Flow and viscous resuspension of a model granular paste in a large gap Couette cell

Transcription

1 Flow and viscous resuspension of a model granular paste in a large gap Couette cell Résumé : O. BLAJ a,c, P. SNABRE a, S. WIEDERSEINER b, C. ANCEY b and B. POULIGNY a a. Centre de Recherche Paul-Pascal CNRS, Pessac, France b. Ecole Polytechnique Fédérale de Lausanne, CH-1015, Switzerland c. MSD, Oss, The Netherlands Wet sand, slurries and fresh concrete are constituted of grains immersed in a viscous fluid. As the grains are large (non Brownian) and in large concentration, such materials behave as very viscous pastes, hence the name granular paste. Since the 1980s, much research has been done to describe the flow properties of such materials from practical and theoretical perspectives. Gravity tends to accumulate heavy particles in highly concentrated sediments that cannot flow (jammed state). Conversely, shear promotes particle dispersion through shear-induced diffusion. Viscous resuspension is the name given to the competition between shear and gravity. Most of the works dedicated to flow properties of non colloidal and granular suspensions have addressed the limiting case of isodense systems, i.e. when the buoyant forces experienced by the particles vanish (g eff 0). In this communication, we report experimental results both in the g eff 0 limit, and with a non isodense system (g eff >0, meaning that the particles are heavier than the fluid). We study the flow and particles spatial distribution within a model granular paste in a large gap Couette cell, with the objective of studying viscous resuspension. In our experiments, both the material and shear cell are transparent, allowing for a complete characterization through optical and video means. We report systematic measurements of the azimuthal velocity and concentration fields, obtained by videotrajectography and laser induced fluorescence photometry. The data are analyzed through the suspension balance model developed by Morris and Boulay (MB) for neutrally buoyant particle suspensions (J. Rheol. 43, 1213 (1999)). In the g eff 0 limit, comparison of the velocity and concentration profiles between the theory and the experiments is found quantitatively satisfactory, using the viscometric functions proposed by MB. In the non isodense case, the comparison is approximate and only qualitative agreement is reached in the description of viscous resuspension patterns. Mots clefs: non colloidal suspensions, wet granular flow, non Newtonian rheology. 1 Introduction The work reported here is a contribution to the general problem of understanding flows of non colloidal suspensions and granular pastes, in the low Reynolds number limit. Such flows can be studied in the laboratory using simple geometries, such as the Couette shear-cell (concentric cylinders). Shearing a suspension in such devices generally leads to a decrease of the concentration in particles (φ ) near the inner cylinder, with a concomitant increase near the outer one, a phenomenon known as migration [1]. Elaboration of predictive models for migration has been the matter of numerous works since the eighties. In a simple representation, the material is assimilated to a continuous effective fluid, characterized by position and time variable concentration φ ( r, t) and velocity v ( r, t) [2-5]. The flow of a granular suspension may then be regarded as a two-field problem, involving coupled velocity and concentration fields. This problem has been documented on a number of experiments, in which the materials consisted of nearly identical spherical particles immersed in a Newtonian fluid (see e.g. [4, 6-8] and references therein). In general, the interstitial fluid was density-matched to the particles, which were then neutrally buoyant. Using magnetic resonance imaging (MRI) has made it possible to measure the φ ( r) and v ( r) distributions [6] in a steady regime, in approximately isodense condition. The neutral buoyancy condition (particle-fluid density difference ρ = 0) simplifies the problem by eliminating the influence of the effective gravity (g eff 0), that otherwise tends to accumulate particles at bottom (sedimentation) or top (creaming) of the shear-cell. 1

