Unifying length-scale-based rheology of dense suspensions

We present a length-scale-based rheology for dense sheared particle suspensions as they transition from inertial- to viscous-dominated. We derive a length-scale ratio using straightforward physics-based considerations for a particle subjected to pressure and drag forces. In doing so, we demonstrate that an appropriately chosen length-scale ratio intrinsically provides a consistent relationship between normal stress and system proximity to its “jammed” or solidlike state, even as a system transitions between inertial and viscous states, captured by a variable Stokes number.

Figure 1

(a) Shear behavior of dense suspensions: The top plate applies a pressure $P_p$ on the particles, inducing shear within the granular assembly as the top plates move at a velocity $v$, while the bottom plate remains fixed. (b) Schematic of a 2D conceptual model of a particle in layer $i+1$ subjected to shear and normal stresses over particles in layer $i$. (c) Relative motion for low $\dot{\gamma }$ (and/or high $P_p$). (d) Relative motion for higher $\dot{\gamma }$ (and/or lower $P_p$). (e) Force analysis of the particle A during shear.

Figure 2

Normalized solid fraction (a) and apparent frictional coefficient (b) plotted vs. data from $G$ for data from Refs. [10, 12, 17, 18, 20]. The solid line in (a) are from Eq. (22b), using $b=0.188$, and the solid line inset of (b) are from Eqs. (22a) and (23) with $a_i=1.0, a_u=4.0$, and $c=41$.

Figure 3

Evolution of fitting parameter (a) $b$ and (b) $a$ vs $I/J$ for data from Refs. [10, 12, 17, 18]. The solid black line in (b) is from Eq. (23) with $a_i=1, a_v=4$, and $c=41$.

Figure 4

(a) $1-\varphi /{\varphi }_c$ vs generalized time-scale ratio $\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ transforms from a inertial number $(2/\sqrt{3})I$ to a viscous number $12J$ and $\alpha =0.5\to 1$ as the fluid-granular system varying from a dry condition to a very viscous condition, the dash line is plotted by $1-\varphi /{\varphi }_c=0.188{\mathrm{\Gamma }}^{\alpha }$ with $\alpha =0.5$, 0.72, and 1. (b) shows $\lambda$ vs $St$, the solid line is plotted by Eq. (11b).