89e

It has been long known in polymer literature that it is not possible to have steady and rectilinear flow of a viscoelastic polymer through non-circular conduits. The non-zero second normal stress difference exhibited by polymers drives a secondary flow within the cross-section, resulting in a helicoidal flow of the polymer. This was first verified experimentally by Geisekus in 1965 for an elliptical channel. However, because of the small amplitude of the second normal stress differences in polymers, the resulting secondary currents bear magnitudes that are two to three orders smaller than the axial velocity.

It has also been observed that concentrated suspensions of non-colloidal particles in Newtonian fluids exhibit a weak, negative first normal stress difference and a strong, negative second normal stress difference [e.g. Zarraga et al. (2000)]. Therefore, analogous to the case of a viscoelastic polymer, the flow of a suspension of non-colloidal particles through a non-circular conduit should be accompanied by a secondary flow that is small in magnitude compared to the axial velocity. What has not been previously recognized is that even though the secondary velocity field is weak, its magnitude could still be much greater than the shear induced migration velocity which scales as *γ a ^{2}* / h, where

(a) Notches and corners in a cross-section are regions of lower shear stress in a geometry due to their smaller length scales. In the absence of secondary currents, the particle concentration in these low shear stress regions is expected to be high. However, the complete solution of the suspension balance equations with the inclusion of secondary flow shows that secondary currents produce a flux that drains particles out of the notches and corners in the conduit at sufficiently high Peclet numbers, resulting in local concentration minima rather than maxima.

(b) In resuspension flow through a tube at steady state, the isotropic model predicts a secondary velocity field that flows downward in the center due to higher particle concentrations (and thus local density) in this region. This convection coupled with an upward flow near the walls results in a clear fluid-suspension interface that is concave downward [e.g. Zhang and Acrivos (1994)]. With the inclusion of the particle stress anisotropy, the secondary velocity field is actually completely reversed: upward (against gravity) in the center and downward near the walls. The clear fluid-suspension interface is thus predicted to be concave upward, a result that agrees with the MRI images of Altobelli et al. (1991). Normal stress induced secondary currents are necessary for a quantitative description of the resuspension process as well. The critical Shields parameter *ψ _{c}* is a reasonable measure of the shear stress required to resuspend a settled bed of particles in a tube. It is defined as the Shields parameter

Historically, suspensions have been modeled as Newtonian fluids with concentration dependent viscosities when calculating velocity distributions due to the tremendous simplification of the governing equations. The results presented in this paper, however, demonstrate that it is critical to consider the complete rheology of a concentrated suspension when modeling flows in complex geometries. While the magnitude of the secondary currents is small, in many cases they are the dominant mechanism governing the resulting particle concentration distribution.

**References**

Altobelli, S. A., R. C. Givler and E. Fukushima, Velocity and concentration measurements of suspensions by nuclear magnetic resonance imaging, *J. Rheol*, **35**(5), 721-734, 1991.

Gadala-Maria F. and A. Acrivos, Shear induced structure in a concentrated suspension of solid spheres, *J. Rheol.*, **24**(6), 799-814, 1980.

Geisekus H., Sekundarstromingen in viskoelastischen flussigkeiten bei stationarer und periodischer bewegung, *Rheol. Acta*, **4**, 85-101, 1965.

Nott P. R. and J. F. Brady, Pressure-driven flow of suspensions: Simulations and theory, *J. Fluid Mech.*, **275**, 157-199, 1994.

Zarraga I. E., D. A. Hill and D. T. Leighton Jr., The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids, *J. Rheol*, **44**(2), 185-220, 2000.

Zhang K. and Acrivos A., Viscous resuspension in fully developed laminar pipe flow, *Intl. J. Multiphase Flow*, **20**, 579-591, 1994.

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