Rheological behavior of vibrated bimodal granular suspensions: a free volume approach

Abstract

In this paper, we investigate the dependence of the viscosity of vibrated bidisperse granular suspensions of spherical particles on the relative fraction of the two populations. At the same total volume fraction ϕ, the viscosity of the bidisperse suspension is lower than that of a monodisperse suspension. A unified viscosity model is derived using an effective free volume approach based on the definition of an equivalent mean diameter and a maximum packing fraction, both depending on the fraction of large particles ξ. The resulting model accounts for the viscosity of the suspending fluid η _f , the amplitude A, and the frequency f of the vibrations, for a constant ratio of the diameters of the beads λ = 5.3 and a fixed value of the total volume fraction ϕ ≈ 0.61. It led to a general equation describing both the rheology of monodisperse and bidisperse suspensions.

Appendix

We study the behavior of the viscosity of the bidisperse suspension in the regime σ > σ _f . In that case, η does not depend on σ _v (Fig. 3a) and increases with the fluid viscosity η _f (Fig. 3b). Experiments carried out with different viscosities of the suspending fluid show that η is a linear function of η _f . Figure 7 displays the evolution of the relative viscosity η _r300 = η/η _f for σ =300 Pa as a function of ξ. The evolution is similar to that observed for σ < σ _f (Fig. 3c) with a significantly lowered viscosity of the bidisperse suspension compared to the monodisperse one. Numerous experimental works (Sweeny and Geckler 1954; Eveson 1959; Chong et al. 1971; Poslinski et al. 1988; Storms et al. 1990; Shapiro and Probstein 1992; Gondret and Petit 1997; Logos and Nguyen 1996) and numerical simulations (Chang and Powell 1993; 1994; He and Ekere 2001; Toivakka and Eklund 1996; Farr and Groot 2009; Biazzo et al. 2009; Clarke and Wiley 1987) have shown that under shear, laws used to describe the viscosity of monodisperse suspensions such as Krieger Dougherty relation could be extended to the case of bidisperse suspensions using a maximum packing fraction ϕ ^∗ that evolves with the fraction of large particles ξ. In the same way, it is reasonable to extend Quemada (1977)’s law, used to describe the viscosity-concentration curve for monodisperse particles (Fig. 2 in Hanotin et al. (2012)), to the case of bidisperse particles by applying Shauly et al. (1998)’s relation (Eq. 9 and Fig. 4) to describe the evolution of ϕ ^∗ with ξ. Figure 7 displays the result of such a treatment and reveals a good agreement between experimental data and Quemada’s law. The agreement is slightly less satisfactory in the range ξ =0.6–0.8, i.e., in the range of minimal viscosity. This may be due either to some inaccuracy in (Shauly et al. 1998)’s relation or in the choice of $\phi ^{\ast }_{mono}$ in (Eq. 9). Still, these results show that in the high shear regime, suspension behaves as a homogeneous suspension of hard spheres, whether the particles are monodisperse or bidisperse.

Fig. 7

Relative high shear viscosity (η _r300 = η/η _f ) with η the suspension viscosity and η _f the fluid viscosity for σ =300 Pa as a function of the fraction of large particles (ξ), for bidisperse particles immersed in a fluid of viscosity η _f = 68 and 165 mPa.s. The line represents the fit of the experimental data by Quemada (1977)’s law with ϕ ^∗ determined by Shauly et al. (1998)’s relation (Eq. 9 and Fig. 4)