Inclined plane rheometry of a dense granular suspension C. Bonnoit, T. Darnige, E. Clement, and A. Lindnera) Laboratoire de Physique et Mécanique des Milieux Hétégogènes (PMMH), UMR 7636, CNRS-ESPCI, Universités Paris 6 et 7, 10 Rue Vauquelin, 75231 Paris Cedex 05, France (Received 24 March 2009; final revision received 9 September 2009; published 25 January 2010兲
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. 关J. Rheol. 50共3兲, 259–292 共2006兲兴. 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. © 2010 The Society of
Rheology. 关DOI: 10.1122/1.3258076兴
I. INTRODUCTION Understanding a dense suspensions’s resistance to flow is of great importance for multiple applications in industry 共food processing, drilling fluids, concrete or cement兲, daily life 共cosmetics, paints兲, and geophysics 共mud or lava flow, land slides兲. 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 关Stickel and Powell 共2005兲兴. Reliable measurements require complicated and costly local measurement techniques as, for example, magnetic resonance imaging 共MRI兲 techniques 关Ovarlez et al. 共2006兲; Huang and Bonn 共2007兲兴. 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 关Macosko 共1994兲; Barnes et al. 共1989兲兴, 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 关GDR 共2004兲兴, submarine flows 关Cassar et al. 共2005兲兴, or yield stress fluids 关Coussot et al. 共2002a兲兴. a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
© 2010 by The Society of Rheology, Inc. J. Rheol. 54共1兲, 65-79 January/February 共2010兲
0148-6055/2010/54共1兲/65/15/$30.00
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We show here that it is also a good choice for dense suspensions, as wall slip effects or fracturation within the sample 关Jana et al. 共1995兲; Barnes 共1995兲; Coussot and Ancey 共1999兲兴 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. 共2006兲. 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 = Vg / Vt, with Vg as the volume occupied by the grains and Vt as the total volume 关Frankel and Acrivos 共1967兲兴. 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 关Coussot 共2005兲兴. 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 关Stickel and Powell 共2005兲兴. Viscous flow of noncolloidal particles has been extensively studied. Einstein 共1906兲 gave the first prediction for the bulk viscosity of a dilute suspension 共 ⱕ 3%兲 of hard spheres in a Newtonian liquid. We found r共兲 = 共兲 / 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 关Batchelor 共1977兲; Batchelor and Green 共1972兲兴 made a significant advance using statistical mechanics arguments to account for hydrodynamic interactions in a semidilute suspension 共 ⱕ 10%兲 of hard spheres in pure shear flow: r共兲 = 共兲 / 0 = 1 + 2.5 + 7.62. 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.
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They are valid in different ranges of volume fractions. Some of the best known models are the Leighton model 关Leighton 共1985兲兴 or the Krieger–Dougherty model 关Krieger and Dougherty 共1959兲; Krieger 共1972兲兴. The latter reads r KD 共兲 = KD共兲/0 = 共1 − /m兲n .
共1兲
The Einstein limit is recovered for n = −2.5m, 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 关Chong et al. 共1971兲; Barnes 共1989兲兴. 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. 共2000兲 who proposed for the whole range of volume fractions the empirical formula
Zr共兲 = Z共兲/0 =
exp共− 2.34兲 , 共1 − /m兲3
共2兲
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 关Gadalamaria and Acrivos 共1980兲兴, flow localization 关Huang et al. 共2005兲; Ovarlez et al. 共2006兲; Coussot 共2005兲兴, and wall slip effects 关Bertola et al. 共2003兲; Jana et al. 共1995兲; Barnes 共1995兲兴. 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 共兲. The microscopic heterogeneities at the origin of the above-mentioned effects are revealed by local viscosity measurements, for example, through MRI techniques 关Huang et al. 共2005兲; Ovarlez et al. 共2006兲; Raynaud et al. 共2002兲; Coussot et al. 共2002b兲; Huang and Bonn 共2007兲兴. With non-intrusive MRI techniques, the local velocity and volume fraction inside the bulk of a given sample can be measured. Ovarlez et al. 共2006兲 performed MRI measurements in a Couette geometry and defined the viscosity as = 共兲␥˙ , 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. 共1兲. 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 关Ovarlez et al. 共2006兲; Huang et al. 共2005兲; Coussot 共2005兲兴: 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 关Chang and Powell 共1994兲兴 or shear thickening 关Barnes 共1989兲兴 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 关Brady and Morris 共1997兲兴, leading to the apparition of normal stresses in the sample 关Zarraga and Leighton 共2001兲兴 with a linear dependence on the shear rate ␥˙ 关Bagnold 共1954兲; Ancey and Coussot 共1999兲; Huang et al. 共2005兲兴.
