^{1}, G. Gauthier

^{1,a)}, J. Martin

^{1,b)}and J. F. Morris

^{2,c)}

### Abstract

Measurements in a cylindrical Taylor–Couette device of the shear-induced radial normal stress in a suspension of neutrally buoyant non-Brownian (noncolloidal) spheres immersed in a Newtonian viscous liquid are reported. The radial normal stress of the fluid phase was obtained by measurement of the grid pressure , i.e., the liquid pressure measured behind a grid which restrained the particles from crossing. The radial component of the total stress of the suspension was obtained by measurement of the pressure, , behind a membrane exposed to both phases. Pressure measurements, varying linearly with the shear rate, were obtained for shear rates low enough to insure a grid pressure below a particle size dependent capillary stress. Under these experimental conditions, the membrane pressure is shown to equal the second normal stress difference, , of the suspension stress whereas the difference between the grid pressure and the total pressure, , equals the radial normal stress of the particle phase, . The collected data show that is about 1 order of magnitude higher than the second normal stress difference of the suspension. The values obtained in this manner are independent of the particle size, and their ratio to the suspension shear stress increases quadratically with , in the range . This finding, in agreement with the theoretical particle pressure prediction of Brady and Morris [J. Fluid Mech. **348**, 103–139 (1997)] for small , supports the contention that the particle phase normal stress is due to asymmetric pair interactions under dilute conditions, and may not require many-body effects. Moreover we show that the values of , normalized by the fluid shear stress, with the suspending fluid viscosity and the magnitude of the shear rate, are well-described by a simple analytic expression recently proposed for the particle pressure.

The authors would like to thank GdR MePhy and GdR MeGe, Dr. A. Deboeuf, Dr. E. Lemaire, Dr. D. Lhuilier, Professor P. Mills, and Professor D. Salin for fruitful discussions, as well as A. Aubertin and R. Pidoux for the design and the realization of the experimental setup and acquisition procedure. This work was partly supported by the Triangle de la Physique, ANR Coliner, and NSF PREM (DMR-0934206) “Dynamics of Heterogeneous and Particulate Materials.”

I. INTRODUCTION

II. EXPERIMENTAL SETUP

A. Particles

B. Viscosity measurements

C. Experimental apparatus

III. EXPERIMENTS

A. Observations

B. Measurements

IV. RESULTS AND DISCUSSION

A. Normal stresses

V. CONCLUSION

### Key Topics

- Suspensions
- 78.0
- Pressure measurement
- 21.0
- Normal stress difference measurements
- 18.0
- Shear rate dependent viscosity
- 16.0
- Viscosity
- 11.0

## Figures

Top: Size distribution of the particles used for the suspensions. (a) Dynoseeds TS-40 ( ) and (b) Dynoseeds TS-140 ( ). Bottom: Ellipticity of the particles. Smaller axis *a* as a function of the larger axis *b* normalized with the average particle diameter. (c) Dynoseeds TS-40 ( ) and (d) Dynoseeds TS-140 ( ). The average aspect ratios *a/b* are equal to 0.924 for the TS-40 and to 0.927 for the TS-140.

Top: Size distribution of the particles used for the suspensions. (a) Dynoseeds TS-40 ( ) and (b) Dynoseeds TS-140 ( ). Bottom: Ellipticity of the particles. Smaller axis *a* as a function of the larger axis *b* normalized with the average particle diameter. (c) Dynoseeds TS-40 ( ) and (d) Dynoseeds TS-140 ( ). The average aspect ratios *a/b* are equal to 0.924 for the TS-40 and to 0.927 for the TS-140.

Rheometry: In a 50 mm parallel plate geometry, (a) viscosity of the pure fluid as a function of the shear rate . Measurements have been performed at ; (b) viscosity of the suspension as a function of the particle volume fraction, for and . The continuous line corresponds to a Krieger–Dougherty law [Krieger (1972)]: , with .

Rheometry: In a 50 mm parallel plate geometry, (a) viscosity of the pure fluid as a function of the shear rate . Measurements have been performed at ; (b) viscosity of the suspension as a function of the particle volume fraction, for and . The continuous line corresponds to a Krieger–Dougherty law [Krieger (1972)]: , with .

Sketch of the Taylor–Couette cell.

Sketch of the Taylor–Couette cell.

Lubrication pressure, calculated in the pure liquid of viscosity , for an axis misalignment and for (- -) , ( ) , (- ) , and the angular gap variation (—) (left axis). Note the cancellation of the lubrication pressure at fixed when averaged over the two directions of rotation.

