Exploring the limits of granular hydrodynamics: A horizontal array of inelastic particles

The limits of granular hydrodynamics are explored in the context of the one-dimensional granular system introduced by Du, Li, and Kadanoff [Phys. Rev. Lett. 74, 1268 (1995)]. The density profile of the characteristic steady state, in which a single particle commutes between the driving wall and a dense cluster, is well captured by a hydrodynamic description provided that the finite size of the particles is incorporated. The temperature, however, is not well described: since all energy exchange is located at the border of the cluster, it is precisely for this quantity that the continuum approach breaks down.

Figure 1

(Color online) (a) Initial condition of a molecular dynamics simulation with $N=20$ identical particles, diameter $d$, randomly distributed over the tube length $L$. Every time the leftmost particle hits the left wall it is given a random velocity. The collisions of the rightmost particle with the right wall are elastic. (b) After many inelastic collisions, a cluster of slow particles is kept close to the right wall by one relatively fast particle commuting between the hot wall and the cluster. (c) Time-averaged number density $\tilde{n}\left(\tilde{x}\right)$ of the steady state and (d) the corresponding granular temperature $\tilde{T}\left(\tilde{x}\right)$ for $L=1000d$. (The tilde above $\tilde{x}$, $\tilde{n}$ and $\tilde{T}$ indicates that these are dimensionless quantities that will be introduced later.)

Figure 2

Example of an inelastic collision of two identical particles. The initial and final velocities are indicated: their sum is constant $\left(v_1+v_2=v_1^{\prime }+v_2^{\prime }\right)$, expressing momentum conservation, whereas the difference becomes smaller and defines the restitution coefficient $e=-\left(v_2^{\prime }-v_1^{\prime }\right)/\left(v_2-v_1\right)=0.9$.

Figure 3

(Color online) (a) Position of the cluster’s center of mass, ${\tilde{x}}_{\text{CM}}\left(t\right)=x_{\text{CM}}/d$, as a function of time (normalized by $t_0=L\sqrt{m/T_0}$) for a MD simulation with $N=20$ particles and a restitution coefficient $e=0.99$. The cluster occasionally expands and recompacts; the expansions occur when the left particle picks up a particularly small velocity from the hot wall. The dotted line (see also inset) marks the maximal value of $x_{\text{CM}}=L-d\left(N-1\right)/2$ corresponding to a close-packed cluster in a tube of length $L=1000d$. (b) For a higher restitution coefficient, $e=0.9999$, $x_{\text{CM}}$ stays much closer to this maximal value.

Figure 4

(Color online) MD vs Hydrodynamics in the near-elastic limit for $N=187$ particles in a tube of length $L=1000d$: [(a) and (b)] Time-averaged dimensionless number density $\tilde{n}\left(\tilde{x}\right)$ and granular temperature $\tilde{T}\left(\tilde{x}\right)$ obtained from MD simulations for three different values of the restitution coefficient $e$. The total duration of each simulation was $t=5\cdot {10}^7{t}_0$ (but we discarded the transient behavior before the onset of the steady state), sampled every 100 time units. [(c) and (d)] Density $\tilde{n}\left(\tilde{x}\right)$ and temperature $\tilde{T}\left(\tilde{x}\right)$ from hydrodynamic theory.

Figure 5

(Color online) Density $\tilde{n}\left(\tilde{x}\right)$ and temperature $\tilde{T}\left(\tilde{x}\right)$ along a tube of length $L=1000d$ for various numbers of particles: [(a) and (b)] $N=187$ and [(c) and (d)] $N=567$. The restitution coefficient is fixed at $e=0.9999$. Dashed red curves represent our MD simulations (total duration was $t/t_0=5\times {10}^7$ with sampling every 100 time units), and solid blue lines the hydrodynamic model (13, 14).

Figure 6

(Color online) Energy transport through the system: hydrodynamics vs MD. The energy transport according to the hydrodynamic model is represented by the thermal conductivity $\tilde{\kappa }\left(\tilde{x}\right)$ (solid blue line); it is calculated via Eq. (15) from the hydrodynamic temperature profile for $N=187$ particles of Figs. 5a, 5b. The energy transport for the MD simulations is expressed by $E_{\text{exch}}\left(\tilde{x}\right)$ (dotted black line).

Figure 7

(Color online) Same MD simulation data as in Figs. 5c, 5d (i.e., for $N=567$ particles, dashed red lines) but here the hydrodynamic curves are calculated with the localized-energy-exchange model of Eqs. (16, 17) (solid black lines). Both the density $\tilde{n}\left(\tilde{x}\right)$ and the temperature $\tilde{T}\left(\tilde{x}\right)$ now show good agreement.