3.1.1. Concentration profiles

First, we estimate the radial distribution of particles inside the sediment. The concentration measurement is based on only 20 images. We have not been able to take more images since it takes approximately one day for the concentration to become steady in the sediment. Thus the statistical error is expected to be as large as $5\,\%$ and only an estimate of $\phi (r)$ can be obtained in the sediment. Figure 3(a) shows this estimation for three heights: $z=h_0/4$, $h_0/2$, $3h_0/4$, where $\phi$ has been averaged over a height of ${\pm }0.15h_0$. Apart from near the rotor and stator where indistinct particle layering is observed, the particle volume fraction is constant along the gap. This expected result indicates in particular that the image processing used to detect particles is reliable. The measurement of the radial concentration profiles is rather important since, in cylindrical Couette flow, outward radial migration is expected in concentrated non-Brownian suspensions (Abbott et al. Reference Abbott, Tetlow, Graham, Altobelli, Fukushima, Mondy and Stephens1991; Graham et al. Reference Graham, Altobelli, Fukushima, Mondy and Stephens1991; Phillips et al. Reference Phillips, Armstrong, Brown, Graham and Abbott1992; Chow et al. Reference Chow, Iwayima, Sinton and Leighton1995; Sarabian et al. Reference Sarabian, Firouznia, Metzger and Hormozi2019). Figures 3(b) and 3(c) show concentration profiles measured at 3 heights: $z=h_s/4$, $h_s/2$, $3h_s/4$, and $z$-averaged over one tenth of the resuspended layer and over 10 000 frames for two selected angular velocities: $\varOmega =0.3$ (b) and $20\ \textrm {r.p.m.}$ (c). For both angular velocities, a particle layering is observed and is all the more pronounced when the angular velocity is low (or the particle volume fraction is high). Particle layering in concentrated non-Brownian suspensions is well documented and has been specifically addressed by numerical simulations (Yeo & Maxey Reference Yeo and Maxey2010; Gallier et al. Reference Gallier, Lemaire, Lobry and Peters2016) and observed in many experiments (Blanc et al. Reference Blanc, Lemaire, Meunier and Peters2013; Metzger, Rahli & Yin Reference Metzger, Rahli and Yin2013; Deboeuf et al. Reference Deboeuf, Lenoir, Hautemayou, Bornert, Blanc and Ovarlez2018; Sarabian et al. Reference Sarabian, Firouznia, Metzger and Hormozi2019). Note that, in the structured zones, the absolute value of $\phi$ is not relevant since the spatial resolution $\delta x= (R_2-R_1)/200\approx 2a/10$ is much smaller than the characteristic distance between particle centres (${O} (2a))$. Thus, since the radial position of the particles is fixed in the layered zone, the determination of the particle number density depends on the sample size. A high spatial resolution has been chosen in order to evidence the layering but the counterpart is that the values of $\phi$ in the structured zones are not proper, only their variation is significant. Outside the layered zones, where particles are randomly distributed, we have checked that the values of $\phi$ do not vary with sampling size (this is also observable in figure 13).

Figure 3.

Radial concentration profiles in the sediment (a) and in the resuspended layer for two values of the rotation speed (b,c) and three heights. The radial profile in the sediment is computed from 20 images with a radial sampling $\delta x=(R_2-R_1)/30$ and a $z$-averaging of ${\pm }0.15h_0$. The radial profiles of (b,c) are computed from 10 000 images with a radial sampling $\delta x=(R_2-R_1)/200$ and a $z$-averaging of ${\pm } 0.05h_s$. Particle layering near the rotor and the stator is observed. Outside these zones (see insets) the measured profiles (solid lines) are almost flat in contrast to the predictions of the suspension balance model (Morris & Boulay Reference Morris and Boulay1999) computed for that is the $r$-averaged volume fraction at a given $z$ (dashed lines).

Outside the structured zones near the rotor and the stator, the concentration profiles are observed to be $z$-dependent but, at a given height, they are almost flat (see insets of figure 3b,c). This trend is verified for all angular velocities, including those that are not shown in the present paper (see supplementary material available at https://doi.org/10.1017/jfm.2020.1074 for more information).

