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P.Evesque / 3-d granular gas and dissipation
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, e-mail:
[email protected] 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[-mv²/(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 to study 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 trends 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 synchronisation 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 reexamined and modified. The typical speed of a ball is found to vary linearly with the piston speed bω, but decreases when the number of balls N is increased (N=12, 24, 36 or 48). The distribution of waiting times τ between ball-gauge collisions is found to follow an exponential distribution experimentally, i.e. p(τ)∝ exp(poτ), 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 "à 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 12vz, no such effect of screening exists; but the collision probability between the wall and the ball still depends on the relative speed vz-bω sin(ωt); so, even with a constant flux of ball , i.e. even when f(p) is independent of time, the term v-bω sin(ωt) generates a net transfer of momentum from the wall to the gas in mean, as soon as v is finite. So, this kind of excitation shall always generate a periodic distribution f(p,q,t) near the moving wall. How deep this distribution modulation may propagate into the cell is not simple: for instance such periodic behaviour is frequently observed far above the bottom in 1-g experiments indicating that the density evolves with time and height. So, in micro-gravity condition, one may ask if f is still modulated with time or not. For instance near the wall, one expects that there are an incoming flux j=∫v>0vf(v)dv of balls and an outgoing flux j'=∫v0f(v)dv and >L1, the forcing shall not be seen deep in the cell in average, while it can still be observed on the collisions rule at the moving walls; hence, it does not mean however that no time modulation is observable on f, since one shall still see a poudres & grains 14 (2), 8-53 (Mai 2004)
P.Evesque / 3-d granular gas and dissipation
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modulation of f(v',t) when selecting a range of v with sufficient accuracy. When collisions happen this randomises more efficiently. Anyhow, as soon as L1
