^{1,a)}, Patrick Snabre

^{1,b)}and Bernard Pouligny

^{1,c)}

### Abstract

We investigate, both experimentally and theoretically, the flow and structure of a slurry when sheared between two horizontal plates. The slurry, otherwise called “wet granular material,” is made of non-Brownian particles immersed in a viscous fluid. The particles are *heavier* than the immersion fluid, in contrast to the so-called “suspensions,” corresponding to density-matched fluid and particles. Consequently, gravity influences the structure and flow profiles of the sheared material. Experiments are carried out in a plane Couette device with a model slurry composed of quasimonodispersed spherical polymethylmetacrylate particles in oil, at high average solid concentration (about 59%). Optical observation reveals a typical two-phase configuration, with a fluidized layer in contact with the upper plate and on top of an amorphous solid phase. We provide data on velocity profiles, wall slip, average shear stress, and average normal stress, versus the angular velocity of the upper plate. To interpret the data, we propose a model for the ideal case of infinite horizontal flat plates (plane Couette flow). The model, of mean-field type, is based on local constitutive equations for the tangential and normal components of the stress tensor and on material expressions relating the material viscometric coefficients (the shear viscosity and the normal stress viscosity) with the local concentration () and the local shear rate. One-, two-, and three-phase configurations are predicted, with nonlinear flow and concentration profiles. We conclude that model equations correctly describe the experimental data, provided that appropriate forms are chosen for the divergences of and near the packing concentration (), namely, a singularity.

This work is financially supported by DGA, SNPE-SME, and Région-Aquitaine. We thank C. Barentin, G. Ovarlez, P. Mills, S. Marchetto, M. Gaudré, and C. Marraud, for fruitful discussions, and the CRPP Instrumentation group for technical support.

I. INTRODUCTION

II. EXPERIMENTAL PROCEDURE

A. Sample

B. Shear apparatus

C. Optics

D. Rheometry

III. RESULTS

A. Flow

B. Rheometry

IV. MODEL

A. Basics

B. Concentration profile and phase sequence

C. Velocity profile and rheological equations

V. DISCUSSION

A. Velocimetry

B. Rheometry

C. Parameter values

D. The singularity

VI. CONCLUSION

### Key Topics

- Shear rate dependent viscosity
- 27.0
- Suspensions
- 19.0
- Shear flows
- 14.0
- Slurries
- 14.0
- Wall slip
- 11.0

## Figures

Scheme of the viscous resuspension problem. The top plate of the shear device is moving, as indicated by the arrow, from left to right. The system under shear adopts (a) three-phase (b), two-phase [(c) or (d)], and one-phase (e) configuration with increasing plate velocity.

Scheme of the viscous resuspension problem. The top plate of the shear device is moving, as indicated by the arrow, from left to right. The system under shear adopts (a) three-phase (b), two-phase [(c) or (d)], and one-phase (e) configuration with increasing plate velocity.

Couette cell and shear experiment preparation. (a) Initially the slurry is prepared with a slight excess of immersion fluid inside the stator (radius ). (b) The rotor (radius ) is brought in contact with the sediment. The excess immersion fluid forms a ring of height between the coaxial cylinders. See Appendix A for definitions of other symbols.

Couette cell and shear experiment preparation. (a) Initially the slurry is prepared with a slight excess of immersion fluid inside the stator (radius ). (b) The rotor (radius ) is brought in contact with the sediment. The excess immersion fluid forms a ring of height between the coaxial cylinders. See Appendix A for definitions of other symbols.

Scheme of the optical setup. In this configuration, the experiment provides a view of fluorescent tracers inside a vertical plane across the Couette device.

Scheme of the optical setup. In this configuration, the experiment provides a view of fluorescent tracers inside a vertical plane across the Couette device.

Fluctuations of the flow and wall slip. (a) The photos are snapshots of the velocity field in a vertical plane. Note the large fluctuations of the DPIV vectors in the flow direction. (b) Time fluctuations of the horizontal component of the velocity, at different depths (, 1, and , from top to bottom), scaled to the plate velocity for and . (c) The coupling of the suspension to the upper plate fluctuates in time. The wall-slip effect is defined through the ratio, where . The coupling is very poor at small speed, but improves up to a plateau value in the high-speed regime.

Fluctuations of the flow and wall slip. (a) The photos are snapshots of the velocity field in a vertical plane. Note the large fluctuations of the DPIV vectors in the flow direction. (b) Time fluctuations of the horizontal component of the velocity, at different depths (, 1, and , from top to bottom), scaled to the plate velocity for and . (c) The coupling of the suspension to the upper plate fluctuates in time. The wall-slip effect is defined through the ratio, where . The coupling is very poor at small speed, but improves up to a plateau value in the high-speed regime.

