Institute of Problems of Mechanical Engineering of the RAS (Russia):

Demidov I. V., Postgraduate

REC «Mekhanobr-tekhnika» (Russia):

Vaisberg L. А., Corresponding Member of the Russian Academy of Sciences (RAS), Doctor of Engineering Sciences, Professor, Company Scientific Advisor

Ivanov K. S., Ph. D. in Engineering Sciences, Researcher

E-mail (common): gornyi@mtspb.com

The article presents a review of the modern methods of computational investigation and modeling of dynamics of granular media under vibration action. Attention is focused on typical states of granular material, realized depending on intensity of vibration action and mechanical characteristics of material. Three «physical states» are distinguished: «elastic-plastic» solid — if mutual arrangement of material particles is changed little, viscous «granular liquid» — if most particles are shifted without breaking contacts, and «granular gaseous» — when time of contact interaction between particles is small in comparison with their free time of flight. Also, three groups of approaches to modeling are distinguished: microscopical — considering material at the level of separate particles, statistical mechanics / kinetic theory — generalizing kinetic theory of gases with respect to inelastic interactions, phenomenological models and continuum mechanical models. A range of applicability and possibilities are specified for each group of models. Application of the described approaches is illustrated by a number of cases of specific behavior of granular materials: Leidenfrost granular effect, granular convection, oscillons, etc.

The work was performed under the Ministry of Education and Science of the Russian Federation grant No. 14.579.21.0023 from 05.06.2014.

1. Jaeger H. M., Nagel S. R., Behringer R. P. Granular solids, liquids, and gases. Rev. of Mod. Phys., 1996, Vol. 68, No. 4, pp. 1259–1273.
2. Duru P., Nikolas M., Hinch J., Guazzelli E. Constitutive laws in liquid-fluidized beds. J. Fluid Mech., 2002, Vol. 452, pp. 371–404.
3. Pastenes J. C., Geminard J.-C., Melo F. Interstitial gas effect on vibrated granular columns. Phys. Rev., 2014, E 89, 062205.
4. Cafiero R., Luding S., Herrmann H. J. Rotationally driven gas of inelastic rough spheres. EPL (Europhysics Letters), 2002, 60, pp. 854–860.
5. http://www.elementy.ru.
6. Burkhardt T. W. Dynamics of inelastic collapse. Phys. Rev. 2000, E 63, 011111.
7. Majumdar S. N., Kearney M. J. Inelastic collapse of a ball bouncing on a randomly vibrating platform. Phys. Rev., 2007, E 76, 031130.
8. Wakou J., Kitagishi H., Sakaue T., Nakanishi H. Inelastic collapse in one-dimensional driven systems under gravity. Phys. Rev., 2013, E 87, 042201.
9. Pöschel T., Herrmann H. J. Size segregation and convection. EPL, 1995, 29 (2), 123.
10. Goldsmith W. Impact. London, Edward Arnold Ltd., 1960, pp. 257–267.
11. Kuwabara G., Kono K. Restitution coefficient in collision between two spheres. Jpn. J. Appl. Phys., 1987, 26, Part 1, pp. 1230–1233.
12. Goldman D., Shattuck M. D., Bizon C., McCormick W. D., Swift J. B., Swinney H. L. Absence of inelastic collapse in a realistic three ball model. Phys. Rev., 1998, E 57, No. 4.
13. Brilliantov N. V., Pöeschel T. Kinetic theory of granular gases. Oxford University Press, 2004, 340 p.
14. Herbst O., Huthmann M., Zippelius A. Dynamics of inelastically colliding spheres with Coulomb friction: relaxation of translational and rotational energy. Granular Matter, 2000, 2, pp. 211–219.
15. Raskin Kh. I. Application of the physical kinetics methods to the problems of vibration effects on granular media. Doklady Akademii Nauk SSSR = Proceedings of the USSR Academy of Sciences, 1975, Vol. 220, No. 1, pp. 54–57.

