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
εG=rg/T0 and
ε=U20/T0[Lt ]1,
and
ε and εG are
used as small parameters in the expansion. Here, g is the acceleration
due to gravity,
U0 is the amplitude of the velocity of the
vibrating surface and T0 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
ρ0∝exp(−gz/T0,
and the velocity
distribution function is given by the Maxwell–Boltzmann distribution
f(u)=(2πT0)−1exp
[−u2/(2T0)],
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
first correction
to the distribution function due to dissipative effects is calculated using
the moment
expansion method, and all non-trivial 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 first
decreases and
then increases for a system where the dissipation is due to viscous drag.
Kompaneets equation for neutrinos: Application to neutrino heating in supernova explosions
