One-dimensional kinetic models of granular flows. (English) Zbl 0981.76098
Summary: We introduce and discuss a one-dimensional kinetic model of Boltzmann equation with dissipative collisions and variable coefficient of restitution. Then, we investigate the behavior of Boltzmann equation in quasi-elastic limit for a wide range of rate functions. By this limit procedure we obtain a class of nonlinear equations classified as nonlinear friction equations. The analysis of the cooling process shows that the nonlinearity of relative velocity is important for the finite time extinction of the solution.
MSC:
| 76T25 | Granular flows |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
Keywords:
granular flows; one-dimensional kinetic model; Boltzmann equation; dissipative collisions; variable coefficient of restitution; quasi-elastic limit; nonlinear frictionReferences:
| [1] | R. Alexandre and C. Villani, On the Boltzmann equation for long range interactions and the Landau approximation in plasma physics. Preprint DMA, École Normale Supérieure (1999). |
| [2] | L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rational Mech. Anal.77 (1981) 11-21. Zbl0547.76085 · Zbl 0547.76085 · doi:10.1007/BF00280403 |
| [3] | G.I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge Univ. Press, New York (1996). |
| [4] | D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér.31 (1997) 615-641. Zbl0888.73006 · Zbl 0888.73006 |
| [5] | D. Benedetto, E. Caglioti and M. Pulvirenti, Erratum: A kinetic equation for granular media [RAIRO Modél. Math. Anal. Numér.31 (1997) 615-641]. ESAIM: M2AN33 (1999) 439-441. Zbl0972.74017 · Zbl 0972.74017 · doi:10.1051/m2an:1999118 |
| [6] | C. Bizon, J.B. Shattuck, M.D. Swift, W.D. McCormick and H.L. Swinney, Pattern in 2D vertically oscillated granular layers: simulation and experiments. Phys. Rev. Lett.80 (1998) 57-60. |
| [7] | A.V. Bobylev, J.A-Carrillo and I. Gamba, On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Statist. Phys.98 (2000) 743-773. Zbl1056.76071 · Zbl 1056.76071 · doi:10.1023/A:1018627625800 |
| [8] | C. Cercignani, R. Illner and M. Pulvirenti, The mathematical theory of dilute gases. Springer Ser. Appl. Math. Sci.106, Springer-Verlag, New York (1994). · Zbl 0813.76001 |
| [9] | L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation. Comm. Math. Phys.168 (1995) 417-440. Zbl0827.76081 · Zbl 0827.76081 · doi:10.1007/BF02101556 |
| [10] | Y. Du, H. Li and L.P. Kadanoff, Breakdown of hydrodynamics in a one-dimensional system of inelastic particles. Phys. Rev. Lett.74 (1995) 1268-1271. |
| [11] | D. Goldman, M.D. Shattuck, C. Bizon, W.D. McCormick, J.B. Swift and H.L. Swinney, Absence of inelastic collapse in a realistic three ball model. Phys. Rev. E57 (1998) 4831-4833. |
| [12] | I. Goldhirsch, Scales and kinetics of granular flows. Chaos9 (1999) 659-672. · Zbl 1055.76569 · doi:10.1063/1.166440 |
| [13] | M. Kac, Probability and related topics in the physical sciences. New York (1959). Zbl0087.33003 · Zbl 0087.33003 |
| [14] | L. Kantorovich, On translation of mass (in Russian). Dokl. AN SSSR37 (1942) 227-229. |
| [15] | L. Landau, Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung. Phys. Z. Sowjet.10 (1936) 154. Trad.: The transport equation in the case of Coulomb interactions, in Collected papers of L.D. Landau, D. ter Haar Ed., Pergamon Press, Oxford (1981) 163-170. Zbl0015.38202 · Zbl 0015.38202 |
| [16] | S. McNamara and W.R. Young, Inelastic collapse and clumping in a one-dimensional granular medium. Phys. Fluids A4 (1992) 496-504. |
| [17] | S. McNamara and W.R. Young, Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A5 (1993) 34-45. |
| [18] | G. Naldi, L. Pareschi and G. Toscani, Spectral methods for a singular Boltzmann equation for granular flows and numerical quasi elastic limit. Preprint (2000). · Zbl 1046.76034 |
| [19] | G. Toscani, The grazing collision asymptotic of the non cut-off Kac equation. RAIRO Modél. Math. Anal. Numér.32 (1998) 763-772. · Zbl 0912.76081 |
| [20] | I. Vaida, Theory of statistical Inference and Information. Kluwer Academic Publishers, Dordrecht (1989). |
| [21] | L.N. Vasershtein, Markov processes on countable product space describing large systems of automata (in Russian). Problemy Peredachi Informatsii5 (1969) 64-73. Zbl0273.60054 · Zbl 0273.60054 |
| [22] | C. Villani, Contribution à l’étude mathématique des équations de Boltzmann et de Landau en théorie cinétique des gaz et des plasmas. Ph.D. thesis, Univ. Paris-Dauphine (1998). |
| [23] | C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal.143 (1998) 273-307. Zbl0912.45011 · Zbl 0912.45011 · doi:10.1007/s002050050106 |
| [24] | C. Villani, Contribution à l’étude mathématique des collisions en théorie cinétique. Ceremade, Paris IX-Dauphine, January 24 (2000). |
| [25] | V.M. Zolotarev, Probability Metrics. Theory Probab. Appl.28 (1983) 278-302. · Zbl 0533.60025 · doi:10.1137/1128025 |
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