16. Kremer G. M., Santos A., Garzó V. Transport coefficients of granular gas of inelastic rough hard spheres. Phys. Rev., 2014, E 90, 022205.
17. Khalil N., Garzó V., Santos A. Hydrodynamic Burnett equations for inelastic Maxwell models of granular gases. Phys. Rev., 2014, E 89, 052201.
18. Rongali R., Alam M. Higher-order effects on orientational correlation and relaxation dynamics in homogeneous cooling of a rough granular gas. Phys. Rev., 2014, E 89, 062201.
19. Khalil N., Garzó V. Transport coefficients for driven granular mixtures at low density. Phys. Rev., 2013, E 88, 052201.
20. Ogawa S. Multitemperature theory of granular materials. Proc. of the US - Japan Seminar on Contin-Mechanical and Statistical Approaches Mechanical Granular Material. Tokyo, Gukujustu Bunken Fukyakai, 1978, p. 208.
21. Warr S., Jacques G. T. H., Huntley J. M. Fluidization of a two-dimensional granular system: Experimental study and scaling behavior. Phys. Rev., 1995, E 52, pp. 5583–5595.
22. Kudrolli A. Size separation in vibrated granular matter. Rep. Prog. Phys., 2004, 67, pp. 209–247.
23. Losert W., Cooper D. G. W., Delour J., Kudrolli A., Gollub J. P. Velocity statistics in vibrated granular media. Chaos, 1999, 9, pp. 682–690.
24. Feitosa K., Menon N. Breakdown of energy equipartition in a 2D dinary vibrated granular gas. Phys. Rev. Lett., 2002, 88, 198301.
25. Wildman R. D., Parker D. J. Coexistence of two granular temperatures in binary vibrofluidized beds. Phys. Rev. Lett., 2002, 88, 064301.
26. Trujillo L., Herrmann H. J. A note on the upward and downward intruder segregation in granular media. Granular Matter, 2003, Vol. 5, No. 2, pp. 85–89.
27. Socolar J. E. S., Schaeffer D. G., Claudin P. Directed force chain networks and stress response in static granular materials. European Physics Journal, 2002, Vol. 7, pp. 353–370.
28. 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. Lett., 2004, 93, 198003.
29. Wakou J., Brito R., Ernst M. Towards a Landau—Ginzburg-type theory for granular fluids. Journal of Statistical Physics, 2002, Vol. 107, Nos. 1/2, pp. 3–22.
30. Eshuis P. Collective phenomena in vertically shaken granular matter. Universiteit Twente, 2008, 186 p.
31. Fouxon I. Inhomogeneous quasistationary state of dense fluids of inelastic hard spheres. Phys. Rev., 2014, E 89, 052210.
32. Chen Y., Hou M., Jiang Y., Liu M. Hydrodynamics of granular gases with a two-peak distribution. Phys. Rev., 2013, E 88, 052204.
33. Brito R., Risso D., Soto R. Hydrodynamic modes in a confined granular fluid. Phys. Rev., 2013, E 87, 022209.
34. Reyes F. V., Santos A., Garzó V. Non-Newtonian granular hydrodynamics. What do the inelastic simple shear flow and the elastic fourier flow have in common? Phys. Rev. Lett., 2010, 104, 028001.
35. Goldhirsch I., Zanetti G. Clustering instability in dissipative gases. Phys. Rev. Lett., 1993, 70, pp.1619–1622.
36. Soto R., Mareschal M., Malek Mansour M. Nonlinear analisys of the shearing instability in granular gases. Phys. Rev., 2000, E 62, 3836.
37. Thoroddsen S. T., Shen A. Q. Granular jets. Phys. Fluids, 2001, 13, pp. 4–6.
38. Ostrovskiy G. M. Prikladnaya mekhanika neodnorodnykh sred (Applied mechanics of inhomogeneous media). St. Petersburg, Nauka, 2000, 359 p.
39. Bouchaud J.-P., Claudin P., Levine D., Otto M. Force chain splitting in granular materials: A mechanism for large scale pseudo-elastic behaviour. Eur. Phys. J., 2001, E 4, pp. 451–457.
40. Digby P. J. The еffective еlastic moduli of porous granular rocks. J. Appl. Mech., 1981, 48 (4), pp. 803–808.
41. Owens E. T., Daniels K. E. Sound propagation and force chains in granular materials. EPL, 2011, 94, 54005.
42. Faraday M. On a perculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic sufaces. Phil. Trans. R. Soc. London, 1831, Vol. 121, pp. 299–340.
43. Umbanhowar P. B., Melo F., Swinney H. L., Localized excitations in a vertically vibrated granular layer. Nature, 1996, 382, pp. 793–796.
44. Lioubashevski O., Hamiel Y., Agnon A., Reches Z., Fineberg J. Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension. Phys. Rev., 1999, 83, pp. 3190–3193.
45. Hunt C. R. The emergence of oscillons in granular media. Physics 569 Emergent States of Matter, 2008. URL: http://guava.physics.uiuc.edu/~nigel/courses/569/Essays_Fall2008/files/hunt.pdf (date accessed 04.08.2015).
46. Tsimring L., Aranson I. Localized and cellular patterns in a vibrated granular layer. Phys. Rev. Lett., 1997, Vol. 79, pp. 213–216.
47. Cerda E., Melo F., Rica S. Model for subharmonic waves in granular materials. Phys. Rev. Lett., 1997, Vol. 79 (23), pp. 4570–4573.
48. Crawford C., Riecke H. Oscillon-type structures and their interaction in a Swift—Hohenberg model. Physica D, 1999, Vol. 129, pp. 83–92.
49. Barashenkov I. V., Alexeeva N. V., Zemlyanaya E. V. Two- and three-dimensional oscillons in nonlinear Faraday resonance. Phys. Rev., 2002, Vol. 89 (10), 104101.1–104101.4.
50. Gleiser M., Sicilia D. General theory of oscillon dynamics. Phys. Rev. D, 2009, 80, 125037.
51. Petrov V., Ouyang Q., Swinney H. Resonant pattern formation in a chemical system, Nature, 1997, Vol. 388, pp. 655–657.
52. Vanag V., Zhabotinsky A., Epstein I. Oscillatory clusters in the periodically illuminated, spatially extended Belousov—Zhabotinsky reaction. Phys. Rev. Lett., 2001, 86, pp. 552–555.

