Effect of friction on dense suspension flows of hard particles

We use numerical simulations to study the effect of particle friction on suspension flows of non-Brownian hard particles. By systematically varying the microscopic friction coefficient ${\mu }_p$ and the viscous number $J$, we build a phase diagram that identifies three regimes of flow: frictionless, frictional sliding, and rolling. Using energy balance in flow, we predict relations between kinetic observables, confirmed by numerical simulations. For realistic friction coefficients and small viscous numbers (below $J\sim {10}^{-3}$), we show that the dominating dissipative mechanism is sliding of frictional contacts, and we characterize asymptotic behaviors as jamming is approached. Outside this regime, our observations support the idea that flow belongs to the universality class of frictionless particles. We discuss recent experiments in the context of our phase diagram.

Figure 1

Phase diagram of dense non-Brownian suspension flow. In the frictionless and rolling regimes, the dominant source of dissipation is viscous drag, whereas in the frictional sliding regime, dissipation is dominated by sliding friction. The dashed line has slope 2.

Figure 2

Ratio of sliding dissipation at rolling contacts to that at sliding contacts.

Figure 3

Ratio of dissipation induced by sliding at contacts to viscous dissipation at indicated viscous numbers. Unity discriminates between flows that are dominated either by frictional or nonfrictional dissipation.

Figure 4

(a,b) Relative fluctuations around the affine velocity field, $\mathcal{L}$, as a function of the viscous number at various particle friction coefficients. Dashed and dot-dashed lines indicate slopes $-1/2$ and $-1/3$, respectively. (c) Fraction of sliding contacts, $\chi$. Dashed and dot-dashed lines indicate slopes 0.1 and 0.13, respectively.

Figure 5

Ratio of mean sliding velocity to velocity fluctuations. The dashed line indicates ${\mathcal{L}}_T/\mathcal{L}\sim J^{0.3}$, suggesting flow inhomogeneity once one enters in the sliding frictional regime.

Figure 6

(a) Stress ratio $\mu$, (b) volume fraction $\varphi$, and (c) ${\varphi }_c-\varphi$ as a function of the viscous number $J$ at indicated particle friction coefficients. In (c), values of ${\varphi }_c$ were determined by fitting curves in (b). In (c), the dashed slope is 0.30 and the dash-dotted slope is $2/3$.

Figure 7

Sketch of the phase diagram when inertia is present, in terms of $J$, ${\mu }_p$, and Stokes number $\text{St}=I^2/J$, where $I=\dot{\epsilon }D\sqrt{\rho /p}$ is the inertial number. Below the colored dome, dissipation is dominated by frictional sliding, while above it is dominated either by viscous dissipation or grain inelasticity. Color corresponds to the value of rescaled confining pressure $\tilde{p}={[I/\left(C_{d}J\right)]}^2$ along the critical surface, distinguishing viscous regimes (yellow, $\tilde{p}\ll 1$) from inertial regimes (dark blue, $\tilde{p}\gg 1$).

Figure 8

Sketch of the phase diagram when inertia is present, in terms of $J$, ${\mu }_p$, and rescaled confining pressure $\tilde{p}=p\rho {[D/\left(C_D{\eta }_0\right)]}^2$. Below the colored dome, dissipation is dominated by frictional sliding, while above it is dominated either by viscous dissipation or grain inelasticity. Color corresponds to the value of $\tilde{p}$ along the critical surface, distinguishing viscous regimes (yellow, $\tilde{p}\ll 1$) from inertial regimes (dark blue, $\tilde{p}\gg 1$).

Figure 9

$({\varphi }_c-\varphi )/{\varphi }_c$ as a function of the viscous number $J$ at indicated particle friction coefficients, in comparison with data from Boyer et al. [7]. The solid line shows ${\varphi }_c/\varphi =1+J^{1/2}$, the fitting form proposed in [7].

Figure 10

Ratio of typical relative velocity to velocity fluctuations. The dashed line ${\mathcal{L}}_R/\mathcal{L}\sim {\mathcal{L}}^{-0.5}$ is a guide to the eye.