# Viscous resuspension of non-Brownian particles: determination of the concentration profiles and particle normal stresses

Enzo d'Ambrosio, Frédéric Blanc, Elisabeth Lemaire

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Enzo d'Ambrosio, Frédéric Blanc, Elisabeth Lemaire. Viscous resuspension of non-Brownian particles: determination of the concentration profiles and particle normal stresses. Journal of Fluid Mechanics, 2021, 911, pp.A22. 10.1017/jfm.2020.1074. hal-02401199

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# Viscous resuspension of non-Brownian particles: determination of the concentration profiles and particle normal stresses.

Enzo d'Ambrosio¹, Frédéric Blanc¹ and Elisabeth Lemaire¹†

¹Université Côte d'Azur, CNRS, InPhyNi-UMR 7010, 06108 Nice Cedex 2, France

(Received xx; revised xx; accepted xx)

We perform local measurements of both the velocity and the particle volume fraction to study viscous resuspension in non-Brownian suspensions for Shields numbers ranging from $10^{-3}$ to 1. To this aim, a suspension of PMMA spherical particles dispersed in a lighter Newtonian fluid (Triton X100) is sheared in a vertical Couette cell where both velocity and particle density mappings are performed. We show that the radial profiles of the velocity and of the particle volume fraction are inconsistent in the framework of local rheology of a Newtonian material and that these discrepancies disappear for a neutrally-buoyant suspension. The vertical concentration profiles are used to deduce the third particle normal stress, $\Sigma_{33}^p$, by solving the Cauchy equation. $\Sigma_{33}^p$ is shown not to vary linearly with shear rate but rather through a power-law with an exponent close to 0.7, irrespective of the value of the particle volume fraction, in accordance with the recent results of Saint-Michel et al. (2019). At last, we compare our results with the results of previous studies where $\alpha_3 = \Sigma_{33}^p/\eta_0\dot{\gamma}$ was deduced from the macroscopic measurement of the height of the resuspended layer. The agreement is fairly good.

**Key words:** non-Brownian suspensions, viscous resuspension, particle normal stresses

## 1. Introduction

Understanding the flow properties of concentrated suspensions is a real challenge in the development of many industrial products (e.g., solid propellant rocket motors and fresh concrete) and in the description of various environmental flows (e.g., torrential lava, mud flows, and submarine slides). Among other transport properties, shear-induced particle migration has received increasing attention in recent decades. Particle migration can be due to inertial effects (Segre & Silberberg 1962) but also occurs at low Reynolds numbers, for instance, in a Poiseuille flow in which the particles tend to migrate towards the centre of the channel (Koh et al. 1994; Hampton et al. 1997; Butler & Bonnecaze 1999; Snook et al. 2016), in wide-gap Couette flow towards the outer cylinder (Graham et al. 1991; Abbott et al. 1991; Chow et al. 1994; Sarabian et al. 2019) and outward in cone-and-plate geometry (Chow et al. 1995).

Another typical example of shear-induced migration is viscous resuspension whereby an initially settled layer of negatively buoyant particles expands vertically when a shear flow is applied. Viscous resuspension has been observed for the first time by Gadala-Maria (1980) and later explained by Leighton & Acrivos (1986) and Acrivos et al. (1993), who

<footnote>† Email address for correspondence: elisabeth.lemaire@unice.fr</footnote>
demonstrated that the height of the resuspended particle layer results from the balance between a downward gravitational flux and an upward shear-induced diffusion flux. The authors studied the resuspension of various particles (different sizes and densities) in two different liquids (different viscosities and densities) sheared in a cylindrical Couette device. They measured the height of the resuspended layer of particles, $h_s$, as a function of the shear rate and showed that the difference between $h_s$ and $h_0$ (i.e., the initial sediment height) normalized by $h_0$ was a function of only the Shields number defined as the ratio between viscous and buoyancy forces:

$$ \frac{h_s - h_0}{h_0} = f(A) \quad \text{with} \quad A = \frac{9 \eta_0 \dot{\gamma}}{2 \Delta \rho g h_0} \qquad (1.1) $$

where $\dot{\gamma}$ is the shear rate, $\Delta\rho$, the density mismatch and $\eta_0$, the viscosity of the suspending fluid. Their experimental results were found to be in very good agreement with the diffusive flux model developed by Leighton & Acrivos (1986). Later, Zarraga et al. (2000) revisited the results of Acrivos et al. (1993) to determine the particle normal stress in the vorticity direction, $\Sigma_{33}^p$, from the height of the resuspended layer of particles by writing the Cauchy momentum balance in the vertical direction:

$$ \frac{\partial \Sigma_{33}^p}{\partial z} = \Delta \rho g \phi \qquad (1.2) $$

Then, a relation between $\Sigma_{33}^p$ and the particle volume fraction at the bottom, $\phi_0$, is obtained by the integration of (1.2) from the interface between the suspended layer and the clear liquid at the bottom together with the equation of particle number conservation. The relationship between particle normal stress and shear-induced migration (or resuspension) has been the subject of several studies and is still an active area of investigation (Nott & Brady 1994; Mills & Snabre 1995; Morris & Brady 1998; Morris & Boulay 1999; Deboeuf et al. 2009; Lhuillier 2009; Nott et al. 2011; Ovarlez & Guazzelli 2013). The suspension balance model proposed by Morris & Boulay (1999) and refined by Lhuillier (2009) and Nott et al. (2011) offers a promising framework for modelling shear-induced particle migration, but it suffers from a relative lack of experimental data on particle normal stresses.

