Macroscopic profiles (transport)

Figure: Snapshots of the 2D Inclined Plane Model with stochastic bottom wall at temperature $T_w=50$ and $T_w=250$. The leftmost inset displays the time-averaged number density profile for both cases. Values of other parameters: $N=500$, $N_w \approx 56$, $r=0.7$, $r_w=0.7$,$g_e=-1$

Figure: Granular (dimensionless) temperature $T_g/(g_e \sigma _B)$ versus dimensionless height $y/\sigma _B$ (above) and versus number density $n$ (bottom) for the for the 2D Inclined Plane Model with stochastic bottom wall, with $N=5000$, $N_w \approx 180$, $r=0.7$, $r_w=0.7$, $g_e=-1$. The solid line is a power-law fit for $T(n)$.

Figure: Snapshot of the 2D Inclined Plane Model with periodically vibrating wall (right) and time-averaged density profile (left) for the following choice of parameters: $N=500$, $N_w \approx 56$, $r=0.5$, $r_w=0.7$, $g_e=-1$, $f_w=400\pi$, $A_w=0.1$

Simulations of the first model, the 2d Inclined Plane Model with a vibrating bottom wall, have been performed for different choices of the number of disks $N$, the normal restitution coefficient $r$, the dimensionless width of the plane $N_w=L_x/\sigma_B$ and the parameter measuring the rate of energy injection from the wall. This is the temperature $T_w$ in the stochastic case and the amplitude and frequency $A_w$, $\omega_w$ in the periodic case.

Here we show how numerical simulations with the Molecular Chaos assumption reproduce the main results obtained in experiments [132] and in high performance computer simulations [116] of inelastic hard disks, in a wide range of values of the volume fraction (here represented by the parameter $N_w$) and of the restitution coefficient $r$.

Snapshots of the system and time-averaged density profiles are shown in Fig. fig:tflip_dens for the case of randomly vibrated wall. We are in the presence of a highly fluidized phase of the type Isobe and Nakanishi call granular turbulent: looking at the time evolution of the density distribution of the system and of the coarse-grained velocity field one observes an intermittency-like behavior with rapid and strong fluctuations of the density, sudden explosions (bubbles) followed by large clusters of particles traveling downward, coherently, under the action of gravity. Of course more dense and ordered phases (that one can expect at lower values of energy injection [116]) are not reproducible with the Direct Simulation Monte Carlo, as strong excluded volume effects appear and the assumption of negligible short range correlations fails.

In Fig. fig:tflip_t it is shown the temperature profile $T_g(y)$. A minimum of the temperature can be observed not far from the bottom wall. From a certain height the strong temperature positive gradient abruptly changes into a very slow increase, which never becomes constant. It is interesting to look at the bottom frame in the same Figure: it shows a parametric plot of the granular temperature $T_g$ versus the number density $n$, similar to those in Fig. fig_tempscaling_1d_MD, Fig. fig_tempscaling_1d_DSMC and Fig. fig_tempscaling_2d_DSMC obtained with the Randomly Driven Granular Gas model. One of the data sets that we analyze is compatible with a power law $T_g \sim n^{-0.88}$ which resembles the algebraic fits in the Randomly Driven model. It must be noted that in a stationary fluid without the gravity field, the pressure is constant and therefore $T_g \sim n^{-1}$.

The case of periodically vibrated wall is illustrated in Fig. fig:vflip_dens where one can see the density profile, together with a snapshot of the system, for a certain choice of the parameters of the model. The snapshot and the time-averaged profile is very similar to that shown for the case of the wall with stochastic vibrations. A rapid sequence of snapshots reproduces the same scenario discussed above, with very strong spatio-temporal gradients near the bottom wall and solitary particles in ballistic flight at the top of the container.