2.1 Set-up

The drop impact protocol is sketched in figure 1. It consists in letting a suspension drop fall and impact perpendicularly onto a plate and observing the first milliseconds following the impact, during which the kinetic energy is dissipated.

The suspensions are prepared by immersing spherical polystyrene particles (Dynoseed TS, Microbeads, with a density of $1049~\text{kg}~\text{m}^{-3}$) in a Newtonian liquid with a close density. Most of the experiments are performed with a mean particle diameter $d=145~\unicode[STIX]{x03BC}\text{m}$ and a relative standard deviation in size below 4 %. A larger diameter ($d=585~\unicode[STIX]{x03BC}\text{m}~\pm 1\,\%$) is also used (see § 4). These ‘large’ particle sizes ensure that interparticle forces are negligible (Morris Reference Morris2020) and that the suspensions do not shear-thicken (which we have also verified, for $d=145~\unicode[STIX]{x03BC}\text{m}$ and up to $\unicode[STIX]{x1D719}=0.50$, during the rheological characterization of the suspensions presented in § 3.2). The liquid is a water solution of PEGPG ($3.9~\text{kg}~\text{mol}^{-1}$ poly(ethylene glycol-ran-propylene glycol)-monobutylether by Sigma-Aldrich), which wets the particles well (the contact angle is typically smaller than $45^{\circ }$) and is sufficiently viscous to avoid particles being ejected from the drop during impact. Its viscosity, $\unicode[STIX]{x1D702}_{0}$, is varied between 0.93 and 192 mPa s by adjusting the PEGPG concentration (from 0 to 50 wt %), whereas its density and surface tension vary from 998 to $1056~\text{kg}~\text{m}^{-3}$ and from 70 to $40~\text{mN}~\text{m}^{-1}$, respectively.

The particle volume fraction of the drop, $\unicode[STIX]{x1D719}$, is varied from 0 to above the critical volume fraction. For moderate concentrations ($\unicode[STIX]{x1D719}\leqslant 0.54$), the formation of the drop is straightforward. The suspension is prepared at the desired $\unicode[STIX]{x1D719}$, infused in a truncated (nozzle-free) plastic syringe with an inner diameter of 4.6 mm and the suspension is slowly extruded until the drop detaches. For $\unicode[STIX]{x1D719}<0.54$, this simple protocol ensures that the volume fraction in the drop is identical (within less than 0.01, as verified a posteriori) to the nominal volume fraction of the preparation. However, it does not allow one to control and reproduce the solid content for more concentrated drops. Those drops, with $\unicode[STIX]{x1D719}>0.54$, are thus obtained by infusing a suspension with a volume fraction of ${\approx}0.55$ in the syringe and concentrating each drop inside the syringe, by letting the particles sediment under gentle tapping or soaking part of the liquid with an absorbing tissue (see figure 1b). In this case the value of $\unicode[STIX]{x1D719}$ is determined post-impact by weighing, for each single drop, both the whole drop and its dry content obtained after washing and drying the particles. This protocol permits volume fractions as high as 0.62 to be reached, with an uncertainty of less than 0.005. The most concentrated drops (above $\unicode[STIX]{x1D719}_{c}\simeq 0.605$) behave like a paste. To avoid them extruding as a long cylinder, their shape is progressively relaxed by gently tapping the syringe. The detached drops remain oblong (see figure 1d) and care is taken to discard the impacts for which the drop has rotated.