2 However gravity is active ( ρ >0 or <0, then g eff 0) in real materials, leading to the phenomenon called viscous resuspension [9]. The goal of the present work is to report experiments on viscous resuspension in a large gap Couette cell, where migration is intense. We present data related to the φ ( r) and v ( r) distributions obtained using novel flow imaging techniques. We investigate both the g eff 0 and g eff 0 cases. Our data are shortly analyzed through the Morris-Boulay (MB) [3] force balance model in the last section. The model material in our experiments is made of transparent acrylate spheres, about 180 µm in diameter, immersed in a fluid with the same index of refraction as that of the particles. The suspension is thus transparent. The average concentrationφ % is 55%. The material is sheared in a transparent Couette cell (Fig. 1), whose cylinders (inner and outer radii R=15 mm and kr =20 mm, respectively) are driven by independent motors (see [4]), at angular velocitiesω 1,2. The device is operated at varyingω 1 and ω 2, but at a constant Ω = ω1 ω2 difference. In our experiments, the fluid is made slightly fluorescent using a dye [10]. The cell is lit by a laser sheet, which can be vertical or horizontal. The vertical configuration is used for concentration measurements: we measure the fluorescence intensity within the fluid, which is simply. Velocity measurements are achieved using the horizontal sheet (in green in Fig. 1). The technique, named multiparticle video trajectography, is based on the analysis of trajectories of fluorescent tracers or of the small volumes of fluorescent fluid between the particles [10, 11]. proportional to 1 φ ( r) FIG.1: Sketch of the Couette cell. The figure shows the vertical diametric cut made by the laser sheet used for concentration measurements. The light brown colour represents the fluorescence from the immersion fluid. In this case, observation is made along the yˆ = xˆ zˆ direction. For velocimetry experiments, we use a horizontal laser sheet, which is coloured green in the figure. Images are captured from below, in the ẑ direction. The same camera is used for both types of measurements. The height occupied by the sample (approximately the vertical dimension of the brown sheet in the sketch) is about 30 mm. The surfaces of both cylinders are made rough, to limit wall slip phenomena. 2 Density matched suspension Here the immersion fluid, a mixture of three liquids provided by Cargille Laboratories, is both index- and density-matched to the particles [12]. Figure 2 shows the velocity and concentration profiles across the gap of the Couette cell, for different values of the average shear rate. Note that the plots give the angular velocity in scaled form, namely ω ( r) = v ( r) Ωr. The data have been recorded at several millimetres above the cell bottom (Fig. 1), to avoid end effects. We found that the profiles exhibit z-invariance, as for an ideal infinitely high cell, provided z 10 mm. The measured ( r) φ profiles provide clear evidence of the migration effect, with φ varying from 0.54 to 0.58 through the gap. The experiment also reveals layer structures next to the gap boundaries, especially over a length of two particle diameters from the inner cylinder wall, see the peak in φ ( r). The angular velocity profile takes a nearly exponential shape in the radial direction r throughout the gap. We find that limr R ω ( r) 1 and limr kr ω ( r) 0 : the particle phase does not slip along the cylinder walls. No shearbanding, i.e. the presence of a jammed phase coexisting with a sheared region, is observed for the average concentration of interest (φ % =55%). Remarkably, φ ( r) and ω ( r) are almost independent of Ω, meaning that the spatial distribution and the angular velocities (in scaled form) of the particles remain the same independently of the average shear rate. This finding confirms the observations of [6], using MRI. Figure 3 shows the scaled local shear rate, & γ ( r) Ω, through the gap. The fact that increasing Ω (the cause) does not result in increased migration (the effect) may seem paradoxical. However this feature is consistent with the suspension force balance model (see section 4). 2

3 FIG. 2 (above): Radial concentration and velocity profiles of the isodense suspension for different angular velocities. Each panel corresponds to a given altitude, whose value is indicated at top. Note that profiles between 15.5 and 19.5 mm are well reproducible, while amplitudes of structural peaks may vary [10]. FIG. 3 (at right): Dimensionless z-averaged shear rate profile in the 55% density-matched suspension for increasing Ω. 3 Non isodense suspensions: effect of gravity Here the interstitial fluid, a mixture of hexadecane and microscope immersion oil, is less dense than the particles: ρ 0.3 g/cm 3. Starting from a pre-mixed suspension, we observe that particles settle within about 10 mn. Operating the shear cell has the consequence of partially resuspending the particles; the effect obviously increases with Ω. We studied resuspension for increasing values of Ω, ranging from rad/s to rad/s. Color maps of concentration and angular velocity fields for Ω = 0.57 rad/s are shown in Fig. 4. Note that both fields, φ ( r, and ω ( r,, strongly depend on z due to the competition between shear and gravity. The spatial distributions also greatly depend on Ω, contrary to the isodense case. Dependence on Ω is illustrated in Figs 5 and 6, for a given z. The graphs in Fig. 5 reveal that a fluid region, similar to that observed throughout the gap in the isodense system, coexists with a nearly jammed phase near the outer cylinder. The latter phase, corresponding to the tails of the velocity profiles, is not strictly jammed, as it still moves, but at a very low velocity. Moreover its concentration decreases with increasing Ω. Note that there is no sharp boundary between both zones, but rather a smooth transition (the graph at right in Fig. 5 indicates a crossover radius r* 17.5 mm). Another feature is the significant slip of the particle phase along the inner cylinder, which also depends on (decreases with) Ω and z. From the data taken at the very bottom of the Couette cell, we estimate φm 0.64 for the jamming concentration in the non isodense suspension. The nearly jammed zone is everywhere well below the jamming limit and is highly fluctuating. Therefore this phase is not a solid; it is in fact a very concentrated 3

