2.1 Fibres and fluids

Four batches of rod-like particles were used in the experiments. They were obtained by using a specially designed device to cut long cylindrical filaments of plastic (Plastinyl 6.6) that were supplied by Plasticfibre S.P.A. (http://www.plasticfibre.com). Images of typical fibres from each batch are shown in figure 1(b). The length and diameter of over 100 fibres were measured with a digital imaging system. The distributions of lengths and diameters were found to be approximately Gaussian for all aspect ratios. The mean value and standard deviation of the fibre aspect ratio $A=L/d$ , length $L$ and diameter $d$ are shown in table 1. Note that batches (II) and (III) have very different lengths and diameters, but roughly the same aspect ratio of $A\approx 6{-}7$ .

The rigid fibres were suspended in a Newtonian fluid that had a matching density of $\unicode[STIX]{x1D70C}_{f}=1056~\text{kg}~\text{m}^{-3}$ . The suspending fluid was a mixture of water (10.72 wt%), Triton X-100 (75.78 wt%) and zinc chloride (13.50 wt%). The fluid viscosity of $\unicode[STIX]{x1D702}_{f}=3~\text{Pa}~\text{s}$ and the density were measured at the same temperature ( $25\,^{\circ }\text{C}$ ) at which the experiments were performed. The suspensions were prepared by adding the fibres to the fluid, where both quantities were weighed, and gently stirring. Little to no settling or creaming was observed.

The rheological measurements were performed at a maximum shear rate of $\dot{\unicode[STIX]{x1D6FE}}\approx 3~\text{s}^{-1}$ , ensuring that a maximum Reynolds number ( $\unicode[STIX]{x1D70C}_{f}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}_{f}$ ) of 0.04 was achieved. The fibres can be considered non-colloidal, owing to their large size, and rigid under the conditions of the experiment. Regarding the latter, the buckling criterion has been characterised by a dimensionless number, $S_{p}=128\,\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}\,A^{4}/E_{Y}\ln (2A)$ , where the Young’s modulus, $E_{Y}$ , is approximately 3000 MPa for PLASTINYL 6.6. The number $S_{p}$ , often called the Sperm number, is a ratio of the viscous and elastic forces acting on the fibre (see e.g. Becker & Shelley Reference Becker and Shelley2001). The values of $S_{p}$ , shown in table 1 for our experiments, were much smaller than the critical Sperm number of 328 for the coil–stretch transition in a cellular flow (Young & Shelley Reference Young and Shelley2007).

Figure 1. (a) Sketch of the experimental apparatus. (b) Images of the plastic fibres. (c) Image of the top plate (the inset is a blowup of the image, showing the nylon mesh).

Table 1. Properties of each batch of fibres. Data shown include the mean value and standard deviation of the aspect ratio $A$ , fibre length $L$ , and fibre diameter $d$ . Values of the dimensionless number $S_{p}$ , characterising the relative strengths of the viscous and elastic forces, are also reported.

2.2 Experimental techniques

The experiments were conducted using a custom rheometer that was originally constructed by Boyer et al. (Reference Boyer, Guazzelli and Pouliquen2011) and then modified by Dagois-Bohy et al. (Reference Dagois-Bohy, Hormozi, Guazzelli and Pouliquen2015). This rheometer, sketched in figure 1(a), provides measurements of both shear and normal stresses. The shearing cell consists of (i) an annular cylinder (of radii $R_{1}=43.95~\text{mm}$ and $R_{2}=90.28~\text{mm}$ ) that is attached to a bottom plate that can be rotated and (ii) a top cover plate that can be moved vertically. This top plate is porous, enabling fluid to flow through it but not particles. The plate was manufactured with holes of sizes 2–5 mm and then was covered by a 0.2 mm nylon mesh (see figure 1 c). The parallel bottom and top plates have also been roughened by positioning regularly spaced strips of height and width 0.5 mm onto their surfaces. A transparent solvent trap covers the cell, hindering evaporation of the suspending fluid.

