ($t_s=0.6$) and $z_b=2\lambda$ (labelled by the dashed line in a). The resulting curves collapse onto the free-fall solution $\eta =\exp ({-\xi })$ given in (3.11) by the dashed line. (c,d) Same as in (a,b) except $\theta _c=19^\circ$ and ${\textit {Bo}}=\!7$ ($\alpha /g=6.2$). The transition parameters used for (d) are $\tau _s=1.7$ ($t_s=4.1$), $z_b=4\lambda$. Panels (a,c) share the same keys. We demonstrate here that, with an appropriately identified boundary for the capillary layer, the ligament radius in the inertial regime is well predicted by the free-fall solution derived from the ${\textit {Bo}}\to \infty$ limit of the SJM.

3.3.3. Lollipop model for tip-drop breakage

Although we have successfully modelled the free-falling region of a ligament in § 3.3.2, the surface tension effects that cause necking near the tip and its ultimate pinch-off are still fundamentally missing in the infinite ${\textit {Bo}}$ theory. In this section, to recover this critical behaviour, we use the insights gained from the geometrical constraints of the dynamics to develop a first-order ‘lollipop’ model solution to the SJM for a ligament at the exact tip breaking instant. We achieve this by solving the conservation form of the SJM given in (3.4) for infinite ${\textit {Bo}}$ over a defined domain that divides a breaking ligament at $t=t_{tip}$ into four parts: (i) a base cone for $0\leq z \leq z_b$, (ii) a free-falling region described by (3.11) for $z_b< z\leq z_c$, (iii) a connecting cone within $z_c\leq z \leq \mathcal {L}-2R_{tip}$ where $\mathcal {L}=L(t_{tip})$ is the breaking jet length and (iv) an ejected spherical tip of radius $R_{tip}$ between $\mathcal {L}-2R_{tip} \leq z \leq \mathcal {L}$. A schematic of the composite geometry that we use to guide the approach, with a lollipop shape, is illustrated in figure 12.

Figure 12.

Schematic of the lollipop model for a tip breaking ligament at $t=t_{tip}$, including a base cone ($0\leq z \leq z_b$), a free-falling ligament ($z_b\leq z \leq z_c$), a connecting cone ($z_c \leq z \leq \mathcal {L}-2R_{tip}$) and a tip sphere ($\mathcal {L}-2R_{tip} \leq z \leq \mathcal {L}$). Here, $\theta _c$ is the equilibrium contact angle, $w_b$ is the boundary flow velocity for the free-fall solution (3.11), $w_c$ is the steady cone velocity (3.20), $\beta =18^\circ$ is the fixed pinch-off angle, $R_{tip}$ is the radius of the ejecting spherical tip of uniform velocity $w_{tip}$. The $h$-profiles of the jet radius are labelled for each region and given in (3.21a,b).

Next, we elucidate specific design of the four parts that contribute to the lollipop model. First, we make the same constant contact-angle assumption as in § 2.3.2 for DNS, and additionally approximate the drop shape in its foot capillary layer to be conical, as generally observed in experiments. This leads to the following shape function $\tilde {h}_{base}$ for $z\in [0,z_b)$:

(3.14)\begin{equation} \tilde{h}(z,\tilde{t})=\tilde{h}_{base}(z,\tilde{t};z_b,\theta_c)\equiv(z_b-z)\cot\theta_c+\tilde{h}_b(\tilde{t};z_b), \end{equation}

where $\tilde {h}_b$ is the boundary condition defined in (3.9) of exponential necking. By conserving the total drop volume, we can then calculate the slender-jet portion of the ligament volume $V_{jet}$ (excluding the base cone) as

(3.15)\begin{equation} V_{jet}(\tilde{t})=\frac{4{\rm \pi}}{3}-\int_0^{z_b}{\rm \pi}\tilde{h}_{base}^2\,{\rm d}z=\frac{\rm \pi}{3}\left[4-{z_c} \left(3 \tilde{h}_b^2+3 \tilde{h}_b {z_b} \cot \theta_c +{z_b}^2 \cot ^2\theta_c \right)\right]. \end{equation}

Substituting $\tilde {h}_b$ and $V_{jet}$ into (3.4a) thus gives the flow velocity across the critical plane $z=z_b$ based on volume flux

(3.16)\begin{equation} w_b(\tilde{t};z_b,\theta_c)=\frac{z_b^{3/2}{\rm e}^{{\tilde{t}}/{2\sqrt{2z_b}}}\cot\theta_c}{2^{3/2}\tilde{h}_0(z_b)}+\left(\frac{z_b}{2}\right)^{1/2}. \end{equation}

Importantly, the boundary velocity $w$ given by (3.16) differs from the steady profile (3.6) evaluated at $z=z_b$. We demonstrate in Appendix D.1 that the time dependence introduced in $w_b$ necessarily corrects the volumetric flow rate $\textrm {d}V_{jet}/\textrm {d}t$.