4 granular paste. These observations suggest that the flow of the granular paste cannot be understood without further consideration of velocity fluctuations, which are generated by the shear itself [13]. FIG. 4: Resuspension patterns, in angular velocity (left panel) and concentration (right panel), for the non isodense suspension. The shear device is operated at Ω = 0.57 rad/s. FIG. 5 (above): Angular velocity and concentration of the non isodense suspension ( ρ > 0 ), at z = 15 mm. Velocity is plotted in semi-log scale, to show the transition between active (gray) and nearly jammed (magenta) zones. Straight lines (red) along the velocity profiles are guides to the eye. FIG. 6 (at right): Dimensionless z-averaged shear rate profile in the non density-matched suspension for increasing Ω. Only data in the active zone are represented. 4 Interpretation based on the force balance model In this model [3, 5, 14], viscous stresses involved in the suspension dynamics are simply proportional to the 4

5 shear rate, with prefactors that only depend on the concentration. τ = η0 ˆs η & γ is the shear stress, and ( p) Σ ii = λη i 0 ˆ ηn & γ are normal stresses exerted by the particle phase, with i =1, 2, 3 in the directions of velocity (θ ), velocity gradient (r) and vorticity (, respectively. η0 is the immersion fluid viscosity, and ˆs η, n ( ϕ ) are the shear and normal viscometric functions. Here ϕ = φ φm is the concentration, scaled to the jamming limitφ m. The coefficient λ1 is conventionally taken =1. Commonly adopted values for λ2 and λ3 are 0.85 and 0.50, based on measurements by Zarraga et al.[15]. ( p) In the case of the isodense system, the force balance equations only involve τ, Σθθ and Σ ( p) rr. The equations can be solved exactly for the azimuthal velocity and concentration profiles. The model generally predicts the coexistence of a sheared zone, near the inner cylinder, with a jammed phase (ϕ =1) at a distance r c > R from the cell axis. When r c > kr the jammed phase is virtual, meaning that the suspension flows throughout the Couette gap. E ϕ = ηn ηs, the ratio of normal to shear viscometric functions. The concentration profile is found given by: We define ( ) ˆ ˆ ( r, G ( r rc ) 1 where the G function is defined as G ( x) = E E ( 1) x σ ( ) 1 In Eq. (1), the -1 exponent stands for functional inversion, and σ ( 1 λ2 ) λ2 ϕ =, (1) G x = for x 1 [10]. = +. The value of r c is inferred from the boundary condition of mass conservation. ϕ given by Eq. (1) only depends on r andϕ% ; it does not depend on Ω, in line with the experimental observations with the isodense suspension,. The azimuthal velocity profile is deduced from the concentration profile, with the boundary conditions on velocity. As illustrated by the graphs in Fig. 7, theoretical profiles depend markedly on the shape of the E ( ϕ ) function. Since our experimental data correspond to concentrations around 55%, comparison of experimental and theoretical profiles bears information on E ( ϕ ) not far below the jamming limit (0.90 ϕ ( r) 0.96, supposing φ m =0.60). We find a good agreement with the function proposed by MB [3], β and similarly with a simple power-law E ( ϕ ) ϕ, with β 12, see the graphs below. The dependence on the value of β illustrates the fact that the profiles are very sensitive to the curvature of E ( ϕ ) in the region of interest. Indeed curvatures of MB and of power-law functions get close to each other for β 12. Note that the above analysis bears no information on the viscometric functions at small concentrations, as in studies by Boyer et al. [5]. FIG. 7: Density-matched suspension. (a) Stationary concentration profile; (b) velocity profile, for different values of Ω, with k=4/3, average concentration=55% and φ m =0.60. Grey circles are experimental data. When gravity is active, as is the case with the non-density matched system, the hydrostatic pressure of the particle phase is balanced by Σ zz = λ3η 0 ˆ ηn & γ. The force balance equations cannot be solved exactly for v( r, and ϕ ( r, if one still supposes that the flow is purely azimuthal. An approximate resolution has been proposed by [10], based on solutions for the isodense case. The calculation leads to a shear-banding structure for which the boundary r c between the shear zone and the jammed phase ( ϕ = 1 ) depends 5

