On granular flows: From kinetic theory to inertial rheology and nonlocal constitutive models

Previous results of discrete simulations of steady, unidirectional particle flows, here collected and critically reanalyzed, permit to make the case that the kinetic theory of granular gases, extended to include the correlations in the velocity fluctuations and the role of friction in collisions, provides the long-sought universal framework to predict the flow of realistic particles over the entire range of solid volume fraction from dilute to very dense—the upper limit being the critical value at which rate-independent components of the stresses arise. The case is made even stronger by the explicit derivation of the popular inertial rheology and its nonlocal extension to deal with heterogeneities based on the granular fluidity concept as special limits of the kinetic theory. In the process, common statements about the frictional-collisional duality in the granular stresses and the importance of long-lasting contacts creating a percolating network are shown to be greatly exaggerated for granular flows in practical applications.

Figure 1

Sketch of the collisional encounter between two hard spheres.

Figure 2

Schematics of a single interaction between two spheres (on the left, the overlap area is indicated in orange) and corresponding time sequence of impulse or overlap for three such events (on the right) for (from top to bottom) hard spheres, soft spheres at subcritical volume fraction, and soft spheres at supercritical volume fraction. Also shown are the time of free flight and the finite duration of the interaction.

Figure 3

Unidirectional flow configurations with the frame of reference: (a) steady, homogeneous (simple) shearing with linear distribution of the mean velocity and uniform distribution of the solid volume fraction; steady, gravity-driven, heterogeneous flow over (b) an erodible bed with lateral confinement and (c) a rigid, bumpy bed without lateral confinement; (d) steady, heterogeneous flow between parallel, rigid, bumpy plates in relative motion (Couette flow) in the absence of gravity. In (c) and (d), the core region, in which the solid volume fraction is uniform, is sandwiched between two conductive boundary layers (B.L.).

Figure 4

(a) Dimensionless pressure as a function of the dimensionless shear rate measured in DEM simulations [80] of volume-controlled, steady, homogeneous shearing flows of spheres with $e_n=0.7$, $e_t=1$, and $\mu =0.5$, at $\nu =0.594$ (solid circles) and $\nu =0.584$ (open circles). (b) Magnitude of the fluctuations in the coordination number as a function of the solid volume fraction measured in DEM simulations [82] of steady, heterogeneous, gravity-driven flows of spheres with $e_n=0.88$, $e_t=1$, $\mu =0.5$, and $k_n\approx 3\times {10}^6{\rho }_{p}g{d}^2$. (c) Same as in (a), but the dimensionless pressure is plotted as a function of the solid volume fraction (only data in the quasistatic limit are shown). (d) Coordination number as a function of the solid volume fraction measured in DEM simulations [79] of volume-controlled, steady, homogeneous shearing flows of spheres with $e_n=0.7$, $e_t=1$, and $\mu =0.5$, for different values of the dimensionless stiffness, $k_n/\left({\rho }_p{d}^3{\dot{\gamma }}^2\right)$, ranging from ${10}^2$ (light gray circles) to ${10}^7$ (black circles). In all plots, the dashed lines represent the coordinates of the critical point for $\mu =0.5$.

Figure 5

Measured critical (a) solid volume fraction and (b) coordination number as functions of the friction coefficient as reported in Ref. [85] (circles), Ref. [84] (triangles), Ref. [80] (squares), and Ref. [83] (diamonds).

Figure 6

(a) Coordination number as a function of the solid volume fraction measured in DEM simulations [82] of steady, heterogeneous, gravity-driven flows of spheres with $e_n=0.88$, $e_t=1$, $\mu =0.5$, and $k_n\approx 3\times {10}^6{\rho }_{p}g{d}^2$. The dashed line indicates the value of ${\nu }_c$. (b) Dimensionless pressure as a function of the solid volume fraction measured in DEM simulations [80] of volume-controlled, steady, homogeneous shearing flows of spheres with $e_n=0.7$, $e_t=1$, and $\mu =0.5$, at different values of the dimensionless particle stiffness, $k_n/\left({\rho }_p{d}^3{\dot{\gamma }}^2\right)$, ranging from 1 to ${10}^{11}$ (the arrow indicates the direction of increasing stiffness). The two dashed lines highlight the data corresponding to 1 mm sand (or glass) and hydrogel spheres sheared at 50 Hz, while the solid line is the prediction of the kinetic theory for hard spheres.

Figure 7

Measurements (DEM simulations, solid circles) [98] and predictions from kinetic theory (lines) of pressure and shear stress in steady, homogeneous flows at $\nu <{\nu }_c$ for $e_t=1$, $\mu =0.5$, $k_n/\left({\rho }_p{d}^3{\dot{\gamma }}^2\right)>{10}^6$ and $e_n=0.7$ (in blue), $e_n=0.8$ (in orange), $e_n=0.9$ (in yellow), $e_n=0.95$ (in purple), and $e_n=0.99$ (in green). The data are plotted using [(a) and (d)] the inertial rheology representation, [(b) and (e)] Bagnold's scalings, and [(c) and (f)] the kinetic theory scalings. Also shown in (d) are DEM data [80] for $\nu \ge {\nu }_c$ (open diamonds).

Figure 8

Measurements (DEM simulations, solid circles) [86, 98] and predictions from kinetic theory (lines) of pressure and shear stress in steady, homogeneous flows at $\nu <{\nu }_c$ for $e_n=0.7$, $e_t=1$, $k_n/\left({\rho }_p{d}^3{\dot{\gamma }}^2\right)>{10}^6$, and $\mu =0.5$ (in blue), $\mu =0.1$ (in orange), and $\mu =0$ (in yellow). The data are plotted using [(a) and (d)] the inertial rheology representation, [(b) and (e)] Bagnold's scaling, and [(c) and (f)] the kinetic theory scaling. Also shown in (d) are DEM data [80] for $\nu \ge {\nu }_c$ (open diamonds).

Figure 9

Measurements (DEM simulations, black circles) [36, 74] and predictions from kinetic theory [Eq. (2), red lines] of pressure in [(a) and (b)] steady, heterogeneous, gravity-driven flows over erodible beds with lateral confinement where $e_n=0.88$, $e_t=1$, $\mu =0.5$, and $k_n\approx 3\times {10}^6{\rho }_{p}g{d}^2$ (angles of inclination between ${24}^{\circ }$ and ${55}^{\circ }$ and channel width between 10 and 30 particle diameters) and [(c) and (d)] Couette flows with $e_n=0.8$, $e_t=1$, $\mu =0$, and $k_n=2\times {10}^5{\rho }_{p}d{V}^2$. The data are plotted using [(a) and (c)] the Bagnold scaling and [(b) and (d)] the kinetic theory scaling.

Figure 10

Cooperativity length as a function of (a) the solid volume fraction and (b) the stress ratio inferred from DEM simulations (black circles) [74] of steady, heterogeneous, gravity-driven flows over erodible beds with lateral confinement at $\nu \ge {\nu }_c$ ($e_n=0.88$, $e_t=1$, $\mu =0.5$, $k_n\approx 3\times {10}^6{\rho }_{p}g{d}^2$, angles of inclination between ${24}^{\circ }$ and ${55}^{\circ }$, and channel width between 10 and 30 particle diameters) and predicted from the kinetic theory [Eq. (31), red lines].