From frictional to viscous behavior: Three-dimensional imaging and rheology of gravitational suspensions

We probe the three-dimensional flow structure and rheology of gravitational (nondensity matched) suspensions for a range of driving rates in a split-bottom geometry. We establish that for sufficiently slow flows, the suspension flows as if it were a dry granular medium, and confirm recent theoretical modeling on the rheology of split-bottom flows. For faster driving, the flow behavior is shown to be consistent with the rheological behavior predicted by the recently developed “inertial number” approaches for suspension flows.

Figure 1

(Color online) (a) The experimental setup used for flow visualization in a “‘split-bottom”’ geometry. (b) Example image of a single cross section, displaying half the box. (c) Geometry used for rheological measurements. (d) Geometry used for flow visualization.

Figure 2

(Color online) Frictional to viscous crossover in flow profiles. (a) The predicted flow field for dry granular flows ${\omega }_D\left(r,z\right)$ from Refs. [19, 20, 21, 22, 23, 24, 25]. (b-e) Measured velocity fields ${\omega }_S\left(r,z\right)$ at driving rates $\Omega =8.3\times {10}^{-5}\text{rps}$ (b), $\Omega =8.3\times {10}^{-4}\text{rps}$ (c), $8.3\times {10}^{-3}\text{rps}$ (d) and $8.3\times {10}^{-2}\text{rps}$ (e). (f) The Newtonian flow field ${\omega }_N\left(r,z\right)$ calculated with the finite element method (see text). Note the similarity of (a) to (b) and (e) to (f). (g) Scatter plot comparison of ${\omega }_D\left(r,z\right)$ (a) and ${\omega }_S\left(r,z\right)$ for $\Omega =8.3\times {10}^{-5}\text{rps}$ (◻) and $\Omega =8.3\times {10}^{-2}$ rps (○). (h) Scatter plot comparison of ${\omega }_S\left(r,z\right)$ and the flow field of a Newtonian flow ${\omega }_N\left(r,z\right)$ for $\Omega =8.3\times {10}^{-5}\text{rps}$ (◻) and $\Omega =8.3\times {10}^{-2}\text{rps}$ (○). (i) ${\chi }^2$ vs $\Omega$ for comparison to granular (△) and Newtonian flow $(\star )$. The flow characteristics change from granular to Newtonian with increasing shear rate.

Figure 3

(Color online) (a) Rheology of 4.6 mm acrylic particles in pure Triton X-100. $T\left(\Omega \right)$ for a $H/R_s=0.24,0.51,0.60,0.67,0.82,0.91,1.0,1.1,1.2$; color (intensity) indicates $H/R_s$. $\Omega =3.0\times {10}^{-4}$ to 0.3. The curve is a fit of the form $T=T_0+C\Omega$ (see text). The four arrows indicate the driving rates where flow profiles were measured. (b) $T_0\left(H\right)$, the plateau values as a function $H$ compared to the prediction from Eq. (2) with $\mu =0.57\pm 0.03$ for the dry (◇) and $\mu =0.59\pm 0.03$ for the suspension $(+)$ case.

Figure 4

(Color online) The rheology of glass beads in a glycerol mixture at different temperatures. The inset shows the same data, with the abscissa rescaled with the viscosity of the glycerol at the given temperatures.