Kinetic theory for a vibro-fluidized bed

The velocity distribution function for a two-dimensional vibro-fluidized bed of particles of radius r is calculated using asymptotic analysis in the limit where (i) the dissipation of energy during a collision due to inelasticity or between successive collisions due to viscous drag is small compared to the energy of a particle and (ii) the length scale for the variation of density is large compared to the particle size. In this limit, it is shown that the parameters $\epsilon_G = rg/T_0$ and $\epsilon = U^2_0/T_0 \ll 1$, and $\epsilon$ and $\epsilon_G$ are used as small parameters in the expansion. Here, g is the acceleration due to gravity, $U_0$ is the amplitude of the velocity of the vibrating surface and $T_0$ is the leading-order temperature (divided by the particle mass). In the leading approximation, the dissipation of energy and the separation of the centres of particles undergoing a binary collision are neglected, and the system is identical to a gas of rigid point particles in a gravitational field. The leading-order particle number density is given by the Boltzmann distribution $\rho_0 \alpha exp(−gz/T_0)$, and the velocity distribution function is given by the Maxwell-Boltzmann distribution ${f(u)= (2 \pi T_0)^{-1}}$ ${exp [-u^2/(2T_0)]}$, where u is the particle velocity. The temperature cannot be determined from the leading approximation, however, and is calculated by a balance between the rate of input of energy at the vibrating surface due to particle collisions with this surface, and the rate of dissipation of energy due to viscous drag or inelastic collisions. The rst correction to the distribution function due to dissipative eects is calculated using the moment expansion method, and all nontrivial first, second and third moments of the velocity distribution are included in the expansion. The correction to the density, temperature and moments of the velocity distribution are obtained analytically. The results show several systematic trends that are in qualitative agreement with previous experimental results. The correction to the density is negative at the bottom of the bed, increases and becomes positive at intermediate heights and decreases exponentially to zero as the height is increased. The correction to the temperature is positive at the bottom of the bed, and decreases and assumes a constant negative value as the height is increased. The mean-square velocity in the vertical direction is greater than that in the horizontal direction, thereby facilitating the transport of energy up the bed. The difference in the mean-square velocities decreases monotonically with height for a system where the dissipation is due to inelastic collisions, but it rst decreases and then increases for a system where the dissipation is due to viscous drag