Dynamical approach to weakly dissipative granular collisions

Granular systems present surprisingly complicated dynamics. In particular, nonlinear interactions and energy dissipation play important roles in these dynamics. Usually (but admittedly not always), constant coefficients of restitution are introduced phenomenologically to account for energy dissipation when grains collide. The collisions are assumed to be instantaneous and to conserve momentum. Here, we introduce the dissipation through a viscous (velocity-dependent) term in the equations of motion for two colliding grains. Using a first-order approximation, we solve the equations of motion in the low viscosity regime. This approach allows us to calculate the collision time, the final velocity of each grain, and a coefficient of restitution that depends on the relative velocity of the grains. We compare our analytic results with those obtained by numerical integration of the equations of motion and with exact ones obtained by other methods for some geometries.

Figure 1

Collision duration for several values of $\gamma$ (from top to bottom, 0.0001, 0.001, 0.01, and 0.1) and two different values of $\alpha$ (0 for lower group and $1/2$ for higher group), obtained via numerical integration of the equations of motion (symbols) and from our theory (line). The line represents the theoretical prediction of Eq. (29). The inset shows the same data on a log-log scale. In this figure, the parameters are as follows: $m_1=m_2=1,R_1=R_2=1$, and $2/\left(5D\right)=1$. As predicted, for these values of $\gamma$ and $\alpha$ the collision durations are essentially independent of these parameters.

Figure 2

Duration of collisions for several values of $R_2$ (from bottom to top, $R_2$ varies from 0.1 to 0.9 in steps of 0.1) obtained via numerical integration of the equations of motion. The lines represent the theoretical prediction Eq. (29). In this figure the parameters are as follows: $m_1=1,R_1=1,\alpha =1/2$, $\gamma =0.001$, and $2/\left(5D_{12}\right)=1$. The densities of the two colliding spheres are equal.

Figure 3

Coefficient of restitution as a function of the initial relative velocity. Each curve corresponds to a different value of $\alpha$ (from bottom to top on the left side, $\alpha$ varies from 0 to 0.9 in steps of 0.1). The other parameters are: $\gamma =0.001$, $m_1=m_2=1$, $R_1=R_2=1$, and $2/\left(5D\right)=1$. The lines correspond to the theoretical prediction of Eq. (33).

Figure 4

Coefficient of restitution as a function of the initial relative velocity. Each curve corresponds to a different value of the radius of granule 2 (from bottom to top, $R_2$ varies from 0.5 to 1.4 in steps of 0.1). The other parameters are: $\gamma =0.001,m_1=1$, $R_1=1$, and $2/\left(5D_{12}\right)=1$. The densities of the two colliding spheres are equal. The lines correspond to the theoretical prediction of Eq. (33).