This finding contrasts with the predictions of the suspension balance model (SBM) (Morris & Boulay Reference Morris and Boulay1999) that are also shown in figures 3(b) and 3(c) (dashed lines). In the framework of a one-dimensional flow (this assumption will be discussed in the next section and in more detail in the supplementary material), according to the SBM, concentration profiles obey the following equation:

(3.1)\begin{equation} q(\phi)=\frac{\eta_N}{\eta_S}=Ar^{{(1+\lambda_2)}/{\lambda_2}}, \end{equation}

where $\eta _N=\varSigma _{11}^p/\eta _0\dot {\gamma }$ is the normal viscosity, $\eta _S$ the relative viscosity and $\lambda _2$ a constant close to 1 (here, we took $\lambda _2=1/1.15)$. The value of $A$ is a constant determined by requiring particle volume conservation. The theoretical profiles of figures 3(b) and 3(c) are obtained with the expressions of $\eta _N$ and $\eta _S$ proposed by Zarraga et al. (Reference Zarraga, Hill and Leighton2000)

(3.2)\begin{equation} \left.\begin{gathered} \eta_S=\frac{\exp(-2.34\phi)}{\left( 1-\dfrac{\phi}{\phi_m}\right)^3},\\ \eta_N=2.17\phi^3\eta_S\exp(2.34\phi). \end{gathered}\right\} \end{equation}

Such discrepancies between SBM predictions and experiments conducted in cylindrical Couette flow have already been reported by several authors (see for instance Ovarlez, Bertrand & Rodts (Reference Ovarlez, Bertrand and Rodts2006); Colbourne et al. (Reference Colbourne, Blythe, Barua, Lovett, Mitchell, Sederman and Gladden2018) and Gholami et al. (Reference Gholami, Rashedi, Lenoir, Hautemayou, Ovarlez and Hormozi2018) and to a lesser extent Sarabian et al. (Reference Sarabian, Firouznia, Metzger and Hormozi2019)). The layering of the particles near the walls probably plays a role, but is not likely to account for the entire discrepancy. As suggested in Gholami et al. (Reference Gholami, Rashedi, Lenoir, Hautemayou, Ovarlez and Hormozi2018), top and bottom boundary effects may also invalidate the use of the SBM in its one-dimensional formulation. Another hypothesis would be that radial size segregation is involved since the particles are not perfectly monodisperse (with a standard deviation of approximately $8\,\%$, see supplementary material for more detail). The larger particles are expected to migrate faster than the smaller ones and then to be focused on the outside. This size segregation effect has been noted by Abbott et al. (Reference Abbott, Tetlow, Graham, Altobelli, Fukushima, Mondy and Stephens1991) in a bimodal suspension containing $3175$ and $780$ $\mathrm {\mu }\textrm {m}$ particles with respective volume fractions of $0.39$ and $0.21$ sheared in a wide-gap Couette rheometer ($R_1=6.4\ \textrm {mm}$ and $R_2=23.8\ \textrm {mm}$). After a few thousand revolutions of the inner cylinder, the larger particles are observed to form hexagonally close-packed sheets near the stator. In our study, even though the particle size distribution (see supplementary materials) has nothing in common with the bimodal distribution used by Abbott et al. (Reference Abbott, Tetlow, Graham, Altobelli, Fukushima, Mondy and Stephens1991), a size gradient across the gap might affect the concentration profiles since, as depicted in § 2.2.3, the particle volume fraction is deduced from the particle number density. More precisely, if the larger particles are preferentially located near the stator, while the region near the rotor is mainly occupied by the smaller ones, the particle volume fraction can increase from the rotor to the stator, even though the particle number density is measured constant across the gap, which would explain why the concentration profiles do not obey SBM predictions. In the supplementary material, we show that there is no evidence in support of the hypothesis. Another explanation would be that migration is modified if there exists an interplay between the resuspension flux associated with density mismatch and the radial migration. Blaj (Reference Blaj2012) measured the radial concentration profiles in suspensions of either density-matched or buoyant non-Brownian suspensions sheared in a Couette rheometer and noticed significant differences between the two cases.

