3.1. A localised and highly accelerated necking

The typical drop detachment dynamics of a highly concentrated suspension is illustrated in figure 1, for a volume fraction $\phi = 0.60$ (see also supplementary movie 1 available at https://doi.org/10.1017/jfm.2026.11678). From its initial cylindrical shape, the drop first stretches over most of its length (albeit for the stress-free lower part). However, as stretching proceeds, the drop develops a localised necking, with a roughly double conical shape, and eventually pinches at a short distance ( $\sim 2h_{{0}}$ ) from the nozzle. Although this dynamics is purely dissipative, it is at odd with that of conventional viscous liquids. Indeed, in contrast to the strongly localised pinching and short breakup distance of concentrated suspensions (already reported for lower volume fractions $0.20\lesssim \phi \lesssim 0.50$ , Furbank & Morris Reference Furbank and Morris2007; Bonnoit et al. Reference Bonnoit, Bertrand, Clement and Lindner2012; Château et al. Reference Château, Guazzelli and Lhuissier2018; Thievenaz & Sauret Reference Thievenaz and Sauret2022), highly viscous Newtonian liquids show a slender necking and a long breakup distance (Wilson Reference Wilson1988; Shi, Brenner & Nagel Reference Shi, Brenner and Nagel1994; Papageorgiou Reference Papageorgiou1995), as illustrated in figure 1(a,right).

Figure 2.

Stress monitoring and effective viscosity at the neck. (a) Drop shape at $t_{\text{break}}-t = 41$ and $4.5\,$ ms ( $\phi = 0.60$ , to disregard the final liquid bridge necking $t_{\text{break}}$ is taken at $2h \approx 0.5d$ ). (b) Velocity profile along the drop obtained from surface PIV. (c) Evolution of the neck bulk-effective viscosity $\eta _{\textit{eff}}$ (scaled by the suspending liquid viscosity $\eta _{{0}}$ ) with current neck diameter $2h$ (scaled by the particle size $d$ ). Thin green lines: experiments (8 replicates). Green thick line: log average. Red line: fitted exponential decay $\eta _{\textit{eff}}=\eta _{{t}}(\eta _{{i}}/\eta _{{t}})^{h/h_{{0}}}$ , with $\eta _{{i}}\simeq 743\eta _{{0}}$ and $\eta _{{t}}\simeq 0.84\eta _{{0}}$ , the initial and terminal effective viscosities, respectively.

Figure 2. Long description

The image contains three graphs labeled (a), (b), and (c). Graph (a) shows the shape of a liquid drop at different time intervals, highlighting the neck region where the drop is thinning. Graph (b) presents a velocity profile along the drop obtained from surface Particle Image Velocimetry (PIV), with velocity (u) in millimeters per second plotted against the vertical position (z) in millimeters. Graph (c) illustrates the evolution of the neck bulk-effective viscosity, scaled by the suspending liquid viscosity, as a function of the current neck diameter, scaled by the particle size. Thin green lines represent experimental data from eight replicates, the green thick line shows the log average, and the red line indicates a fitted exponential decay. The initial and terminal effective viscosities are marked on the graph.

Besides its spatial focusing, the necking of concentrated suspensions is characterised by its dramatic acceleration, illustrated in figure 1(b) through the time evolution of the neck diameter $2h(t)$ . While deformation is initially slow, the stretching rate $-2\dot {h}/h$ increases by almost four decades as the neck radius shrinks from $h/d \approx 8$ to $h/d \approx 1$ . Importantly, this increase largely exceeds the ${\sim}2$ -decade decrease in the neck cross section $h^2$ , which suggests that the bulk-effective viscosity at the neck also decays as pinching proceeds.

