Inclined plane rheometry of a dense granular suspension

Transcription

1 Inclined plane rheometry of a dense granular suspension C. Bonnoit, T. Darnige, E. Clement, and A. Lindner a) Laboratoire de Physique et Mécanique des Milieux Hétégogènes (PMMH), UMR 7636, CNRS-ESPCI, Universités Paris 6 et 7, 10 Rue Vauquelin, Paris Cedex 05, France (Received 24 March 2009; final revision received 9 September 2009; published 25 January 2010 Synopsis We present a new method to measure the viscosity of a dense model suspension using an inclined plane rheometer. The suspension is made of mono-disperse, spherical, non-brownian polystyrene beads immersed in a density matched silicon oil. We show that with this simple set-up, the viscosity can be directly measured up to volume fractions of =61% and that particle migration can be neglected. The results are in excellent agreement with local viscosity measurements obtained by magnetic resonance imaging techniques by Ovarlez et al. J. Rheol. 503, In the high density regime, we show that the viscosity is within the tested range of parameters, independent of the shear rate and the confinement pressure. Finally, we discuss deviations from the viscous behavior of the suspensions The Society of Rheology. DOI: / I. INTRODUCTION Understanding a dense suspensions s resistance to flow is of great importance for multiple applications in industry food processing, drilling fluids, concrete or cement, daily life cosmetics, paints, and geophysics mud or lava flow, land slides. However, the rheology of dense suspensions remains a difficult subject and classical rheology often fails to predict the viscosity as a function of a global volume fraction Stickel and Powell Reliable measurements require complicated and costly local measurement techniques as, for example, magnetic resonance imaging MRI techniques Ovarlez et al. 2006; Huang and Bonn In this paper we present a simple technique to determine the viscosity of a suspension with volume fractions up to 61%. We work with a model suspension of non-brownian spherical particles, density matched with the suspending fluid. We use an inclined plane rheometer which, in contrast to classical rheometry Macosko 1994; Barnes et al. 1989, fixes the ratio of normal stresses to tangential stresses on the suspension layer. The thickness of the layer is however not fixed and directly gives the normal stress distribution. It has been shown that this set-up is a good choice to study the rheology of dry granular materials GDR 2004, submarine flows Cassar et al. 2005, or yield stress fluids Coussot et al. 2002a. a Author to whom correspondence should be addressed; electronic mail: anke.lindner@espci.fr 2010 by The Society of Rheology, Inc. J. Rheol. 541, January/February /2010/541/65/15/$

2 66 BONNOIT et al. We show here that it is also a good choice for dense suspensions, as wall slip effects or fracturation within the sample Jana et al. 1995; Barnes 1995; Coussot and Ancey 1999 do not play a role. We measure in our set-up the global viscosity of a layer of a dense suspension that can, under our experimental conditions, be considered as homogeneous. The limits of validity of the approach are determined without the use of another technique. The obtained results are in excellent agreement with local viscosity measurements by Ovarlez et al The inclined plane geometry differs substantially from classical rheometry; the flow behavior of dense suspensions can thus be tested under different conditions. We show, for example, that there is no dependence on the confinement pressure in the range of values tested. The paper is organized as follows. In Sec. II, we present a short introduction to granular suspensions. In Sec. III, the materials and methods used in the present experimental investigation are described. Section IV presents the characterization of our model suspension by means of classical rheology. In Sec. V, the experimental results obtained on the inclined plane rheometer are presented and discussed. In Sec. VI, we conclude. II. BACKGROUND A suspension consists of solid particles completely immersed in a viscous liquid. In this paper, we will focus on suspensions of noncolloidal hard spheres in a Newtonian fluid. Brownian motion, deformability of spheres, and colloidal interactions can thus be neglected. The macroscopic properties of these suspensions are essentially determined by the volume fraction =V g /V t, with V g as the volume occupied by the grains and V t as the total volume Frankel and Acrivos The relative motion of two spheres in a viscous liquid implies flow of the interstitial fluid and thus creates viscous friction, called hydrodynamic interaction between the particles. With increasing volume fraction, the distance between particles decreases and hydrodynamic interactions are dominated by lubrication forces, which originate from the shearing of fluid between close spheres. For even higher volume fractions, the typical separation distance between the surfaces of two neighboring elements is about the particle roughness, solid contacts are established, and a force network builds in the suspension Coussot This behavior is reflected when considering the shear viscosity of a granular suspension. The viscosity is defined as the ratio between shear stress and shear rate in the bulk of the fluid. The presence of spheres induces additional energy dissipation leading to an increase of viscosity with volume fraction. A divergence of viscosity is observed when the solid fraction tends toward the maximum packing fraction m, which depends on the particle characteristics Stickel and Powell Viscous flow of noncolloidal particles has been extensively studied. Einstein 1906 gave the first prediction for the bulk viscosity of a dilute suspension 3% of hard spheres in a Newtonian liquid. We found r =/ 0 =1+2.5, where 0 is the viscosity of the interstitial fluid. The rheology of suspensions was further developed extrapolating the approach of Einstein to concentrated systems. Batchelor and Green Batchelor 1977; Batchelor and Green 1972 made a significant advance using statistical mechanics arguments to account for hydrodynamic interactions in a semidilute suspension 10% of hard spheres in pure shear flow: r =/ 0 = Models attempting to extend the work of Batchelor to higher volume fractions are always semi-empirical. They recover the Einstein limit at low concentration and try to account for the divergence of the viscosity close to the maximum packing fraction m.

