Understanding an instability in vibrated granular monolayers

We investigate the dynamics of an ensemble of smooth inelastic hard spheres confined between two horizontal plates separated by a distance smaller than twice the diameter of the particles, in such a way that the system is quasi-two-dimensional. The bottom wall is vibrating and, therefore, it injects energy into the system in the vertical direction and a stationary state is reached. It is found that if the size of the plates is small enough, the stationary state is homogeneous. Otherwise, a cluster of particles is developed. The instability is understood by using some effective hydrodynamic equations in the horizontal plane. Moreover, the theoretical prediction for the size of the system above which it becomes unstable agrees very well with molecular dynamics simulation results without any fitting parameter.

Figure 1

Schematic representation of the model.

Figure 2

Snapshot of a configuration of a MD simulation where the cluster shows up (the system is observed from above). The values of the parameters of the system are given in the main text.

Figure 3

Ratio of the stationary temperatures, $\gamma \equiv \frac{T_{z,s}}{T_s}$, measured in the simulations as a function of the inelasticity for $\epsilon =0.5$ (circles) and $\epsilon =0.2$ (squares). The solid and dashed lines are the corresponding theoretical predictions given by Eq. (3).

Figure 4

Stationary horizontal temperature scaled with $m{v}_p^2$ as a function of the inelasticity for $\epsilon =0.5$. The circles are the simulation results and the solid line is the theoretical prediction given by (4). The same is represented in the inset for $\epsilon =0.2$.

Figure 5

Real part of the eigenvalues of $\mathbf{L}$ as functions of $k$ for $\epsilon =0.5$ and $\alpha =0.9$. $k$ and $\lambda$ are measured in the dimensionless units defined in the main text.