Is Dissipative Granular Gas in Knudsen Regime Excited by Vibration Biphasic?

P. Evesque: Lab MSSM, UMR 8579 CNRS, Ecole Centrale Paris, 92295 Châtenay-Malabry, France , evesque@mssmat.ecp.fr

Abstract: Investigation is pursued of the simple model I have proposed recently to describe a granular gas in Knudsen regime, in micro-gravity, and excited by a vibrating piston (vibration direction along Oz according to z=bw cos(wt)). This model predicts a probability distribution function f(v) of speed v in the direction of vibration whose tail varies approximately as f(v) µ (1/v) exp(-v/v_o), for which v_o obeys v_o=bbw/(an_l); here n_l is the number of granular layers in the cell at rest and b/a is a constant coefficient whose range is 0.06< b/a<2/3. This model results from a specific non local coupling between dissipation that occurs during a roundtrip due to ball-ball collisions and speed amplification due to ball-piston collision. It explains the main trends of the distribution p(I) of impacts I with a fix target. This trend has been obtained in recent experiments in board of the Airbus A300-0g of CNES, giving p(I) µ exp(-I/I_o). It predicts also the rate N_c of collisions with a fix gauge perpendicular to vibration; it finds N_c varies linearly with the gauge surface S and with the piston speed bw, but is independent of the number N of balls; the theory leads to a correct estimate of the experimental N_c. However, as the experimental N_c depends slightly on N, a second phase of balls “nearly at rest” is assumed to exist in order to explain the N_c vs N dependence. This phase describes balls “merely at rest”, which are in excess compared to the f(v) prediction; the dependence of v_o on this second phase is discussed. Compatibility between results from granular gas experiments in micro-gravity and experiments on Maxwell’s demon in 1-g is discussed. The main result of the paper is that the probability distribution function of speed v along z, i.e. f(v) µ (1/v) exp(-v/v_o), diverges as 1/v at small speed and is quite non Boltzmannian at large speed. Hence this makes the granular gas in Knudsen regime a peculiar problem, completely different from classic statistical mechanics. The main idea which allows understanding these results is to consider the piston playing the role of an impact generator or of a “velostat” instead of a thermostat. It is shown also that the model predicts completely different behaviour in 1g.

Pacs # : 05.45.-a, 45.50.-j, 45.70.-n, 81.70.Bt, 81.70.Ha, 83.10.Pp

poudres & grains 15 (2), 18-34 (1^er mars 2005)