In a typical experiment, the annular cell was filled with suspension and the porous plate was lowered into the fluid to a position $h$ . We have checked that the initial filling procedure does not influence the measurement obtained in the steady shear regime. The height $h$ , measured independently by a position sensor (Novotechnik T-50), ranges between 10.8 and 18 mm, corresponding to 13–25 fibre diameters depending on the fibre batch. The height measurement enables calculation of the fibre volume fraction, $\unicode[STIX]{x1D719}$ . The bottom annulus was rotated at a rate $\unicode[STIX]{x1D6FA}$ by an asynchronous motor (Parvalux SD18) regulated by a frequency controller (Omron MX2 0.4 kW), while the torque exerted on the top plate was measured by a torque transducer (TEI-CFF401). The shear stress $\unicode[STIX]{x1D70F}$ was deduced from these torque measurements after calibration with a pure fluid to subtract undesired contributions resulting from the friction at the central axis and the shear in the thin gap between the top plate and the cell walls; the calibration method is described by Dagois-Bohy et al. (Reference Dagois-Bohy, Hormozi, Guazzelli and Pouliquen2015). A precision scale (Mettler-Toledo XS6002S) was placed on a vertical translation stage driven by a LabVIEW code in order to measure the apparent weight of the top plate. These measurements, after correcting for buoyancy, provided the determination of the normal force that the particles exert on the porous plate in the gradient direction. Dividing by the area of the plate gives the gradient component of the normal stress, which is referred to simply as the particle pressure, $P$ . A normal viscosity in the gradient direction can be defined as $P/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ , as was done by Morris & Boulay (Reference Morris and Boulay1999).

The rheometer can be run in a pressure-imposed mode or in a volume-imposed mode, and measurements were recorded as a function of the mean shear rate, $\dot{\unicode[STIX]{x1D6FE}}=\unicode[STIX]{x1D6FA}(R_{2}+R_{1})/2h$ , once a steady state was achieved. In pressure-imposed rheometry, the particle pressure $P$ is maintained at a set value that is measured by the precision scale; the volume fraction $\unicode[STIX]{x1D719}$ and the shear stress $\unicode[STIX]{x1D70F}$ are measured as functions of the shear rate $\dot{\unicode[STIX]{x1D6FE}}$ and pressure $P$ . In volume-imposed rheometry, the height $h$ , and consequently the volume fraction $\unicode[STIX]{x1D719}$ , are maintained at a fixed value, while the shear stress, $\unicode[STIX]{x1D70F}$ , and particle pressure, $P$ , are measured as functions of the shear rate, $\dot{\unicode[STIX]{x1D6FE}}$ . Errors in the measurements of $\unicode[STIX]{x1D70F}$ , $P$ and $\unicode[STIX]{x1D719}$ for the suspensions depend upon the calibration experiments, the preparation of the suspension samples and the precision of the height, torque and scale measurements. Estimates, based upon tests with independently created samples of suspension, suggest errors of $\pm 6~\text{Pa}$ , $\pm 5~\text{Pa}$ and $\pm 0.005$ for $\unicode[STIX]{x1D70F}$ , $P$ and $\unicode[STIX]{x1D719}$ , respectively.

Figure 2. Apparent relative (a) shear ( $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ ) and (b) normal ( $P/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ ) viscosities as well as relative (c) shear ( $(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ ) and (d) normal ( $(P-P_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ ) viscosities (after subtraction of the yield stresses) versus volume fraction, $\unicode[STIX]{x1D719}$ , for the fibres of batch (II) in pressure-imposed (▴) and volume-imposed (▵) configurations.

3.1 Rheological observations

Typical rheological data for the apparent relative shear and normal viscosities, $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ and $P/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ , are plotted against volume fraction, $\unicode[STIX]{x1D719}$ , in figure 2(a,b). The data were collected for fibres of batch (II) using pressure-imposed and volume-imposed measurements. As expected, both quantities increase with increasing $\unicode[STIX]{x1D719}$ . However, multiple values of the apparent viscosities are measured for any given $\unicode[STIX]{x1D719}$ . Plotting the shear stress, $\unicode[STIX]{x1D70F}$ , and the particle pressure, $P$ , against the shear rate for different values of $\unicode[STIX]{x1D719}$ demonstrates that $\unicode[STIX]{x1D70F}$ and $P$ are linear in $\dot{\unicode[STIX]{x1D6FE}}$ , but have a non-zero value at $\dot{\unicode[STIX]{x1D6FE}}=0$ , see figure 3(a,b). This seems to suggest that a yield stress exists for both the shear stress and the particle pressure, $\unicode[STIX]{x1D70F}_{0}$ and $P_{0}$ , respectively. Their values can be determined using a linear fit of the stress and pressure data as functions of $\dot{\unicode[STIX]{x1D6FE}}$ , as indicated by the lines in figure 3(a,b). Both yield stresses, $\unicode[STIX]{x1D70F}_{0}$ and $P_{0}$ , increase with increasing $\unicode[STIX]{x1D719}$ , as shown in figure 3(c,d) for all four batches of fibres. The growth in $\unicode[STIX]{x1D70F}_{0}$ and $P_{0}$ with respect to $\unicode[STIX]{x1D719}$ is more pronounced for larger aspect ratios $A$ .