6 explicitly on z. Theoretical profiles qualitatively reproduce the main features of the resuspension maps (Fig. 4), but the calculated velocities and concentrations markedly differ from the experimentally measured profiles. In conclusion, a quantitative model of resuspension is still lacking. 5 Conclusion We reported detailed descriptions of concentration and velocity maps of a model non colloidal suspension in a shear Couette device, in a steady regime, for an average concentrationφ % =0.55. In zero effective gravity, the experimental maps can be faithfully reproduced using the suspension force balance model, when an appropriate form of the ratio between normal and shear viscometric functions is adopted. We studied the response of the flow to gravity in non-isodense suspensions. The model still captures the main features of the experimental maps, but the agreement is qualitative and partial. How the suspension precisely responds to combined shear and gravity is still an open problem. References [1] Leighton D. and Acrivos A. The shear induced migration of particles in concentrated suspensions, Journal of Fluid Mechanics 181, , [2] Nott P. R. and Brady J. F. Pressure-driven suspension flow: simulation and theory, Journal of Fluid Mechanics 275, , [3] Morris J. F. and Boulay F. Curvilinear flows of non colloidal suspensions: the role of normal stresses, Journal of Rheology 43, , [4] Lenoble M., Snabre P. and Pouligny B. The flow of a very concentrated slurry in a parallel-plate device: influence of gravity-driven threshold, Physics of Fluids 17, , [5] Boyer F., Pouliquen O. and Guazzelli E. Dense suspensions in rotating-rod flows: normal stresses and particle migration, Journal of Fluid Mechanics 686, 5 25, [6] Ovarlez G., Bertrand F. and Rodts S. Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging, Journal of Rheology 50, , [7] Boyer F., Guazzelli E. and Pouliquen O. Unifying Suspension and Granular Rheology, Physical Review Letters 107, , [8] Dijksman J.A., Wandersman E., Slotterback S., Berardi C.R., Updegraff W.D., van Hecke and Losert W. From frictional to viscous behavior: Three-dimensional imaging and rheology of gravitational suspensions, Physical Review E 82, , [9] Leighton D. and Acrivos A. Viscous resuspension, Chemical Engineering Science 41, , [10] Blaj O. Comment coule une pâte granulaire? Etude des composantes primaire et secondaire et des fluctuations de l écoulement, PhD Dissertation, Université Bordeaux 1, [11] Snabre P., Blaj O. and Pouligny B. Shear induced diffusion of particles in a granular paste sheared in a large gap Couette cell, 2 nd IMA conference on dense granular flows, Cambridge, UK, 1-4 july [12] Wiederseiner S., Andreini N., Epely-Chauvin G. and Ancey C. Refractive index and density matching in concentrated particle suspension: a review, Experiments in Fluids 50, , [13] Bocquet L., Lorset W., Schalk D., Lubensky T.C. and Gollub J.P. Granular shear flow dynamics and forces: Experiments and continuum theory, Physical Review Letters E 65 : , [14] Garland S., Gauthier G., Martin J. and Morris J.F. Normal stress measurements in sheared non- Brownian Suspensions, Journal of Rheology 57, 71-88, [15] Zarraga I. E., Hill D. A. and Leighton D. T. The characterization of the total stress of concentrated suspensions of non colloidal spheres in Newtonian fluids, Journal of Rheology 44, ,