3 INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS 67 They are valid in different ranges of volume fractions. Some of the best known models are the Leighton model Leighton 1985 or the Krieger Dougherty model Krieger and Dougherty 1959; Krieger The latter reads r KD = KD / 0 = 1 / m n. 1 The Einstein limit is recovered for n= 2.5 m, but this choice does not often fit the experimental data at high volume fractions. Allowing a second free parameter n fits many experimental data over a limited range of volume fractions and leads to a widely used model for high volume fractions Chong et al. 1971; Barnes The appropriate values of n and m are however still subject to debate. The fact that it is not possible to describe the viscosity over the whole range of volume fractions with one value of n within the framework of the Krieger Dougherty model reflects the change from an isotropic to an anisotropic particle distribution with increasing volume fraction. One of the most recent models is the model of Zarraga et al who proposed for the whole range of volume fractions the empirical formula Z r = Z / 0 = exp / m 3, 2 with m =62% without any adjustable parameters. There is no general agreement on the exact behavior of the suspension viscosity close to the maximal packing fraction m so far. Measuring viscosity experimentally is difficult mainly due to particle migration Gadalamaria and Acrivos 1980, flow localization Huang et al. 2005; Ovarlez et al. 2006; Coussot 2005, and wall slip effects Bertola et al. 2003; Jana et al. 1995; Barnes As a consequence, global viscosity measurements performed by means of classical rheology, which rely on the assumption of a homogeneous material, might not yield a meaningful determination of. The microscopic heterogeneities at the origin of the above-mentioned effects are revealed by local viscosity measurements, for example, through MRI techniques Huang et al. 2005; Ovarlez et al. 2006; Raynaud et al. 2002; Coussot et al. 2002b; Huang and Bonn With non-intrusive MRI techniques, the local velocity and volume fraction inside the bulk of a given sample can be measured. Ovarlez et al performed MRI measurements in a Couette geometry and defined the viscosity as =, where and are local values. In this way, they obtained viscosities that can be linked in an unambigous way to a volume fraction. These measurements are in good agreement with models as given by Eq. 1. Ovarlez et al. also showed that particle migration in their Couette device leads to gradients in the concentration throughout the sheared suspension. For low shear rates, shear banding or flow localization occurs Ovarlez et al. 2006; Huang et al. 2005; Coussot 2005: the flow starts for a critical shear rate = c, progressively invades the gap, and completely occupies the gap for = 1. Dense suspensions also show non-newtonian properties. The shear viscosity might depend on the shear rate and show shear thinning Chang and Powell 1994 or shear thickening Barnes 1989 at high shear rates. One also observes the existence of normal stresses. Due to hard spheres repulsion, the microstructure of the sample becomes anisotropic under shear Brady and Morris 1997, leading to the apparition of normal stresses in the sample Zarraga and Leighton 2001 with a linear dependence on the shear rate Bagnold 1954; Ancey and Coussot 1999; Huang et al

4 68 BONNOIT et al. Dense suspensions exhibit a yield stress: below a critical stress c, no flow is observed in the bulk Coussot 2005; Huang et al Huang et al showed that this yield stress is associated with a viscosity bifurcation: above the critical shear stress c, the shear rate is higher than a critical shear rate c; below c no steady flow exists. This behavior resembles in a number of points the behavior of dry granular media GDR 2004; Jop et al At low shear rates the shear stress is nearly constant, as observed for dry granular media. The rheology of dry granular media depends strongly on the confinement pressure or normal stress. One thus defines an effective friction coefficient eff that is given by the ratio of normal to shear stresses GDR 2004; Jop et al For dense suspensions the role of the confinement pressure is still an open question Fall et al In conclusion, suspensions might behave either as dry granular materials, where frictional contacts are dominant low shear rates and low viscosity of the interstitial fluid, or as viscously dominated systems high shear rates and high viscosity of the interstitial fluid, where the contacts are lubricated Ovarlez et al. 2006; Huang et al The two main challenges when attempting to measure the viscosity of dense suspensions are thus to perform meaningful global measurements that can be validated without the use of difficult local techniques and to make sure to work within the limits of the viscous behavior. III. MATERIALS AND METHODS A. Granular model suspension Our suspensions are made of monodisperse spherical polystyrene beads from Dynoseeds with diameter d=405 m. The density of the spheres is = g cm 3. As interstitial fluid, we use a modified silicone oil Shin Etsu SE. KF-6011 S =116 mpa s at T=20 C, which is density matched at a value =1.07 g cm 3. The Stokes velocity of a single sphere in oil is about 0.6 mm/h; as our experiments take place on a typical time scale of 10 min, sedimentation can be neglected. We vary the volume fraction from =35% up to =61%. Special care is taken when preparing the suspensions: we control the volume fraction by measuring the weight and the material is degassed above =58% to prevent trapping of air bubbles and to obtain a homogeneous concentration. This procedure guaranties a precision of =1%. Before each experiment, the suspension is well mixed. The mixing plays the role of a preshear and reduces the effect of the slight density mismatch that can lead to creaming of the suspension when at rest. In this way, we obtain a reproducible initial state. B. Classical rheology Classical rheometry Macosko 1994; Barnes et al uses typically Couette or parallel plate devices. In this geometry the distance between the two cylinders or the two plates is fixed during the experiment. To measure a shear viscosity, one can either apply a constant shear rate or a constant shear stress allowing. Due to the fixed gap, the normal stress is not constant and can develop and vary during the experiments as a function of the material or the shear rate. Depending on the geometry, the shear rate is constant throughout the whole gap typically observed for a Couette device with a small gap or there are gradients of the shear rate due to nonlinear velocity profiles. Our classical rheological measurements are performed in a parallel-plate geometry with gap width b=0.5 mm and radius R=35 mm on a commercial rheometer Haake RS100. The geometry has rough surfaces of roughness 0.4 mm to avoid slip of the