Figure 3. (a) Shear stress ( $\unicode[STIX]{x1D70F}$ ) and (b) particle pressure ( $P$ ) versus shear rate, $\dot{\unicode[STIX]{x1D6FE}}$ , for the fibre suspension of batch (II) at different $\unicode[STIX]{x1D719}$ values of 0.26 (lightest grey shade), 0.30, 0.35, 0.38 and 0.41 (black). The lines represent the linear fit for each different $\unicode[STIX]{x1D719}$ value. Yield stress (c) for the shear stress ( $\unicode[STIX]{x1D70F}_{0}$ ) and (d) particle pressure ( $P_{0}$ ) versus $\unicode[STIX]{x1D719}$ for fibres of batches (I), (II), (III) and (IV) shown using the symbols ▫, ▵, ♢ and ○, respectively (see table 1). The insets of graphs (c,d) are log–log plots versus $\unicode[STIX]{x1D719}/\unicode[STIX]{x1D719}_{m}$ where $\unicode[STIX]{x1D719}_{m}$ is the maximum flowable volume fraction given in figure 5(a).

The data of figure 3(a,b) demonstrate that the stresses scale linearly with the rate of shear, as expected. Furthermore, the slopes of $\unicode[STIX]{x1D70F}$ and $P$ with $\dot{\unicode[STIX]{x1D6FE}}$ increase with $\unicode[STIX]{x1D719}$ , which is evidence of the increase of the shear and normal viscosities with $\unicode[STIX]{x1D719}$ . These shear and normal viscosities can be collapsed into a single function of $\unicode[STIX]{x1D719}$ by removing the yield stresses. Figure 2(c,d) shows the results of $(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ and $(P-P_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ as functions of $\unicode[STIX]{x1D719}$ . In all of the following analysis, the yield stresses are subtracted systematically from the raw data.

Figure 4. Rheological data: (a) $\unicode[STIX]{x1D702}_{s}=(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ and (b) $\unicode[STIX]{x1D702}_{n}=(P-P_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ versus $\unicode[STIX]{x1D719}$ as well as (c) $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D702}_{s}/\unicode[STIX]{x1D702}_{n}$ and (d) $\unicode[STIX]{x1D719}$ versus $J=\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}/(P-P_{0})$ , for fibre batches (I), (II), (III) and (IV) as represented by the symbols ▫, ▵, ♢ and ○, respectively (see table 1). The insets of graphs (c,d) are log–log and semilogarithmic plots.

3.2 Constitutive laws

Figure 4(a,b) shows $\unicode[STIX]{x1D702}_{s}=(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ and $\unicode[STIX]{x1D702}_{n}=(P-P_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ , the relative shear viscosity and relative normal viscosity, for all of the fibre batches. Both quantities increase with $\unicode[STIX]{x1D719}$ and seem to diverge at a maximum volume fraction that depends on the aspect ratio $A$ . The influence of the aspect ratio is also seen on the rheological functions as $\unicode[STIX]{x1D702}_{s}(\unicode[STIX]{x1D719})$ and $\unicode[STIX]{x1D702}_{n}(\unicode[STIX]{x1D719})$ shift towards lower values of $\unicode[STIX]{x1D719}$ with increasing $A$ . An interesting observation is that the data for batches (II) and (III), corresponding to similar values of $A$ but different sizes, collapse onto the same curve. This indicates that finite-size effects are not significant. Also, the decrease of $\unicode[STIX]{x1D702}_{n}$ is much stronger than that of $\unicode[STIX]{x1D702}_{s}$ for $\unicode[STIX]{x1D719}\lesssim 0.35$ .

