New corner stones in
dissipative granular gases On some theoretical implication of Liouville’s Equation in the physics of loose granular dissipative gases
P.
Evesque : Lab MSSMat, UMR
8579 CNRS, Ecole Centrale Paris 92295 CHATENAY-MALABRY, France, evesque@mssmat.ecp.fr
Abstract: The dynamics of a granular dissipative gas
is discussed starting from the Liouville’s equation to derive a Boltzmann’s
equation, taking account of the inelasticity of collisions with the walls and
between balls . It is recalled that the Boltzmann’s distribution, i.e. exp[-mv˛/(2kT)], is a steady solution of
Boltzmann’s equation only when collisions are elastic; hence it is not
applicable in the case of dissipative granular gas.
Then experiments on non interacting balls in a vibrated cylindrical
box are re-examined using cells containing 1 ball or 2 balls. They allow
studying the effect on the dynamics of the dissipation during ball-wall
collisions. In a first experiment with an electromagnetic vibrator on earth or
in board of Airbus A300 –0g of CNES, the 1-ball dynamics exhibit little
transverse motion and an intermittent quasi periodic motion along the direction
parallel to the vibration. It is quite different from the erratic motion
predicted for the Fermi case, with no dissipation. The reported behaviour
proves a significant reduction of the phase space dimension of this
billiard-like system from 11-d to 3-d or 1-d. It is caused by dissipation,
which generates non ergodic dynamics. It exemplifies the coupling between
translation and rotation degrees of freedom during the collisions with the
walls, due to solid friction at contacts. This eliminates ball rotation and
freezes transverse velocity fluctuations. This trend is confirmed by 3-d
simulations with JJ Moreau discrete element code, and by a two-ball experiment
performed under zero-g conditions in the Maxus 5 flight. For this second
experiment, the quasi-periodicity is found much greater, which is probably due
to an improvement of experimental conditions. The two balls are not in perfect
synchronization showing the effect of small random noise; but the two particles
have never collided. This is then the normal dynamics of a gas of
non-interacting dilute spherical grains in a vibrated rectangular container.
The dynamics of interacting particles with dissipation is then
studied experimentally in the case of a small number of grains, i.e. of small
ball-ball interaction, in a cell with a vibrating piston. The interpretation is
re-examined and modified. The typical speed of a ball is found to vary linearly
with the piston speed bw, but
decreases when the number of balls N is
increased (N=12, 24, 36 or 48).
The distribution of waiting times t between ball-gauge collisions is
found to follow an exponential distribution experimentally, i.e. p(t)µ
exp(-pot), proving the uncorrelated motion
of balls. The amplitude I of
the ball-gauge impacts has been determined from the signal response of the
sensor. This requires to determine a transfer function and to proceed to a
deconvolution. The N=12 balls case is used for this purpose. The distribution f(v) of ball speed v exhibits an exponential trend f(v)=exp(-v/vo) in the case N=24, 36, 48. This is temptatively
explained using a model "ŕ la Boltzmann" associated with the notion
of "velostat". Also a second
model is proposed, which describes the fast speed tail of the distribution
which is determined to leading order. It is found experimentally that both, vo and po, depend on N, and a scaling law is
proposed, although the scaling is tested in a very small range 12<N<48 .
At last a general discussion is tempted in the framework of
Boltzmann’s equation.
It turns out that coupling between rotation and translation cannot
be neglected in the collisions, because it generates efficient dissipation;
that makes the system quite sensitive to even a small number of collisions.
This shall be introduced in simulations codes.
Pacs # : 05.45.-a, 45.50.-j, 45.70.-n, 81.70.Bt, 81.70.Ha,
83.10.Pp