The boundary conditions of the 2D Inclined Plane Model

Figure 4.1: A sketch of the first model where the granular assembly is driven by gravity plus a (periodically or stochastic) vibrating wall

The first model that we study is called ``2D Inclined Plane Model''. It is illustrated in Fig. fig:a_sketch and inspired to recent laboratory experiments [132] and numerical simulations [116]. We have reviewed those experiments in section 1.2.We have shown that in many of them it has been used a closely packed array of grains (for example see [76,181]). Our inspiration has come, instead, from the more dilute setup of Kudrolli and Henry [132], where the velocity distributions and the density correlations have been measured. We have considered that the dilute setup and the measurement performed where better suited to be investigated with our numerical tools, e.g. Direct Simulation Monte Carlo in two dimensions.

Similar models have been previously studied in the one-dimensional case, that is a vibrated column of grains under the force of gravity [147,155] and the transition or the coexistence of different phases (gas, partially fluidized and condensed) was investigated. In two dimensions experiments [223], simulations [156] and theories [134,204] have analyzed a vertical system of grains with gravity and a vibrating bottom wall (with different kinds of vibration) searching for a simple scaling relation between global variables as the global granular temperature $T_g$ or the center of mass height $h_{cm}$ as function of the size of the system $N$, the typical velocity of the vibrating wall $V$ or the restitution coefficient $r$. In all these calculations the authors did not pay too much attention to the hydrodynamic profiles of the system, always assuming a constant granular temperature (``isothermal atmosphere'') and a density profile exponentially decaying with the height, as in the case of a Boltzmann elastic gas under gravity.

The ``apparatus'' consists of a plane of dimension $L_x \times L_y$ inclined by an angle $\theta$ with respect to the horizontal. The particles are constrained to move in such a plane under the action of an effective gravitational force $g_e=g\sin{\theta}$ pointing downward. In the horizontal direction there are periodic boundary conditions: each particle going out from the left or the right border enters at the same altitude and with the same velocity on the opposite border. The particles are confined by walls on top and bottom: both walls are inelastic with a restitution coefficient $r_w$. We use, in general, different restitution coefficients for the particle-particle interaction and the wall-particle interaction: this is reasonable also if the wall is covered by stuck grains (as often happens in experiments), because the microscopic dynamics of a collision with a stuck grain is completely different from that with a free grain.

The bottom wall vertically vibrates and therefore injects energy and momentum into the system. The vibration can have two different behaviors:

a)

harmonic vibration (as in Kudrolli). In this case, the wall oscillates vertically with the law $Y_w(t)=A_w \sin(\omega_w t)$ ($Y_w(t)$ is the vertical position of the wall at time $t$) and the particles collide with it as with a body of infinite mass, so that the vertical component of their velocity after the collision is

$$v_y'=-r_wv_y+(1+r_w)V_w$$ (4.4)

where $V_w=A_w \omega_w \cos(\omega_w t)$ is the velocity of the vibrating wall.

b)

stochastic vibration, with thermal properties (as in Isobe). In this case we assume that the vibration amplitude is negligible and that the particle colliding with the wall have, after the collision, new random velocity components $v_x \in (-\infty,+\infty)$ and $v_y\in (0,+\infty)$ with the following probability distributions:

$$P(v_y)=\frac{v_y}{T_w}\exp(-\frac{v_y^2}{2 T_w})$$ (4.5)

$$P(v_x)=\frac{1}{\sqrt{2 \pi T_w}}\exp(-\frac{v_x^2}{2 T_w})$$ (4.6)