Simulations of vibrated granular medium with impact-velocity-dependent restitution coefficient

Sean McNamara and Eric Falcon

Phys. Rev. E 71, 031302 – Published 24 March 2005

Abstract

We report numerical simulations of strongly vibrated granular materials designed to mimic recent experiments performed in both the presence and the absence of gravity. The coefficient of restitution used here depends on the impact velocity by taking into account both the viscoelastic and plastic deformations of particles, occurring at low and high velocities, respectively. We show that this model with impact-velocity-dependent restitution coefficient reproduces results that agree with experiments. We measure the scaling exponents of the granular temperature, collision frequency, impulse, and pressure with the vibrating piston velocity as the particle number increases. As the system changes from a homogeneous gas state at low density to a clustered state at high density, these exponents are all found to decrease continuously with increasing particle number. All these results differ significantly from classical inelastic hard sphere kinetic theory and previous simulations, both based on a constant restitution coefficient.

Received 30 November 2004

DOI:https://doi.org/10.1103/PhysRevE.71.031302

Authors & Affiliations

Sean McNamara^1,* and Eric Falcon^2,†

¹Centre Européen de Calcul Atomique et Moléculaire, 46, allée d’Italie, 69 007 Lyon, France
²Laboratoire de Physique, École Normale Supérieure de Lyon, UMR 5672, 46, allée d’Italie, 69 007 Lyon, France

^*Permanent address: I.C.P., Universität Stuttgart, 70569 Stuttgart, Germany.
^†Email address: Eric.Falcon@ens-lyon.fr;URL:http//perso.ens-lyon.fr/eric.falcon/

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Vol. 71, Iss. 3 — March 2005

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Images

Figure 1
The restitution coefficient $r$ as a function of impact velocity $v$ , as given in Eq. (1) (solid line). The dashed lines show $v_{0} = 0.3 m ∕ s$ and $r_{0} = 0.95$ . Experimental points (●) for steel spheres were extracted from Fig. 1 of Ref. 25.Reuse & Permissions
Figure 2
Time averaged pressure $P$ on the top of the cell as a function of particle layer $n$ for various vibration frequencies $f$ . (a) Experimental results from [10] for stainless steel beads $2 mm$ in diameter, with $A = 25 mm$ , $10 ⩽ f ⩽ 20 Hz$ with a $1 Hz$ step (from bottom to top) and $h - h_{0} = 5 mm$ . (b) Numerical simulation where the coefficient of restitution $r (v)$ is given by Eq. (1). (c) Numerical simulation with a coefficient of restitution of 0.95, independent of impact velocity. (d) Numerical simulation with a coefficient of restitution of 0.7, independent of impact velocity. The simulations (b), (c), and (d) are 2D with gravity, done for $2 mm$ disks, with $A = 25 mm$ , $10 ⩽ f ⩽ 30 Hz$ with a $2 Hz$ step (from bottom to top) and $h - h_{0} = 15 mm$ . In the simulations, the two-dimensional pressure is given in N/m.Reuse & Permissions
Figure 3
Snapshots from the simulations with $n = 3$ , gravity $g \neq 0$ , driving frequency $f = 30 Hz$ , and $h - h_{0} = 15 mm$ . The upper wall is stationary, and the lower wall is the piston, which is at its lowest position in both snapshots. The horizontal boundaries are periodic (indicated by dashed lines). Gravity points downward. (a) $r = r (v)$ , as given in Eq. (1), and (b) constant $r = 0.7$ . In (b) we see a tight cluster which was not observed in the experiments.Reuse & Permissions
Figure 4
The exponents $θ$ as a function of $n$ which give the scaling of the granular temperature $T$ (◇), collision frequency $N_{c}$ (∗), mean impulsion $Δ I$ (▵), and pressure $P$ (엯). All these quantities are proportional to $V^{θ (n)}$ . Without gravity, (a) for $r = 0.95$ and (b) for $r = r (v)$ . With gravity, (c) for $r = 0.95$ and (d) for $r = r (v)$ . The exponents are obtained by fixing $n$ and performing 11 simulations, varying $f$ from $10 to 30 Hz$ . Then $\ln (X)$ (where $X$ is the quantity being considered) is plotted against $\ln (V)$ . The resulting curve is always nearly a straight line [except for $n > 3$ in (d)—see text], and the exponent is calculated from a least squares fit. The pressure scaling point (●) on (b) is from the experiment [11] performed in low gravity. See Fig. 3a for typical snapshot corresponding to $n = 3$ , $g \neq 0$ and $r = r (v)$ .Reuse & Permissions
Figure 5
Experimental data performed under gravity from Ref. 10: The exponents $θ (n)$ of time averaged pressure (◻) [see Fig. 2a], and kinetic energy extracted from density profile (엯) or volume expansion (◇) measurements. These data should be compared with the simulations of Fig. 4d.Reuse & Permissions

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