In addition to the above-cited work of Zarraga et al. (2000), who used the viscous resuspension experiment of Acrivos et al. (1993) to deduce $\Sigma_{33}^p$ for particle volume fractions ranging from 0.3 to 0.5, Deboeuf et al. (2009) determined $\Sigma_{33}^p$ for particle volume fractions ranging from 0.3 to 0.5 through the measurement of the pore pressure in a cylindrical Couette flow. Boyer et al. (2011) used a pressure-imposed shear cell to measure $\Sigma_{22}^p$ in the range $\phi \in [0.4, 0.585]$, and Dbouk et al. (2013) determined $\Sigma_{22}^p$ in the range $\phi \in [0.3, 0.47]$ through the measurement of both the total stress $\Sigma_{22}$ and the pore pressure. See (Guazzelli & Pouliquen 2018) for a review. All of these studies show a quasi-linear relationship between the particle normal stress components and the shear rate, but recently, Saint-Michel et al. (2019) performed X-ray radiography experiments on viscous resuspension that revealed a non-linear relationship with the shear rate.

In this paper, we present the experimental results of viscous resuspension in a Couette device in which the local particle volume fraction and the local shear rate are measured by optical imaging. $\Sigma_{33}^p$ is obtained by integrating (1.2) from the interface between the clear fluid and the resuspended layer to any height $z$ below the interface. These experiments present the dual advantage that $\Sigma_{33}^p$ can be determined for a wide range of particle fractions and that the local shear rate can be measured to accurately test the scaling of particle normal stresses with shear rate. In section 2, we present the experimental device and the methods used to compute both the velocity field and the particle concentration
<figure><img src="image_2.png" /><figcaption>FIGURE 1. a) Sketch of the experimental device. b) View from above. The vertical laser sheet is shifted by an offset of length $y_0 < R_1$ from the radial position (dashed line). $x$ is the horizontal position in the laser sheet, and $z$ is the vertical position. $z = 0$ is set by the mercury/suspension interface.</figcaption></figure>

field, for Shields number ranging from $10^{-3}$ to 1. Section 3 is devoted to the results. We first discuss the radial profiles of velocity and of particle concentration. Then we present the vertical concentration profiles from which $\Sigma_{33}^p$ is deduced. We finish with some concluding remarks (section 4).

## 2. Materials and Methods

### 2.1. Suspension and device

PMMA spheres (Arkema BS572), $2a = 268 \pm 25$ µm in diameter and $1.19 \ 10^3 \pm 10$ kg/m$^3$ in density, are used. The particles are dispersed in Triton X 100 to which a small amount of a fluorescent dye (Nile Blue A, Sigma-Aldrich) is added. This mixture is Newtonian with a viscosity of $\eta_0 = 0.34 \pm 0.02$ Pa.s and a density $1.06 \ 10^3 \pm 10$ kg/m$^3$ at $T = 23^\circ C$. The characteristic settling velocity of the particles is then $V_S = 2/9 \Delta\rho g/a^2 \eta \approx 20$ µm/s. The liquid and the particles are chosen to have roughly the same refractive index, 1.49, and accurate index matching is achieved by tuning the temperature of the chamber that contains the rheometer.

The resuspension experiments are conducted in a Couette cell made of PMMA mounted on a controlled-stress rheometer (Mars II, Thermofisher) (see figure 1(a)). The rotor has a radius $R_1 = 19$ mm, and the stator has a radius $R_2 = 24$ mm. Thus, the gap is much larger than the particle diameter (($R_2 - R_1$)/a $\approx$ 37) and the variation of the shear stress over the gap is expected to be of the order of 1.6 ($\Sigma_{12}(R_1)/\Sigma_{12}(R_2) = R_2^2/R_1^2 \approx$ 1.6). The impact of the stress variation on particle migration will be discussed in section 3.1.

The bottom of the Couette cell is filled with mercury to prevent the particles from migrating out of the gap (under the cup) and to maximize slip at the suspension/bottom interface in order to have a shear rate as homogeneous as possible inside the gap. The suspension is poured into the rheometer cell and illuminated by a thin vertical laser sheet (thickness $\approx$ 50 µm) offset by $y_0 = 16.2$ mm from the radial plane (see figure 1(b)). A camera (IDS, nominal frequency 33 Hz, full resolution $4104 \times 2174$ px$^2$) is placed at $90^\circ$ to the enlightened plane. The accurate matching of the refractive index, the thinness of the laser sheet and the resolution of the camera allow the recording of high-quality images with a resolution of 30 px per particle.
## 2.2. Experimental procedure and measurement method