Above $z>z_b$, as discussed in § 3.3.2, we proceed with the assumption that a discontinuous transition into the free-fall regime occurs, acquiring for the lollipop model the ligament shape and velocity given by (3.11) and (3.6), respectively. However, to capture breakup in this model, the tip end of the free-fall solution is replaced by an ejecting drop. Informed by the experiments as well as the recent self-similar solution derived by Contò et al. (Reference Contò, Marín, Antkowiak, Castrejón-Pita and Gordillo2019), we propose that at breakup time $t_{tip}$ the ejecting drop is spherical in shape and uniform in velocity. This gives for $z\in [\mathcal {L}-2R_{tip}, \mathcal {L}]$

(3.17)\begin{equation} h(z,t_{tip})=h_{tip}(z;\mathcal{L},R_{tip})\equiv \sqrt{R_{tip}^2-(z+R_{tip}-\mathcal{L})^2}, \quad w(z)=w_{tip}, \end{equation}

where the tip radius $R_{tip}$, the total ligament length at breakup $\mathcal {L}=L(t_{tip})$ and the tip velocity $w_{tip}$ are to be determined in the next section. Note that a spatially uniform $w_{tip}$ ensures the tip drop remains spherical after breakup.

To link the free-falling ligament and the ejecting tip, we further devise a connecting cone in between so that the combined lollipop model reasonably emulates an end-pinching filament, e.g. Day, Hinch & Lister (Reference Day, Hinch and Lister1998). Specifically, Day et al. (Reference Day, Hinch and Lister1998) obtained a self-similar solution to the Euler equations for axisymmetric inviscid drop breakup, and revealed a double-cone structure at breaking point with an inner angle of $18^\circ$ made by the ligament and an outer angle of $113\,^\circ$ by the breaking tip. These angles were numerically confirmed again by Leppinen & Lister (Reference Leppinen and Lister2003). Here, we adopt the inner cone of fixed angle $\beta =18^\circ$ for $z\in [z_c,\mathcal {L}-2R_{tip}]$ as follows:

(3.18)\begin{equation} h(z,t_{tip})=h_{cone}(z;\mathcal{L},R_{tip})\equiv{-}(z-\mathcal{L}+2R_{tip})\cot\beta +\delta, \end{equation}

where the matching point $z=z_c$ is determined by requiring

(3.19)\begin{equation} \tilde{h}_{free\text{-}fall}(z_c,t_{tip}-t_s)=h_{cone}(z_c), \end{equation}

with $\tilde {h}_{free\textrm {-}fall}$ given in (3.11). A minimum neck radius $\delta \sim O(10^{-3})$ at $z=\mathcal {L}-2R_{tip}$ is formally introduced in (3.18) to indicate the end of inviscid pinch off before a viscous solution takes over (Eggers & Villermaux Reference Eggers and Villermaux2008). However, because $\delta$ is small, its effect on the cone shape and momentum is negligible. We consider the velocity field within this connecting cone steady for simplicity, hence computed using (3.2a), i.e. $\partial _z(wh^2)=0$, as

(3.20)\begin{equation} w(z) = w_c(z;\mathcal{L},R_{tip}) \equiv \frac{2z_c h_{cone}(z_c)^2}{h_{cone}(z)^2}. \end{equation}

Markedly, having $\delta \to 0$ introduces a singularity of $w_c$ as $z\to \mathcal {L}-2R_{tip}$, which is acceptable for calculating momentum because $wh^2$ remains constant throughout the cone, but is problematic for the energy estimation if $\delta =0$. We address this point in detail in Appendix D.2.

In summary, using (3.14), (3.11), (3.18), (3.17), (3.16) and (3.20), we have approximated the breaking jet's shape and velocity at $t=t_{tip}$, (equivalently, $\tilde {t}=\tilde {t}_{tip}=t_{tip}-t_s$) by