In order to discriminate between these two hypotheses (radial size segregation or buoyancy effects), we measured the radial concentration profile in a neutrally buoyant suspension made of the same particles dispersed in a mixture of water, Triton X100 and zinc chloride (density $1.19\ \textrm {g}\ \textrm {cm}^{-3}$, viscosity $4.43\ \textrm {Pa}\ \textrm {s}$). The results, described in appendix B, show that, contrary to the case of a buoyant suspension, particle migration clearly takes place. Furthermore, the agreement with the SBM predictions is fairly satisfactory (see figure 13). Note that, to allow a more quantitative comparison between our experimental results and the SBM predictions, we also present the variation of the viscosity of the neutrally buoyant suspension with particle volume fraction and shear stress in appendix C. Thus, since SBM appears to very accurately predict the concentration in the neutrally buoyant suspension, the very weak migration observed in figure 3 for non-neutrally buoyant suspensions is likely to come from the density mismatch between the particles and the fluid that probably leads to an interplay between the vertical and the radial particle fluxes.

3.1.2. Velocity profiles

Figure 4 shows the radial velocity profiles that correspond to the concentration profiles of figure 3 ($v(r)$-profiles $z$-averaged over one tenth of the resuspended layer height for $\varOmega =0.3$ and $20\ \textrm {r.p.m.}$). The velocity profiles clearly demonstrate a non-Newtonian behaviour of the suspension as well as wall slip, especially in the case of the lower angular velocity, i.e. for the larger particle volume fractions. The wall slip phenomenon in concentrated non-Brownian suspensions is well known (Jana, Kapoor & Acrivos Reference Jana, Kapoor and Acrivos1995; Ahuja & Singh Reference Ahuja and Singh2009; Blanc, Peters & Lemaire Reference Blanc, Peters and Lemaire2011; Korhonen et al. Reference Korhonen, Mohtaschemi, Puisto, Illa and Alava2015) and is probably related to particle layering near the walls.

Figure 4.

Azimuthal velocity profiles measured for angular velocities: $\varOmega =0.3\ \textrm {r.p.m.}$ and $20\ \textrm {r.p.m.}$ at three different heights: $h_s/4$, $h_s/2$ and $3h_s/4$. Also shown are the theoretical profiles corresponding to the mean particle volume fraction measured at the corresponding heights. Solid lines: Newtonian profiles without wall slip, dashed lines: Newtonian profiles with wall slip evaluated from Jana et al. (Reference Jana, Kapoor and Acrivos1995) (3.3), dotted lines: profiles calculated according to (3.1) and (3.2) without wall slip, dash-dotted lines: profiles calculated according to (3.1) and (3.2) with wall slip (3.3).

In figure 4 we also plot some theoretical velocity profiles. The solid lines correspond to Newtonian profiles, the dashed line to Newtonian profiles with wall slip, the dotted lines to profiles computed from the predictions of the SBM (3.1) and the constitutive law of Zarraga (3.2) with $\phi _m=0.58$ and the dash-dotted lines are the profiles obtained from (3.1), (3.2) and slip boundary conditions. We used the slip boundary conditions proposed by Jana et al. (Reference Jana, Kapoor and Acrivos1995) where the apparent slip velocities at the inner and outer cylinders are given by

(3.3)\begin{equation} u_s(R_{1,2})= \beta_{1,2}a \dot{\gamma}_{1,2} \quad \mathrm{with} \ \beta_{1,2}=\beta(\phi_{1,2}) \approx \frac{\eta_S(\phi_{1,2})}{8}, \end{equation}

where $\phi _{1,2}$ and $\dot {\gamma }_{1,2}$ are respectively the particle volume fraction and the shear rate at the rotor and at the stator.