These behaviours are observed for all volume fractions we have investigated ( $0.55\leq \phi \leq 0.605$ ). As shown in figure 1(b), a dramatic acceleration of the breakup is systematically observed, although the overall breakup duration and the magnitude of the acceleration both increase with increasing $\phi$ (for the lower $\phi$ , necking initiates during extrusion, to facilitate comparison all dynamics are shown from the same initial neck diameter $2h=10\,$ mm, slightly smaller than $2h_{{0}}=12.03\,$ mm, and breakup duration is measured from this time). Similarly, the drop shape at breakup is less and less slender as $\phi$ is increased (with pinching distances to nozzle decreasing from $\approx 4h_{{0}}$ for $\phi =0.55$ to $\approx 2h_{{0}}$ for $\phi =0.605$ ).

3.2. Stress monitoring and neck effective viscosity

The observations of § 3.1 confirm that highly concentrated suspension drops detach differently from Newtonian ones. To specify this difference, we systematically monitor the effective strain rate $-{\dot {h^2}}/h^2$ as well as the mean tensile stress $\sigma$ at the neck. This requires to consider inertia, which eventually alleviates the drop weight in the last instants before pinch-off. Since the flow through the neck is in practice small, the mass $M$ below the neck can be considered as constant throughout detachment (the volume below the neck varies by typically less than 2 % between $2h/d \approx 12$ and $h/d \approx 0$ ). This allows us to express the mean neck tensile stress as

(3.1) \begin{equation} \sigma = \frac {M(g - a)}{\pi h^2}, \end{equation}

with $a$ the mean vertical acceleration of the drop part below the neck. To obtain $a$ , we use surface PIV to measure the drop vertical velocity. Figure 2(a,b) presents two typical velocity profiles along the drop, $u(z)=\boldsymbol{u}(z)\boldsymbol{\cdot }\boldsymbol{e_z}$ . They show that deformation is localised over a short region around the neck. Above the neck, $u\approx 0$ , whereas below the neck, the velocity plateaus at a constant value $u\approx \dot {Z}$ . This means that most of the drop below the neck experiences solid body acceleration $\ddot {Z}$ . Therefore, the mean acceleration follows $a\simeq \ddot {Z}$ , which we obtain by time deriving the measured plateau velocity $\dot {Z}$ .

By definition, the neck stress $\sigma$ is also related to the neck stretching rate $\partial _z u=-\dot {h^2}/h^2$ , through the neck extensional bulk-effective viscosity, $3\eta _{\textit{eff}}$ , as

(3.2) \begin{equation} \sigma = -3 \eta _{\textit{eff}} \frac {\dot {h^2}}{h^2}, \end{equation}

where we have assumed a slender neck flow (which is reasonably well verified in experiments where $\partial _z h \lesssim 0.25$ ) and neglected capillary stresses, since $({\gamma }/{\sigma h})\ll 1$ (for viscous necking, $({\gamma }/{\sigma h})$ decreases from the initial value $\sim ({\gamma }/{\rho g h_{{0}}^2}) \lt 10^{-3}$ . Inertia tends to increase the ratio, but from the model of § 3.6, it remains below $10^{-3}$ down to $h=d$ ).

By combining (3.1) and (3.2), we obtain an expression for the neck effective viscosity

(3.3) \begin{equation} \eta _{\textit{eff}} = -\frac {1}{3\pi } \frac {M\bigl(g - \ddot {Z}\bigr)}{\dot{h^2}}, \end{equation}

which can be computed from the post-pinch-off measurements of $M$ , obtained by weighing the detached drop, and from those of $h(t)$ , $\dot {h}(t)$ and $\ddot {Z}(t)$ , at all times. Importantly, the effective viscosity of the suspension is presumably not uniform within and around the neck, and this measurement must be interpreted as the neck effective resistance to flow.