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Dense suspensions exhibit a yield stress: below a critical stress c, no flow is observed in the bulk 关Coussot 共2005兲; Huang et al. 共2005兲兴. Huang et al. 共2005兲 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 关GDR 共2004兲; Jop et al. 共2006兲兴. 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 关GDR 共2004兲; Jop et al. 共2006兲兴. For dense suspensions the role of the confinement pressure is still an open question 关Fall et al. 共2008兲兴. In conclusion, suspensions might behave either as dry granular materials, where frictional contacts are dominant 共low shear rates and low viscosity of the interstitial fluid兲, or as viscously dominated systems 共high shear rates and high viscosity of the interstitial fluid兲, where the contacts are lubricated 关Ovarlez et al. 共2006兲; Huang et al. 共2005兲兴. 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 = 40⫾ 5 m. The density of the spheres is = 1.05– 1.06 g cm−3. As interstitial fluid, we use a modified silicone oil 共Shin Etsu SE. KF-6011兲 共S = 116 mPa s at T = 20 ° C兲, 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 关Macosko 共1994兲; Barnes et al. 共1989兲兴 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 共typically observed for a Couette device with a small gap兲 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 共Haake RS100兲. The geometry has rough surfaces of roughness 0.4 mm to avoid slip of the
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granular material 关see a schematic representation of the geometry in Fig. 2共b兲兴. 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 关Chevalier et al. 共2007兲兴 were performed in a double Couette geometry rheometer 共Haake-RS600兲 of gap width 2 ⫻ 0.25 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 关Coussot and Ancey 共1999兲兴 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 = g共h − y兲sin , where is the density of the liquid and 共Oy兲 is the direction perpendicular to the plane. Local momentum balance follows from the Navier–Stokes equation and can be written as −g sin + 共dxy / dy兲 = 0. For flow of a Newtonian viscous fluid, the constitutive equation is xy = ␥˙ = 共dvx / dy兲, leading to a velocity profile v共y兲 = 共g sin / 2兲y共2h − y兲 关Landau and Lifshitz 共1966兲兴. The surface velocity at y = h reads vs =
gh2 sin . 2
共3兲
Measuring the thickness of the layer h and the surface velocity vs yields the viscosity of the viscous fluid as follows:
=
gh2 sin . 2vs
共4兲
The normal stress yy is given by the hydrostatic pressure and reads yy = −g cos 共h − y兲. The thickness of the layer and as a consequence the tangential stress is adjusted by the system itself 共in contrast to typical set-ups used in classical rheology兲. 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 关GDR 共2004兲兴, submarine flows 关Cassar et al. 共2005兲兴, or yield stress fluids 关Coussot et al. 共2002a兲兴. It is also an appropriate rheometer for different types of complex fluids such as snow 关Rognon et al. 共2008兲兴 or viscoplastic fluids 关Chambon et al. 共2009兲; Cochard and Ancey 共2009兲兴. 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 关Fig. 1共a兲兴. 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.