### 2.2.1. Experimental procedure

In this paper, we will focus on the steady state of resuspension obtained for various angular velocities of the rotor, $\Omega$: 0.3, 0.5, 1, 2, 5, 10, 20, 30, 40 and 60 rotations per minute (rpm). For all these angular velocity values, the Reynolds number and the Taylor number are less than 1 (for the highest angular velocity, $Re = \rho\Omega R_1(R_2 - R_1)/\eta \approx 1$ and $Ta = 4\rho^2\Omega^2(R_2 - R_1)^4/\eta^2 \approx 1$) and the Péclet number is very large ($Pe = 6\pi\eta a^3 \dot{\gamma}/k_B T > 10^8$). Note that the inertial effects have also to be compared to the gravity forces. The ratio of inertial to gravitational forces can be written as $R_i\Omega^2/g$. In our experiment this number is comprised between $10^{-6}$ and $10^{-1}$, meaning that inertial effects can indeed be neglected. Furthermore, beyond this simple evaluation, Saint-Michel et al. (2019) performed 2D calculations to evaluate the role of the centrifugal forces that arise due to geometry curvature and showed that inertia effects do not alter the concentration profiles for $R_i\Omega^2/g < 1$.

To reach the steady state, the suspension is first sheared with an angular velocity of the rotor equal to 5 rpm for one hour. Then, the speed is set to the desired value for a period until the steady state is reached; the steady state is considered attained when the torque applied by the rheometer becomes constant. The time duration necessary to achieve the steady state is approximately a few hours. Figure 2 shows the viscous resuspension observed for a few rotor angular velocity values. As $\Omega$ increases, the resuspended height increases and the bulk particle concentration decreases.

### 2.2.2. Settled layer

The first (left upper) picture of the figure 2 shows an image of the suspension in the settled state. The sediment height is $h_0 = 21.3$ mm $\approx 4(R_2 - R_1)$. It is really important to determine as precisely as possible the value of the packing fraction in the sediment, $\phi_0$, since it will subsequently be used to deduce the particle volume fraction during the resuspension experiments (see 2.2.3). $\phi_0$ is the ratio of volume of the particles that belong to the sediment to the total volume of the sediment. Thus, to evaluate $\phi_0$, it is necessary to know the exact quantity of particles in the sediment and to measure the total volume of the sediment. Unfortunately, it is difficult to accurately control the mass of particles that are poured in the Couette cell. Thus, we decided to measure $\phi_0$ in a separated vessel. To this aim, a known mass of particles, $m_p$, is introduced in a graduated cylinder of known cross section (approximately 1 cm$^2$), $S$, that contains the suspending liquid (Triton X100). After the particles have settled (approximately after 24 hrs), the sediment height, $h_{sed}$ is carefully measured and $\phi_0$ is deduced ($\phi_0 = m_p/(\rho_p h_{sed} S)$). We took three measurements and obtained $\phi_0 = 0.574 \pm 0.003$.

### 2.2.3. Concentration field determination

The concentration field is determined through the measurement of the particle number density, $n_{ij}$ in the $(x, z)$ vertical laser plane. To this aim, each image is binarized with a local threshold. The local threshold value $T(x, z)$ is calculated individually for each pixel $(x, z)$ of the image $I(x, z)$ where $T(x, z)$ is a weighted sum (cross-correlation with a gaussian window) of $171 \times 171$ px$^2$ neighborhood of the pixel $(x, z)\dagger$. The particles are detected through a watershed segmentation process (Vincent & Soille 1991). The position of the barycentre of each segmented zone gives the position of each particle centre in the $(x, z)$ plane sampled with rectangular cells $(i, j)$ of size $\delta x = (R_2 - R_1)/8$ and $\delta z = 2a$. In each cell, the number of particle centres, $N_{ij}$, is measured. The particle
<figure><img src="image_3.png" /><figcaption>FIGURE 2. First line: typical images recorded for different rotor rotation speeds. A photo of the settled layer ($\Omega = 0$) is also presented (height 21.3 mm). Second line: mapping of the particle volume fraction averaged over 10000 images (with the exception of the sediment for which only 20 images have been used). Third line: azimuthal velocity normalized by the rotor velocity $\Omega R_1$ and averaged over 100 velocity fields. Fourth line: vertical velocity normalized by $\Omega R_1$ and averaged over 100 velocity fields. The rotor is on the left of each frame, the stator is on the right, and the mercury/suspension interface corresponds to the bottom of each frame.</figcaption></figure>
density $n_{ij} = N_{ij}/(\delta x \delta z)$ is reconstructed in the $(r, z)$ plane, making the change of variable $r = \sqrt{y_0^2 + x^2}$. Due to the non-zero thickness of the laser sheet and of the slight polydispersity of the particles, $n_{ij}$ is not the absolute particle density, and to compute the true particle volume fraction, we use the particle volume conservation from the sediment to the resuspended state:

$$ \phi(r, z) = \chi n(r, z) \text{ with } \chi = \frac{\phi_0 \pi (R_2^2 - R_1^2) h_0}{\int_0^{h_s} \int_{R_1}^{R_2} n_{ij} 2\pi r \, dr dz} \quad (2.1) $$

To compute the mean particle volume fraction, this procedure is repeated over 10000 decorrelated images (with the exception of the sediment for which only 20 images were used). Note that the acquisition time can be as long as 100 hrs for the lowest rotation speed of the rotor. Examples of the concentration field are given in figure 2, which calls some comments that will be discussed in more detail in section 3:

(i) Near the walls, the particle fraction is lower than in the bulk of the suspension, which should stem from the layering of the particles near the walls (Suzuki et al. 2008; Yeo & Maxey 2010; Blanc et al. 2013; Gallier et al. 2016; Deboeuf et al. 2018).