3.3. Typical evolution of the neck effective viscosity

The typical evolution of $\eta _{\textit{eff}}$ is shown in figure 2(c), for $\phi =0.60$ . It is presented as a function of the current neck diameter to highlight how the drop in $\eta _{\textit{eff}}$ occurs. Surprisingly, the decay is not sharp, but very progressive in magnitude. Consistently with the large volume fraction in the drop, the neck effective viscosity is initially much larger than the suspending liquid viscosity, $\eta _{{0}}$ . As necking proceeds, it is found to decay by over two decades. On average (thick green line), the decay follows a close to exponential dependence on the current neck diameter, which can be fitted as follows:

(3.4) \begin{equation} \eta _{\textit{eff}} \simeq \eta _{{t}}\Big (\frac {\eta _{{i}}}{\eta _{{t}}}\Big )^{\!h/h_{{0}}}, \end{equation}

and which is fully characterised by the initial and terminal effective viscosities, $\eta _{{i}}$ and $\eta _{{t}}$ , extrapolated in $h=h_{{0}}$ and $h=0$ , respectively. The terminal value, $\eta _{{t}}$ , is found to compare with the suspending liquid viscosity, $\eta _{{0}}$ . The reproducibility of the scaling is highlighted by its consistency across the eight experimental replicates (thin green lines).

3.4. Dependences of the neck effective viscosity

Figure 3.

Dependence of the neck effective viscosity. (a) $\eta _{\textit{eff}}/\eta _{{0}}$ vs $2h/d$ for different volume fractions ( $2h_{{0}}/d = 21$ ). (b) For different drop sizes ( $8.3d \lt 2h_{{0}} \lt 47d$ , $\phi =0.60$ ). Grey solid line: $4\alpha ^2\phi (\phi -\phi _{{c}})^2h_{{0}}^2/3K$ , with dilatancy factor $\alpha =4$ , permeability $K=10^{-3}d^2$ and critical volume fraction $\phi _{{c}}=0.58$ . (c) Initial and terminal effective viscosity vs $\phi$ . Grey solid line: steady uniform bulk-effective law, $\eta _{\textit{eff}}/\eta _{{0}}\sim (\phi _{{c}} -\phi )^{-2}$ .

Figure 3. Long description

The image contains three line graphs labeled (a), (b), and (c). Graph (a) shows the value of neck effective viscosity (η_eff/η_0) for different volume fractions (ϕ) ranging from 0.550 to 0.605, plotted against the ratio of neck height to particle diameter (2h_0/d). Graph (b) illustrates the neck effective viscosity for different drop sizes (2h_0/d) ranging from 8.3 to 47, plotted against the same ratio. A grey solid line represents the equation 4α²ϕ(ϕ−ϕ_c)²h_0²/3K. Graph (c) compares the initial and terminal effective viscosity (η_i/η_0) against the volume fraction (ϕ), with a grey solid line representing the steady uniform bulk-effective law (ϕ_c−ϕ)−². The graphs show how these viscosities vary with different parameters, highlighting the influence of volume fraction and drop size on the suspension rheology.

The exponential decay in effective viscosity is robust. As shown in figure 3(a), which displays the mean value of $\eta _{\textit{eff}}$ between experimental replicates, all volume fractions investigated ( $0.55\leq \phi \leq 0.605$ ) show a close to exponential decay (the variations between experiment replicates are similar to those shown in figure 2(c) for $\phi =0.60$ , except for $\phi =0.605$ , where fluctuations become larger towards pinch-off). However, the initial value $\eta _{{i}}$ is not fixed, but seems to increase with increasing $\phi$ . As reported in figure 3(b), the exponential decay in $\eta _{\textit{eff}}$ is also observed for different system sizes. We ranged $2h_{{0}}$ from 8.3 to $47d$ , using syringes of different inner radius $h_{{0}}$ , while keeping the same particles, suspending liquid, initial aspect ratio of the drop and volume fraction fixed at $\phi =0.60$ (thus changing the initial neck-to-particle size ratio, $2h_{{0}}/d$ , and initial stress scale, $\sim \kern-2pt \rho g h_{{0}}$ ). For all $h_{{0}}/d$ , the decay in $\eta _{\textit{eff}}$ is close to exponential. However, here also, the initial value varies.