5 INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS 69 granular material see a schematic representation of the geometry in Fig. 2b. We work at constant shear rate with a fixed thickness of the layer. We use the following experimental procedure to obtain reproducible measurements. First, a pre-shear of =200 s 1 is applied during 60 s, then the shear rate is increased from =0.1 to =300 s 1 and decreased again during 120 s. Complementary measurements Chevalier et al were performed in a double Couette geometry rheometer Haake-RS600 of gap width mm and mean radius 20 mm. The radius being large compared to the gap width, the velocity profile can be considered as linear. Thus the local shear rate in the gap can be considered as uniform in this case. C. Inclined plane rheometry An inclined plane rheometer Coussot and Ancey 1999 consists of a large plane inclined by a given angle with respect to the horizontal. The test liquid flows down the plane with a free surface and can thus adjust its height h. When changing the angle, the tangential stress xy applied on a layer of the test liquid at position y is changed: xy =gh ysin, where is the density of the liquid and Oy is the direction perpendicular to the plane. Local momentum balance follows from the Navier Stokes equation and can be written as g sin +d xy /dy=0. For flow of a Newtonian viscous fluid, the constitutive equation is xy = =dv x /dy, leading to a velocity profile vy=g sin /2y2h y Landau and Lifshitz The surface velocity at y=h reads v s = gh2 sin. 3 2 Measuring the thickness of the layer h and the surface velocity v s yields the viscosity of the viscous fluid as follows: = gh2 sin. 4 2v s The normal stress yy is given by the hydrostatic pressure and reads yy = g cos h y. The thickness of the layer and as a consequence the tangential stress is adjusted by the system itself in contrast to typical set-ups used in classical rheology. On the inclined plane, the normal stress is well controlled during the experiments and is zero at the free surface. The inclined plane rheometer is thus a good choice to establish constitutive equations in situations where the confinement pressure might play a role as for dry granular materials GDR 2004, submarine flows Cassar et al. 2005, or yield stress fluids Coussot et al. 2002a. It is also an appropriate rheometer for different types of complex fluids such as snow Rognon et al or viscoplastic fluids Chambon et al. 2009; Cochard and Ancey Our set-up consists of an inclined plane of length L=1 m, width l=38 cm, and a tank of section A connected to the inclined plane by a variable aperture e at its bottom Fig. 1a. The plane is smooth on the scale of the particle size. The tank is filled with the suspension that flows out of the aperture onto the inclined plane. As the aperture has to be sufficiently large, to avoid filtration effects at the tank exit, all our experiments were performed at e=5 mm. Some larger trap apertures where tested and lead to identical results. Note that we have explicitly tested that the volume fraction of the suspension collected at the bottom of the inclined plane is identical to the one of the suspension initially filled in the tank.

6 70 BONNOIT et al. (a) (c) (b) (d) FIG. 1. Experimental set-up and measurements a Side view. Schematic representation of the suspension flow on the inclined plane. b Top view. Snapshot of the suspension surface taken by the top camera. The deviation of the laser slice is used to measure the height h of the layer. Colored tracer particles allow us to measure the surface velocity v s via CIV techniques. c Flow rate Q as a function of time. d Typical measurements at =15 for a volume fraction =53.6%: surface velocity and thickness of the layer as a function of time. Our experimental control parameters are the volume fraction and the angle. Here we work with two different angles =5 and =15 and volume fractions ranging from =35% to =61%. We measure the input flux by monitoring the height of the suspension H in the tank with a charge coupled device camera and calculate the flow rate at the outlet of the tank as a function of time Q = A dh 5 dt. We use a second camera above the experiment to visualize the flow snapshot of the experiment on Fig. 1b. The thickness h of the layer is determined with using a laser slice reflection technique: the deviation of the laser beam is proportional to the thickness of the layer. Adding colored particles makes it possible to determine the surface velocity v s with a correlation image velocimetry CIV technique using DaVis software. The two cameras are synchronized and coupled to computers for direct image acquisition. In this way, we measure h and v s at every moment of the experiment see Fig. 1d. At the beginning of the experiment, the suspension front passes the camera field and we can simultaneously measure the velocity of the front v front corresponding to the mean velocity v and the surface velocity v s. We continuously monitor the surface velocity v s, the thickness of the layer h, and the height in the tank H. The flux being proportional to the time derivative of H, its value decreases during the experiment Fig. 1c leading to a decrease of h and v s during one