BONNOIT et al.
flo w
ra te Q
(c m
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/s )
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4 0 3 0 2 0 1 0 0
4 0 0
8 0 tim e ( s )
(c)
1 2 0
1 6 0
2 5 1 2
s u r fa c e v e lo c ity V
s
2 0 1 5
5 0
(d)
(b)
® ¬
1 0
0
8 4
1 0 0
2 0 0 3 0 0 tim e ( s )
4 0 0
5 0 0
th ic k n e s s h ( m m )
(m m /s )
(a)
5 0
0
FIG. 1. Experimental set-up and measurements 共a兲 Side view. Schematic representation of the suspension flow on the inclined plane. 共b兲 Top view. Snapshot of the suspension surface taken by the top camera. The deviation of the laser slice 䉭 is used to measure the height h of the layer. Colored tracer particles allow us to measure the surface velocity vs via CIV techniques. 共c兲 Flow rate Q as a function of time. 共d兲 Typical measurements at = 15° for a volume fraction = 53.6%: surface velocity 共䉱兲 and thickness of the layer 共쎲兲 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 共H兲 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
冏 冏
dH . dt
共5兲
We use a second camera above the experiment to visualize the flow 关snapshot of the experiment on Fig. 1共b兲兴. 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 ® vs with a correlation image velocimetry 共CIV兲 technique 共using DaVis software兲. The two cameras are synchronized and coupled to computers for direct image acquisition. In this way, we measure h and vs at every moment of the experiment 关see Fig. 1共d兲兴. At the beginning of the experiment, the suspension front passes the camera field and we can simultaneously measure the velocity of the front vfront corresponding to the mean velocity 具v典 and the surface velocity vs. We continuously monitor the surface velocity vs, 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 关Fig. 1共c兲兴 leading to a decrease of h and vs during one
h
Z
r e la tiv e v is c o s ity h /h
v is c o s ity h
3 .0
1 0
p p
(P a .s )
INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS
4 2
1 4 2
0 .1 1
2
4
6
2
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6
1 0 -1 1 0 0 s h e a r ra te (s )
(a)
2
4
p p
h h
2 .0
71
c 1 c 2
1 .0 0 .0
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
v o lu m e fr a c tio n f
(b)
(c)
FIG. 2. Classical rheology. 共a兲 Shear viscosity obtained in a parallel-plate geometry: pp as a function of the shear rate ␥˙ for different grain fractions at T = 20 ° C 关 = 48% 共 䉳 兲, = 46% 共 兲, = 44% 共〫兲, = 40% 共쎲兲, = 30% 共䊏兲, = 20% 共䊊兲, = 0% 共䊐兲兴. 共b兲 Schematic representation of the parallel-plate geometry with rough surfaces and the double Couette geometry. 共c兲 Normalized shear viscosity 共兲 / Z共兲 for different grain fractions obtained in different geometries: pp 共parrallel plate兲, c1 共Couette geometry, d = 40 m兲, and c2 共Couette geometry, d = 80 m兲. c measurements from Chevalier et al. 共2007兲.
experiment and a varying shear rate applied on the layer. But even if the flow rate is not constant, the quantity 关共hvs兲 / t兴 / 共gh sin 兲 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共兲 using the parallel-plate geometry and the experimental protocol described in Sec. III. The results for different grain fractions are displayed in Fig. 2共a兲 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 关Chang and Powell 共1994兲兴. Note that shear thickening occurs at even higher volume fractions and is observed to be very weak for spherical particles 关Barnes 共1989兲兴. Below a volume fraction of = 48%, we can thus define a viscosity 共兲 independent from ␥˙ in a relatively large range of shear rates. Above a volume fraction = 48%, measurements 共with our experimental protocol兲 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. 共2007兲 using a double Couette geometry. The results for c共兲 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. 2共c兲. The figure shows the ratio of the measured viscosity 共兲 to the viscosity Z共兲 predicted by the model of Zarraga 关see
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4 0 (f ) (P a .s )
1 2 0 8 0
h
0
h
ip
4 0
(a)
0
1 0 V
2 0 S
(m /s )
3 0
4 0 x 1 0
2 0
(b)
h
1 0 0
-3
*
3 0
h
2
(m
2
)
1 6 0 x 1 0
0
h *
1 0 0
2 0 0
*
3 0 0 ¬
tim e
h /d
FIG. 3. 共a兲 Viscous behavior on the inclined plane. The square of the layer thickness h2 as a function of surface velocity vs for the angle = 15° and three different volume fractions: 3 = 57.8% 共䉭兲, 2 = 53.6% 共쎲兲, 1 = 48.8% 共䊐兲. 共b兲 Viscosity as a function of dimensionless thickness h / d. Dashed line represents the mean value of the plateau and the measured viscosity ip共兲. hⴱ is the thickness at which the measured viscosity deviates from the plateau value.