To highlight the change in the range of neck effective viscosity spanned during detachment, figure 3(c) reports the initial (triangles) and terminal values (circles) inferred by fitting the data with (3.4). For all $\phi$ and $h_{{0}}/d$ , the inferred terminal value $\eta _{{t}}$ remains around $\eta _{{0}}$ , without systematic dependence, which indicates that the neck effective viscosity progressively decreases towards the pure liquid value $\eta _{{0}}$ . By contrast, the inferred initial value $\eta _{{i}}$ increases with increasing $\phi$ (approximately fivefold between $\phi =0.55$ and 0.605), as well as with increasing $h_{{0}}/d$ (approximately tenfold between $2h_{{0}}/d=8.3$ and 47).

These dependences indicate that both the system size and history influence the resistance to flow. Indeed, the initial value $\eta _{{i}}$ depends on the $h_{{0}}/d$ and does not follow the diverging law, $\sim \kern-1pt \eta _{{0}}(\phi _{{c}}-\phi )^{-2}$ , reported for steady uniform flows as $\phi$ approaches $\phi _{{c}}$ (see figure 3 c, the law is plotted for the conventionally reported value $\phi _{{c}}\approx 0.58$ (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018); note that a similar divergency exponent, $-2$ , and a similar critical value, $\phi _{{c}} \simeq 0.575$ , are reported for extensional flow of frictional suspensions (Cheal & Ness Reference Cheal and Ness2018)). Furthermore, the experiments with different $h_{{0}}/d$ show that $\eta _{\textit{eff}}$ is not even a function of $\phi$ and the current neck size $h/d$ only. For fixed $\phi$ and $h/d$ , the effective viscosity also depends on the previous necking dynamics (for instance, in figure 3 b, the effective viscosity at $2h/d \approx 15$ is over one decade larger for $2h_{{0}}/d = 21$ , than for $2h_{{0}}/d = 47$ ). These evolutions of $\eta _{\textit{eff}}$ stress that the rheological response of the neck depends not only on finite size effects, but also on transient and presumably non-uniform deformation effects.

The last remark suggests an important role of the dilation of the particle phase, at least at the beginning of the necking. This idea can be tested with the observed dependence of $\eta _{{i}}$ on the system size. Indeed, for $\phi$ sufficiently above $\phi _{{c}}$ , the initial neck deformation is expected to be slaved to dilation through a kinematic relation $\dot {\phi }/\phi \approx -\alpha (2\dot {\varepsilon })(\phi -\phi _{{c}})$ , where $\alpha \approx 2\,$ – $4$ is a geometrical dilation factor (Pailha & Pouliquen Reference Pailha and Pouliquen2009), and $\dot {\varepsilon }= -\dot {h}^2/{h^2}$ is the strain rate. Because of incompressibility, the dilatant particle motion imposes an opposite suspending liquid flow through the pore network between particles. This Darcy-like relative flow builds up a negative pore pressure, which is set by the particle phase permeability $K\approx 10^{-3}d^2$ , through $\boldsymbol{\nabla} ^2 {p_{{pore}}} \approx -(\eta _{{0}}/K)\dot {\phi }/\phi$ (where gradients in $\phi$ have been neglected following Jerome, Vandenberghe & Forterre Reference Jerome, Vandenberghe and Forterre2016; Jørgensen et al. Reference Jørgensen, Forterre and Lhuissier2020). Identifying the Darcy flow scale with the neck radius $h_{{0}}$ , we obtain ${p_{{pore}}} \approx -\eta _{{0}} (h_{{0}}^2/K)\dot {\phi }/\phi$ . Last, assuming the volume dissipation in the suspension, $3\eta _{\textit{eff}}\dot {\varepsilon }^2$ , is dominated by the work of dilation against the pore pressure, $-\dot {\phi }{p_{{pore}}}$ , we obtain that the initial effective viscosity should follow

(3.5) \begin{equation} \frac {\eta _{{i}}}{\eta _{{0}}} \approx \frac {4}{3}\alpha ^2\phi (\phi -\phi _{{c}})^2\frac {h_{{0}}^2}{K} \approx 1.3\times 10^3\alpha ^2\phi (\phi -\phi _{{c}})^2\left (\frac {h_{{0}}}{d}\right )^2\!. \end{equation}