7 INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS 71 (a) (b) (c) FIG. 2. Classical rheology. a Shear viscosity obtained in a parallel-plate geometry: pp as a function of the shear rate for different grain fractions at T=20 C =48%, =46%, =44%, =40%, =30%, =20%, =0%. b Schematic representation of the parallel-plate geometry with rough surfaces and the double Couette geometry. c Normalized shear viscosity / Z for different grain fractions obtained in different geometries: pp parrallel plate, c1 Couette geometry, d=40 m, and c2 Couette geometry, d=80 m. c measurements from Chevalier et al experiment and a varying shear rate applied on the layer. But even if the flow rate is not constant, the quantity hv s /t/gh sin is very small compared to 1, meaning that temporal variations are small compared to a typical velocity in our experiments. Moreover, when changing the initial height in the tank for different experiments, we change the history for a given flow, but no difference is observed. We can thus consider that our experiments are always in a quasi-stationary regime. Note also that variations in the layer thickness of the layer h/x are below 0.1%. IV. VISCOSITY BY CLASSICAL RHEOMETRY First we characterize our suspensions classical rheology. We validate the results by using different geometries and by comparing to existing models for the viscosity of suspensions as given in Sec. II. We measure the viscosity pp using the parallel-plate geometry and the experimental protocol described in Sec. III. The results for different grain fractions are displayed in Fig. 2a as a function of shear rate. From =0% to =40% and in the range of shear rates tested, the suspensions behave as a Newtonian fluid. From =40% to =48% slight shear thinning occurs at higher shear rates but the measurements are still reproducible. No difference between a first and a second ramp in shear rate is observed. Slight shear-thinning has also been observed by other authors Chang and Powell Note that shear thickening occurs at even higher volume fractions and is observed to be very weak for spherical particles Barnes Below a volume fraction of =48%, wecan thus define a viscosity independent from in a relatively large range of shear rates. Above a volume fraction =48%, measurements with our experimental protocol imposing the shear rate on a parallel-plate geometry are not reproducible anymore. Subsequent ramps in shear rate lead to different results. Visual observation reveals localization and fracturation in the sample; it is no longer possible to measure a global viscosity of the suspension. The same suspensions were tested by Chevalier et al using a double Couette geometry. The results for c obtained for two different grain sizes d=40 m and d =80 m and concentrations up to =40% are displayed together with the results obtained in the parallel-plate geometry in Fig. 2c. The figure shows the ratio of the measured viscosity to the viscosity Z predicted by the model of Zarraga see

8 72 BONNOIT et al. (a) (b) FIG. 3. a Viscous behavior on the inclined plane. The square of the layer thickness h 2 as a function of surface velocity v s for the angle =15 and three different volume fractions: 3 =57.8%, 2 =53.6%, 1 =48.8%. b Viscosity as a function of dimensionless thickness h/d. Dashed line represents the mean value of the plateau and the measured viscosity ip. h is the thickness at which the measured viscosity deviates from the plateau value. Eq. 2 for the given grain fraction. First, one can conclude that the results obtained by the three different experiments are in agreement within 10%. Second, the representation in Fig. 2c is a very strong test of the semi-empirical model of Zarraga. It shows on a linear scale the ratio of experimentally obtained viscosities and the semi-empirical model of Zarraga. The Zarraga model faithfully describes the data up to a volume fraction =48% with a precision better than 10% without any adjustable parameter. The good agreement between the different experimental results, obtained using different geometries, and the good agreement between our data and the Zarraga model tend to prove that below a volume fraction of 50%, particle migration is indeed negligible. It also proves that our experiment determines reliably the viscosity up to grain fractions of =48% in the case of the parallel-plate geometry and up to =40% in the case of the Couette geometry. Above a volume fraction =50%, classical rheometry fails to measure a global viscosity when imposing the shear rate. It has been shown that it might be more appropriate to impose a constant shear stress when measuring for concentrated suspensions Huang et al and that classical rheology might need to account for particle migration Huang and Bonn 2007; Stickel and Powell In the following section, we will show that with the inclined plane set-up, suspension viscosities for dense suspensions can be measured up to volume fractions of =60%. V. VISCOSITY ON THE INCLINED PLANE A. Viscous behavior of the suspensions Now we characterize the flow of suspensions in the context of inclined plane rheometry. For a Newtonian fluid the surface velocity v s is proportional to the second power of the thickness of the layer h 2 see Eq. 3. In Fig. 3a we show experimental observations for three typical experiments: at the beginning of the experiments, when h is high, the relation between h 2 and v s is indeed linear. The linearity implies that the flow of the suspensions can be described by a viscous model for a Newtonian fluid with a parabolic flow profile and that the slope is directly proportional to a macroscopic viscosity of the suspension. We can thus naturally define a viscosity of the suspension measured on the inclined plane as

