Rheology of a granular gas under a plane shear

The rheology of a two-dimensional granular gas under a plane shear is investigated. From the comparison among the discrete element method, the simulation of a set of hydrodynamic equation, and the analytic solution of the steady hydrodynamic equations, it is confirmed that the fluid equations derived from the kinetic theory give us accurate results even in relatively high density cases.

Figure 1

The configuration of our simulation.

Figure 2

The time evolution of the particles’ configurations from (a) to (b) for $\overline{\nu }=0.121$.

Figure 3

The time evolution of the kinetic energy, where the units of time and the energy are $2\sigma ∕U$ and $m{U}^2∕2$, respectively.

Figure 4

The time evolution of dimensionless $u_y$ with (a) at $t=60$ and (b) at $t=380$, where the units of the time and $u_y$ are $(2\sigma ∕U)$ and $U∕2$, respectively.

Figure 5

The initial conditions of hydrodynamic equations (solid lines) and the corresponding data of DEM (open circles) at $t=20$, where $\nu$, $\theta$, $u_x$, $u_y$, and $y$ are all dimensionless quantities (see text). The solid lines for $\nu$ and $\theta$ are the polynomials of even powers of $y$ until $y^6$, while the lines for $u_x$ and $u_y$ are the polynomials of the odd powers of $y$ until $y^5$.

Figure 6

The comparison of the data for the area fraction for $\overline{\nu }=0.121$ obtained by DEM (open circles) at $t=20$ (label 1), 60 (label 3), and 380 (label 5), and the result of hydrodynamic equations (25, 26, 27) (solid lines). Here $y$ is the dimensionless coordinate.

Figure 7

The time evolution of the dimensionless temperature (a), the dimensionless velocity fields $u_x$ (b) and $u_y$ (c) in hydrodynamic simulations shown in solid lines, where the data in (a) and (b) are obtained by DEM. The numbers 1,3,5 in these figures correspond to results at $t=20$, 60, and 380, respectively. The DEM data with the solid squares and open circles correspond to the result at $t=20$ and 60, respectively. Note that comparison of the theory and DEM in the steady values of $\theta$ and $u_x$ will be shown in the next section, while we do not include DEM data for $u_y$ because of the disagreement in the scale (see Fig. 4).

Figure 8

The time evolution of the area fraction (a), dimensionless temperature (b), the dimensionless velocity fields $u_x$ (c) and $u_y$ (d) in the simulation of a simplified model (42) of hydrodynamic equations. The numbers 1,3,5 in these figures correspond to the data at $t=20$, 60, and 380, respectively.

Figure 9

The comparison between theory (solid lines) and DEM simulation without the tangential interaction (open circles) for area fraction (a), dimensionless temperature (b), and $u_x$ (c). The mean area fraction is $\overline{\nu }=0.121$.

Figure 10

The comparison between theory (solid lines) and DEM simulation with the tangential interaction (open circles) for area fraction (a), dimensionless temperature (b), and $u_x$ (c). Here we use $e=0.798$ and the mean area fraction $\overline{\nu }=0.121$.

Figure 11

The comparison of Eq. (7) with (A4) (solid line) and the data obtained from DEM with $\mu =0.2$ (open circles). Here $\beta$ and $\gamma$ are respectively the tangential restitution coefficient and the collision angle.