Macroscopic profiles (transport)

Figure: Normalized number density $n$, dimensionless horizontal velocity $v_x/\sqrt {g_x\sigma _B}$ and dimensionless granular temperature $T/\sqrt {g_x\sigma _B}$ versus dimensionless height $y/\sigma _B$ for the 2D Inclined Channel Model: $N=500$, $N_w \approx 56$, $g_x=1$, $g_y=-2$ (i.e.: the inclination angle $\phi =\pi /6$), $r=0.95$, $r_w=0.95$

Figure: Normalized number density $n$, dimensionless horizontal velocity $v_x/\sqrt {g_x\sigma _B}$ and dimensionless granular temperature $T/\sqrt {g_x\sigma _B}$ versus dimensionless height $y/\sigma _B$ for the 2D Inclined Channel Model: $N=500$, $N_w \approx 56$, $g_x=1$, $g_y=-2$ (i.e.: the inclination angle $\phi =\pi /6$), $r=0.95$, $r_w=0.4$

Figure: Cooling rate, as defined in the text, versus dimensionless height $y/\sigma _B$ for the 2D Inclined Channel Model: $N=500$, $N_w \approx 56$, $g_x=1$, $g_y=-2$ (i.e.: the inclination angle $\phi =\pi /6$), $r=0.95$, $r_w=0.95$ or $r_w=0.4$

In Fig. fig:c_prof1 and Fig. fig:c_prof2 we show some profiles of mesoscopic coarse grained physical quantities: the hydrodynamical fields $n(y)$ (number density), $v_x(y)$ (velocity component parallel to the flow), $T_g(y)$ (granular temperature) are shown as functions of the distance from the bottom wall $y$. The velocity, the temperature and the height are made dimensionless dividing them by $\sqrt{g_x\sigma_B}$, $g_x\sigma_B$ and $\sigma_B$ respectively. This means that we consider unitary the potential energy (in the gravity field) of a point of mass $1$ at the height of a ``Bird radius'' $\sigma_B$.

The profiles well reproduce those measured experimentally by Azanza et al. [6]: they show a critical height $H$ of about six times the radius $\sigma_B$ which corresponds to the separation between two different regimes of the dynamics: under this critical height, the profiles are almost linear (in particular the density and velocity profiles); above this height $H$ those profiles rapidly change and become almost constant.

This change of behavior can be explained studying the local dissipation rate due to inelastic collisions. In a mean field framework the local rate of dissipation due to the inelastic collisions (as already stated before) is $\zeta \propto nT_g^{3/2}$. This has been discussed in section 2.4.1 and can be simply understood noting that the collision rate is proportional to the local density and to the local relative velocity of the particles ( $\sqrt{T_g}$), while the change in the granular temperature induced by every collision is proportional to the temperature $T_g$. The quantity $\tilde{\zeta}=nT_g^{3/2}$ as a function of $y$ is shown in Fig. fig:c_cool. The cooling rate decreases exponentially and is reduced under $1/100$ of its maximum value at about the observed critical height $H \approx 6\sigma_B$. This means that under the critical height $H$ the transport is mainly due to collisional exchange, while above $H$ it is due to ballistic flights.

With respect to the velocity and temperature profiles in Fig. fig:c_prof1, we note here that quite nonphysical features appear: in particular the strong slipping effect near the bottom wall is in contrast with the experimental findings. We check ahead the possibility that this could be due to a wrong modeling of the particle-wall collision events.

The restitution coefficient used in our model has to be considered as an effective parameter describing the energetics of collisions. It should depend on the details of the collision event, in principle even on the relative velocities of the colliding particles. In the experiment the bottom wall was covered with particles identical to the flowing ones with a spacing bounded between 0 and $0.8$ mm: however the particles are stuck to the bottom wall so that the collision event is completely different from a two-particles collision.

Using a lower effective restitution coefficient for the wall $r_w$ (see Fig. fig:c_prof1) we obtain a better agreement with the experimental profiles. In particular, both temperature and velocity profiles seems to go to zero near the bottom, although we cannot really rule out slipping effects ( $v_x(y=0)\neq 0$) (similar slipping effects are reported in other simulations, e.g. see [52]).