9 INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS 73 FIG. 4. Relative shear viscosity r =/ 0 as a function of the volume fraction. Results from inclined plane rheometry,, classical rheometry, and local measurements made by Ovarlez et al The line represents the prediction of the Zarraga model Zarraga et al ip = gh2 sin, 6 2v s with as the volumic mass of the interstitial fluid, as the angle of the inclined plane, and as the volume fraction. In Sec. V B we will discuss the observed velocity profile in detail. Systematically, for lower values of h, there is a deviation from the viscous behavior, as h 2 is not proportional to v s anymore. This becomes clearer when plotting the viscosity ip as a function of h/d as can be seen in Fig. 3b. In the beginning of the experiments high h, the viscosity ip is constant at a certain plateau value. Therefore in this regime the viscosity does not depend on the shear rate nor the confinement pressure given by yy = g cos h y. Toward the end of the experiments low h we observe a systematic decrease in the viscosity. This deviation occurs at a given value of the layer thickness h. h is large compared to the grain size in all cases. This deviation from a classical viscous behavior will be discussed in more detail at the end of this section. Note that the deviations occur for low h and thus low shear rates. They can thus not be due to shear thinning, which is observed at high shear rates. The ranges of shear rates and confinement pressures observed during one experiment depend on the volume fraction. For high volume fractions, the typical range of pressure is Pa and the typical range of shear rate is 1 20 s 1 estimated by the ratio v s /h. For low volume fractions, the typical range of pressure is Pa and the typical range of shear rate is s 1. The classical rheological measurements have been performed for shear rates of s 1. The data from the inclined plane do thus fall in a comparable range of shear rates. The range of shear rates accessible on the inclined plane is however limited and cannot be tuned independently. We define the macroscopic viscosity ip measured with our inclined plane set-up for all our experiments as the mean plateau value in the viscosity observed at the beginning of the experiments see Fig. 3b. The results for r as a function of are represented in Fig. 4 for two different angles =5 or =15. The measured viscosity is independent of the tilt angle. This shows once again that the viscosity is independent of the shear rate and the confinement pressure for the whole range of volume fraction tested from 35% to 61%.

10 74 BONNOIT et al. FIG. 5. Normalized shear viscosity ip / Z as a function of grain fraction for different angles obtained on the inclined rheometer: =15, =5. Local measurements made by Ovarlez et al Dashed line represents the error on the Zarraga viscosity Z due to the error =0.01 on the volume fraction. The line is a fit to a Krieger Dougherty law using the parameters by Ovarlez et al The results obtained on the inclined plane are compared to the experiments performed on the classical rheometers. Good agreement is observed in the range of volume fractions accessible by classical rheometry. With the inclined plane set-up, we can reach volume fractions not accessible via classical rheometry. In this range of volume fractions 50% our results can only be compared to local measurements as those obtained by Ovarlez et al by MRI in a Couette device. On this log-scale, very good agreement between their data and our measurements is obtained. Our simple set-up thus yields reliable results for suspension viscosities in a range of volume fractions that are normally only accessible via very sophisticated local techniques. Our results are not only in good agreement with the existing experimental results but also with the Zarraga model up to high volume fractions. Our data can be fitted reasonably well to a Krieger Dougherty law see Eq. 1 when allowing two free fit parameters. We find a maximum packing fraction m =62% and an exponent n=2.35 fit not shown. This fit does not recover the Einstein limit. Ovarlez et al obtained good agreement between their data and the Krieger Dougherty model using n=2 and m =60.5% in the range of volume fractions =50% 60%. Note that they obtained m directly from their local measurements; their only fit parameter was n. It is known that the value of m depends strongly on the properties of the particles used Stickel and Powell The Zarraga model attempts to describe the data over the whole range of volume fractions. In Fig. 5 we compare our measured viscosities more precisely to the Zarraga model by displaying ip / Z on a normal scale. The error bars dashed line on Fig. 5 take into account an error of =1% for the Z value. Below a volume fraction =55%, there is very good agreement of our data with the Zarraga model. Above =55% systematic deviations from the Zarraga model are observed but our data are, within the errors bars, still in good agreement with the Zarraga model. We also represent the local rheology measurements by Ovarlez et al. that show quantitatively the same tendency when compared to the Zarraga model. The Zarraga model is a semi-empirical model and the exact slope at high volume fractions is not necessarily given by this model. The Krieger Dougherty model is widely used in the literature but the exact values of the fit parameters are still subject to debate. We chose to compare our data to this model using the fit parameters obtained by Ovarlez et al Their adjustment has the