The drop impacts a smooth glass plate (washed and rinsed with micro-filtered water before each impact). However, a series of impacts has also been performed on a rough surface (flat sand paper with a grain size ${\approx}160~\unicode[STIX]{x03BC}\text{m}\approx d$) to verify the influence of potential sliding of the particles on the surface (see § 2.2). The impact velocity $V$ is varied from 1.6 to $10~\text{m}~\text{s}^{-1}$. In order to achieve the largest velocities while keeping aerodynamic effects on the drop negligible, the drop is released from a small height (${\approx}20~\text{cm}$) and the plate is moved vertically towards the drop. To this end, the plate is stuck on a fast translating stage (electromagnetic actuator, Copley STA2506S). The stage motion is triggered by the occulting of a laser beam by the falling drop and the displacement versus time law is tuned so as to ensure that the impact occurs at the fixed observation spot with the desired impact velocity $V$ ($=|V_{drop}|+|V_{stage}|$). (The stage velocity and its constancy during the impact are verified in the movies made to obtain the measurements.) The impact is imaged with a fast camera facing an intense collimated light source, which yields a sufficiently short time sampling ($62~\unicode[STIX]{x03BC}\text{s}$ with a shutter time of $2~\unicode[STIX]{x03BC}\text{s}$) and high spatial resolution ($16~\unicode[STIX]{x03BC}\text{m}$). A grazing view of the plate is used in order to access the whole deformation of the drop, including at the contact with the plate.

2.2 General observations

The typical impact dynamics for a concentrated suspension drop is illustrated in figure 1(d,e) for $\unicode[STIX]{x1D719}=0.556$, $\unicode[STIX]{x1D702}_{0}=50.5~\text{mPa}~\text{s}$ and $V=4~\text{m}~\text{s}^{-1}$. As the drop makes contact with the plate, its lower surface flattens and the remainder of the drop widens while contracting in the vertical direction. The deformation is initially restricted to the region neighbouring the plate and progressively extends up to the top of the drop. The whole sequence lasts only a few hundreds of microseconds, after which all the kinetic energy is dissipated and fast deformations stop (the subsequent capillary–gravitational relaxation of the shape, lasting ≳1 s and not shown in figure 1(d,e), is orders of magnitude slower).

Figure 2.

(a) Relative spreading as a function of the particle volume fraction for a suspension drop with $d=145~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D702}_{0}=50.5~\text{mPa}~\text{s}$ ($V=4~\text{m}~\text{s}^{-1}$), as measured from the maximal deformed diameter $D_{f}$ ($\unicode[STIX]{x1D6FD}\equiv D_{f}/D_{i}-1$, filled symbols) and from the contact diameter $D_{c}$ ($\unicode[STIX]{x1D6FD}_{c}\equiv D_{c}/D_{eq}$, open symbols). The error bars for $\unicode[STIX]{x1D719}=0$, 0.1, 0.2, 0.3, 0.4, 0.45, 0.50 and 0.54 represent the standard deviation over three drop impacts. The other points represent a single impact event. (b) Spreading versus $\unicode[STIX]{x1D719}$ for different suspending liquids and the same ratio $V/\unicode[STIX]{x1D702}_{0}$.

We characterize the final spreading by measuring both the maximal diameter $D_{f}$ and the contact diameter $D_{c}$ (as observed from the side view, see figure 1d). This yields two different measurements of the deformation:

(2.1a,b)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}\equiv \frac{D_{f}}{D_{i}}-1\quad \text{and}\quad \unicode[STIX]{x1D6FD}_{c}\equiv \frac{D_{c}}{D_{eq}}, & & \displaystyle\end{eqnarray}$$

where $D_{i}$ and $D_{eq}$ are, respectively, the maximal diameter before impact and the diameter of the equivalent-volume sphere. These definitions satisfy $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{c}=0$ for a vanishing deformation. To obtain the fairest comparison between the slightly different drop shapes, as well as with the spherical geometry assumed below, we choose to scale $D_{c}$ by $D_{eq}$ rather than $D_{i}$ (which cannot be done for $\unicode[STIX]{x1D6FD}$ since we require that it vanishes in the absence of deformation).

The dependence of the spreading on the particle volume fraction is presented in figure 2(a), for the same suspension and impact velocity as in figure 1(d). For both $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6FD}_{c}$ the spreading decreases continuously and significantly from $\unicode[STIX]{x1D719}=0$ up to the largest concentration explored here ($\unicode[STIX]{x1D719}=0.587$). Importantly, the spreading is found to be almost unchanged upon using an impact plate with roughnesses of the order of $d$ rather than a smooth glass plate. This suggests that, even with the smooth plate, there is no significant slipping of the particles at the plate surface.