For $\varOmega =20\ \textrm {r.p.m.}$, the combination of wall slip and SBM allows for a rough description of the experimental profiles. But we have to keep in mind that the particle volume fraction profiles predicted by the SBM appreciably differ from the experimental profiles. Recent work of Colbourne et al. (Reference Colbourne, Blythe, Barua, Lovett, Mitchell, Sederman and Gladden2018) shows the same trend with velocity profiles compatible with the SBM predictions while the particle volume fraction profiles are much flatter than expected from the SBM. For $\varOmega =0.3\ \textrm {r.p.m.}$, none of the tested models (Newtonian, Newtonian $+$ wall slip, SBM or SBM $+$ wall slip) fits the experimental profiles and the shear seems to localize near the rotor, even for particle fractions lower than the expected jamming fraction (approximately 0.58). Such localization has already been observed by many authors (Huang et al. Reference Huang, Ovarlez, Bertrand, Rodts, Coussot and Bonn2005; Ovarlez et al. Reference Ovarlez, Bertrand and Rodts2006; Blaj Reference Blaj2012) and is generally attributed to the existence of a critical shear rate below which no steady flow exists (Ovarlez et al. Reference Ovarlez, Bertrand and Rodts2006). Finally, it should be noted that the shear-thinning behaviour that is observed in most concentrated non-Brownian suspensions (Dai et al. Reference Dai, Bertevas, Qi and Tanner2013; Vázquez-Quesada, Tanner & Ellero Reference Vázquez-Quesada, Tanner and Ellero2016; Vázquez-Quesada et al. Reference Vázquez-Quesada, Mahmud, Dai, Ellero and Tanner2017; Tanner et al. Reference Tanner, Ness, Mahmud, Dai and Moon2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) cannot alone explain the shape of the profiles. In our experiment, the shear stress varies by a 1.6 factor only, leading to a variation of the viscosity that is far too small to explain the highly non-Newtonian velocity profiles, even for large particle volume fractions (see supplementary materials for a comparison with experimental results).

We also measured the velocity profile for the neutrally buoyant suspension at $\phi =0.52$ (see figure 14 in appendix B) and we obtained experimental velocity profiles that are in good agreement with the velocity profiles computed from the predictions of the SBM (3.1) and the constitutive law of Zarraga (3.2) with slip boundary conditions (3.3). As is the case for concentration profiles, the density mismatch between particles and suspending liquid appears to significantly affect the velocity profiles.

3.1.3. Summary on radial profiles

1. (i) Particle layering is observed and the size of the layered zones increases with the volume fraction and is of the order of $8a\approx (R_2-R_1)/5$ for $\phi \approx 0.55$.

2. (ii) Outside these zones, the particle number density hardly varies along $r$ at given $z$ (in contrast with what is observed for a neutrally buoyant suspension, as shown in appendix B).

3. (iii) The examination of the velocity profiles shows highly non-Newtonian flow characteristics which are not consistent with the measured concentration profiles in the framework of a local Newtonian (or quasi-Newtonian) rheology. Again, the discrepancy between the velocity profiles that are measured or computed from the SBM vanishes for a neutrally buoyant suspension.

3.3.1. Local shear rate

As shown in § 3.1.2, the velocity profiles are far from what is expected. Thus, to estimate $\alpha _3=\varSigma _{33}^p/\eta _0\dot {\gamma }$, it is necessary to measure the local shear rate (at least in a wide gap) since it may differ significantly from the macroscopic expected shear rate, called hereafter the nominal shear rate

(3.4)\begin{equation} \dot{\gamma}_N(r)=2\varOmega \dfrac{R_1^2 R_2^2}{R_2^2-R_1^2} \dfrac{1}{r^2}. \end{equation}

Under the assumption that the main component of the shear rate is $\dot {\gamma }_{r\theta }$ and that all the other components are much smaller, the true local shear rate can be deduced from the PIV measurements

(3.5)\begin{equation} \dot{\gamma}(r,z)\approx \dot{\gamma}_{r \theta}=r \dfrac{\partial (v_\theta(r,z)/r)}{\partial r} . \end{equation}

However, (3.5) presumes that the flow is essentially one-dimensional while two-dimensional characteristics are clearly observed in figure 2, especially for the lowest angular velocity. We have shown that $v_z$ was much smaller than $v_\theta$ which makes it valid to neglect the terms that depend on the spatial variation of $v_z$, but the variation of $v_\theta$ with $z$ may not be completely negligible and the component $\partial v_\theta /\partial z$ has to be evaluated before it can be neglected. In the supplementary material, we show that, in most cases, $\partial v_\theta /\partial z\ll r \partial (v_\theta /r)/\partial r$ and that for the lowest angular velocity values and highest particle concentrations, using (3.5) rather than the invariant shear rate introduces an error of, at most, $10\,\%$.