Lubrication pressure, calculated in the pure liquid of viscosity , for an axis misalignment and for (- -) , ( ) , (- ) , and the angular gap variation (—) (left axis). Note the cancellation of the lubrication pressure at fixed when averaged over the two directions of rotation.

(- -) Ramp of shear rate (right hand axis) against time (*s*) and (—) pressure transducer signal (left hand axis) measured (a) behind one grid and (b) (—) behind one impermeable membrane, for a suspension of particles of diameter at volume fraction .

(- -) Ramp of shear rate (right hand axis) against time (*s*) and (—) pressure transducer signal (left hand axis) measured (a) behind one grid and (b) (—) behind one impermeable membrane, for a suspension of particles of diameter at volume fraction .

(a) Image of the experimental cell with a suspension of TS-140 of volume fraction for taken 10 s after the beginning of the shear. (b) Spatiotemporal diagram built by plotting the light intensity of a vertical line along time (horizontal direction). The white strip on the top of the suspension corresponds to a “cream” of particles that have been ejected from the suspension due to shear.

(a) Image of the experimental cell with a suspension of TS-140 of volume fraction for taken 10 s after the beginning of the shear. (b) Spatiotemporal diagram built by plotting the light intensity of a vertical line along time (horizontal direction). The white strip on the top of the suspension corresponds to a “cream” of particles that have been ejected from the suspension due to shear.

Evolution of the grid (a) and membrane (b) pressure as a function of the shear rate for a suspension with volume fraction . ( ) Dynoseeds TS-40, ( ) Dynoseeds TS-140. The two dashed lines correspond to the asymptotic values of the pressure (i.e., 540 Pa for TS-140 and 1050 Pa for TS-40).

Evolution of the grid (a) and membrane (b) pressure as a function of the shear rate for a suspension with volume fraction . ( ) Dynoseeds TS-40, ( ) Dynoseeds TS-140. The two dashed lines correspond to the asymptotic values of the pressure (i.e., 540 Pa for TS-140 and 1050 Pa for TS-40).

Evolution of the dimensionless grid (a) and membrane (b) pressure as a function of the particle volume fraction. Pressures are normalized by . ( ) Dynoseeds TS-40 and ( ) Dynoseeds TS-140. The insets display in logarithmic scale the pressures normalized with the shear stress of the suspension ( ).

Evolution of the dimensionless grid (a) and membrane (b) pressure as a function of the particle volume fraction. Pressures are normalized by . ( ) Dynoseeds TS-40 and ( ) Dynoseeds TS-140. The insets display in logarithmic scale the pressures normalized with the shear stress of the suspension ( ).

Variation of the normalized (relative to the shear stress of the pure liquid) second normal suspension stress difference , with the volume fraction . ( ) Dynoseeds TS-40, ( ) Dynoseeds TS-140, (+) ^{ Yeo and Maxey (2010) } , ( ) ^{ Sierou and Brady (2002) } , ( ) ^{ Zarraga et al. (2000) } , ( ) ^{ Singh and Nott (2003) } , and ( ) ^{ Couturier et al. (2011) } .

Variation of the normalized (relative to the shear stress of the pure liquid) second normal suspension stress difference , with the volume fraction . ( ) Dynoseeds TS-40, ( ) Dynoseeds TS-140, (+) ^{ Yeo and Maxey (2010) } , ( ) ^{ Sierou and Brady (2002) } , ( ) ^{ Zarraga et al. (2000) } , ( ) ^{ Singh and Nott (2003) } , and ( ) ^{ Couturier et al. (2011) } .

, normalized with the suspension shear stress, as a function of (a) and of (b). Our data, ( ) Dynoseeds TS-40 and ( ) Dynoseeds TS-140, are compared to the data ( ) of ^{ Sierou and Brady (2002) } and the fit (solid line) of ^{ Zarraga et al. (2000) } in (a), and to (solid straight line) in (b).

, normalized with the suspension shear stress, as a function of (a) and of (b). Our data, ( ) Dynoseeds TS-40 and ( ) Dynoseeds TS-140, are compared to the data ( ) of ^{ Sierou and Brady (2002) } and the fit (solid line) of ^{ Zarraga et al. (2000) } in (a), and to (solid straight line) in (b).

Comparison of , normalized with the fluid shear stress, , with the expression for the particle pressure, (see text).

Comparison of , normalized with the fluid shear stress, , with the expression for the particle pressure, (see text).

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