Before seeking to understand the dependence to the particle content, a last crucial remark should be made by comparing the deformation obtained by varying both the velocity and suspending liquid such that the ratio $V/\unicode[STIX]{x1D702}_{0}$ remains constant, i.e. keeping the Reynolds number at fixed volume fraction unchanged. As shown in figure 2(b), the deformation is found to be essentially unchanged for a sixfold increase in $V$ (with $\mathit{Re}_{0}\equiv \unicode[STIX]{x1D70C}VD_{i}/\unicode[STIX]{x1D702}_{0}=350\pm 20$) over the whole range of $\unicode[STIX]{x1D719}$. This suggests that, similarly to the Newtonian case and in contrast with the Bagnold scaling invoked in Boyer et al. (Reference Boyer, Sandoval-Nava, Snoeijer, Dijksman and Lohse2016), the dissipation of the motion of the suspension drops follows a viscous scaling and is directly controlled by the Reynolds number.

To go beyond these findings and precisely assess the similarity and differences with the Newtonian case, a clear Newtonian reference is required, in particular in the limit of large viscosities (low Reynolds numbers), where the deformation is expected to be small.

3.1 Newtonian calibration and low-deformation limit

Figure 3.

(a) Spreading for a viscous Newtonian drop as a function of the impact Reynolds number, $\unicode[STIX]{x1D70C}D_{i}V/\unicode[STIX]{x1D702}_{0}$ (respectively, $\unicode[STIX]{x1D70C}D_{eq}V/\unicode[STIX]{x1D702}_{0}$), for $\unicode[STIX]{x1D6FD}$ (respectively, $\unicode[STIX]{x1D6FD}_{c}$). Six different PEGPG mixtures and pure UCON Oil are used. From large to low values of $\mathit{Re}$, the viscosity, velocity and Weber-number range, respectively, from 0.012 to 41.5 Pa s, from 4 to $0.4~\text{m}~\text{s}^{-1}$ and from 2250 to 15 (the raw data are available at https://doi.org/10.1017/jfm.2020.394). The orange triangles represent the measurements of Lagubeau et al. (Reference Lagubeau, Fontelos, C, Maurel, Pagneux and Petitjeans2012) (the value of $\unicode[STIX]{x1D6FD}=D_{f}/D_{0}-1=(3h/2D_{0})^{-1/2}$ is obtained from the relative thickness ($h/D_{0}$) they measure and from the assumption of a flat cylindrical shape of the deformed drop). The green squares represent those obtained by Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) for the largest viscosity they have used ($\unicode[STIX]{x1D702}=51~\text{mPa}~\text{s}$). The lines represent equation (3.2). (b) The same data as in figure 2(b) plotted versus the effective Reynolds number $\unicode[STIX]{x1D70C}D_{i}V/\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})$ (respectively, $\unicode[STIX]{x1D70C}D_{eq}V/\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})$) for $\unicode[STIX]{x1D6FD}$ (respectively, $\unicode[STIX]{x1D6FD}_{c}$) defined according to (3.3) with the critical volume fraction, $\unicode[STIX]{x1D719}_{c}=0.605$, obtained with the indentation set-up (figure 4). The red line (respectively, envelope) represents a sliding geometrical average (respectively, standard deviation) of the spreading for the suspensions (the average is taken over $\pm$ one-third of a decade).