53. Trizac E., Barrat A. Free cooling and inelastic collapse of granular gases in high dimensions. Eur. Phys. J., 2000, E 3, pp. 291–294.
54. Burton J. C., Lu P. Y., Nagel S. R. Collision dynamics of particle clusters in a two-dimensional granular gas. Phys. Rev., 2013, E 88, 062204.
55. Blekhman I. I., Dzhanelidze G. Yu. Vibratsionnoye peremeshcheniye (Vibrational displacement). Moscow, Nauka, 1964, 410 p.
56. Palmov V. A. Description of complex dynamic systems high frequency vibration using the theory of heat conduction. Izbrannyye problemy prikladnoy mekhaniki. Sbornik, posvyashchennyy 60-letiyu akademika V. N. Chelomeya (Selected problems of applied mechanics. Collection, dedicated to the 60^th anniversary of academician V. N. Chelomey). Moscow, VINITI, 1974, pp. 535–542.
57. Blekhman I. I. Chto mozhet nauka? O «vibratsionnoy mekhanike» i vibratsionnoy tekhnike (What can the vibration? About «vibration mechanics» and vibration engineering). Moscow, Nauka, 1988, 208 p.
58. Arsentyev V. A., Vaisberg L. A., Ustinov I. D. Trends in development of low-water-consuption technologies and machines for finely ground mineral materials processing. Obogashchenie Rud, 2014, No. 5, pp. 3–9.
59. Blekhman I. I., Blekhman L. I., Vaisberg L. A., Ivanov K. S. Revisiting the models of vibration screening process. Vibroengineering PROCEDIA, 2014, Vol. 3, pp. 169–174.
60. Blekhman I. I. Teoriya vibratsionnykh protsessov i ustroystv (Theory of vibration processes and devices). St. Petersburg, «Ore and Metals» Publishing House, 2013, 640 p.
61. Kremer E. B., Fidlin A. Ya. Dimensional dynamic continuum model of granular material. Doklady Akademii Nauk SSSR = Proceedings of the USSR Academy of Sciences, 1989, Vol. 309, No. 4.
62. Eshuis P., van der Weele K., Calzavarini E., Lohse D., van der Meer D. Exploring the limits of granular hydrodynamics: A horizontal array of inelastic particles. Phys. Rev., 2009, E 80, 011302.
63. Nepomnyashchiy E. A. Mathematical description of bulk materials separation kinetics. Trudy VNIIZ (VNIIZ Proceedings). Moscow, 1967, Iss. 61–62.
64. Blekhman I. I., Khaynman V. Ya. On the theory of granular mixtures separation under vibration. Inzhenernyy Zhurnal. Mekhanika Tverdogo Tela = Mekhanics of Solids, 1968, No. 1, pp. 5–13.
65. Möbius M. E., Cheng X., Eshuis P., Karczmar G. S., Nagel S. R., Jaeger H. M. The effect of air on granular size separation in a vibrated granular bed. Phys. Rev., 2005, E 72, 011304.
66. Eshuis P., van der Weele K., van der Meer D., Lohse D. Granular Leidenfrost effect: Experiment and theory of floating particle clusters. Phys. Rev. Lett., 2005, 95, 258001.