The prediction is expected to be valid at the onset of motion (for a deformation $\lesssim 1$ , when the volume fraction is still close to the initial value). It is compared with measurements in figure 3(b). Using $\phi _{{c}}=0.58$ and a dilatancy prefactor $\alpha =4$ within the range 2–4 reported previously (Pailha & Pouliquen Reference Pailha and Pouliquen2009; Jerome et al. Reference Jerome, Vandenberghe and Forterre2016), equation (3.5) captures reasonably well the magnitude and increase of $\eta _{{i}}$ with $h_{{0}}/d$ (albeit for the largest system size, where the measured value is somewhat below). This agreement suggests that for $\phi$ above $\phi _{{c}}$ the initial detachment dynamics is essentially slaved to the viscosity-limited particle dilation.

The above description is limited to the initial deformation and effective viscosity $\eta _{{i}}$ . Nonetheless, it is important because we will see below (§ 3.6) that $\eta _{{i}}$ sets the overall breakup duration. As for the subsequent exponential decay of $\eta _{\textit{eff}}$ , a more involved analysis is needed, which would consider large accumulated strains (Athani et al. Reference Athani, Metzger, Forterre and Mari2022) and non-uniformity around the neck. These difficulties prevent us from providing a quantitative model, but we present below additional local observations, which highlight the qualitative evolution of the suspension during pinching.

3.5. Symmetry breaking and capillary collapse

Figure 4.

Symmetry breaking and capillary interface. (a–b) Asymmetry of the pinching ( $\phi =0.60$ ). (a) Surface velocity field. Main: absolute velocity $\boldsymbol{u}$ . Close up: deviation $\boldsymbol{u} - \langle u\rangle |_z\boldsymbol{e_z}$ to extensional flow. (b) Average evolution of the neck apparent off-centring for different volume fractions (5 experiments each). (c) Mid-plane image of the drop, obtained by three-phase refractive-index matching, showing the suspension conformation and the capillary meniscus mid-plane curvatures $\kappa _\parallel$ . The suspension detaches in a lighter, non-miscible liquid. The suspending liquid is fluorescent, the other two phases appear as black ( $\phi = 0.60$ , $d = 1025\,\unicode {x03BC}$ m). (d) Meniscus curvature, at the neck and bottom of the drop, vs $2h$ (symbols, $\kappa =2\kappa _\parallel$ , bars, $\kappa =\kappa _\parallel$ ). Pink line: critical total curvature $\kappa _{{c}} \simeq 6.6/d$ (see characterisation in § 2), above which capillary confinement collapses.

Figure 4. Long description

The image contains four subfigures related to the breaking-up of liquid jets loaded with solid particles. Subfigure (a) shows the surface velocity field and asymmetry of pinching. The main part of the subfigure displays the absolute velocity, while the close-up shows the deviation to extensional flow. Subfigure (b) displays the average evolution of the neck apparent off-centring for different volume fractions, with five experiments conducted for each volume fraction. Subfigure (c) presents a mid-plane image of the drop, obtained by three-phase refractive-index matching, showing the suspension conformation and capillary meniscus mid-plane curvatures. The suspension detaches in a lighter, non-miscible liquid. The suspending liquid is fluorescent, while the other two phases appear as black. Subfigure (d) illustrates the meniscus curvature at the neck and bottom of the drop versus a variable, with symbols representing the data points and bars indicating the uncertainty. A pink line represents the critical total curvature above which capillary confinement collapses.

A first important observation is that deviations to axial symmetry increase as necking develops. This deviation can be evidenced in the surface velocity field (obtained by correlating particle patterns over the visible drop surface, see typical field in figure 4 a). The deviation $\boldsymbol{u} - \langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{e_z}\rangle |_z\boldsymbol{e_z}$ to a purely vertical $z$ -dependent flow shows an asymmetric zip-like structure, akin to a crack-opening flow (see inset). This flow is associated with an increasingly asymmetric neck shape, as revealed by the increasing off-centring $\delta$ between the neck and nozzle axes. The off-centring is realisation-dependent (and our side observation only provides one out of two off-centring components), but measurements in figure 4(b) show it is on average increasing with decreasing $h$ as well as with increasing $\phi$ .