11 INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS 75 advantage that they obtained the maximum packing fraction from independent measurements. Note that they find m =60.5%, whereas we attempt to measure viscosities up to volume fractions of =61%. Within our experimental resolution this is however not a contradiction. One can see from Fig. 5 that at high volume fractions the Krieger Dougherty model captures the dependence of the measured viscosity on the volume fraction slightly better than the Zarraga model. Our measurements, in agreement with local measurements, could be helpful to develop more accurate models for high volume fractions. B. Discussion of particle migration The good agreement of our measured viscosities and local viscosity measurements indicates that migration of particles can be neglected in our set-up and that the volume fraction remains homogeneous within the layer of the granular suspension. It has however been shown by a number of experimental Timberlake and Morris 2005; Lyon and Leal 1998 and theoretical studies Gadalamaria and Acrivos 1980; Morris and Brady 1998; Brady and Morris 1997; Leighton and Acrivos 1987; Acrivos 1995 that particles migrate from regions of high shear rate to regions of low shear rate in geometries as Couette devices, capillary flows, or Hele-Shaw cells. As a consequence the volume fraction is not constant anymore and the flow profile is modified. The modification of the flow profile can be measured experimentally in our set-up and we will verify that changes are indeed small in our system. The parabolic flow profile observed for a Newtonian fluid vy=g sin /2y2h y=v s y/h 2 2h y=v s 1 y h/h 2 evolves toward flatter profiles when migration of particles takes place Lyon and Leal This change can be described by the use of an index n larger than 2 and the flow profile is then defined as vy = v s1 y h h n. 7 The mean velocity can be written as v=n/n+1v s leading to the following expression for the index n: 1 n = v s /v 1. 8 Experimentally, at the early stage of the flow we can simultaneously measure the velocity of the suspension front, corresponding to the mean velocity, and the surface velocity when the suspension front passes the camera field see Fig. 6a. From these measurements we can deduce n and thus obtain information on the velocity profile. The results obtained for different angles and different trap apertures on the plane are displayed in Fig. 6b. We always find values of n close to 2 as one would expect for Poiseuille flow. Furthermore, we compare our results to results obtained by Lyon and Leal 1998 who measured flow profiles for suspension flow in a Hele-Shaw cell. They found a deviation from the parabolic flow profile directly linked to particle migration. We have made an adjustment of their data with a power law profile as given above and have extracted values for the exponent n for three different volume fractions. We see a strong increase in n with the volume fraction and n much higher than the values we observe in our experiments Fig. 6b. Within our experimental resolution, we cannot distinguish between a small systematic error and a real deviation toward values of n slightly below 2, but we clearly observe a constant value of n for all experimental conditions and volume fractions. Furthermore, n is significantly lower than the values observed for

12 76 BONNOIT et al. (a) (b) FIG. 6. Flow profile. a Schematic representation of the surface velocity v s and the mean velocity v front when the flow is passing through the camera field. b Power law index n=1/v s /v 1 as a function of volume fraction : =15, =5, obtained on the inclined plane and deduced from a fully developed flow profile in a Hele-Shaw cell obtained by Lyon and Leal Dashed line is a guide for the eyes. The full line represents the expected value n=2 for a parabolic flow profile. steady flow profiles in the presence of particle migration. This directly proves that we can neglect restructuration in the sample. The reason why particle migration can be neglected in our set-up might be that we work at very low deformation of the material. When the suspension passes the camera field, located at 10 cm from the outlet of the tank, the height of the suspension layer is typically h =2 10 mm. Typical deformations are found to be of the order Moreover, the deformation is constant at our measuring point and does not evolve with time in sharp contrast to classical rheometry. A number of theoretical studies predict that particle migration is linked to the normal stress distribution Morris and Boulay 1999; Nott and Brady 1994; Mills and Snabre 1995; Morris and Brady The normal stress distribution on the inclined plane differs from the normal stress distribution of flow in capillary tubes or Hele-Shaw cells. This might be one of the reasons why the effect of migration is negligible in our measurements. Timberlake and Morris 2005 reported particle migration in inclined plane flow using a geometry similar to ours. They work however with much thinner layer thicknesses typically below 50 particle diameters, whereas we work with layer thicknesses typically higher than 100 particle diameters. As a consequence, the effect of migration is much more pronounced in their study. In conclusion, we can confirm that the flow profile remains in our experiments close to a parabolic flow profile and that we are working under experimental conditions where we can consider our suspension as homogeneous. This validates our method to obtain the viscosity from the plateau values of the viscosity measurements, as described in Sec. V. This also explains that our measurements are in excellent agreement with local rheology measurements. C. Limits of the viscous behavior of the suspension flow We have pointed out before see Fig. 3 that there is a systematic deviation from the Newtonian viscous behavior at low layer thicknesses and toward the end of the experiments. Below a certain layer thickness h, the viscosity is no longer constant anymore and we observe a decrease in the measured viscosity. We can thus define a cross-over thickness of the layer h which defines the limit of validity of our experimental method. Below h the measurements should not be considered when measuring the viscosity of a dense suspension on an inclined plane. We define h as the lowest value of the thickness of the layer for which the viscosity is still constant, i.e., within a deviation of 5% from the mean value of in the constant range see Fig. 3. This thickness h depends strongly on