(ii) Outside of the structured zones, no or very weak radial particle migration is observed: the maximum difference in the particle volume fraction is evaluated to be less than 2%.

(iii) Along the vertical direction, a negative concentration gradient is observed as expected in the case of resuspension flows with a sharp interface separating the suspension and the pure fluid (Acrivos et al. 1994).

(iv) Near the bottom on the rotor side, particle volume fraction decreases by approximately 2%. This will be discussed in more detail in section 3.2 but from this observation we can evaluate the resulting uncertainty on $\chi$ and thus on the absolute value of $\phi(r, z)$. The area affected by the concentration change is roughly one tenth of the sediment leading an uncertainty of 0.3% on the particle volume fraction.

### 2.2.4. Velocity fields

The aim of the present study is to investigate resuspension and to link it to particle normal stresses. Because $\Sigma_{33}^p$ is a function of the shear rate, it is essential that the shear rate is known as precisely as possible. For this purpose, we measured the velocity field in the gap. The shift in the laser sheet out of the radial plane allows particle image velocimetry (PIV) measurements (Manneville et al. 2004) in the $(x, z)$ plane. Under the assumption that the radial component of the velocity is zero or much smaller than the azimuthal component, $v_\theta$ can be deduced from a simple projection of $v_x$ along the orthoradial direction (see figure 1(b)):

$$ v_{\theta}(x, z) = v_x(x, z) \frac{\sqrt{x^2 + y_0^2}}{y_0} \quad (2.2) $$

The velocity field $v(v_x(x, z), v_z(x, z))$ is computed using the open source software DPIVSOFT† (Meunier & Leweke 2003). Each image is divided into correlation windows of size $128 \times 128$ px$^2$. Each correlation window contains approximately 10 particles that are the PIV tracers. The cross correlation of the corresponding windows from two successive images yields the mean velocity of the particles in the window. The in-plane loss of pairs error is decreased by translating the correlation windows in a second run (Westerweel 1997), thus reducing the correlation windows size to $64 \times 64$ px$^2$. The same procedure performed on all the windows gives the velocity field, which is averaged over 100 images.

<footnote>† Available on the web (https://www.irphe.fr/meunier/)</footnote>
The mapping of the θ-component of the velocity field in the plane (x, z) is then obtained and used to reconstruct the velocity field in the (r, z) plane. Velocity maps are shown in figure 2, in which the velocity normalized by the velocity of the rotor is represented for several values of Ω.

Note that the PIV measurements also enable estimation of the z-component of the velocity, particularly to check that there is no significant secondary flow. Second normal stress difference is known to be responsible for secondary flows arising in non-axisymmetric conduits resulting in non-trivial concentration distributions (Ramachandran & Leighton 2008). Some evidence of secondary flows in cylindrical Couette flow has been given by Blaj et al. (2011) who show examples of 3D trajectories of a tracer introduced in a concentrated non-Brownian suspension. It is difficult to deduce some general features of any secondary flow from these trajectories which furthermore are likely to depend on the disturbances created at the bottom. Then, in order to attempt to characterize any potential secondary flow, the axial velocity, $v_z$ is registered and mapped, as showed in figure 2 (fourth line). As expected, the axial component of velocity is very small compared to the azimuthal component and, more interestingly, no peculiar spatial correlation is detected, perhaps with the exception of a slight tendency of an upward motion of the particles near the rotor and a downward motion near the stator.

# 3. Results

## 3.1. Radial profiles

### 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 around 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 $\pm0.15h_0$. Apart from near the rotor and stator where somehow indistinct particle layering is observed, 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 (Graham et al. 1991; Abbott et al. 1991; Phillips et al. 1992; Chow et al. 1995; Sarabian et al. 2019). Figures 3(b) and (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: $\Omega = 0.3$ (b) and 20 rpm (c). For both angular velocities, a particle layering is observed and is all the most pronounced that 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 2010; Gallier et al. 2016) and observed in many experiments (Metzger et al. 2013; Blanc et al. 2013; Sarabian et al. 2019; Deboeuf et al. 2018).

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 inserts of figures 3(b) and (c)). This trend is verified for all angular velocities including those that are not shown in the present paper (see supplementary material for more information).

