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 a2 / h, where γ = U / h is the characteristic shear rate in the cross-section, a is particle radius, h is the characteristic dimension of the cross-section and U is the axial velocity scaling. More precisely, the secondary velocity field is linked to the concentration distribution equation through the convective term which scales as the Peclet number Pe = h2 / a2. In order to model the suspension as a continuum, the particle size a must be much smaller than conduit dimension h. Therefore, operating Peclet numbers in suspension experiments are high, usually much greater than 100. Thus, even though the numerical magnitude of the secondary currents is small, the effect of these currents on the concentration distribution can be strong. In this paper, we explore the impact of the secondary currents on the steady state concentration and velocity fields in different geometries. The suspension is modeled using the suspension balance approach of Nott and Brady (1994) coupled with the constitutive equations of Zarraga et al. (2000). Some surprising effects of the second normal stress difference induced secondary flow on the concentration distribution in a conduit are:
(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 ψc 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 ψ = (U μ0 / R2 Δρg) for which the particle average velocity in the tube at steady state is equal to the fluid average velocity. Here μ0 is the viscosity of the suspending fluid, R is the radius of the tube and Δρ is the density difference between the particles and the suspending fluid. The isotropic model underpredicts the experimentally measured critical Shields parameter for resuspension by a factor of 4 at 40% average concentration, but the anisotropic model with the second normal stress difference effects included captures ψc to within 15%.
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.
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