Forcing and Velocity Correlations in a Vibrated Granular Monolayer

The role of forcing on the dynamics of a vertically shaken granular monolayer is investigated. Using a flat plate, surprising negative velocity correlations are measured. A mechanism for this anticorrelation is proposed with support from both experimental results and molecular dynamics simulations. Using a rough plate, velocity correlations are positive, and the velocity distribution evolves from a Gaussian at very low densities to a broader distribution at high densities. These results are interpreted as a balance between stochastic forcing, interparticle collisions, and friction with the plate.

Figure 1

Longitudinal velocity correlations normalized by the granular temperature $T$ on the smooth plate: $\Gamma =1.5$, $\rho =0.4$ ($•$), and $\rho =0.5$ ($\circ$). Inset: transverse velocity correlations. Data represent averages over approximately 8000 pairs of images with about 180 balls per image.

Figure 2

$⟨\mathbf{v}(0)·\mathbf{v}(r)⟩$ (as described in the text) normalized by $T$ for $\Gamma =1.5$, $\rho =0.4$ ($•$), and $\rho =0.5$ ($\circ$).

Figure 3

Longitudinal velocity correlations normalized by the granular temperature on the rough plate: $\Gamma =1.5$, $\rho =0.125$ ($•$), and $\rho =0.6$ ($\circ$). Inset: transverse velocity correlations. Data are averaged over 16 000 pairs of images ($\sim 80$ balls per image) for $\rho =0.125$ and 8000 pairs ($\sim 420$ balls) for $\rho =0.6$.

Figure 4

Scaled longitudinal and transverse velocity correlations normalized by $T$. $C_{||}/(\alpha T)$ (symbols with solid lines) and $C_{\perp }/(\alpha T)$ (symbols with dashed lines) for $\Gamma =1.5$, $\rho =0.125$ ($\circ$), $\rho =0.5$ ($\times$), and $\rho =0.6$ ($•$). Inset: Log-linear plot of the scaled velocity correlations (the straight lines are exponentials with decay lengths of $2\sigma$ and $0.6\sigma$).

Figure 5

Normalized velocity distributions $P(v_x)$, on the rough plate for $\Gamma =1.5$, $\rho =0.014$ ($△$), $\rho =0.125$ ($\circ$), and $\rho =0.6$ ($•$). The solid line is a Gaussian. [Results for $P(v_y)$ are identical.]