13 INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS 77 (a) (b) FIG. 7. Limits of the viscous behavior. a Cross-over thickness h and b critical shear rate delimiting the validity of the viscous regime: =15, =5. Dashed line is a guide for the eyes. the volume fraction see Fig. 7a. The bigger the volume fraction, the sooner the deviation occurs during an experiment and the bigger is h. We show the average shear rates observed at the cross-over on Fig. 7b. They depend strongly on the volume fraction and also show a dependence on the angle. We compare this result to observations made by Huang et al who define a critical shear rate below which flow localization takes place. For viscosities comparable to the viscosity of our interstitial fluid, Huang et al. found a value around 0.02 s 1. This is significantly lower than the values we observe. This seems to indicate that the transition we observe cannot be explained by the same mechanism. The deviations from the viscous regime we observe below a certain layer thickness might be a signature of non-local effects as described by Goyon et al VI. CONCLUSIONS In this paper we present a very simple experimental method of measuring the viscosity of dense suspensions up to volume fractions =61%. This range of volume fractions is difficult to reach by classical rheology and has until now only been accessible by local rheological measurements requiring sophisticated techniques as MRI. We work with a very simple set-up, using flow of dense suspensions on an inclined plane. Analyzing the thickness and velocity of the layer and using a model for flow of a purely viscous liquid, we measure a global viscosity. The limits of validity of this approach are directly obtained from our measurements and do not require the additional use of other techniques such as, for example, local viscosity measurements. By the use of a model suspension, we have thus validated the inclined plane rheometer as a reliable method of measuring the viscosity of dense suspensions up to volume fractions of =61%. This easy method could in the future be used to obtain the viscosity of more complex dense suspensions, formed by non-spherical, rough, or polydisperse particles. This technique might also be used directly in the field. We show that our set-up guarantees the condition of a homogeneous volume fraction throughout the sample. Thus we can demonstrate that the viscosity we measure is independent of the shear rate and the confinement pressure, which is given by the height of the layer of the suspension and the inclination of the plane. Our results are in good agreement with the results found by local measurements and recent models as the Zarraga model. We show that for thin layers the behavior of the suspension deviates from a purely viscous behavior and might behave more like a granular material. Collective effects as described by Goyon et al might be responsible of this effect.

14 78 BONNOIT et al. ACKNOWLEDGMENTS We thank Jose Lanuza for help with the experimental set-up and Guylaine Ducouret PPMD-ESPCI for help with the rheological measurements. They acknowledge many interesting discussions with Guillaume Ovarlez Institut Navier, Université Paris Est and thank him for a critical reading of the paper. This work is supported by the ANR PIGE : Physique des Instabilités Gravitaires et Érosives. References Acrivos, A., Shear-induced particle diffusion in concentrated suspensions of a noncolloidal particles, J. Rheol. 395, Ancey, C., and P. Coussot, Transition frictionnelle visqueuse pour une suspension granulaire, C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron. 327, Bagnold, R. A., Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear, Proc. R. Soc. London, Ser. A 225, Barnes, H. A., Shear-thickening dilatancy in suspensions of nonaggregating solid particles dispersed in Newtonian liquids, J. Rheol. 332, Barnes, H. A., A review of the slip wall depletion of polymer-solutions, emulsions and particle suspensions in viscometers-its cause, character, and cure, J. Non-Newtonian Fluid Mech. 563, Barnes, H. A., J. F. Hutton, and K. Walters, An Introduction to Rheology Elsevier, Amsterdam, Batchelor, G. K., Effect of Brownian motion on bulk stress in a suspension of spherical particles, J. Fluid Mech. 83, Batchelor, G. K., and T. Green, The determination of the bulk stress in a suspension of spherical particles to order c, J. Fluid Mech. 56, Bertola, V., F. Bertrand, H. Tabuteau, D. Bonn, and P. Coussot, Wall slip and yielding in pasty materials, J. Rheol. 475, Brady, J. F., and J. F. Morris, Microstructure of strongly sheared suspensions and its impact on rheology and diffusion, J. Fluid Mech. 348, Cassar, C., M. Nicolas, and O. Pouliquen, Submarine granular flows down inclined planes, Phys. Fluids 1710, Chambon, G., A. Ghemmour, and D. Laigle, Gravity-driven surges of a viscoplastic fluid: An experimental study, J. Non-Newtonian Fluid Mech , Chang, C. Y., and R. L. Powell, Effect of particle-size distributions on the rheology of concentrated bimodal suspensions, J. Rheol. 381, Chevalier, C., A. Lindner, and E. Clement, Destabilization of a saffman-taylor fingerlike pattern in a granular suspension, Phys. Rev. Lett. 9917, Chong, J. S., E. Christia, and A. D. Baer, Rheology of concentrated suspensions, J. Appl. Polym. Sci. 158, Cochard, S., and C. Ancey, Experimental investigation of the spreading of viscoplastic fluids on inclined planes, J. Non-Newtonian Fluid Mech , Coussot, P., Rheometry of Pastes Suspensions and Granular Materials Wiley, New York, Coussot, P., and C. Ancey, Rhéophysique des Pâtes et des Suspensions EDP Sciences, Paris, Coussot, P., Q. D. Nguyen, H. T. Huynh, and D. Bonn, Avalanche behavior in yield stress fluids, Phys. Rev. Lett. 8817, a. Coussot, P., J. S. Raynaud, F. Bertrand, P. Moucheront, J. P. Guilbaud, H. T. Huynh, S. Jarny, and D. Lesueur, Coexistence of liquid and solid phases in flowing soft-glassy materials, Phys. Rev. Lett. 8821, b. Einstein, A., Zur theorie der Brownschen bewegung, Ann. Phys. 324, Fall, A., N., Huang, F. Bertrand, G. Ovarlez, and D. Bonn, Shear thickening of cornstarch suspensions as a

