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Homogeneous Cooling with Repulsive and Attractive Long-Range Potentials

Published online by Cambridge University Press:  18 July 2011

M. K. Müller  and
S. Luding
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Abstract

The interplay between dissipation and long-range repulsive/attractive forces in homogeneous, dilute, mono-disperse particle systems is studied. The pseudo-Liouville operator formalism, originally introduced for hard-sphere interactions, is modified such that it provides very good predictions for systems with weak long-range forces at low densities, with the ratio of potential to fluctuation kinetic energy as control parameter. By numerical simulations, the theoretical results are generalized with empirical, density dependent correction-functions up to moderate densities.

The main result of this study on dissipative cooling is an analytical prediction for the reduced cooling rate due to repulsive forces and for the increased rate due to attractive forces. In the latter case, surprisingly, for intermediate densities, similar cooling behavior is observed as in systems without long-range interactions. In the attractive case, in general, dissipation leads to inhomogeneities earlier and faster than in the repulsive case.

Keywords

granular gashydrodynamicslong-range forcesnumerical simulations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 4: Granular hydrodynamics , 2011 , pp. 118 - 150
Copyright
© EDP Sciences, 2011

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References

M. P. Allen, D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, 1987.
Almekinders, J. C., Jones, C.. Multiple jet electrohydrodynamic spraying and applications. J. Aerosol Sci., 30 (1999), 969–971. CrossRefGoogle Scholar
Barnes, J., Hut, P.. A hierarchical O(NlogN) force calculation algorithm. Nature, 324 (1986), 446–449. CrossRefGoogle Scholar
D. L. Blair, A. Kudrolli. Magnetized granular materials. In H. Hinrichsen and D. Wolf, editors, The Physics of Granular Media., pages 281–296, Weinheim, 2004. Wiley-VCH.
Blum, J., Bruns, S., Rademacher, D., Voss, A., Willenberg, B., Krause, M.. Measurement of the translational and rotational Brownian motion of individual particles in a rarefied gas. Phys. Rev. Lett., 97 (2006), 230601. CrossRefGoogle Scholar
Blum, J., Wurm, G., Kempf, S., Poppe, T., Klahr, H., Kozasa, T., Rott, M., Henning, T., Dorschner, J., Schräpler, R., Keller, H.U., Markiewicz, W.J., Mann, I., Gustafson, B.A.S., Giovane, F., Neuhaus, D., Fechtig, H., Grün, E., Feuerbacher, B., Kochan, H., Ratke, L., El Goresy, A., Morfill, G., Weidenschilling, S.J., Schwehm, G., Metzler, K., Ip, W.-H.. Growth and form of planetary seedlings: Results from a microgravity aggregation experiment. Phys. Rev. Lett., 85 (2000), 2426–2429. CrossRefGoogle ScholarPubMed
Bode, P., Ostriker, J. P.. Tree particle-mesh: An adaptive, efficient, and parallel code of collisionless cosmological simulation. Astrophys. J. Supplem. Series, 145 (2003), No. 1, 1–13. CrossRefGoogle Scholar
Brahic, A.. Systems of colliding bodies in a gravitational field: I - numerical simulation of the standard model. Astronomy and Astrophysics, 54 (1977), 895–907. Google Scholar
Bridges, F. G., Hatzes, A., Lin, D. N. C.. Structure, stability and evolution of Saturn’s rings. Nature, 309 (1984), 333–335. CrossRefGoogle Scholar
Brilliantov, N. V., Pöschel, T.. Deviation from maxwell distribution in granular gases with constant restitution coefficient. Phys. Rev. E, 61 (2000), 2809–2814. CrossRefGoogle Scholar
N. V. Brilliantov, T. Pöschel. Kinetic Theory of Granular Gases. Oxford University Press, Oxford, 2004.
Brilliantov, N. V., Saluena, C., Schwager, T., Pöschel, T.. Transient structures in a granular gas. Phys. Rev. Let., 93 (2004), No. 13, 134301. CrossRefGoogle Scholar