Figure 6 shows the ratio of the measured shear rate (and deduced from (3.5)) to the nominal shear rate deduced from (3.4). It is observed that, for the lowest angular velocities (i.e. the largest $\phi$), the shear rate is much lower than expected from (3.4) (except near the rotor). To quantify the difference between $\dot {\gamma }$ and $\dot {\gamma }_N$, we plot the ratio of the measured shear rate averaged over the central third of the gap to the nominal shear rate, $\dot {\gamma }_N$, calculated at the middle of the gap as a function of $\phi$ for all the values of $\varOmega$ (figure 7). A few comments on this figure are needed. First, it is observed that all the data collapse onto a unique curve regardless of the angular velocity of the rotor. Second, for low particle volume fractions, $\dot {\gamma }$ tends to $\dot {\gamma }_N$ and $\dot {\gamma }\approx \dot {\gamma }_N$ for $\phi \approx 0.2$. In contrast, for higher concentrations, the local shear rate can substantially deviate from $\dot {\gamma }_N$; for the smallest values of $\varOmega$ (the largest values of $\phi$), the true shear rate can be as small as one fifth of the apparent macroscopic shear rate, making it necessary to measure the velocity field in the gap.

Figure 6.

Maps of the measured shear rate $\dot {\gamma }(r,z)$ divided by the expected shear rate $\dot \gamma (r)$ for a Newtonian fluid (3.4) for several angular velocities. It is observed that, except near the rotor, the measured shear rate is lower than expected for a Newtonian fluid. The dashed rectangles indicate the areas that are used to measure $\varSigma _{33}^p$ (figure 9) and to determine the variation of $\varSigma _{33}^p/\eta _0\dot {\gamma }$ with $\phi$ (figure 10).

Figure 7.

Ratio of the local shear rate to the nominal shear rate vs. the local volume fraction. Each colour corresponds to a given value of $\varOmega$. Each point was obtained by averaging $\dot {\gamma }(r,z)$ and $\phi (r,z)$ over the central third of the gap for a given height $z\in [h_s/4,h_s]$. Here, $\dot {\gamma }_N$ is the nominal shear rate calculated at the middle of the gap $r=(R_1 + R_2)/2$.

3.3.2. Third particle normal stress, $\varSigma _{33}^p$

To determine $\varSigma _{33}^p$, (1.2) is integrated from the interface between the resuspended layer and the clear fluid to the height $z(\phi )$

(3.6)\begin{equation} \varSigma_{zz}^p(r,z)=-\int_{z(\phi)}^{z(\phi=0)}\Delta \rho g \phi(r,\zeta) \,\textrm{d}\zeta \end{equation}

with the boundary condition

(3.7)\begin{equation} \varSigma_{zz}^p\left(z(\phi=0)\right)=\varSigma_{zz}^p(h_s)=0. \end{equation}

Then, making the assumption that the flow is essentially one-dimensional, $\varSigma _{zz}^p$ can be equated to $\varSigma _{33}^p$. Rigorously speaking, since the $(z,\theta )$ component of the shear rate is not exactly zero (see § 3.3.1), $\varSigma _{22}^p$ also contributes to $\varSigma _{zz}^p$. This point is discussed in the supplementary material, where we show that the subsequent error is completely negligible compared with the scatter of the data.

Figure 8 displays the maps of $\varSigma _{33}^p$ for four values of the angular velocity. It can be seen that, near the walls, in the regions where particle layering is observed, $\varSigma _{33}^p$ is lower than in the bulk at a given $z$. Outside these zones, $\varSigma _{33}^p$ hardly varies with $r$.

Figure 8.