The Newtonian reference is obtained by impacting drops of Newtonian liquids (PEGPG and UCON Oil (Dow) aqueous solutions, with a viscosity $\unicode[STIX]{x1D702}$ between 0.012 and 41.5 Pa s) for values of $\mathit{Re}=\unicode[STIX]{x1D70C}D_{i}V/\unicode[STIX]{x1D702}$ ranging from $10^{-1}$ to $10^{3}$. As shown in figure 3(a), for large $\mathit{Re}$ we recover the large-deformation limit, $\unicode[STIX]{x1D6FD}\sim \mathit{Re}^{1/5}$, derived by Madejski (Reference Madejski1976). However, for larger viscosities, the deformation is so small that it is restricted to the surroundings of the contact with the plate (and no measurable oscillation of the diameter is observed). In this case we expect a different law. Assuming, in the spirit of Hertz’s model for elastic deformations (Hertz Reference Hertz1896), that the deformed drop remains close to a sphere except for the flattened contact area with diameter $D_{c}\ll D_{f}\simeq D_{i}$ where it is indented with a typical depth $\unicode[STIX]{x1D6FF}\sim D_{c}^{2}/D_{i}$, one expects that the deformation is restricted to a volume $\unicode[STIX]{x1D6FA}\sim D_{c}^{3}$ and has a typical magnitude $\unicode[STIX]{x1D700}\sim \unicode[STIX]{x1D6FF}/D_{c}$. (This assumption is relevant in the present limit of $Re\lesssim 1$, for which the thickness of the viscous boundary layer,$\sim \sqrt{(\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70C})t}\sim \sqrt{(\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70C})D_{c}/V}$, is expected to be larger than the contact width $D_{c}$.) Equating the initial kinetic energy of the drop, ${\sim}\unicode[STIX]{x1D70C}V^{2}D_{i}^{3}$, with the viscous dissipation, ${\sim}\unicode[STIX]{x1D702}_{0}\dot{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}\sim \unicode[STIX]{x1D702}_{0}\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D6FA}/(\unicode[STIX]{x1D6FF}/V)$, gives us $D_{c}/D_{i}\sim \mathit{Re}^{1/3}$. Last, making use of mass conservation, which imposes that the volume increase of the spherical part of the drop, ${D_{f}}^{3}-{D_{i}}^{3}\sim {D_{i}}^{2}(D_{f}-D_{i})$, be equal to the indented volume, ${\sim}\unicode[STIX]{x1D6FF}D_{c}^{2}$, we obtain

(3.1a,b)$$\begin{eqnarray}\unicode[STIX]{x1D6FD}\equiv \frac{D_{f}}{D_{i}}-1\sim \mathit{Re}^{4/3}\quad \text{and}\quad \unicode[STIX]{x1D6FD}_{c}\equiv \frac{D_{c}}{D_{eq}}\sim \mathit{Re}^{1/3}\quad \text{for }\mathit{Re}\ll 1,\end{eqnarray}$$

which is found to be in good agreement with the measurements for small deformations (figure 3a). Note that, although the most viscous drops do not relax to a spherical shape before impact (similarly to the concentrated suspension drops), the two different definitions of the deformation yield the same agreement between (3.1) and the measurements, which suggests that the small departures from sphericity do not significantly affect the characterization of the deformation.

In order to allow the comparison between the suspension case and the Newtonian case for any deformation, we fit the Newtonian data with the following cross-over functions:

(3.2)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\mathit{Re}\equiv F^{-1}(\unicode[STIX]{x1D6FD})=[(2\unicode[STIX]{x1D6FD})^{3/16}+(1.4\unicode[STIX]{x1D6FD})^{5/4}]^{4},\\ \mathit{Re}\equiv {F_{c}}^{-1}(\unicode[STIX]{x1D6FD}_{c})=[(1.3\unicode[STIX]{x1D6FD}_{c})^{6}+(1.1\unicode[STIX]{x1D6FD}_{c})^{10}]^{1/2}.\end{array}\right\}\end{eqnarray}$$

These are plotted in figure 3(a) together with the measurements. Their functional form is chosen so as to recover the scaling laws of both the large- and the low-deformation limits.

In order to serve as a reference for the suspension case, this Newtonian formalism has to be complemented with a characterization of the effective rheology of the suspensions.

3.2 Rheological characterization of the suspensions

Figure 4.

Rheological characterization of the suspensions. (a) Relative effective viscosity measured by Palma & Lhuissier (Reference Palma and Lhuissier2019) for the same suspensions ($d=145~\unicode[STIX]{x03BC}\text{m}$, PEGPG solution) as those used in the present study. The black dashed line is equation (3.3) with $a=2.2$. The red dashed line indicates $\unicode[STIX]{x1D719}_{c}$ as measured in (b,c). (b) Indentation set-up used to measure the critical volume fraction $\unicode[STIX]{x1D719}_{c}$. (c) Inset: decay of the indenter initial velocity for slightly different volume fractions of the sedimented particle pile ($t=0$ indicates the contact instant). Main: maximal post-impact deceleration versus $\unicode[STIX]{x1D719}$. The horizontal and inclined dashed lines are fitted to the data. Their intersection sets $\unicode[STIX]{x1D719}_{c}\simeq 0.605$.