An alternative representation of the rheological data plots the friction coefficient $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D702}_{s}/\unicode[STIX]{x1D702}_{n}$ and the volume fraction $\unicode[STIX]{x1D719}$ as functions of the dimensionless shear rate, $J=\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}/(P-P_{0})$ (Boyer et al. Reference Boyer, Guazzelli and Pouliquen2011); note that $J=1/\unicode[STIX]{x1D702}_{n}$ and it is a function of $\unicode[STIX]{x1D719}$ as shown in figure 4(b). The rheology is then described by the two functions $\unicode[STIX]{x1D707}(J)$ and $\unicode[STIX]{x1D719}(J)$ as shown in figure 4(c,d) for the same data as in figure 4(a,b). A striking result is that a complete collapse of all the data is observed for $\unicode[STIX]{x1D707}(J)$ , indicating that the friction coefficient is independent of the aspect ratio $A$ . The volume fraction $\unicode[STIX]{x1D719}$ is a decreasing function of the dimensionless number $J$ . There is a clear shift of $\unicode[STIX]{x1D719}(J)$ towards the lower values of $\unicode[STIX]{x1D719}$ when $A$ is increased. The data for batches (II) and (III), having similar aspect ratios, again collapse onto the same curve.

This frictional approach is particularly well suited to study the jamming transition, because it circumvents the divergence of the viscosities. From the semilogarithmic plot of $\unicode[STIX]{x1D719}(J)$ , shown in the inset of figure 4(d), the critical (or maximum flowable) volume fraction $\unicode[STIX]{x1D719}_{m}$ can be determined from the limiting value of $\unicode[STIX]{x1D719}$ as $J$ goes to zero. Similarly, the semilogarithmic plot of $\unicode[STIX]{x1D707}(J)$ in the inset of figure 4(c) shows that the friction coefficient tends to a finite value $\unicode[STIX]{x1D707}_{s}$ at the jamming point.

Figure 5. Critical values (a) $\unicode[STIX]{x1D719}_{m}$ (○) and (b) $\unicode[STIX]{x1D707}_{s}$ (○) at the jamming point versus fibre aspect ratio, $A$ , together with the data ( $\star$ ) obtained by Boyer et al. (Reference Boyer, Guazzelli and Pouliquen2011) for suspensions of spheres ( $A=1$ ). Estimated errors in the values of $\unicode[STIX]{x1D719}_{m}$ are smaller than the symbols. Comparisons with experimental data from Rahli, Tadrist & Blanc (Reference Rahli, Tadrist and Blanc1999) (▫) on the dry packing of rigid fibres and the simulations of Williams & Philipse (Reference Williams and Philipse2003) (▵) for the maximum random packing of spherocylinders are given in panel (a).

Figure 6. Rescaled rheological data: (a) $\unicode[STIX]{x1D702}_{s}=(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ , (b) $\unicode[STIX]{x1D702}_{n}=(P-P_{0})/\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}$ and (c) $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D702}_{s}/\unicode[STIX]{x1D702}_{n}$ versus $\unicode[STIX]{x1D719}/\unicode[STIX]{x1D719}_{m}$ as well as (d) $\unicode[STIX]{x1D719}/\unicode[STIX]{x1D719}_{m}$ versus $J=\unicode[STIX]{x1D702}_{f}\dot{\unicode[STIX]{x1D6FE}}/(P-P_{0})$ , for all the data of the different batches (I), (II), (III) and (IV) shown using the symbols ▫, ▵, ♢ and ○, respectively (see table 1). The insets of graphs (a–d) are log–log plots. The red solid curves correspond to the rheological laws given by equations (3.1)–(3.3).

The critical values $\unicode[STIX]{x1D719}_{m}$ and $\unicode[STIX]{x1D707}_{s}$ are plotted against the fibre aspect ratio $A$ in figures 5(a) and 5(b), respectively. Again, the similar results for batches (II) and (III) indicate that confinement is not influencing the measurements, and the values obtained by Boyer et al. (Reference Boyer, Guazzelli and Pouliquen2011) for suspensions of poly(methyl methacrylate) spheres are also plotted on these graphs (for $A=1$ , although strictly speaking a sphere is not a cylinder of aspect ratio one). Clearly, $\unicode[STIX]{x1D719}_{m}$ decreases with increasing $A$ . This follows the general trends of a decrease in volume fraction with the aspect ratio for processes such as dry packing, as shown in figure 5(a). A comparison is also made in figure 5(a) between the values of $\unicode[STIX]{x1D719}_{m}$ and estimates from simulations (Williams & Philipse Reference Williams and Philipse2003) of the maximum concentration at which the orientation distribution remains random. The critical friction $\unicode[STIX]{x1D707}_{s}$ does not vary significantly with $A$ in the explored range and its value ( ${\approx}0.47$ ) is larger than that obtained for spheres ( ${\approx}0.32$ ) by Boyer et al. (Reference Boyer, Guazzelli and Pouliquen2011).

