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,


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

poudres & grains 14 (2), 8-53 (Mai 2004)