Figure 4(a) reports the bulk effective shear viscosity as a function of the particle volume fraction up to $\unicode[STIX]{x1D719}=0.50$, as measured in one of our previous works for the same particles and liquids (Palma & Lhuissier (Reference Palma and Lhuissier2019), the sketch of the dedicated shear cell used in this study is shown in the inset). To characterize the suspension at much higher volume fractions that cannot be accessed with the shear cell, we use a complementary protocol inspired by Jerome, Vandenberghe & Forterre (Reference Jerome, Vandenberghe and Forterre2016): a protocol of dynamic indentation into a sedimented state of the suspension, which allows us to measure accurately the critical volume fraction $\unicode[STIX]{x1D719}_{c}$ by probing the transient rheology both below and above $\unicode[STIX]{x1D719}_{c}$. A sedimented particle pile is prepared at a volume fraction between ≃0. 565 and ≃0. 632 by letting the particle sediment gently at the loose random packing fraction ≃0. 565 and increasing the packing by a controlled tapping on the container (see figure 4b). The pile is indented dynamically with a solid cylinder impacting at $1.4~\text{m}~\text{s}^{-1}$. The post-impact velocity of the indenter is measured (with high precision by applying conventional image correlation methods to the speckle-patterned surface of the indenter), as shown in the inset of figure 4(c). For each value of $\unicode[STIX]{x1D719}$, the maximal deceleration of the indenter is extracted. As shown in figure 4(c), the deceleration is almost constant (at typically $100~\text{m}~\text{s}^{-2}$) up to $\unicode[STIX]{x1D719}\simeq 0.605$, and increases significantly and almost linearly (by a factor up to 40), above.

This steep change in the transient rheology of the suspension indicates the onset of a dilatant response to deformation (Jerome et al. Reference Jerome, Vandenberghe and Forterre2016), which implies an enhanced dissipation because the increase in pore volume between the particles forces a Darcy flow of the suspending liquid through the particle phase. By definition, the onset of dilatancy is also the maximal volume fraction at which a constant-volume deformation is possible, i.e. the volume fraction $\unicode[STIX]{x1D719}_{c}$ at which the steady effective rheology of the suspension diverges. We therefore obtain a direct and accurate characterization of $\unicode[STIX]{x1D719}_{c}$ for our suspensions by determining the onset of the dilatancy from the fitting procedure defined in the caption of figure 4(c), which yields $\unicode[STIX]{x1D719}_{c}\simeq 0.605$. This exhaustive characterization is obtained for $d=145~\unicode[STIX]{x03BC}\text{m}$ in pure water. However, by measuring the volume fraction of the loose random packing obtained by slow sedimentation (without tapping) we could verify that the value is almost the same (between 0.554 and 0.567) for all the different particles and suspending liquids that we have used, which indicates that the values of $\unicode[STIX]{x1D719}_{c}$ should also be almost the same. Similarly quasi-static measurements of the starting angle of avalanche in a rotating drum yielded very similar values for particles in water ($25.5^{\circ }$) and in a 25 wt % PEGPG solution ($27^{\circ }$).

To obtain an effective viscosity reference for the whole range of $\unicode[STIX]{x1D719}$, we choose to summarize the above rheological characterizations by the following functional form:

(3.3)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})}{\unicode[STIX]{x1D702}_{0}}=1+a-\frac{5\unicode[STIX]{x1D719}_{c}}{2}+\left(\frac{5\unicode[STIX]{x1D719}_{c}}{2}-2a\right)\left[1-\frac{\unicode[STIX]{x1D719}}{\unicode[STIX]{x1D719}_{c}}\right]^{-1}+a\left[1-\frac{\unicode[STIX]{x1D719}}{\unicode[STIX]{x1D719}_{c}}\right]^{-2},\end{eqnarray}$$