This finding contrasts with the predictions of the suspension balance model (SBM)
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Radial concentration profiles in the sediment (a) and in the resuspended layer for two values of the rotating 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 figures (b, c) is computed from 10000 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 inserts) the measured profiles (solid lines) are almost flat in contrast to the predictions of the Suspension Balance Model (Morris & Boulay 1999) computed for $\Phi$ that is the r-averaged volume fraction at a given z (dashed lines).</figcaption></figure>

(Morris & Boulay 1999) that are also shown in figures 3(b) and (c) (dashed lines). In the framework of a 1D-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:

$$q(\phi) = \frac{\eta_N}{\eta_S} = Ar^{\frac{1+\lambda_2}{\lambda_2}} \qquad (3.1)$$

where $\eta_N = \Sigma_{11}^p/\eta_0\dot{\gamma}$ is the normal viscosity, $\eta_S$ the relative viscosity and $\lambda_2$, a constant close to 1 (here we took $\lambda_2 = 1/1.15$). A is a constant determined by requiring particle volume conservation. The theoretical profiles of figures 3(b) and (c) are obtained with the expressions of $\eta_N$ and $\eta_S$ proposed by Zarraga et al. (2000):

$$\eta_S = \frac{\exp(-2.34\phi)}{\left(1 - \frac{\phi}{\phi_m}\right)^3} \qquad (3.2)$$

$$\eta_N = 2.17\phi^3\eta_S \exp(2.34\phi)$$

Such discrepancies between SBM predictions and experiments conducted in cylindrical Couette flow have already been reported by several authors (see for instance (Ovarlez et al. 2006; Colbourne et al. 2018; Gholami et al. 2018)) and to a lesser extent (Sarabian et al. 2019)). 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. (2018), top and bottom boundary effects may also invalidate the use of the SBM in its 1D formulation. Another hypothesis would be that radial size segregation is involved since the particles are not perfectly monodisperse (with a standard deviation of about 8%). The larger particles are expected to migrate faster than the smaller ones and then to be focused at the outside. This size segregation effect has been noted by Abbott et al. (1991) in a bimodal suspension containing 3175– and 780– $\mu$m particles with respective volume fractions of 0.39 and 0.21 sheared in a wide-gap Couette rheometer ($R_1 = 6.4$ mm and $R_2 = 23.8$ mm). After a few thousands revolutions of the inner cylinder, the larger particles are observed to form hexagonally close-packed sheets near the stator. In our study, as depicted in section 2.2.3, we deduce the particle volume fraction from the
particle number density. Thus, a size gradient across the gap could possibly affect the concentration profiles. 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. 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 (2012) 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 hypothesis (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 g/cm³, viscosity 4.43 Pa.s). The results, described in the appendix, show that, contrarily to the case of a buoyant suspension, particle migration clearly takes place. Furthermore, the agreement with the SBM predictions is fairly satisfactory (see figure 11). Thus, the very weak migration observed in figure 3 appears to stem 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 $\Omega = 0.3$ and 20 rpm). 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 et al. 1995; Ahuja & Singh 2009; Blanc et al. 2011; Korhonen et al. 2015) and is probably related to particle layering near the walls.

In figure 4 are also plotted 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, at last, 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. (1995) where the apparent slip velocities at the inner and outer cylinders are given by:

$$ u_s(R_{1,2}) = \beta_{1,2}a\dot{\gamma}_{1,2} \qquad (3.3) \\ \text{with } \beta_{1,2} = \beta(\phi_{1,2}) \approx \frac{\eta_S(\phi_{1,2})}{8} $$

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 $\Omega = 20$ rpm, 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. (2018) 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 $\Omega = 0.3$ rpm, none of the tested models (Newtonian, Newtonian + wall slip, SBM or SBM + wall slip) fits the experimental
<figure><img src="image_5.png" /><figcaption>FIGURE 4. Azimuthal velocity profiles measured for angular velocities: $\Omega = 0.3$ rpm and 20 rpm at three different heights: $h_s/4$, $h_s/2$ and $3h_s/4$. Are also shown 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. (1995) (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)</figcaption></figure>

profiles and the shear seems to localize near the rotor, even for particle fractions lower than the expected jamming fraction (around 0.58). Such localization has already been observed by many authors (Huang et al. 2005; Ovarlez et al. 2006; Blaj 2012) and is generally attributed to the existence of a critical shear rate below which no steady flow exists (Ovarlez et al. 2006). At last, it should be noted that the shear-thinning behavior that is observed in most concentrated non-Brownian suspensions (Dai et al. 2013; Vázquez-Quesada et al. 2016, 2017; Tanner et al. 2018; Lobry et al. 2019) 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 12 in Appendix) 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 in
the case of concentration profiles, the density mismatch between particles and suspending liquid appears to significantly affect the velocity profiles.

### 3.1.3. Summary on radial profiles

(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 $\phi \approx 0.55$.

(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).

(iii) The examination of the velocity profiles show highly non-Newtonian flow characteristics which is 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 SBM vanishes for a neutrally buoyant suspension.

### 3.2. Vertical concentration profiles

To study the vertical variation of concentration, we restrict ourselves to the gap region outside the layered zones (where furthermore possible secondary flows may be present) and we focus on the central third of the gap. Figure 5(a) shows two concentration profiles averaged over the central third of the gap at low ($\Omega = 0.3$ rpm) and high ($\Omega = 20$ rpm) angular velocities. It is observed that the concentration is almost constant in the resuspended layer and drops to zero quite sharply, even for the highest angular velocity. This sharp interface between the resuspended layer and the clear fluid was already predicted by Acrivos et al. (1993) when interpreting their experiments in light of a diffusive flux model. Figure 5(a) also shows the profiles predicted by Acrivos et al. (1993). The agreement is quite good even though the resuspension height that we measured at low angular velocity is slightly larger than that obtained by Acrivos et al. (1993) and marginally smaller at high angular velocity. This trend is seen in figure 5(b), where, as in Acrivos et al. (1993), we observe a power-law dependence of the sediment expansion with the Shields number (1.1) but with an exponent slightly lower than $1/3$ (Zarraga et al. 2000).