15 INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS 79 reentrant jamming transition, Phys. Rev. Lett. 1001, Frankel, N. A., and A. Acrivos, On the viscosity of a concentrated suspension of solid spheres, Chem. Eng. Sci. 22, Gadalamaria, F., and A. Acrivos, Shear-induced structure in a concentrated suspension of solid spheres, J. Rheol. 24, MiDi, GDR, On dense granular flows, Eur Phys J. E Soft Matter 144, Goyon, J., A. Colin, G. Ovarlez, A. Ajdari, and L. Bocquet, Spatial cooperativity in soft glassy flows, Nature London , Huang, N., and D. Bonn, Viscosity of a dense suspension in Couette flow, J. Fluid Mech. 590, Huang, N., G. Ovarlez, F. Bertrand, S. Rodts, P. Coussot, and D. Bonn, Flow of wet granular materials, Phys. Rev. Lett. 942, Jana, S. C., B. Kapoor, and A. Acrivos, Apparent wall slip velocity coefficients in concentrated suspensions of noncolloidal particles, J. Rheol. 396, Jop, P., Y. Forterre, and O. Pouliquen, A constitutive law for dense granular flows, Nature London , Krieger, I. M., Rheology of monodisperse lattices, Adv. Colloid Interface Sci. 3, Krieger, I. M., and T. J. Dougherty, A mechanism for non-newtonian flow in suspensions of rigid spheres, Trans. Soc. Rheol. 3, Landau, L. D., and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics Pergamon Press, Oxford, 1966, Vol.6. Leighton, D., and A. Acrivos, Measurement of shear-induced self-diffusion in concentrated suspensions of spheres, J. Fluid Mech. 177, Leighton, D. T., The shear induced migration of particulates in concentrated suspensions, Ph.D. thesis, Standford University, Lyon, M. K., and L. G. Leal, An experimental study of the motion of concentrated suspensions in twodimensional channel flow. Part 1. Monodisperse systems, J. Fluid Mech. 363, Macosko, C. H., Rheology: Principles, Measurements, and Apllications Wiley, New York, Mills, P., and P. Snabre, Rheology and structure of concentrated suspensions of hard-spheres-shear-induced particle migration, J. Phys. I 510, Morris, J. F., and F. Boulay, Curvilinear flows of noncolloidal suspensions: The role of normal stresses, J. Rheol. 435, Morris, J. F., and J. F. Brady, Pressure-driven flow of a suspension: Buoyancy effects, Int. J. Multiphase Flow 241, Nott, P. R., and J. F. Brady, Pressure-driven flow of suspensions Simulation and theory, J. Fluid Mech. 275, Ovarlez, G., F. Bertrand, and S. Rodts, Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging, J. Rheol. 503, Raynaud, J. S., P. Moucheront, J. C. Baudez, F. Bertrand, J. P. Guilbaud, and P. Coussot, Direct determination by nuclear magnetic resonance of the thixotropic and yielding behavior of suspensions, J. Rheol. 463, Rognon, P. G., F. Chevoir, H. Bellot, F. Ousset, M. Naaim, and P. Coussot, Rheology of dense snow flows: Inferences from steady state chute-flow experiments, J. Rheol. 523, Stickel, J. J., and R. L. Powell, Fluid mechanics and rheology of dense suspensions, Annu. Rev. Fluid Mech. 37, Timberlake, B. D., and J. F. Morris, Particle migration and free-surface topography in inclined plane flow of a suspension, J. Fluid Mech. 538, Zarraga, I. E., and D. T. Leighton, Normal stress and diffusion in a dilute suspension of hard spheres undergoing simple shear, Phys. Fluids 133, Zarraga, I. E., D. A. Hill, and D. T. Leighton, The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids, J. Rheol. 442,