Further information can be obtained about the actual conformation of the suspension at the neck by implementing a transparent set-up offering undistorted visualisation inside the drop. This set-up (detailed in § 2) is similar to the original one, except that the suspension drop detaches inside a lighter, immiscible and low-viscosity liquid bath, rather than air. This allows us to match the three phases’ refractive indices and obtain undistorted images in a plane illuminated with an (undistorted) laser sheet. Importantly, the detachment conditions and the suspension are similar to the in-air set-up (initially viscous detachment, particles wetted by suspending liquid, $\phi = 0.60$ , $h_{{0}}/d=11.7$ and similar Bond numbers, see § 2). A typical snapshot of the drop mid-plane is shown in figure 4(c), for $2h/d\simeq 3$ , with the dyed suspending liquid appearing as bright, the other two phases as black (see also supplementary movie 2). It reveals that the suspension changes along the drop. It seems more diluted at the neck than at the bottom. Concomitantly, the capillary menisci between the particles at the drop surface are much more curved inward at the neck, which indicates a much more negative liquid pressure there. While sufficiently accurate local volume fraction measurements cannot be achieved because of the small number of particles involved and limitations of particle edge detection, we report in figure 4(b) the evolution of the maximal meniscus curvature observed in the neck region (red symbols) and at the bottom (blue ones). The value at the neck continuously increases as pinching proceeds, and rapidly compares with the critical curvature, $\kappa _{{c}}\simeq 6.6/d$ , at which menisci break down and allow the outer phase to permeate the suspension (the value $\kappa _{{c}} d\simeq 6.6$ is experimentally characterised on a flat, non-moving, concentrated suspension interface, see § 2).

These local observations suggest that the solid phase at the neck dilutes gradually. Concomitantly, deviations to axial symmetry grow and the outer phase presumably permeates the suspension at some point. All these evolutions are expected to decrease the neck resistance to flow, but it remains unclear which contribution dominates. In spite of this limitation, we now turn to see how the detachment dynamics can actually be modelled if the observed rheological evolution at the neck is taken as granted, and how the breakup duration depends on $\eta _{{i}}$ .

3.6. Model of the detachment dynamics

Figure 5.

Model of the pinching dynamics. (a) The almost constant cone angle $2\beta$ of the pinching drop relates the drop and necking accelerations, $\ddot {Z}$ and $\ddot {h}$ ( $\phi = 0.60$ ). (b) Relative neck radius vs relative time to breakup. Thick solid lines: solution of (3.7), with $\eta _{{i}}/\eta _{{0}}=743$ , for different inertial numbers (grey lines, $\mathcal{I}\!=\!0$ , $6.4\times 10^{-8}$ and $4.0\times 10^{-5}$ , from left to right), including the experimental value (green line, $\mathcal{I}\!=\!1.6\times 10^{-6}$ ). Thin solid lines: experiments (same data as in figure 2). Inset: breakup time obtained experimentally (black-outlined circles) and from the model fed with values of table 1 (red-outlined circles) vs $\phi$ .

Figure 5. Long description

The image contains two main parts. On the left, labeled as (a), three diagrams show the pinching process of a drop with varying cone angles. The diagrams highlight the relationship between the drop and necking accelerations, with annotations indicating angles and directions. On the right, labeled as (b), a log-log plot displays the relative neck radius versus relative time to breakup. The plot includes thick solid lines representing the solution of a specific equation for different inertial numbers, with colors ranging from grey to green. Thin solid lines represent experimental data. An inset within this plot shows the breakup time obtained experimentally and from the model versus a variable, with black-outlined and red-outlined circles representing the data points.