where $a$ is a fitting parameter. Similarly to Eilers’ law (Stickel & Powell Reference Stickel and Powell2005), equation (3.3) verifies both Einstein relation ($\unicode[STIX]{x1D702}/\unicode[STIX]{x1D702}_{0}\rightarrow 1+5\unicode[STIX]{x1D719}/2$) in the dilute limit and the inverse quadratic divergence ($\unicode[STIX]{x1D702}/\unicode[STIX]{x1D702}_{0}\propto (\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{-2}$) reported experimentally near $\unicode[STIX]{x1D719}_{c}$. This ensures that (3.3) (with the fitted value $a=2.2$) accurately represents the experimental data over the range $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.50$, as shown in figure 4(a), but also close to the critical volume fraction $\unicode[STIX]{x1D719}_{c}$.

3.3 Effective Newtonian deformation

From the viscous spreading law, $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70C}VD_{i}/\unicode[STIX]{x1D702})$ (3.2), and the model for the steady effective viscosity of the suspensions, $\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})$ (3.3), we obtain a prediction for the effective Newtonian spreading of a suspension drop as a function of the particle volume fraction:

(3.4a,b)$$\begin{eqnarray}\unicode[STIX]{x1D6FD}=F\left(\frac{\unicode[STIX]{x1D702}_{0}}{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})}\mathit{Re}_{0}\right)\quad \text{and}\quad \unicode[STIX]{x1D6FD}_{c}=F_{c}\left(\frac{\unicode[STIX]{x1D702}_{0}}{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})}\mathit{Re}_{0}\right),\end{eqnarray}$$

with $\mathit{Re}_{0}=\unicode[STIX]{x1D70C}VD_{i}/\unicode[STIX]{x1D702}_{0}$ and $\unicode[STIX]{x1D70C}$ stands for the density of the suspension. From the monotonic increase of $\unicode[STIX]{x1D6FD}(\mathit{Re})$ and $\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})$ and their low-Reynolds-number limits ($\unicode[STIX]{x1D6FD}\sim \mathit{Re}^{4/3}$, $\unicode[STIX]{x1D6FD}_{c}\sim \mathit{Re}^{1/3}$ and $\unicode[STIX]{x1D702}\propto (\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{-2}$), equation (3.4) predicts that the deformation decreases with increasing $\unicode[STIX]{x1D719}$ and vanishes rapidly as the particle volume fraction approaches $\unicode[STIX]{x1D719}_{c}$, according to $\unicode[STIX]{x1D6FD}\propto (\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{8/3}$ and $\unicode[STIX]{x1D6FD}_{c}\propto (\unicode[STIX]{x1D719}_{c}-\unicode[STIX]{x1D719})^{2/3}$. Consistently, equation (3.4) implies that the drop is not deformed at all above $\unicode[STIX]{x1D719}_{c}$ since the steady effective viscosity is infinite above $\unicode[STIX]{x1D719}_{c}$.

Figure 5.

Spreading versus $\unicode[STIX]{x1D719}$ around the critical volume fraction $\unicode[STIX]{x1D719}_{c}$ for (a) different viscosities of the suspending liquid (at $V=4~\text{m}~\text{s}^{-1}$) and (b) varied impact velocities (for $\unicode[STIX]{x1D702}_{0}=12.0~\text{mPa}~\text{s}$). The particle size is $d=145~\unicode[STIX]{x03BC}\text{m}$. The coloured solid lines show the effective viscous law (2.1). The vertical dashed line indicates $\unicode[STIX]{x1D719}_{c}$.

We are now able to assess the relevance of the effective Newtonian approach to the suspension case. Figure 3(b) compares the same data as in figure 2 to equation (3.4). At low $\unicode[STIX]{x1D719}$ (large deformations and effective Reynolds numbers) the spreading of the suspensions is found to follow closely the effective Newtonian law. By contrast, a significant departure is observed on the low-deformation side when the volume fraction approaches $\unicode[STIX]{x1D719}_{c}$. Instead of continuing to decrease with decreasing effective Reynolds number, the spreading of the suspension seems to plateau or decrease much more slowly. (Although the spreading of purely Newtonian drops is still experimentally measurable and in agreement with the Newtonian law over this range of deformations; see figure 3a.)