Average shear flow. (a) Velocity profiles, scaled to , at constant and variable . (b) Averaged video sequences at variable and constant ( in this example) used to determine the thickness of the flowing region.

Average shear flow. (a) Velocity profiles, scaled to , at constant and variable . (b) Averaged video sequences at variable and constant ( in this example) used to determine the thickness of the flowing region.

Thickness of the flowing region vs local plate velocity . (a) The graph is composed of three curves corresponding to three rotors speeds. Finite-size effects due to coupling to the outer boundary of the shear device are responsible for the decrease of at the cell periphery. (b) vs (filled symbols) and vs (open symbols), in log-log representation. The dotted straight line is the prediction of the model in the asymptotic regime [Eq. (19)], with , , , , , and .

Thickness of the flowing region vs local plate velocity . (a) The graph is composed of three curves corresponding to three rotors speeds. Finite-size effects due to coupling to the outer boundary of the shear device are responsible for the decrease of at the cell periphery. (b) vs (filled symbols) and vs (open symbols), in log-log representation. The dotted straight line is the prediction of the model in the asymptotic regime [Eq. (19)], with , , , , , and .

Average shear stress (a) and average normal stress (b) vs angular velocity of the rotor in controlled shear mode (c.sh.) or controlled stress mode (c.str.) for mean particle volume fraction and gap . (c) Average normal stress vs average shear stress . The dashed line is the prediction of the model, according to Eq. (22), with , , , , , , , and . The straight line (mixed) corresponds to the asymptotic relation Eq. (23) with .

Average shear stress (a) and average normal stress (b) vs angular velocity of the rotor in controlled shear mode (c.sh.) or controlled stress mode (c.str.) for mean particle volume fraction and gap . (c) Average normal stress vs average shear stress . The dashed line is the prediction of the model, according to Eq. (22), with , , , , , , , and . The straight line (mixed) corresponds to the asymptotic relation Eq. (23) with .

Scheme of the theoretical infinite-plate geometry. and are the tangential and normal stresses exerted by the sheared slurry upon the upper plate.

Scheme of the theoretical infinite-plate geometry. and are the tangential and normal stresses exerted by the sheared slurry upon the upper plate.

Graphical construction to derive concentration fields. (a) Shape of the master curve [Eq. (9)]. The inflexion point is inside the [0,1] physical concentration range only if is large. (b) The sample initial state, in theory, is a two-phase configuration, with a small layer (thickness ) of immersion fluid on top of a sediment . [(c)–(e)] Sequence of three-, two-, and one-phase configurations corresponding to increasing values of .

Graphical construction to derive concentration fields. (a) Shape of the master curve [Eq. (9)]. The inflexion point is inside the [0,1] physical concentration range only if is large. (b) The sample initial state, in theory, is a two-phase configuration, with a small layer (thickness ) of immersion fluid on top of a sediment . [(c)–(e)] Sequence of three-, two-, and one-phase configurations corresponding to increasing values of .

Scaled velocity fields. (a) Data recorded at constant and variable . (b) Constant and variable .

Scaled velocity fields. (a) Data recorded at constant and variable . (b) Constant and variable .

Average shear stress vs rotor velocity for pure oil filling the space between parallel plates and for , 2, 5, . The corresponding straight lines are the predictions from Eq. (A3) with and . The bold solid line indicates the dependence for the parallel-disk geometry without edge effects.

Average shear stress vs rotor velocity for pure oil filling the space between parallel plates and for , 2, 5, . The corresponding straight lines are the predictions from Eq. (A3) with and . The bold solid line indicates the dependence for the parallel-disk geometry without edge effects.

Theoretical concentration profiles (a), velocity profiles (b), and thickness of the flowing region vs the tangential stress (c) and vs the slurry boundary velocity (d), in infinite-plate geometry. Solid and dotted lines correspond to the full numerical solution and to asymptotic expressions Eqs. (15)–(20), respectively, with , , , , , , , and .

Theoretical concentration profiles (a), velocity profiles (b), and thickness of the flowing region vs the tangential stress (c) and vs the slurry boundary velocity (d), in infinite-plate geometry. Solid and dotted lines correspond to the full numerical solution and to asymptotic expressions Eqs. (15)–(20), respectively, with , , , , , , , and .

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