67. Meerson B., Pöschel T., Bromberg Y. Close-packed floating clusters: Granular hydrodynamics beyond the freezing point? Phys. Rev. Lett., 2003, 91, 024301.
68. Grossman E. L., Zhou T., Ben-Naim E. Towards granular hydrodynamics in two-dimensions. Phys. Rev., 1997, E 55, 4200.
69. Hsiau S. S., Chen C. H. Granular convection cells in a vertical shaker. Powder Technology, 2000, 111, pp. 210–217.
70. Wildman R. D., Huntley J. M., Parker D. J. Convection in highly fluidized three-dimensional granular beds. Phys. Rev. Lett., 2001, 86, pp. 3304–3307.
71. Garcimartin A., Maza D., Ilquimiche J. L., Zuriguel I. Convective motion in a vibrated granular layer. Phys. Rev., 2002, E 65, 031303.
72. Hsiau S. S., Wang P. C., Tai C. H. Convection cells and segregation in a vibrated granular bed. AIChE J, 2002, 48, pp. 1430–1438.
73. Tai C. H., Hsiau S. S. Dynamics behaviors of powders in a vibrating bed. Powder Technology, 2004, 139, pp. 221–132.
74. Rodriguez-Linan G. M., Nahmad-Molinari Y. Granular convection driven by shearing interial forces. Phys. Rev., 2006, E 73, 011302.
75. Ramírez R., Risso D., Cordero P. Thermal convection in fluidized granular systems. Phys. Rev. Lett., 2000, 85, pp. 1230–1233.
76. Sunthar P., Kumaran V. Characterization of the stationary states of a dilute vibrofluidized granular bed. Phys. Rev., 2001, E 64, 041303.
77. Talbot J., Viot P. Wall-enhanced convection in vibrofluidized granular systems. Phys. Rev. Lett., 2002, 89, 064301.
78. Cordero P., Ramirez R., Risso D. Buoyancy driven convectino and hysteresis in granular gases: Numerical solution. Physica A, 2003, 327, pp. 82–87.
79. Risso D., Soto R., Godoy S., Cordero P. Friction and convection in a vertically vibrated granular system. Phys. Rev., 2005, E 72, 011305.
80. Hayakawa H., Yue S., Hong D. C. Hydrodynamic description of granular convection. Phys. Rev. Lett., 1995, 75, pp. 2328–2331.
81. He X., Meerson B., Doolen G. Hydrodynamics of thermal granular convection. Phys. Rev., 2002, E 65, 030301.
82. Ohtsuki T., Ohsawa T. Hydrodynamics for convection in vibrating beds of cohesionless granular materials. J. Phys. Soc. Jpn., 2003, 72, pp. 1963–1967.
83. Khain E., Meerson B. Onset of thermal convection in a horizontal layer of granular gas. Phys. Rev., 2003, E 67, 021306.
84. Miao G., Huang K., Yun Y., Wei R. Active thermal convection in vibrofluidized granular systems. Eur. Phys. J., 2004, B 40, pp. 301–304.
85. Eshuis P., van der Meer D., Alam M., van Gerner H. J., van der Weele K., Lohse D. Onset of convection in strongly shaken granular matter. Phys. Rev., 2010, 104, 038001.