Maps of $\varSigma _{33}^p$ for several angular velocities. The dashed rectangles indicate the areas that are used to measure $\varSigma _{33}^p$ (figure 9) and to determine the variation of $\varSigma _{33}^p/\eta _0\dot {\gamma }$ with $\phi$ (figure 10). It is observed that, in these zones, $\varSigma _{33}^p$ hardly varies along $r$.

For each point $(r,z)\in [R_1+(R_2-R_1)/3 < r < R_2-(R_2-R_1)/3; z > h_s/4]$, the third particle normal stress, the local shear rate and the particle volume fraction are computed. Figure 9 shows the variation of $-\varSigma _{33}^p$ as a function of $\dot {\gamma }$ for several particle volume fractions. The particle volume fraction range is restricted to $[0.3\text {--}0.5]$ because, in our experiment, for higher concentrations, the shear rate does not vary enough to obtain a reliable dependence between $\varSigma _{33}^p$ and $\dot {\gamma }$ and, for lower concentrations, there are too few data (all contained in the very upper region of the resuspended layer) to extract the variation of $\varSigma _{33}^p$ with $\dot {\gamma }$. From figure 9, it appears that, for a given value of $\phi$, $\varSigma _{33}^p$ varies with $\dot \gamma$ following a power law with an exponent close to 0.7. This result is in perfect agreement with the recent measurements of Saint-Michel et al. (Reference Saint-Michel, Manneville, Meeker, Ovarlez and Bodiguel2019). Shear thinning (i.e. a decrease of the viscosity with shear rate or shear stress) is an attribute of most non-Brownian concentrated suspensions (Acrivos et al. Reference Acrivos, Fan and Mauri1994; Vázquez-Quesada et al. Reference Vázquez-Quesada, Tanner and Ellero2016, Reference Vázquez-Quesada, Mahmud, Dai, Ellero and Tanner2017; Tanner et al. Reference Tanner, Ness, Mahmud, Dai and Moon2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019). Recently, it has been proposed that shear thinning comes from a non-Coulomb friction between particles (Chatté et al. Reference Chatté, Comtet, Niguès, Bocquet, Siria, Ducouret, Lequeux, Lenoir, Ovarlez and Colin2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019). Thus the nonlinear scaling of $\varSigma _{33}^p$ with $\dot {\gamma }$ is not surprising since particle normal stresses are expected to be scaled by the shear stress (Boyer et al. Reference Boyer, Guazzelli and Pouliquen2011; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) leading to a nonlinear scaling law between $\varSigma _{33}^p$ and $\dot {\gamma }$. More surprising is that there is no clear variation of the exponent value with $\phi$ whereas shear thinning is known to be all the more pronounced where the particle volume fraction is large. This last observation would suggest that $\varSigma _{33}^p$ might not vary linearly with shear stress. This could seem odd but can be justified by the fact that there is a fraction of $\varSigma _{33}^p$ that is rate independent. Indeed, in a fully settled state, $\varSigma _{33}^p$ is the result of the weight of particles borne by solid contact at the bottom wall. Thus, under shear, there will likely always be a fraction of $\varSigma _{33}^p$ transmitted by this mechanism which is rate independent and may affect the apparent dependence on shear rate. To answer this question, it is probably necessary to perform combined measurements of both $\varSigma _{33}^p$ and $\varSigma _{12}$, which is not possible in the present work since the vertical gradient of particle volume fraction prevents us from deducing $\varSigma _{12}$ from the value of the torque applied to the rotor.

Figure 9.

Variation of the third particle normal stress with shear rate for several particle volume fractions. Each set of data is fitted with a power law $\varSigma _{33}^p\propto \dot {\gamma }^n$.

3.3.3. Determination of $\alpha _3=\varSigma _{33}^p/\eta _0\dot {\gamma }$

The variation of $-\varSigma _{33}^p$ normalized by $\eta _0 \dot {\gamma }$ as a function of $\phi$ is shown in figure 10 where we observe a reasonable collapse of the data onto a single curve for a wide range of $\phi$ between $0.15$ and $0.55$ with variation in $\varSigma _{33}^p/\eta _0 \dot {\gamma }$ over more than five decades.

Figure 10.

