Shock propagation in locally driven granular systems

We study shock propagation in a system of initially stationary hard spheres that is driven by a continuous injection of particles at the origin. The disturbance created by the injection of energy spreads radially outward through collisions between particles. Using scaling arguments, we determine the exponent characterizing the power-law growth of this disturbance in all dimensions. The scaling functions describing the various physical quantities are determined using large-scale event-driven simulations in two and three dimensions for both elastic and inelastic systems. The results are shown to describe well the data from two different experiments on granular systems that are similarly driven.

Figure 1

Moving (red) and stationary (blue) particles at times (a) $t=500$, (b) $t=1000$, (c) $t=1500$, and (d) $t=2000$. Energetic particles are injected at the center. All collisions are elastic with $r=1$. The data are for the conserved model.

Figure 2

Moving (red) and stationary (blue) particles at times (a) $t=1000$, (b) $t=2000$, (c) $t=4000$, and (d) $t=8000$. Energetic particles are injected at the center. All collisions are inelastic with $r=0.1$. The data are for the conserved model.

Figure 3

Radial momentum as a function of time $t$ for two- and three-dimensional inelastic systems, showing a linear increase. The inset shows the data on a log-log scale, which show an initial transient regime before the linear growth is attained. The data are for the conserved model.

Figure 4

Simulation results for the elastic system ($r=1$) for the temporal variation of (a) radius $R_t$, (b) kinetic energy $E_t$, and (c) number of moving particles $N_t$ in two and three dimensions. The solid lines are power laws with exponents as predicted by the scaling arguments presented in the text. The data are for the conserved model.

Figure 5

Scaled radial distribution functions against scaled distances $r/t^{\alpha }$ for the elastic gas: (a) $\rho (r,t)$, (c) $v(r,t)$, (e) $e(r,t)$, and (g) $\langle cos\theta (r,t)\rangle$ in two dimensions and (b), (d), (f), and (h) corresponding quantities in three dimensions. Here $\alpha =3/(d+2)$, as in Eq. (4), and $\beta =2(1-\alpha )$. The data are for the conserved model.

Figure 6

Simulation results for the inelastic system ($r=0.1$) for the temporal variation of (a) radius $R_t$, (b) kinetic energy $E_t$, and (c) number of moving particles $N_t$ in two and three dimensions. The solid lines are power laws with exponents as predicted by the scaling arguments presented in the text. The data are for the conserved model.

Figure 7

Scaled radial distribution functions against scaled distances $r/t^{\alpha }$ for the inelastic gas: (a) $\rho (r,t)$, (c) $v(r,t)$, (e) $e(r,t)$, and (g) $\langle cos\theta (r,t)\rangle$ in two dimensions and (b), (d), (f), and (h) corresponding quantities in three dimensions. Here $\alpha =2/(d+1)$, as in Eq. (5), and $\beta =2(1-\alpha )$. The data are for the conserved model.

Figure 8

Experimental data (taken from Ref. [12]) for the scaled radius $R$ of the longest finger from the center, as a function of normalized time $t/t_0$. Here $R_0=R\left(t_0\right)$. The data have been plotted for different gas overpressures. The solid lines are power laws $t^{1/2}$, $t^{2/3}$, and $t^1$ and are shown for reference.

Figure 9

Experimental data (taken from Ref. [15]) for the growth of maximum radial coordinate of the central zone of disturbance with time for two different values of injection pressures. The solid lines are power laws $t^{1/2}$, $t^{2/3}$, and $t^1$ and are shown for reference.