Figure 6 displays the same data as figure 4, but with $\unicode[STIX]{x1D719}$ scaled by $\unicode[STIX]{x1D719}_{m}$ . This simple rescaling leads to a good collapse of the data for all of the fibre batches, indicating that the aspect ratio principally impacts the maximum volume fraction, $\unicode[STIX]{x1D719}_{m}$ . Another remarkable result is that the relative shear and normal viscosities, $\unicode[STIX]{x1D702}_{s}$ and $\unicode[STIX]{x1D702}_{n}$ , diverge near the jamming transition with a scaling close to $(\unicode[STIX]{x1D719}_{m}-\unicode[STIX]{x1D719})^{-1}$ , as clearly evidenced by the insets of figure 6(a,b). This contrasts starkly with the divergence of $(\unicode[STIX]{x1D719}_{m}-\unicode[STIX]{x1D719})^{-2}$ observed for suspensions of spheres (Boyer et al. Reference Boyer, Guazzelli and Pouliquen2011).

A constitutive law for $\unicode[STIX]{x1D707}$ can be generated by fitting the data to a linear combination of powers of $(\unicode[STIX]{x1D719}_{m}-\unicode[STIX]{x1D719})/\unicode[STIX]{x1D719}$ ,

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D707}(\unicode[STIX]{x1D719})=\unicode[STIX]{x1D707}_{s}+\unicode[STIX]{x1D6FC}\left(\frac{\unicode[STIX]{x1D719}_{m}-\unicode[STIX]{x1D719}}{\unicode[STIX]{x1D719}}\right)+\unicode[STIX]{x1D6FD}\left(\frac{\unicode[STIX]{x1D719}_{m}-\unicode[STIX]{x1D719}}{\unicode[STIX]{x1D719}}\right)^{2},\end{eqnarray}$$

as was done by Dagois-Bohy et al. (Reference Dagois-Bohy, Hormozi, Guazzelli and Pouliquen2015). The red curve in figure 6(c) shows the result, with $\unicode[STIX]{x1D707}_{s}=0.47$ , $\unicode[STIX]{x1D6FC}=2.44$ and $\unicode[STIX]{x1D6FD}=10.20$ . As noted previously, the value for $\unicode[STIX]{x1D707}_{s}$ is larger than that obtained for suspensions of spheres ( $\unicode[STIX]{x1D707}_{s}=0.3$ ). The values for $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ also differ from those obtained for suspensions of spheres ( $\unicode[STIX]{x1D6FC}=4.6$ and $\unicode[STIX]{x1D6FD}=6$ ). The best fit for $\unicode[STIX]{x1D702}_{s}$ was found to be

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{s}(\unicode[STIX]{x1D719})=14.51\left(\frac{\unicode[STIX]{x1D719}_{m}-\unicode[STIX]{x1D719}}{\unicode[STIX]{x1D719}_{m}}\right)^{-0.90},\end{eqnarray}$$

as seen in figure 6(a). Note that the best-fit exponent is $-0.9$ rather than $-1$ . The rheological law for $\unicode[STIX]{x1D702}_{n}$ is then just given by

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{n}(\unicode[STIX]{x1D719})=\unicode[STIX]{x1D702}_{s}(\unicode[STIX]{x1D719})/\unicode[STIX]{x1D707}(\unicode[STIX]{x1D719}),\end{eqnarray}$$

which is represented by the red curve in figure 6(b). The variation of $\unicode[STIX]{x1D719}$ with $J$ can be deduced from this last law since $J=1/\unicode[STIX]{x1D702}_{n}(\unicode[STIX]{x1D719})$ ; this result is shown in figure 6(d).