Granular Matter

, Volume 15, Issue 6, pp 893–911

Buoyancy driven convection in vertically shaken granular matter: experiment, numerics, and theory

  • Peter Eshuis
  • Ko van der Weele
  • Meheboob Alam
  • Henk Jan van Gerner
  • Martin van der Hoef
  • Hans Kuipers
  • Stefan Luding
  • Devaraj van der Meer
  • Detlef Lohse
Original Paper

Abstract

Buoyancy driven granular convection is studied for a shallow, vertically shaken granular bed in a quasi 2D container. Starting from the granular Leidenfrost state, in which a dense particle cluster floats on top of a dilute gaseous layer of fast particles (Meerson et al. in Phys Rev Lett 91:024301, 2003; Eshuis et al. in Phys Rev Lett 95:258001, 2005), we witness the emergence of counter-rotating convection rolls when the shaking strength is increased above a critical level. This resembles the classical onset of convection—at a critical value of the Rayleigh number—in a fluid heated from below. The same transition, even quantitatively, is seen in molecular dynamics simulations, and explained by a hydrodynamic-like model in which the granular material is treated as a continuum. The critical shaking strength for the onset of granular convection is accurately reproduced by a linear stability analysis of the model. The results from experiment, simulation, and theory are in good agreement. The present paper extends and completes our earlier analysis (Eshuis et al. in Phys Rev Lett 104:038001, 2010).

