Skip to main content Accessibility help

Normal stress differences in suspensions of rigid  fibres

  • Braden Snook (a1) (a2), Levi M. Davidson (a1), Jason E. Butler (a1), Olivier Pouliquen (a2) and Élisabeth Guazzelli (a2)...


Measurements of normal stress differences are reported for suspensions of rigid, non-Brownian fibres for concentrations of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}nL^2d=1.5\text {--}3$ and aspect ratios of $L/d=11\text {--}32$ , where $n$ is the number of fibres per unit volume, $L$ is the fibre length and $d$ is the diameter. The first and second normal stress differences are determined experimentally from measuring the deformation in the free surface in a tilted trough and in a Weissenberg rheometer. Simulations are performed as well, and the hydrodynamic and contact contributions to the normal stresses are calculated. The experiments and simulations indicate that the second normal stress difference is negative and that its magnitude increases as the concentration is raised and the aspect ratio is lowered. The first normal stress difference is positive and its magnitude is approximately twice that of the second normal stress difference. Simulation results indicate that, for the concentrations and aspect ratios studied, contact forces between fibres form the dominant contribution to the normal stress differences.


Corresponding author

Email address for correspondence:


Hide All
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419440.
Batchelor, G. K. 1971 The stress generated in a non-dilute suspension of elongated particles by pure straining motion. J. Fluid Mech. 46, 813829.
Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order $c^2$ . J. Fluid Mech. 56, 401427.
Beavers, G. S. & Joseph, D. D. 1975 The rotating-rod viscometer. J. Fluid Mech. 69, 475512.
Boyer, F., Pouliquen, O. & Guazzelli, E. 2011 Dense suspensions in rotating-rod flows: normal stresses and particle migration. J. Fluid Mech. 686, 525.
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284304.
Couturier, É., Boyer, F., Pouliquen, O. & Guazzelli, É. 2011 Suspensions in a tilted trough: second normal stress difference. J. Fluid Mech. 686, 2639.
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44, 791810.
Dai, S.-C., Bertevas, E., Qi, F. & Tanner, R. I. 2013 Viscometric functions for noncolloidal sphere suspensions with Newtonian matrices. J. Rheol. 57, 493510.
Dbouk, T., Lobry, L. & Lemaire, E. 2013 Normal stresses in concentrated non-Brownian suspensions. J. Fluid Mech. 715, 239272.
Dinh, S. M. & Armstrong, C. R. 1984 A rheological equation of state for semiconcentrated fiber suspensions. J. Rheol. 28, 207227.
Einstein, A. 1905 Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. (Berlin) (4) 17, 549560; English translation by A. D. Cowper reprinted in 1956 in Investigations on the Theory of Brownian Movement, Dover Publications.
Forgacs, O. L. & Mason, S. G. 1959 Particle motions in sheared suspensions. Part 9. Spin and deformation of thread-like particles. J. Colloid Interface Sci. 14, 457472.
Franceschini, F., Filippidi, E., Guazelli, É. & Pine, D. J. 2011 Transverse alignment of fibers in a periodically sheared suspension: an absorbing phase transition with a slowly-varying control parameter. Phys. Rev. Lett. 107, 250603.
Goto, S., Nagazono, H. & Kato, H. 1986 The flow behavior of fiber suspensions in Newtonian fluids and polymer solutions. I. Mechanical properties. Rheol. Acta 25, 119129.
Hinch, E. J. 2011 The measurement of suspension rheology. J. Fluid Mech. 686, 14.
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161179.
Keshtkar, M., Heuzey, M. C. & Carreau, P. J. 2009 Rheological behavior of fiber-filled model suspensions: effect of fiber flexibility. J. Rheol. 53, 631650.
Kitano, T. & Kataoka, T. 1981 The rheology of suspension of vinyl on fibers in polymer liquids. I. Suspensions in silicone oil. Rheol. Acta 20, 390402.
Kuo, Y. & Tanner, R. I. 1974 On the use of open-channel flows to measure the second normal stress difference. Rheol. Acta 13, 443456.
Mackaplow, M. B. & Shaqfeh, E. S. G. 1996 A numerical study of the rheological properties of suspensions of rigid, non-Brownian fibres. J. Fluid Mech. 329, 155186.
Mason, S. G. & Manley, R. J. 1956 Particle motions in sheared suspensions: orientation and interactions of rigid rods. Proc. R. Soc. Lond. A 238, 117131.
Petrich, M. P., Koch, D. L. & Cohen, C. 2000 An experimental determination of the stress microstructure relationship in semi-concentrated fiber suspensions. J. Non-Newtonian Fluid Mech. 95, 101133.
Petrie, C. J. S. 1999 The rheology of fibre suspensions. J. Non-Newtonian Fluid Mech. 87, 369402.
Sepehr, M., Carreau, P. J., Moan, M. & Ausias, G. 2004 Rheological properties of short fiber model suspensions. J. Rheol. 48, 10231048.
Shaqfeh, E. S. G. & Fredrickson, G. F. 1990 The hydrodynamic stress in a suspension of rods. Phys. Fluids 2, 724.
Sierou, A. & Brady, J. F. 2002 Rheology and microstructure in concentrated non-colloidal suspensions. J. Rheol. 46, 10311056.
Singh, A. & Nott, P. R. 2000 Normal stresses and microstructure in bounded sheared suspensions via Stokesian Dynamics simulations. J. Fluid Mech. 412, 279301.
Singh, A. & Nott, P. R. 2003 Experimental measurements of the normal stresses in sheared Stokesian suspensions. J. Fluid Mech. 490, 293320.
Snook, B., Guazzelli, É. & Butler, J. E. 2012 Vorticity alignment of rigid fibers in oscillatory shear flow: role of confinement. Phys. Fluids 24, 121702.
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129149.
Stover, C. A., Koch, D. L. & Cohen, C. 1992 Observations of fibre orientation in simple shear flow of semi-dilute suspensions. J. Fluid Mech. 238, 277296.
Sundararajakumar, R. R. & Koch, D. L. 1997 Structure and properties of sheared fiber suspensions with mechanical contacts. J. Non-Newtonian Fluid Mech. 73 (3), 205239.
Tanner, R. I. 1970 Some methods for estimating the normal stress functions in viscometric flows. Trans. Soc. Rheol. 14 (4), 483507.
Trevelyan, B. J. & Mason, S. G. 1951 Particle motions in sheared suspensions. I. Rotations. J. Colloid Sci. 6, 354367.
Wineman, A. & Pipkin, A. 1966 Slow viscoelastic flow in tilted troughs. Acta Mechanica 2, 104115.
Zarraga, I. E., Hill, D. A. & Leighton, D. T. 2000 The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J. Rheol. 44, 185220.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Normal stress differences in suspensions of rigid  fibres

  • Braden Snook (a1) (a2), Levi M. Davidson (a1), Jason E. Butler (a1), Olivier Pouliquen (a2) and Élisabeth Guazzelli (a2)...


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.