(a) Third particle normal stress normalized by the product of the fluid viscosity and the local shear rate versus the local particle volume fraction. The different colours correspond to different angular velocities of the rotor (see figure 7). The agreement with the correlation proposed by Zarraga et al. (Reference Zarraga, Hill and Leighton2000) where $\phi _0=0.574$ is very good (red line). In contrast, it is not possible to represent the experimental results by the correlation obtained by Boyer et al. (Reference Boyer, Guazzelli and Pouliquen2011) for $\varSigma _{22}^p$ together with $\lambda _3=1/2$ (blue line). (b) Zoom of (a) for $\phi \in [0.3-0.5]$. (c) Variation of $\varSigma _{33}^p$ divided by $\eta _0\dot {\gamma }^{0.7}$ versus $\phi$. The collapse of the data is much better when the nonlinear power law is used.

In figure 10(a), we restricted the data to particle volume fractions greater than $0.15$ because, as shown in figure 5, below this value, the concentration profile is very sharp, which makes it difficult to measure the concentration. The data are somewhat scattered, especially for the largest values of $\phi$. This finding may have different origins. First, it can stem from experimental issues because, as observed in figure 5, for the lowest angular velocity values (i.e. the larger particle volume fractions), the vertical concentration profile is nearly flat, which means that $\varSigma _{33}^p$ given by the integral of $\phi (z)$ varies greatly with $\phi$. Thus, even a small error in $\phi$ is likely to cause a large error in the computation of $\varSigma _{33}^p$. For lower concentrations, a better collapse can be obtained if, rather than $\varSigma _{33}^p/\dot {\gamma }$, $\varSigma _{33}^p/\dot {\gamma }^{0.7}$ is plotted against $\phi$ in the range $\phi \in [0.3-0.5]$ within which the power-law scaling has been obtained for $\varSigma _{33}^p$ (see figure 10b,c). This is in agreement with the recent results of Saint-Michel et al. (Reference Saint-Michel, Manneville, Meeker, Ovarlez and Bodiguel2019).

The red and black lines in figure 10(a) represent the correlation proposed by Zarraga et al. (Reference Zarraga, Hill and Leighton2000)

(3.8)\begin{equation} \varSigma_{33}^p=-\eta_0 \dot{\gamma} \dfrac{\phi^3}{\left(1-\dfrac{\phi}{\phi_0}\right)^3}. \end{equation}

The black curve is obtained with the original value of $\phi _0$ proposed by Zarraga et al. (Reference Zarraga, Hill and Leighton2000) ($\phi _0=0.62$), while the red curve has been obtained for $\phi _0=0.574$: the value of the particle volume fraction inside the settled layer that we measured. We observe a very good agreement between the experimental data and the correlation from Zarraga et al. (Reference Zarraga, Hill and Leighton2000). Furthermore, Zarraga et al. (Reference Zarraga, Hill and Leighton2000) established the correlation for a particle volume fraction ranging from $0.3$ to $0.5$, while our results show that this correlation can be expanded to a wider range of concentrations. The blue curve is obtained by using the correlation obtained by Boyer et al. (Reference Boyer, Guazzelli and Pouliquen2011) for $\varSigma _{22}^p$ (with $\phi _0=0.574$)

(3.9)\begin{equation} \varSigma_{22}^p=-\eta_0 \dot{\gamma}\dfrac{\phi^2}{\left(\phi_0-\phi\right)^2}, \end{equation}

and assuming that $\lambda _2=\varSigma _{22}^p/\varSigma _{11}^p\approx 1$ and $\lambda _3=\varSigma _{33}^p/\varSigma _{11}^p=0.5$, as suggested by Morris & Boulay (Reference Morris and Boulay1999).

The agreement between the blue curve and our data is not satisfactory. In our opinion, this discrepancy does not call into question the results obtained by Boyer et al. (Reference Boyer, Guazzelli and Pouliquen2011) but rather the lack of variability in $\lambda _{3}$ with $\phi$. This last result has already been noted by Gallier et al. (Reference Gallier, Lemaire, Peters and Lobry2014) and was previously suggested by Morris & Boulay (Reference Morris and Boulay1999) themselves.