Finally, it should be noted that near the bottom of the Couette cell, the particle concentration tends to decrease. This finding may be related either to a problem of particle detection near the interface with mercury, which reflects light and may downgrade the image quality in its vicinity or to bottom boundary effects which locally modify the flow and particle concentration. In the next section, in which the concentration will be used to evaluate $\Sigma_{33}^p$, we will not consider this zone.

### 3.3. Determination of $\alpha_3 = \Sigma_{33}^p/\eta_0\dot{\gamma}$

To estimate $\alpha_3 = \Sigma_{33}^p/\eta_0\dot{\gamma}$, we utilize the region outside the layered zone and well above the suspension/mercury interface: $R_1 + (R_2 - R_1)/3 < r < R_2 - (R_2 - R_1)/3$; $z > h_s/4$.

#### 3.3.1. Local shear rate

As shown in section 3.1.2, velocity profiles are far from expected. Thus, to estimate $\alpha_3 = \Sigma_{33}^p/\eta_0\dot{\gamma}$, 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 nominal shear rate:

$$ \dot{\gamma}_N(r) = 2\Omega \frac{R_1^2 R_2^2}{R_2^2 - R_1^2 r^2} \qquad (3.4) $$
<figure><img src="image_6.png" /><figcaption>FIGURE 5. a) Examples of the vertical concentration profiles obtained by averaging $\phi(r)$ over the central third of the gap. The corresponding Shields numbers are computed using the local shear rate (averaged over the central third), and the results are compared to the predictions of Acrivos et al. (1993) (red lines). b) Relative expansion of the particle layer versus the Shields number. $h_s$ is arbitrary defined as the height in which $\phi = 0.1$, and the results are compared with the correlation proposed by Acrivos et al. (1993).</figcaption></figure>

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:

$$ \dot{\gamma}(r, z) \approx \dot{\gamma}_{r\theta} = r \frac{\partial(v_{\theta}(r, z)/r)}{\partial r} \qquad (3.5) $$

However (3.5) presumes that the flow is essentially 1D while 2D 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 $\partial v_{\theta}/\partial z$ have to be evaluated before it can be neglected. In supplementary material, we show that, in most cases, $\partial v_{\theta}/\partial z \ll r \partial v_{\theta}/\partial r$ 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 $\Omega$ (figure 7). A few comments on this figure are needed. First, it is observed that all the data collapse on 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 $\Omega$ (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.

### 3.3.2. Third particle normal stress, $\Sigma_{33}^p$

To determine $\Sigma_{33}^p$, (1.2) is integrated from the interface between the resuspended layer and the clear fluid to the height $z(\phi)$:
<figure><img src="image_7.png" /><figcaption>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 $\Sigma_{33}^p$ (figure 9) and to determine the variation of $\Sigma_{33}^p/\eta_0\dot{\gamma}$ with $\phi$ (figure 10).</figcaption></figure>

$$ \Sigma_{zz}^p(r, z) = - \int_{z(\phi)}^{z(\phi=0)} \Delta\rho g \phi(r, \zeta) d\zeta \quad (3.6) $$

with the boundary condition

$$ \Sigma_{zz}^p(z(\phi=0)) = \Sigma_{zz}^p(h_s) = 0 \quad (3.7) $$

Then, making the assumption that the flow is essentially 1D, $\Sigma_{zz}^p$ can be equated to $\Sigma_{33}^p$. Rigorously speaking, since the $(z, \theta)$ component of the shear rate is not exactly zero (see section 3.3.1), $\Sigma_{22}^p$ also contributes to $\Sigma_{zz}^p$. This point is discussed in the supplementary material where we show that the subsequent error is completely negligible compared to the scattering of the data.

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

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 $-\Sigma_{33}^p$ as a function of $\dot{\gamma}$ for several particle volume fractions. The particle volume fraction range is restricted to $[0.3 - 0.5]$ because, in our experiment, for higher concentrations, the shear rate does not vary enough to obtain reliable dependence between $\Sigma_{33}^p$ 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 $\Sigma_{33}^p$ with $\dot{\gamma}$. From figure 9, it appears that, for a given value of $\phi$, $\Sigma_{33}^p$ 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. (2019). 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. 1994; Lobry et al. 2019;
<figure><img src="image_8.png" /><figcaption>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 $\Omega$. 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 $z \in [h_s/4, h_s]$. $\dot{\gamma}_N$ is the nominal shear rate calculated at the middle of the gap $r = (R_1 + R_2)/2$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIGURE 8. Maps of $\Sigma_{33}^p$ for several angular velocities. The dashed rectangles indicate the areas that are used to measure $\Sigma_{33}^p$ (figure 9) and to determine the variation of $\Sigma_{33}^p/\eta_0\dot{\gamma}$ with $\phi$ (figure 10). It is observed that, in these zones, $\Sigma_{33}^p$ hardly varies along $r$.</figcaption></figure>

Vázquez-Quesada et al. 2016, 2017; Tanner et al. 2018) and particle normal stresses are expected to be scaled by the shear stress (Boyer et al. 2011; Lobry et al. 2019) leading to a nonlinear scaling law between $\Sigma_{33}^p$ 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 that particle volume fraction is large.
<figure><img src="image_10.png" /><figcaption>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 $\Sigma_{33}^p \propto \dot{\gamma}^n$.</figcaption></figure>

### 3.3.3. Determination of $\alpha_3 = \Sigma_{33}^p / \eta_0 \dot{\gamma}$

The variation of $-\Sigma_{33}^p$ 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 on a single curve for a wide range of $\phi$ between 0.15 and 0.55 with variation in $\Sigma_{33}^p / \eta_0 \dot{\gamma}$ over more than five decades.

