The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids

Abstract

The total stress of a concentrated suspension of noncolloidal spheres in a Newtonian fluid was characterized by independent measurements in viscometric flows. Using a suspension balance formulation, the normal stress in the vorticity direction (Σ33) for a suspension undergoing simple shear was extracted from Acrivos et al.’s [Int. J. Multiphase Flow 19, 797 (1993)] resuspension data in a Couette device. Employing a new correlation for the relative viscosity μr which obeys the Einstein relation in the dilute limit while diverging at random close packing, it was found that Σ33/τ (where τ is the magnitude of the shear stress) was a strong function of the solid volume fraction φ, scaling as φ3e2.34φ. The relative viscosity, measured in a parallel plate viscometer, was in good agreement with the proposed correlation, while the normal stress differences N1 and N2 for concentrated suspensions (φ=0.30–0.55) were characterized using parallel plate and cone-and-plate geometries, as well as laser profilometry measurements of the suspension surface deflection in a rotating rod geometry. The normal stresses were proportional to the shear stress τ, and with β≡N1/τ and δ≡N2/τ, the parameter combinations resulting from the three experimental geometries, β−δ, β, and δ+12 β, were all seen to increase with φ according to the derived scaling φ3e2.34φ. Furthermore, the best-fit N1 and N2 values consistent with the set of experiments were both negative, with |N2| > |N1| at any given concentration and shear rate. Taken together, the results obtained allow a complete determination of the total stress of a sheared suspension and in particular enabled us to compute the shear-induced particle-phase pressure Π, as defined in Jeffrey et al. [Phys. Fluids A 5, 2317 (1993)].