Brilliantov, N. V., Spahn, F., Hertzsch, J.-M., Pöschel, T.. A model for collisions in granular gases. Phys. Rev. E, 53 (1996), 5382–5392. CrossRefGoogle ScholarPubMed
Carnahan, N. F., Starling, K. E.. Equation of state for nonattractive rigid spheres. J. Chem. Phys., 51 (1969), No. 2, 635–636. CrossRefGoogle Scholar
J. A. Cross. Electrostatics: Principles, Problems and Applications. Adam Hilger, Bristol, 1987.
S. M. Dammer, J. Werth, H. Hinrichsen. Electrostatically charged granular matter. In H. Hinrichsen D. Wolf, editors, The Physics of Granular Media., pages 255–280, Wiley-VCH, Weinheim, 2004.
Dammer, S. M., Wolf, D. E.. Self-focusing dynamics in monopolarly charged suspensions. Phys. Rev. E, 93 (2004), No. 15, 150602. Google ScholarPubMed
Du, J.. Hydrostatic equilibrium and Tsallis’ equilibrium for self-gravitating systems. Central European Journal of Physics, 3 (2005), No. 3, 376–381. Google Scholar
J. Duran. Sands, Powders and Grains. Springer-Verlag, New York, 2000.
Eastwood, J. W., Hockney, R. W., Lawrence, D.. P3M3DP - the 3-dimensional periodic particle-particle-particle-mesh program. Comp. Phys. Commun., 19 (1980), 215–261. CrossRefGoogle Scholar
Ernst, M. H.. Nonlinear model-Boltzmann equations and exact solutions. Physics Reports, 78 (1981), No. 1, 1–171. CrossRefGoogle Scholar
Ernst, M. H., Dorfman, J. R., Hoegy, W. R., van Leeuwen, J. M. J.. Hard-sphere dynamics and binary-collision operators. Physica, 45 (1969), No. 1, 127–146. CrossRefGoogle Scholar
Ernst, M. H., Trizac, E., Barrat, A.. The Boltzmann equation for driven systems of inelastic soft spheres. J. Stat. Phys., 124 (2006), No. 2–4, 549–586. CrossRefGoogle Scholar
Eshuis, P., van der Weele, K., van der Meer, D., Lohse, D.. The granular Leidenfrost effect: Experiment and theory of floating particle clusters. Phys. Rev. E, 95 (2005), 258001.
Esipov, S. E., Pöschel., T.. The granular phase diagram. J. Stat. Phys., 86 (1997), No. 5/6, 1385–1395. CrossRefGoogle Scholar
L. W. Esposito, J. N. Cuzzi, J. B. Holberg, E. A. Marouf, G. L. Tyler, C. C. Porco. Saturn’s rings, structure, dynamics and particle properties. In Saturn, pages 463–545, Tucson, AZ, Univ. of Arizona Press, 1984.
Ewald, P. P.. The calculation of optical and electrostatic grid potential. Ann. d. Physik, 64 (1921), 253–287. CrossRefGoogle Scholar
K. B. Geerse. Application of Electrospray: from people to plants. Ph.D. thesis, Technische Universiteit Delft, 2003.
Goldhirsch, I., Zanetti, G.. Clustering instability in dissipative gases. Phys. Rev. Lett., 70 (1993), No. 11, 1619–1622. CrossRefGoogle ScholarPubMed
Goldreich, P.. The dynamics of planetary rings. Ann. Rev. Astron. Astrophys., 20 (1982), 249–283. CrossRefGoogle Scholar
R. Greenberg, A. Brahic. Planetary Rings. Arizona University Press, Tucson, AZ, 1984.
Greengard, L., Rokhlin, V.. A fast algorithm for particle simulations. J. of Comp. Phys., 73 (1987), 325–348. CrossRefGoogle Scholar
Haff, P. K.. Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech., 134 (1983), 401–430. CrossRefGoogle Scholar
J.-P. Hansen, I. R. McDonald. Theory of Simple Liquids. Academic Press Ltd., London, San Diego, 1990.
Henderson, D.. A simple equation of state for hard discs. Molec. Phys., 30 (1975), No. 3, 971–972. CrossRefGoogle Scholar
Herbst, O., Cafiero, R., Zippelius, A., Herrmann, H. J., Luding, S.. A driven two-dimensional granular gas with coulomb friction. Phys. of Fluids, 17 (2005), No. 10, 107102. CrossRefGoogle Scholar
Herbst, O., Müller, P., Zippelius, A.. Local heat flux and energy loss in a two-dimensional vibrated granular gas. Phys. Rev. E, 72 (2005) No. 4, 041303. CrossRefGoogle Scholar
Hernquist, L.. Hierarchical N-body methods. Comp. Phys. Commun., 48 (1988), 107–115. CrossRefGoogle Scholar
R. Hoffmann. Modeling and Simulation of an Electrostatic Image Transfer. (Ph.D. thesis) Shaker Verlag, Aachen, 2004.
Høye, J. S.. Dynamical pair correlations of classical and quantum fluids perturbed with long-range forces. Physica A, 389 (2010), 1380–1390. CrossRefGoogle Scholar