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 $\Sigma_{33}^p$ 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 $\Sigma_{33}^p$. For lower concentrations, a better collapse can be obtained if, rather than $\Sigma_{33}^p / \dot{\gamma}$, $\Sigma_{33}^p / \dot{\gamma}^{0.7}$ is plotted against $\phi$ in the range $\phi \in [0.3 - 0.5]$ within which the power-law scaling has been obtained for $\Sigma_{33}^p$ (see figures 10(b) and (c)). This is in agreement with the recent results of Saint-Michel et al. (2019).

The red and black lines in figure 10(a) represent the correlation proposed by Zarraga et al. (2000):

$$ \Sigma_{33}^p = -\eta_0 \dot{\gamma} \frac{\phi^3}{\left(1 - \frac{\phi}{\phi_0}\right)^3} \qquad (3.8) $$

The black curve is obtained with the original value of $\phi_0$ proposed by Zarraga et al. (2000) ($\phi_0 = 0.62$), 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. (2000). Furthermore, Zarraga et al. (2000) established the correlation for a particle
<figure><img src="image_11.png" /><figcaption>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 colors correspond to different angular velocities of the rotor (see figure 7). The agreement with the correlation proposed by Zarraga et al. (2000) 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. (2011) for $\Sigma_{22}^p$ together with $\lambda_3 = 1/2$ (blue line). b) zoom of figure a) for $\phi \in [0.3 - 0.5]$. c) variation of $\Sigma_{33}^p$ 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.</figcaption></figure>

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. (2011) for $\Sigma_{22}^p$ (with $\phi_0 = 0.574$):

$$ \Sigma_{22}^p = -\eta_0 \dot{\gamma} \frac{\phi^2}{(\phi_0 - \phi)^2} \qquad (3.9) $$

and assuming that $\lambda_2 = \Sigma_{22}^p/\Sigma_{11}^p \approx 1$ and $\lambda_3 = \Sigma_{33}^p/\Sigma_{11}^p = 0.5$, as suggested by Morris & Boulay (1999).

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. (2011) but rather the lack of variability in $\lambda_3$ with $\phi$. This last result has already been noted by Gallier et al. (2014) and was previously suggested by Morris & Boulay (1999) themselves.
4. Concluding remarks

With the aim of studying viscous resuspension, we conducted local measurements
of both the velocity and the particle volume fraction in a suspension of negatively
buoyant non-Brownian spheres sheared in a cylindrical Couette flow. We observed a
strong layering of the particles near the rotor and the stator. Outside these regions, the
particle number density does almost not vary along r. If the suspension is considered as
monodisperse (i.e. size segregation effects can be neglected), this result contrasts with the
predictions of the Suspension Balance Model (Morris & Boulay 1999). Furthermore, the
measured radial concentration profiles are inconsistent with the radial velocity profiles in
the framework of local rheology of a (quasi-)Newtonian material. These discrepancies
between predictions and experimental results on concentration and velocity profiles
vanish for a neutrally buoyant suspension, as already noted by Blaj (2012) who showed
that buoyant effects may be at the origin of shear localization even for particle volume
fractions lower than the jamming fraction. This observation appears quite important
and probably deserves further theoretical work to explore the role of buoyancy effects on
suspension flows.

From a practical point of view and in the context of the present study where a wide-
gap has been used, the unexpected velocity profiles lead to no less unexpected values of
the local shear rate that, consequently, has to be measured in order to deduce correctly
the value of α₃ = Σ₃₃<sup>p</sup>/η₀γ̇.

The vertical concentration profiles are less startling. We reached almost the same
results as Acrivos et al. (1993): the concentration slowly decreases from the bottom to
the interface with the clear fluid and drops sharply to zero. The resuspension height is
controlled by the Shields number (ratio between viscous and buoyancy forces) through a
power-law relation with an exponent close to 1/3, as already proposed by Acrivos et al.
(1993). The vertical concentration profiles are used to deduce Σ<sub>33</sub><sup>p</sup> by solving the Cauchy
equation. We observed that Σ<sub>33</sub><sup>p</sup> does not vary linearly with γ̇ but follows a power-law
with an exponent close to 0.7. The value of the exponent hardly varies with particle
concentration in the range φ ∈ [0.3 – 0.5]. This result is in agreement with the recent
findings of Saint-Michel et al. (2019) but raises questions. On one hand, this non linear
behavior is anticipated since Σ<sub>33</sub><sup>p</sup> is expected to be proportional to the shear stress which
itself varies less rapidly than linearly with shear rate (see for instance (Tanner et al. 2018)
or (Lobry et al. 2019)). But, on the other hand, shear-thinning is known to be all the
more pronounced that the particle volume fraction is high. This should lead to exponents
of the power law that decrease when φ increases, in contrast with what is observed.