From the observed evolution of the neck effective viscosity we propose a simplified effective model for the drop detachment dynamics, which accounts for inertial effects, by taking advantage of the close-to-conical shape of the detaching drop. Indeed, for a drop breaking with a double cone shape with constant half-angle $\beta$ (see figure 5 a), the vertical velocity $\dot {Z}$ of the solid-like detaching part must relate to the necking speed $-\dot {h}$ through

(3.6) \begin{equation} \dot {Z} \approx -\frac {\dot {h}}{\tan \beta }. \end{equation}

In this case, the vertical acceleration of the drop, $\ddot {Z}$ , also relates to the neck radius acceleration, $-\ddot {h}$ , through $\ddot {Z} \approx -\ddot {h}/\tan \beta$ . Together with the vertical momentum balance at the neck, (3.1,3.2), this allows us to close the problem and obtain the following evolution equation for the neck radius:

(3.7) \begin{align} \mathcal{I}\frac {\tau ^2}{h_{{0}}}\ddot {h} = -1 - \left (\frac {\eta _{{i}}}{\eta _{{0}}}\right )^{\!h/h_{{0}}-1}\!\frac {\tau }{h_{{0}}^2}\dot {h^2},&& \text{with}&& \tau = 3\pi \frac {\eta _{{i}}h_{{0}}^2}{\textit{Mg}},\qquad \mathcal{I} = \frac {1}{\tan \hspace {-.08em}\beta }\frac {h_{{0}}}{g\tau ^2}. \end{align}

In this equation, the left-hand term corresponds to the drop (vertical) inertia. The other two correspond, from left to right, to gravity and to the viscous stress, which involves the effective viscosity $\eta _{\textit{eff}}/\eta _{{i}} = (\eta _{{i}}/\eta _{{t}})^{h/h_{{0}}-1} \simeq (\eta _{{i}}/\eta _{{0}})^{h/h_{{0}}-1}$ , where we have used the experimental observation $\eta _{{t}}\simeq \eta _{{0}}$ . The visco-gravity time scale $\tau$ is the typical viscous detachment time, based on the initial viscosity $\eta _{{i}}$ . Last, the inertial number $\mathcal{I} \sim \rho ^2 \eta _{{i}}^2/gh_{{0}}^3$ (inverse Galilei number) embeds the initial magnitude of inertia and determines from which time the acceleration eventually alleviates the drop weight.

Table 1.

Mean measured quantities ( $\eta _{{i}}/\eta _{{0}}$ , $\beta$ ) and computed parameters ( $\tau$ , $\mathcal{I}$ , $\eta _{\textit{eff}}(2h=\!10\,\textrm {mm})/\eta _{{0}}$ , $\tau _{10\,\text{mm}}$ , $t_{\textit{break},10\,\textrm{mm}}$ ) for the different volume fractions ( $d=570\,\unicode {x03BC}$ m, $2h_{{0}}=12.03\,$ mm, $\eta _{{0}}=2.8$ Pa s, $M=1.50$ g). The last three values are for comparison with the experimental breakup time $t_{\textit{break},10\,\textrm{mm}}$ , which is measured from $2h=10\,$ mm. They are computed as $\eta _{\textit{eff}}(2h=\!10\,\textrm {mm})/\eta _{{0}} = (\eta _{{i}}/\eta _{{0}})^{10\,\textrm {mm}/2h_{{0}}}$ using (3.4), and by evaluating equations (3.7) and (3.8) for $\tau$ and $t_{\text{break}}$ , with $2h_{{0}}=10\,$ mm and $\eta _{{i}}=\eta _{\textit{eff}}(2h=\!10\,\textrm {mm})$ , respectively.

Table 1. Long description

The table presents mean measured quantities and computed parameters for various volume fractions. It includes columns for volume fraction, ratio of viscosities, contact angle, surface tension, and inertial number. Additionally, it shows the ratio of viscosity to density, breakup time at 10 millimeters, and breakup time at 10 millimeters for different volume fractions. The data is organized in rows, each representing a specific volume fraction with corresponding values for the measured and computed parameters. Notable trends include variations in inertial numbers and breakup times across different volume fractions, providing insights into the dynamics of pinching drops.