Huthmann, M., Zippelius, A.. Dynamics of inelastically colliding rough spheres: Relaxation of translational and rotational energy. Phys. Rev. E, 56 (1997), No. 6, R6275–R6278. CrossRefGoogle Scholar
Kleber,, W.. Lang, A.. Triboelectrically charged powder coatings generated by running through holes and slits. J. of Electrostatics, 40&41 (1997), 237–240. CrossRefGoogle Scholar
Krause, M., Blum, J.. Growth and form of planetary seedlings: Results from a sounding rocket microgravity aggregation experiment. Phys. Rev. Lett., 93 (2004), 021103. CrossRefGoogle ScholarPubMed
C. W. J. Lemmens. An Investigation, Implementation and Comparison of 3 important Particle Simulation Techniques: PP: Particle-Particle PM: Particle-Mesh TC: Tree-Code. Report 97-46, Faculty of Technical Mathematics and Informatics, Delft, 1997.
Linsenbühler, M., Werth, J. H., Dammer, S. M., Knudsen, H. A., Hinrichsen, H., Wirth, K.-E., Wolf, D. E.. Cluster size distribution of charged nanopowders in suspensions. Powder Technology, 167 (2006), No. 3, 124–133. CrossRefGoogle Scholar
Lohse, D., Bergmann, R., Mikkelsen, R., Zeilstra, C., van der Meer, D., Versluis, M., van der Weele, K., van der Hoef, M., Kuipers, H.. Impact on soft sand: Void collapse and jet formation. Phys. Rev. Let., 93 (2004), No. 19, 198003. CrossRefGoogle ScholarPubMed
Lowell, J., Rose-Innes, A. C.. Contact electrification. Adv. in Phys., 29 (1980), No. 6, 947–1023. CrossRefGoogle Scholar
Luding, S.. Clustering instabilities, arching, and anomalous interaction probabilities as examples for cooperative phenomena in dry granular media. T.A.S.K. Quarterly, Scientific Bulletin of Academic Computer Centre of the Technical University of Gdansk., 2 (1998), No. 3, 417–443. Google Scholar
S. Luding. Collisions & contacts between two particles. In H. J. Herrmann, J.-P. Hovi, S. Luding, editors, Physics of dry granular media - NATO ASI Series E350, page 285, Dordrecht, 1998. Kluwer Academic Publishers.
Luding, S.. Structure and cluster formation in granular media. Pranama-J. Phys., 64 (2005), No. 6, 893–902. CrossRefGoogle Scholar
Luding, S.. Cohesive frictional powders: Contact models for tension. Granular Matter, 10 (2008), No. 4, 235–246. CrossRefGoogle Scholar
Luding, S.. Towards dense, realistic granular media in 2d. Nonlinearity, 22 (2009), R101–R146. CrossRefGoogle Scholar
Luding, S., Goldshtein, A.. Collisional cooling with multi-particle interactions. Granular Matter, 5 (2003), No. 3, 159–163. CrossRefGoogle Scholar
Luding, S., Herrmann, H. J.. Cluster growth in freely cooling granular media. Chaos, 9 (1999), No. 3, 673–681. CrossRefGoogle ScholarPubMed
Luding, S., Huthmann, M., McNamara, S., Zippelius, A.. Homogeneous cooling of rough, dissipative particles: Theory and simulations. Phys. Rev. E, 58 (1998), 3416–3425. CrossRefGoogle Scholar
Luding, S., McNamara, S.. How to handle the inelastic collapse of a dissipative hard-sphere gas with the TC model. Granular Matter, 1 (1998), No. 3, 113–128. CrossRefGoogle Scholar
McNamara, S.. Hydrodynamic modes of a uniform granular medium. Phys. of Fluids A, 5 (1993), 3056–3070. CrossRefGoogle Scholar
McNamara, S. Young, W. R.. Dynamics of a freely evolving, two-dimensional granular medium. Phys. Rev. E, 53 (1996), 5089–5100. CrossRefGoogle Scholar
S. Miller. Clusterbildung in Granularen Gasen. (in German). Ph.D. thesis, Universität Stuttgart, 2003.
Miller, S., Luding, S.. Cluster growth in two- and three-dimensional granular gases. Phys. Rev. E, 69 (2004), No. 3, 031305. CrossRefGoogle ScholarPubMed
Montanero, J. M., Garzò, V., Alam, M., Luding, S.. Rheology of 2d and 3d granular mixtures under uniform shear flow: Enskog kinetic theory versus molecular dynamics simulations. Granular Matter, 8 (2006), No. 2, 103–115. CrossRefGoogle Scholar
M.-K. Müller. Untersuchung von Akkretionsscheiben mit Hilfe der Molekulardynamik. (in German). Diploma thesis, Universität Stuttgart, 2001.
