Velocity distribution for a dilute vibrated granular material

The velocity distribution for a vibrated granular material is determined in the dilute limit where the frequency of particle collisions with the vibrating surface is large compared to the frequency of binary collisions. The particle motion is driven by the source of energy due to particle collisions with the vibrating surface, and two dissipation mechanisms–inelastic collisions and air drag—are considered. In the latter case, a general form for the drag force is assumed. First, the distribution function for the vertical velocity for a single particle colliding with a vibrating surface is determined in the limit where the dissipation during a collision due to inelasticity or between successive collisions due to drag is small compared to the energy of a particle. In addition, two types of amplitude functions for the velocity of the surface, symmetric and asymmetric about zero velocity, are considered. In all cases, differential equations for the distribution of velocities at the vibrating surface are obtained using a flux balance condition in velocity space, and these are solved to determine the distribution function. It is found that the distribution function is a Gaussian distribution when the dissipation is due to inelastic collisions and the amplitude function is symmetric, and the mean square velocity scales as $[〈U^2{〉}_S/(1\mathrm{}-e^2\left)\right],$ where $〈U^2{〉}_S$ is the mean square velocity of the vibrating surface and e is the coefficient of restitution. The distribution function is very different from a Gaussian when the dissipation is due to air drag and the amplitude function is symmetric, and the mean square velocity scales as $(〈U^2{〉}_{S}g/{\mu }_m{)}^{1/(m+2\mathrm{})\mathrm{}}$ when the acceleration due to the fluid drag is $-{\mu }_m{u}_y|u_y{|}^{m-1},$ where g is the acceleration due to gravity. For an asymmetric amplitude function, the distribution function at the vibrating surface is found to be sharply peaked around $[\pm 2〈U{〉}_S/(1\mathrm{}-e\left)\right]$ when the dissipation is due to inelastic collisions, and around $\pm \left[\mathrm{}\right(m+2\mathrm{})\mathrm{}〈U{〉}_{S}g/{\mu }_m{]}^{1/(m+1\mathrm{})\mathrm{}}$ when the dissipation is due to fluid drag, where $〈U{〉}_S$ is the mean velocity of the surface. The distribution functions are compared with numerical simulations of a particle colliding with a vibrating surface, and excellent agreement is found with no adjustable parameters. The distribution function for a two-dimensional vibrated granular material that includes the first effect of binary collisions is determined for the system with dissipation due to inelastic collisions and the amplitude function for the velocity of the vibrating surface is symmetric in the limit ${\delta }_I\equiv \mathrm{}\left(2nr\right)/\left(1\mathrm{}-e\right)\ll 1.$ Here, n is the number of particles per unit width and r is the particle radius. In this limit, an asymptotic analysis is used about the limit where there are no binary collisions. It is found that the distribution function has a power-law divergence proportional to $|u_x{|}^{\left(c{\delta }_I-1\right)}$ in the limit $u_x\to 0,$ where $u_x$ is the horizontal velocity. The constant c and the moments of the distribution function are evaluated from the conservation equation in velocity space. It is found that the mean square velocity in the horizontal direction scales as $O\left({\delta }_{I}T\right),$ and the nontrivial third moments of the velocity distribution scale as $O\left({\delta }_I{\epsilon }_I{T}^{3/2}\right)$ where ${\epsilon }_I=(1\mathrm{}-{e)}^{1/2}.$ Here, $T=[2〈U^2{〉}_S/(1\mathrm{}-e\left)\right]$ is the mean square velocity of the particles.