Eq. 共2兲兴 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. 2共c兲 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 关Huang et al. 共2005兲兴 and that classical rheology might need to account for particle migration 关Huang and Bonn 共2007兲; Stickel and Powell 共2005兲兴. 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 vs is proportional to the second power of the thickness of the layer h2 关see Eq. 共3兲兴. In Fig. 3共a兲 we show experimental observations for three typical experiments: at the beginning of the experiments, when h is high, the relation between h2 and vs 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
r e la tiv e v is c o s ity h
r
INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS
4
1 0
h
ip
q = 1 5 °
3
1 0
2
1 0
1
1 0
0
q = L o c a l C la s s Z a rra
1 0
h
0 .1
ip
73
5 ° v is c o s ity b y M R I ( O v a r le z e t a l.) ic a l r h e o m e tr y ( h p p ,h c 1 ,h c 2 ) g a m o d e l
0 .2
0 .3
0 .4
0 .5
0 .6
v o lu m e fr a c tio n f FIG. 4. Relative shear viscosity r共兲 = 共兲 / 0 as a function of the volume fraction . Results from inclined plane rheometry 共䊐, 䉭兲, classical rheometry 共 兲, and local measurements 共쎲兲 made by Ovarlez et al. 共2006兲. The line represents the prediction of the Zarraga model 共Zarraga et al. 2000兲.
ip共兲 =
gh2 sin , 2vs
共6兲
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 h2 is not proportional to vs anymore. This becomes clearer when plotting the viscosity ip共兲 as a function of h / d as can be seen in Fig. 3共b兲. In the beginning of the experiments 共high h兲, 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 关given by yy = −g cos 共h − y兲兴. Toward the end of the experiments 共low h兲 we observe a systematic decrease in the viscosity. This deviation occurs at a given value of the layer thickness hⴱ. hⴱ 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 10–150 Pa and the typical range of shear rate is 1 – 20 s−1 estimated by the ratio vs / h. For low volume fractions, the typical range of pressure is 100–150 Pa and the typical range of shear rate is 0.1– 1 s−1. The classical rheological measurements have been performed for shear rates of 1 – 300 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 关see Fig. 3共b兲兴. The results for r共兲 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 共from 35% to 61%兲.
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0 .5 0
0 .6 0
v o lu m e fr a c tio n f FIG. 5. Normalized shear viscosity ip共兲 / Z共兲 as a function of grain fraction for different angles obtained on the inclined rheometer: = 15° 共䉭兲, = 5° 共䊐兲. Local measurements 共쎲兲 made by Ovarlez et al. 共2006兲. 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. 共2006兲.