M.-K. Müller. Long-Range Interactions In Dilute Granular Systems. Ph.D. thesis, Universiteit Twente/Enschede, 2007.
M.-K. Müller, S. Luding. Long-range interactions in ring-shaped particle aggregates. In R. García-Rojo, H.J. Herrmann, S. McNamara, editors, Powders & Grains, pages 1119–1122, Balkema, Leiden, 2005.
M.-K. Müller, S. Luding. Homogeneous cooling with repulsive and attractive long-range interactions. In M. Nakagawa S. Luding, editors, Powders & Grains, pages 697–700, AIP Conf. Procs. #1145, 2009.
M.-K. Müller, T. Winkels, K.B. Geerse, J.C.M. Marijnissen, A. Schmidt-Ott, S. Luding. Experiment and simulation of charged particle sprays. In Proceedings PARTEC 2004, Nuremberg, 2004.
B. Muth, M.-K. Müller, P. Eberhard, S. Luding. Contacts between many bodies. In W. Kurnik, editor, Machine Dynamics Problems, pages 101–114, Warsaw, 2004.
E. Németh. Triboelektrische Aufladung von Kunststoffen. (in German). Ph.D. thesis, Technische Universität Bergakademie Freiberg, 2003.
F. Niermöller. Ladungsverteilung in Mineralgemischen und elektrostatische Sortierung nach Triboaufladung. (in German). Ph.D. thesis, Technische Universität Clausthal, 1988.
Olafsen, J. S., Urbach, J. S.. Clustering, order, and collapse in a driven granular monolayer. Phys. Rev. Lett., 81 (1998), 4369. CrossRefGoogle Scholar
Orza, J. A. G., Brito, R., van Noije, T. P. C., Ernst, M. H.. Patterns and long range correlations in idealized granular flows. Int. J. of Mod. Phys. C, 8 (1997), No. 4, 953–965. CrossRefGoogle Scholar
T. Pöschel S. Luding, editors. Granular Gases, Lecture Notes in Physics 564. Springer, Berlin, 2001.
Ramírez, R., Pöschel, T., Brilliantov, N. V., Schwager, T.. Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. E, 60 (1999), 4465–4472. CrossRefGoogle ScholarPubMed
Ree, F. H., Hoover, W. G.. Fifth and sixth virial coefficients for hard spheres and hard disks. J. Chem. Phys., 40 (1964), No. 4, 939–950. CrossRefGoogle Scholar
T. N. Scheffler. Kollisionskühlung in elektrisch geladener granularer Materie. (in German). Ph.D. thesis, Gerhard-Mercator-Universität Duisburg, 2000.
Scheffler, T. N., Wolf, D. E.. Collision rates in charged granular gases. Granular Matter, 4 (2002), No. 3, 103–113. CrossRefGoogle Scholar
Schwager, T., Pëoschel, T.. Coefficient of restitution of viscous particles and cooling rate of granular gases. Phys. Rev. E, 57 (1998), 650–654. CrossRefGoogle Scholar
Spahn, F., Schmidt, J.. Saturn’s bared mini-moons. Nature, 440 (2006), 614–615. CrossRefGoogle ScholarPubMed
Springel, V., White, S. D. M., Jenkins, A., Frenk, C. S., Yoshida, N., Gao, L., Navarro, J., Thacker, R., Croton, D., Helly, J., Peacock, J. A., Cole, S., Thomas, P., Couchman, H., Evrard, A., Colberg, J., Pearce, F.. Simulating the joint evolution of quasars, galaxies and their large-scale distribution. Nature, 435 (2005), 629–636. CrossRefGoogle Scholar
U. Trottenberg, C. W. Oosterlee, A. Schüller. Multigrid. Academic Press, San Diego, 2001.
van Noije, T. P. C., Ernst, M. H., Brito, R.. Ring kinetic theory for an idealized granular gas. Physica A, 251 (1998), 266–283. CrossRefGoogle Scholar
Werth, J. H., Dammer, S. M., Farkas, Z., Hinrichsen, H., Wolf, D. E.. Agglomeration in charged suspensions. Computer Physics Communications, 147 (2002), 259–262. CrossRefGoogle Scholar
Werth, J. H., Knudsen, H., Hinrichsen, H.. Agglomeration of oppositely charged particles in nonpolar liquids. Phys. Rev. E, 73 (2006), 021402. CrossRefGoogle ScholarPubMed
Werth, J. H., Linsenbuhler, M., Dammer, S. M., Farkas, Z., Hinrichsen, H., Wirth, K.-E., Wolf, D. E.. Agglomeration of charged nanopowders in suspensions. Powder Technology, 133 (2003), 106–112. CrossRefGoogle Scholar
Wolf, D. E., Scheffler, T. N., Schäfer, J.. Granular flow, collisional cooling and charged grains. Physica A, 274 (1999), 171–181. CrossRefGoogle Scholar