At last concerning the variation of α₃ = Σ₃₃<sup>p</sup>/η₀γ̇ with φ, we would like to make two
comments. The first one is that our findings are in good agreement with the results of
Zarraga et al. (2000) who deduced α₃ from the resuspension height measured by Acrivos
et al. (1993). The second comment is that our results cannot be predicted by using the
well-established correlation of Boyer et al. (2011) for Σ₂₂<sup>p</sup> with the assumption that λ₃/λ₂
is constant, which should likely be revisited. It would probably be interesting to measure
accurately the three particle normal stresses and their variation with φ, in particular in
order to refine the Suspension Balance Model that is a powerful framework for capturing
migration.

We thank L. Lobry, F. Peters and B. Saint-Michel for fruitful discussions, and D.
Gilbert for the 3D sketch of the experimental device. We are grateful to the referees for
thoughtful comments and helpful suggestions.
<figure><img src="image_12.png" /><figcaption>FIGURE 11. Radial concentration profiles measured for a neutrally-buoyant suspension at $\phi = 0.52$. Blue line: spatial resolution has been fixed to one two-hundredth of the gap width (($R_2 - R_1$/200, orange line: volume fraction average over ($R_2 - R_1$/14). Dark dashed line: Suspension Balance Model prediction (Morris & Boulay 1999) computed for a bulk volume fraction of 0.52.</figcaption></figure>

Appendix

In this appendix we want to examine the reasons why concentration and velocity profiles are much different from what is expected from Suspension Balance Model (Morris & Boulay 1999). As a reminder, we identified two possible causes of this discrepancy: buoyancy or segregation effects. To discriminate between these two reasons, we performed additional experiments with a neutrally buoyant suspension. The experimental device is exactly the same as that which was used to study resuspension except for the suspending liquid that has now the same density as the particles. We used a mixture of water (11.90 wt%), Triton X-100 (73.86 wt%) and Zinc Chloride (14.24 wt%) that matches both the density and the refractive index of PMMA particles (Souzy et al. 2015, 2016). PMMA particles (Arkema BS572, $2a = 268 \pm 25$ µm) are added to this Newtonian mixture ($\eta_0 = 4.43$ Pa·s) in order to obtain a suspension at $\phi = 0.52$.

Figure 11 displays the radial concentration profiles that have been measured. In blue, the spatial resolution has been fixed to one two-hundredth of the gap width and the orange line corresponds to the concentration profile averaged over one fourteenth of the gap width. As well as in the case of non-neutrally suspensions a marked layering is observed but outside these structured zones, the concentration varies along $r$, in contrast with what was observed for a non-neutrally buoyant suspension. The dark dashed line corresponds to the SBM predictions. We note that the slopes of the experimental and theoretical profiles are almost the same. The experimental profiles are somehow shifted toward higher volume fractions because, as indicated by the orange line, the concentration in the layered zones is smaller than the averaged volume fraction and thus higher outside these zones. Notwithstanding this difference, the agreement between the experimental and predicted profiles is good, suggesting that density mismatch between fluid ans particles is responsible for the discrepancy observed between experimental and predicted concentration profiles displayed in figure 3. We also undertook the measurement of the
<figure><img src="image_13.png" /><figcaption>FIGURE 12. Normalized azimuthal velocity profiles. Blue line: measured for neutrally-buoyant suspension at $\Omega = 0.3$ rpm and $\phi = 0.52$. Are also shown the theoretical profiles. Solid line: Newtonian profile without wall slip, dashed line: Newtonian profile with wall slip evaluated from Jana *et al.* (1995) (3.3), dotted line: profile calculated according to (3.1) and (3.2) without wall slip, dash-dotted line: profile calculated according to (3.1) and (3.2) with wall slip (3.3).</figcaption></figure>

velocity profile. Figure 12 shows the results obtained for $\Omega = 0.3$ rpm (blue curve). For the sake of comparison the Newtonian profiles without (solid line) and with (dashed line) slipping conditions (3.3) have been plotted as well as the profiles computed from the predictions of the SBM ((3.1) and (3.2)), without (dotted line) and with (dash-dotted line) (3.3). The experimental velocity profile is very close to what was expected from the combination of wall slip conditions and SBM.

To conclude, the comparison of the velocity and concentration profiles obtained either for a neutrally or a non-neutrally buoyant suspension presents significant differences. Furthermore, the Suspension Balance Model appears to reliably predict concentration profiles only if there is no density mismatch between the fluid and the particles. In this latter case, the SBM (Morris & Boulay 1999) computed with the correlation of Zarraga *et al.* (2000) for the variation of normal and shear viscosity with particle concentration together with the slip boundary conditions proposed by Jana *et al.* (1995) (3.3) works well to describe the velocity profiles. On the opposite, in the case of density mismatch between particles and fluid, velocity profiles are much more heterogeneous.

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