Viscous resuspension in a tube: The impact of secondary flows resulting from second normal stress differences

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The viscous resuspension of non-neutrally buoyant particles has been modeled as a competition between the rate of shear-induced diffusion and sedimentation or as a balance between gradients in the particle stress and gravity. Typically, however, the rheology of the suspension has been modeled using a simple concentration-dependent Newtonian viscosity. In this paper, we demonstrate through theory and comparison with existing experimental results that the anisotropy of the total stress tensor must be included to accurately describe the resuspension process in a tube. At steady state, the isotropic model predicts a secondary current within the tube cross section that flows downwards at the center and upwards near the sidewalls, resulting in a concave upward interface between the clear suspending fluid and the particles [K. Zhang and A. Acrivos, Int. J. Multiphase Flow 20, 579 (1994)]. In contrast, with the inclusion of the known non-Newtonian suspension rheology, the secondary current profile is reversed: upwards near the center and downwards near the walls. This leads to a concave downward shape of the interface between the suspending fluid and the suspension and is in quantitative agreement with the experimental measurements of Altobelli et al. [J. Rheol. 35, 721 (1991)].