Viscous to Inertial Transition in Dense Granular Suspension

Granular suspensions present a transition from a Newtonian rheology in the Stokes limit to a Bagnoldian rheology when inertia is increased. A custom rheometer that can be run in a pressure- or a volume-imposed mode is used to examine this transition in the dense regime close to jamming. By varying systematically the interstitial fluid, shear rate, and packing fraction in volume-imposed measurements, we show that the transition takes place at a Stokes number of 10 independent of the packing fraction. Using pressure-imposed rheometry, we investigate whether the inertial and viscous regimes can be unified as a function of a single dimensionless number based on stress additivity.

Figure 1

(a) Image of the poly(methyl methacrylate) spheres. (b) Sketch of the four-ball tribometer. (c) Sketch of the rheometer.

Figure 2

Rheological data coming from volume-imposed rheometry: (a) ${\eta }_s=\tau /{\eta }_f\dot{\gamma }$, (b) ${\eta }_n=P/{\eta }_f\dot{\gamma }$, and (c) $\mu =\tau /P={\eta }_s/{\eta }_n$ versus $\mathrm{St}={\rho }_p{d}^2\dot{\gamma }/{\eta }_f$ for four suspensions having suspending fluids of variable viscosity, ${\eta }_f=11$ (orange), 20 (yellow), 50 (green), 86 (blue) mPa s, at four different volume fractions, $\varphi =0.520$ ($○$), 0.545 ($△$), 0.567 ($\square$), 0.577 ($☆$). The data ($◊$) from Fig. 7 of [3] for spheres having $d=1.32\mathrm{mm}$ and two suspending fluids ${\eta }_f=1$ (magenta) and 7 (cyan) mPa s are also reported. Insets of graphs (a),(b): ${\eta }_s/{\eta }_s(\mathrm{St}\to 0)$ and ${\eta }_n/{\eta }_n(\mathrm{St}\to 0)$ versus St showing a transition from a Newtonian (slope 0) to a Bagnoldian (slope 1) rheology (the white solid lines are the simplest power-law fit of the whole data). Red lines in (c): Constitutive law [Eq. (3)] for the different $\varphi$ with ${\varphi }_c=0.615$, ${\mu }_c=0.31$, $a_{\mu }=11.29$, $a_{\varphi }=0.66$, ${\alpha }_{\mu }=0.0088$, ${\alpha }_{\varphi }=0.1$ coming from the pressure-imposed rheology.

Figure 3

Rheological data coming from pressure-imposed rheometry: (a) $\mu /{\mu }_c$ and (b) $\varphi /{\varphi }_c$ versus $K=J+\alpha I^2$ with $\alpha =1/S{t}_{v\to i}$ and (c) $\mu /{\mu }_c$ versus $\varphi /{\varphi }_c$ for the same neutrally buoyant suspensions as in Fig. 2 and two additional negatively buoyant systems consisting of the sole dry spheres and of the spheres immersed in pure water. The data coming from Fig. 5 of [11] using slightly rough polystyrene sphere (with $d=580\mu \mathrm{m}$) in the dry case as well as immersed in a very viscous Newtonian fluid of same density (with ${\eta }_f=2.01\mathrm{Pa}\mathrm{s}$) are also reported. Inset of (a): $\mu /{\mu }_c$ versus $J+{\alpha }_{\mu }{I}^2$; the black solid line corresponds to the best fit: $\mu /{\mu }_c=1+a_{\mu }(J+{\alpha }_{\mu }{I}^2{)}^{1/2}$ with $a_{\mu }=11.29$ and ${\alpha }_{\mu }=0.0088$.

Figure 4

Rheological data coming from pressure-imposed rheometry: (a) $\tau$ and (b) $P$ normalized by ${\eta }_f\dot{\gamma }+\alpha {\rho }_p{d}^2{\dot{\gamma }}^2={\eta }_f\dot{\gamma }(1+\alpha \mathrm{St})$ versus $\varphi /{\varphi }_c$ (same symbols as in Fig. 3). The insets show these quantities versus the rescaled volume fraction, $1-\varphi /{\varphi }_c$. Red solid line in (b): Constitutive law [Eq. (5)] with $a_{\varphi }=0.66$.