(3.21a,b)\begin{align} h(z)=\begin{cases} \tilde{h}_{base}(z,\tilde{t}_{tip}), & z\in[0,z_b],\\ \tilde{h}_{free\text{-}fall}(z,\tilde{t}_{tip}), & z\in [z_b,z_c],\\ h_{cone}(z), & z\in [z_c,\mathcal{L}-2R_{tip}],\\ h_{tip}(z), & z\in [\mathcal{L}-2R_{tip},\mathcal{L}] \end{cases} \quad w(z)=\begin{cases} w_b(z,t_{tip}), & z=z_b,\\ \sqrt{2z}, & z\in [z_b,z_c],\\ w_c(z), & z\in [z_c,\mathcal{L}-2R_{tip}],\\ w_{tip}, & z\in [\mathcal{L}-2R_{tip},\mathcal{L}], \end{cases} \end{align}

where $t_{tip}$, $R_{tip}$, $\mathcal {L}$ and $w_{tip}$ are the four unknown variables to be calculated next. Note that we assumed in the lollipop model that tip ejection always precedes foot breaking, which is not always experimentally observed for large ${\textit {Bo}}$. Nonetheless, we will show in § 3.4 and Appendix D.1 that, if foot breaking should occur first at $t=t_{foot}< t_{tip}$, because $w_b$ decays rapidly in our lollipop model, only negligible mass, momentum or energy can be transported from the foot into the slender jet during $t_{foot}< t< t_{tip}$.

3.3.4. Solution process

Here, we determine the four SJM lollipop parameters, $t_{tip}$, $R_{tip}$, $\mathcal {L}$ and $w_{tip}$, using the weak form equations (3.4). However, without solving the evolution system as a continuous function of time, closure relies on additional access to at least one of the four unknowns. We choose to estimate tip breakup time $t_{tip}$ based on the empirical relation (3.1) and show in the following that the resulting lollipop structure, particularly the tip size $R_{tip}$, depends weakly on $t_{tip}$ within a reasonable range of its values.

To proceed, the ${\textit {Bo}}\to \infty$ limit of (3.4) is integrated over $0 \leq \tilde {t} \leq \tilde {t}_{tip}$, generating

(3.22a)$$\begin{gather} {\rm \pi}\int_{z_b}^\mathcal{L} h^2\,{\rm d}z=V_{jet}(\tilde{t}_{tip}), \end{gather}$$

(3.22b)$$\begin{gather}{\rm \pi}\int_{z_b}^\mathcal{L} wh^2\,{\rm d}z=P(\tilde{t}_{tip}) = P(0)+\int_0^{\tilde{t}_{tip}} \left(V_{jet}+ {\rm \pi}\tilde{h}_b^2w_b^2\right)\,{\rm d}\tilde{t}, \end{gather}$$

(3.22c)$$\begin{gather}{\rm \pi}\int_{z_b}^\mathcal{L} \left(\frac{w^2}{2}-z\right)h^2\,{\rm d}z = E(\tilde{t}_{tip}) = E(0) + {\rm \pi}\int_0^{\tilde{t}_{tip}} \left[\left(\frac{w_b^2}{2}-z\right)\tilde{h}_b^2w_b\right]\,{\rm d}\tilde{t}, \end{gather}$$

where $h$ and $w$ are piecewise and given by the lollipop model (3.21a,b); $\tilde {h}_b$ and $w_b$ are respectively found in (3.9) and (3.16); the slender-jet volume $V_{jet}$ is defined in (3.15), and its shifted initial momentum $P$ and net mechanical energy $E$ follow as

(3.23a,b)\begin{equation} P(0)={\rm \pi}\int_{z_b}^{\tilde{L}_0} w(z) \tilde{h}_0(z)^2\,{\rm d}z,\quad E(0)= {\rm \pi}\int_{z_b}^{\tilde{L}_0} \left(\frac{w(z)^2}{2}-z\right)\tilde{h}_0(z)^2\,{\rm d}z=0, \end{equation}

where $w(z)=\sqrt {2z}$ from (3.6) is used, and $\tilde {h}_0$ is the shifted initial shape (3.7a–c) of length $\tilde {L}_0$. All integrals involved in (3.22) can be carried out exactly. The resulting expressions are lengthy and therefore omitted for brevity. Nevertheless, for any given initial drop shape $\tilde {h}_0$ and estimate of the break time $t_{tip}$, (3.22) forms an explicit algebraic system from which $R_{tip}$, $\mathcal {L}$ and $w_{tip}$ can be solved, under the physical constraints that $0< R_{tip}<1$, $\tilde {L}_0<\mathcal {L}<\zeta (\tilde {L}_0,\tilde {t}_{tip})$. With $\zeta$ given by (3.12), the latter condition forbids the ligament tip to travel at free fall for all time without capillary retraction. In Appendix D.2 we further simplify the energy equation (3.22c) and establish a reduced expression, shown to be highly accurate, which reads

(3.24)\begin{equation} w_{tip}^2 = 2\left(\mathcal{L}-R_{tip}\right), \end{equation}

and expresses that the tip drop is ejected at a speed given by its free-falling centre, as per (3.6). Also, (3.24) eliminates the minimum neck radius $\delta$ introduced in (3.20) and its corresponding energy singularity. Hence, we can safely proceed with $\delta =0$ for mathematical convenience.