Quantitative study of the rheology of frictional suspensions: Influence of friction coefficient in a large range of viscous numbers

The rheology of dense suspensions is studied by discrete-element method simulation, focusing on the interplay of the solid fraction, confining pressure, shear rate, and viscosity. Using a minimal model based on lubrication and contact forces, we are able to recover experimental results available in the literature, in a very large range of solid fractions. We show that bulk friction is only weakly dependent on contact friction when a normalized shear rate, the so-called viscous number $I_v$, is kept constant. In contrast, contact friction has a strong influence on the jamming solid fraction ${\varphi }_m$. We provide empirical proof that all the rheology could be accounted for using $I_v$ and $\varphi /{\varphi }_m$. By separating the contributions of lubrication and contact forces on the total shear stress, it is shown that contacts dominate at a solid fraction above 0.77 of jamming solid fraction. Universal expressions of macroscopic friction and solid fraction as functions of the viscous number are finally offered.

Figure 1

Sensitivity analysis of the (left) contact stiffness and (right) roughness on the relative viscosity, for various solid fractions. Simulations at imposed solid fraction, $N=5000, {\mu }_m=0.4, \epsilon =0.02$ (for stiffness), and $k/\left({\eta }_f\dot{\gamma }\right)={10}^7$ (for roughness).

Figure 2

Snapshot of a slice of the system, normal to the vorticity direction. The arrows indicate the particle velocities, and the colors correspond to the magnitude of the pressure gradient, normalized by the global confining pressure (see color bar). This example has been obtained at $I_v={10}^{-2}$ for ${\mu }_m=0.3$.

Figure 3

Comparison of the shear viscosity obtained with and without pore pressure feedback. For this comparison, ${\mu }_m=0.3$. The solid line corresponds to equality.

Figure 4

Relative viscosity as a function of solid fraction, for various microscopic friction coeficients (see legend). Present data (open symbols) are compared to experimental results of Boyer et al. [11] and Dbouk et al. [32] and numerical results of Mari et al. [9] and Gallier et al. [10].

Figure 5

Effective friction coefficient $\mu =\tau /P_p$ as a function of $I_v$, for various microscopic friction coefficients ${\mu }_m$, as indicated in the legend. Data from Refs. [10, 11] are superimposed for comparison.

Figure 6

(a) Solid fraction as a function of the viscous number for various value of ${\mu }_m$, as indicated in the legend. The solid line represents the correlation proposed in Boyer et al. [11], $\varphi ={\varphi }_m/\left(1+I_v^{0.5}\right)$, which accounts well for the experimental data, for $I_v<0.1$. The dashed line is simply the extrapolation of this correlation at higher viscous numbers, where no experimental data are available. (b) Same data, but the solid fraction is normalized by its asymptotic value ${\varphi }_m$ at low $I_v$. The solid line represents the correlation $1/\left(1+I_v^{0.5}\right)$. In insert are shown the different values of ${\varphi }_m$ as a function of the microscopic friction coefficient. The data are fitted by an exponential function, which accounts well for the data, and displayed in Eq. (13).

Figure 7

(a) Reduced viscosity (same data as in Fig. 4), plotted as a function of the distance to jamming, $1-\varphi /{\varphi }_m$. Solid and dashed lines represent power law functions of exponents $-2$ and $-1$, respectively. (b) $1-\varphi /{\varphi }_m$, plotted vs the viscous number $I_v$ for the whole range of ${\mu }_m$ tested. The solid line represents $I_v^{1/2}$.

Figure 8

Contact and lubrication contribution to the effective friction coefficient. (Left) All the components (circles for contact contribution, squares for normal lubrication contribution, and crosses for shear lubrication contribution) are plotted as a function of $I_v$. (Right) Magnified plot of (top) the contact contribution and (bottom) the sum of lubrication contribution $\left({\mu }_L={\mu }_L^N+{\mu }_L^S\right)$. Discontinuous lines are models proposed by different authors. Contact contribution proposed by Gallier et al. [10] is the same as that proposed by Boyer et al. [11]. Solid lines represents the model proposed in Eqs. (14) and (15).