Stresses and orientational order in shearing flows of granular liquid crystals

We perform discrete element simulations of homogeneous shearing of frictionless cylinders and show that the particles are characterized by orientational order and form a granular liquid crystal. For elongated and flat cylinders, the alignment is in the plane of shearing, while cylinders having an aspect ratio equal to 1 and 0.8 show no orientational order. We show that the particle pressure is insensitive to the cylinder aspect ratio and well predicted by the kinetic theory of granular gases, with a singularity in the radial distribution function at contact different from that for frictionless spheres. The numerical results quantitatively agree with physical experiments on different geometries. The particle shear stress is affected by orientational anisotropy. We postulate that, for frictionless cylinders, the viscosity is roughly due to the motion of the orientationally disordered fraction of the particles, and show that it is proportional, through the order parameter, to the expression of kinetic theory. Finally, we suggest that the orientational order is the result of the competing effects of the shear rate, which induces alignment, and the granular temperature, which ramdomizes.

Figure 1

Snapshots of the shear flows of cylinders at $\nu =0.4$ for (a) $L/d=0.25$, (b) $L/d=0.8$, and (c) $L/d=4$. $x,y$, and $z$ are the flow, velocity gradient, and vorticity directions, respectively. Periodic boundary conditions along the $x$ and $z$ directions, and Lees-Edwards [20] boundary conditions along the $y$ direction have been employed.

Figure 2

(a) Cylinder orientation in terms of the two angles $\alpha$ and $\beta$. (b) Order parameter as a function of the solid volume fraction for $L/d=0.17$ (solid lower triangles), $L/d=0.25$ (solid upper triangles), $L/d=0.5$ (solid squares), $L/d=0.8$ (diamonds), $L/d=1$ (circles), $L/d=2$ (hollow squares), $L/d=4$ (hollow upper triangles), and $L/d=6$ (hollow lower triangles). (c) Average alignment angle as a function of the order parameter [same symbols as in Fig. 2]. Also shown are the experimental values for glass cylinders and wooden pegs [3] (crosses).

Figure 3

Dimensionless pressure versus solid volume fraction obtained from the present numerical simulations [same symbols as in Fig. 2] and experiments (crosses) on collisional suspensions of plastic cylinders in water. The solid line represents Eq. (1).

Figure 4

(a) Dimensionless viscosity versus solid volume fraction obtained from the present numerical simulations [same symbols as in Fig. 2]. (b) Same as in Fig. 4, but the dimensionless viscosity is scaled with $1-S$. The solid line represents Eq. (5).

Figure 5

Order parameter as a function of the ratio of the shear rate to the square root of the granular temperature [same symbols as in Fig. 2]. The solid line is a tentative fitting using the logistic function $S=1/[1+exp(6-4\dot{\gamma }{d}_v/T^{1/2})]$.