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 共 ⬎ 50%兲 our results can only be compared to local measurements as those obtained by Ovarlez et al. 共2006兲 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 关see Eq. 共1兲兴 when allowing two free fit parameters. We find a maximum packing fraction m = 62% and an exponent n = 2.35 共fit not shown兲. This fit does not recover the Einstein limit. Ovarlez et al. 共2006兲 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 关Stickel and Powell 共2005兲兴. 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 共dashed line on Fig. 5兲 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. 共2006兲. Their adjustment has the
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 关Timberlake and Morris 共2005兲; Lyon and Leal 共1998兲兴 and theoretical studies 关Gadalamaria and Acrivos 共1980兲; Morris and Brady 共1998兲; Brady and Morris 共1997兲; Leighton and Acrivos 共1987兲; Acrivos 共1995兲兴 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 v共y兲 = 共g sin / 2兲y共2h − y兲 = vs共y / h2兲共2h − y兲 = vs共1 − 共共y − h兲 / h兲2兲 evolves toward flatter profiles when migration of particles takes place 关Lyon and Leal 共1998兲兴. This change can be described by the use of an index n larger than 2 and the flow profile is then defined as
冉 冉 冊冊
v共y兲 = vs 1 −
y−h h
n
.
共7兲
The mean velocity can be written as 具v典 = 共n / 共n + 1兲兲vs leading to the following expression for the index n: n=
1 . 共vs/具v典兲 − 1
共8兲
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 关see Fig. 6共a兲兴. 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. 6共b兲. 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 共1998兲 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 共as given above兲 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 关Fig. 6共b兲兴. 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
76
BONNOIT et al. 4 3 n
2 1 0
(a)
(b)
0 .3 0
0 .4 0
0 .5 0
v o lu m e fr a c tio n
0 .6 0
f
FIG. 6. Flow profile. 共a兲 Schematic representation of the surface velocity vs and the mean velocity vfront when the flow is passing through the camera field. 共b兲 Power law index n = 1 / 共共vs / 具v典兲 − 1兲 as a function of volume fraction : = 15° 共䉭兲, = 5° 共䊐兲, obtained on the inclined plane and 共䊊兲 deduced from a fully developed flow profile in a Hele-Shaw cell obtained by Lyon and Leal 共1998兲. 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 10–50. 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 关Morris and Boulay 共1999兲; Nott and Brady 共1994兲; Mills and Snabre 共1995兲; Morris and Brady 共1998兲兴. 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 共2005兲 reported particle migration in inclined plane flow using a geometry similar to ours. They work however with much thinner layer thicknesses 共typically below 50 particle diameters兲, 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 共see Fig. 3兲 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 hⴱ, 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 hⴱ which defines the limit of validity of our experimental method. Below hⴱ the measurements should not be considered when measuring the viscosity of a dense suspension on an inclined plane. We define hⴱ 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 共see Fig. 3兲. This thickness hⴱ depends strongly on
)
INCLINED PLANE RHEOMETRY OF DENSE SUSPENSIONS
2 .5
.
a v e ra g e s h e a r ra te g (s
-1
1 2
77
1 0 h (m m )
8
*
6 4 2 0 0 .4 5
(a)
0 .5 0 0 .5 5 v o lu m e fr a c tio n f
0 .6 0
(b)
2 .0 1 .5 1 .0 0 .5 0 .0
0 .4 5
0 .5 0
0 .5 5
0 .6 0
v o lu m e fr a c tio n f
FIG. 7. Limits of the viscous behavior. 共a兲 Cross-over thickness hⴱ and 共b兲 critical shear rate delimiting the validity of the viscous regime: = 15° 共䉭兲, = 5° 共䊐兲. Dashed line is a guide for the eyes.
the volume fraction 关see Fig. 7共a兲兴. The bigger the volume fraction, the sooner the deviation occurs during an experiment and the bigger is hⴱ. We show the average shear rates observed at the cross-over on Fig. 7共b兲. 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. 共2005兲 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. 共2008兲. 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. 共2008兲 might be responsible of this effect.
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ACKNOWLEDGMENTS We thank Jose Lanuza for help with the experimental set-up and Guylaine Ducouret 共PPMD-ESPCI兲 for help with the rheological measurements. They acknowledge many interesting discussions with Guillaume Ovarlez 共Institut Navier, Université Paris Est兲 and thank him for a critical reading of the paper. This work is supported by the ANR PIGE 2006-2009: “Physique des Instabilités Gravitaires et Érosives.”
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