Figure 5(b) represents integrations of (3.7) for small inertial numbers ( $\mathcal{I}\ll 1$ ), which are relevant for highly concentrated drops. Initially, the dynamics follows the purely viscous evolution ( $\mathcal{I}=0$ , given by $[1+(kh/h_{{0}}-1)\mathrm{e}^{kh/h_{{0}}}]/[1+(k-1)\mathrm{e}^{k}]=1-t/t_{\text{break}}$ , with $k=\ln (\eta _{{i}}/\eta _{{0}})$ and $t_{\text{break}}/\tau =2(k-1+\mathrm{e}^{-k})/k^2$ ). However, as pinching proceeds, inertia eventually matters. Acceleration alleviates the drop weight, and pinching becomes comparatively slower, with a ballistic terminal evolution $h\propto t_{\text{break}}-t$ . The larger the value of $\mathcal{I}$ , the earlier the viscous to inertial cross-over. For large viscosity ratio ( $\eta _{{i}}/\eta _{{0}}\gg 1$ ), this cross-over takes place when $h/h_{{0}}\sim k^{-1}\ln (\mathcal{I} k^3\mathrm{e}^{2k}/4)$ . For our experiments this happens for $h$ not much below $h_{{0}}$ , which is why inertia must be considered. However, because of the tremendous acceleration of the dynamics, this only corresponds to the very last instants before pinch-off. Therefore, while the final breakup dynamics is largely influenced by inertia, the total breakup duration is almost undistinguishable from the viscous limit

(3.8) \begin{align} \frac {t_{\text{break}}}{\tau }=2\frac {\ln \eta _{{i}}/\eta _{{0}}-1+\eta _{{0}}/\eta _{{i}}}{(\ln \eta _{{i}}/\eta _{{0}})^2},&& (\mathcal{I}\ll 1), \end{align}

which converges to $2(\ln \eta _{{i}}/\eta _{{0}})^{-1}$ , for $\eta _{{i}}\gg \eta _{{0}}$ .

As shown in figure 5(b) with $\phi =0.60$ , this simple model captures the average evolution of $h$ towards pinch-off. The integration of (3.7), with the experimental values $\eta _{{i}}/\eta _{{0}}=743$ and $\mathcal{I}=1.6\times 10^{-6}$ matches the ensemble evolution across the different replicates of the experiment ( $\mathcal{I}$ is computed from the weighted drop mass, $M=1.5\,$ g, and the apparent neck angle, $\beta = 9.4^\circ$ , measured on images before pinch-off. A similar value, $\textrm {atan}(-\dot {h}/\dot {Z}) \sim 10^\circ$ , is obtained from the instantaneous velocity ratio $-\dot {h}/\dot {Z}$ ). From the experimental values of $\eta _{{i}}$ and $\beta$ reported in table 1, the model also recovers the breakup time across the different volume fractions $\phi$ . Since the experimental value, $t_{\textit{break},10\,\textrm{mm}}$ , is measured from $2h=10$ mm, expression (3.8) for $t_{\text{break}}$ must be evaluated with the neck size and effective viscosity at $2h=10\,$ mm. This is done by using (3.4) to express $\eta _{\textit{eff}}(2h=\!10\,\textrm {mm}) = \eta _{{0}}(\eta _{{i}}/\eta _{{0}})^{10\,\textrm {mm}/2h_{{0}}}$ and by evaluating (3.7) and (3.8) with $2h_{{0}}=10\,$ mm and $\eta _{{i}}=\eta _{\textit{eff}}(2h=\!10\,\textrm {mm})$ . The computed breakup times, reported in table 1 and figure 5(b), are found to match well the magnitude and trend of the experimental data. Part of this match is natural, since the model is fed with the observed effective decay in viscosity. Nonetheless, the simplified geometric input of the model (the relation (3.6) between $\ddot {Z}$ and $\ddot {h}$ ) allows us to account for inertia and explains that, for the present case of a low inertial number $\mathcal{I}$ , while inertia largely influences the late dynamics, the breakup time is essentially set by the initial viscous time scale $\tau$ and the logarithmic viscous correction of (3.8).