Keywords

Shaken granular matter Granular gas Leidenfrost state 

References

  1. 1.
    Jenkins, J.T., Savage, S.B.: A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187 (1983)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Haff, P.K.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401 (1983)ADSCrossRefMATHGoogle Scholar
  3. 3.
    Jenkins, J., Richman, M.: Boundary conditions for plane flows of smooth nearly elastic circular discs. J. Fluid Mech. 171, 53 (1986)Google Scholar
  4. 4.
    Campbell, C.S.: Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57 (1990)ADSCrossRefGoogle Scholar
  5. 5.
    Jaeger, H.M., Nagel, S.R., Behringer, R.P.: Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259 (1996)ADSCrossRefGoogle Scholar
  6. 6.
    Behringer, R.P., Jaeger, H.M., Nagel, S.R.: The physics of granular materials. Phys. Today 49, 32 (1996)Google Scholar
  7. 7.
    Sela, N., Goldhirsch, I.: Hydrodynamic equations for rapid flows of smooth inelastic spheres to Burnett order. J. Fluid Mech. 361, 41 (1998)MathSciNetADSCrossRefMATHGoogle Scholar
  8. 8.
    Brey, J.J., Dufty, J.W., Kim, C.S., Santos, A.: Hydrodynamics for granular flow at low density. Phys. Rev. E 58, 4638 (1998)ADSCrossRefGoogle Scholar
  9. 9.
    Kadanoff, L.P.: Built upon sand: theoretical ideas inspired by granular flows. Rev. Mod. Phys. 71, 435 (1999)ADSCrossRefGoogle Scholar
  10. 10.
    Goldhirsch, I.: Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267 (2003)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Goldhirsch, I., Noskowicz, S., Bar-Lev, O.: Nearly smooth granular gases. Phys. Rev. Lett. 95, 068002 (2005)ADSCrossRefGoogle Scholar
  12. 12.
    Du, Y., Li, H., Kadanoff, L.P.: Breakdown of hydrodynamics in a one-dimensional system of inelastic particles. Phys. Rev. Lett. 74, 1268 (1995)ADSCrossRefGoogle Scholar
  13. 13.
    Sela, N., Goldhirsch, I.: Hydrodynamics of a one-dimensional granular medium. Phys. Fluids 7, 507 (1995)ADSCrossRefMATHGoogle Scholar
  14. 14.
    Duran, J.: Sand, Powders and Grains: An Introduction to the Physics of Granular Materials. Springer, New-York (1999)Google Scholar
  15. 15.
    Aranson, I.S., Tsimring, L.S.: Patterns and collective behavior in granular media: theoretical concepts. Rev. Mod. Phys. 78, 641 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    Goldhirsch, I., Zanetti, G.: Clustering instability in dissipative gases. Phys. Rev. Lett. 70, 1619 (1993)ADSCrossRefGoogle Scholar
  17. 17.
    Kudrolli, A., Wolpert, M., Gollub, J.P.: Cluster formation due to collisions in granular material. Phys. Rev. Lett. 78, 1383 (1997)Google Scholar
  18. 18.
    Eggers, J.: Sand as Maxwell’s demon. Phys. Rev. Lett. 83, 5322 (1999)ADSCrossRefGoogle Scholar
  19. 19.
    van der Weele, K., van der Meer, D., Versluis, M., Lohse, D.: Hysteretic custering in granular gas. Europhys. Lett. 53, 328 (2001)Google Scholar
  20. 20.
    van der Meer, D., van der Weele, K., Lohse, D.: Sudden death of a granular cluster. Phys. Rev. Lett. 88, 174302 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    Alam, M., Nott, P.R.: Stability of plane couette flow of a granular material. J. Fluid Mech. 377, 99 (1998)MathSciNetADSCrossRefMATHGoogle Scholar
  22. 22.
    Forterre, Y., Pouliquen, O.: Stability analysis of rapid granular chute flows: formation of longitudinal vortices. J. Fluid Mech. 467, 361 (2002)ADSCrossRefMATHGoogle Scholar
  23. 23.
    Alam, M.: Streamwise vortices and density patterns in rapid granular couette flow: a linear stability analysis. J. Fluid Mech. 553, 1 (2006)MathSciNetADSCrossRefMATHGoogle Scholar
  24. 24.
    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. 93, 198003 (2004)ADSCrossRefGoogle Scholar
  25. 25.
    Royer, J.R., Corwin, E.I., Flior, A., Cordero, M.L., Rivers, M.L., Eng, P.J., Jaeger, H.M.: Formation of granular jets observed by high-speed x-ray radiography. Nat. Phys. 1, 164 (2005)Google Scholar
  26. 26.
    Kuipers, J.A.M.: Multilevel modelling of dispersed multiphase flows. Oil Gas Sci. Technol. Rev. IFP 55, 427 (2000)CrossRefGoogle Scholar
  27. 27.
    Eshuis, P., van der Meer, D., Alam, M., Gerner, H.J., van der Weele, K., Lohse, D.: Onset of convection in strongly shaken granular. Matter Phys. Rev. Lett. 104, 038001 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    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. 95, 258001 (2005)ADSCrossRefGoogle Scholar
  29. 29.
    Eshuis, P., van der Weele, K., van der Meer, D., Bos, R., Lohse, D.: Phase diagram of vertically shaken granular matter. Phys. Fluids 19, 123301 (2007)ADSCrossRefGoogle Scholar
  30. 30.
    Normand, C., Porneau, Y., Velarde, M.G.: Convective instability: a physicist’s approach. Rev. Mod. Phys. 49, 581 (1977)ADSCrossRefGoogle Scholar
  31. 31.
    Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. E 15, 319 (1977)ADSCrossRefGoogle Scholar
  32. 32.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover, New-York (1981)Google Scholar
  33. 33.
    Bodenschatz, E., Pesch, W., Ahlers, G.: Recent developments in rayleigh-bénard convection. Annu. Rev. Fluid Mech. 32, 709 (2000)MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    Rogers, J.L., Schatz, M.F., Bougie, J.L., Swift, J.B.: Rayleigh-bénard convection in a vertically oscillated fluid layer. Phys. Rev. Lett. 84, 87 (2000)ADSCrossRefGoogle Scholar
  35. 35.
    Bormann, A.S.: The onset of convection in the rayleigh-bénard problem for compressible fluids. Cont. Mech. Thermodyn. 13, 9 (2001)CrossRefMATHGoogle Scholar
  36. 36.
    Oh, J., Ahlers, G.: Thermal-noise effect on the transition to rayleigh-bénard convection. Phys. Rev. Lett. 91, 094501 (2003)ADSCrossRefGoogle Scholar
  37. 37.
    Mutabazi, I., Guyon, E., Wesfreid, J.E.: Dynamics of Spatio-Temporal Cellular Structures, Henri Bénard Centenary Review, vol. 207. Springer, New York (2006)CrossRefGoogle Scholar
  38. 38.
    Knight, J.B., Jaeger, H.M., Nagel, S.R.: Vibration-induced size separation in granular media: the convection connection. Phys. Rev. Lett. 70, 3728 (1993)ADSCrossRefGoogle Scholar
  39. 39.
    Clément, E., Rajchenbach, J.: Fluidization of a bidimensional powder. Europhys. Lett. 16, 133 (1991)ADSCrossRefGoogle Scholar
  40. 40.
    Gallas, J.A.C., Herrmann, H.J., Sokolowski, S.: Convection cells in vibrating granular media. Phys. Rev. Lett. 69, 1371 (1992)ADSCrossRefGoogle Scholar
  41. 41.
    Taguchi, Y.-H.: Taguchi, New origin of a convective motion: Elastically induced convection in granular materials. Phys. Rev. Lett. 69, 1367 (1992)ADSCrossRefGoogle Scholar
  42. 42.
    Luding, S., Clément, E., Blumen, A., Rajchenbach, J., Duran, J.: The onset of convection in molecular dynamics simulations of grains. Phys. Rev. E 50, R1762 (1994)ADSCrossRefGoogle Scholar
  43. 43.
    Hayakawa, H., Yue, S., Hong, D.C.: Hydrodynamic description of granular convection. Phys. Rev. Lett. 75, 2328 (1995)ADSCrossRefGoogle Scholar
  44. 44.
    Ehrichs, E.E., Jaeger, H.M., Karczmar, G.S., Knight, J.B., Kuperman, V.Y., Nagel, S.R.: Granular convection observed by magnetic resonance imaging. Science 267, 1632 (1995)ADSCrossRefGoogle Scholar
  45. 45.
    Bourzutschky, M., Miller, J.: Granular convection in a vibrated fluid. Phys. Rev. Lett. 74, 2216 (1995)ADSCrossRefGoogle Scholar
  46. 46.
    Aoki, K.M., Akiyama, T., Maki, Y., Watanabe, T.: Convective roll patterns in vertically vibrated beds of granules. Phys. Rev. E 54, 874 (1996)ADSCrossRefGoogle Scholar
  47. 47.
    Knight, J.B., Ehrichs, E.E., Kuperman, V.Y., Flint, J.K., Jaeger, H.M., Nagel, S.R.: Experimental study of granular convection. Phys. Rev. E 54, 5726 (1996)ADSCrossRefGoogle Scholar
  48. 48.
    Lan, Y., Rosato, A.D.: Convection related phenomena in granular dynamics simulations of vibrated beds. Phys. Fluids 9, 3615 (1997)ADSCrossRefGoogle Scholar
  49. 49.
    Aoki, K.M., Akiyama, T.: Control parameter in granular convection. Phys. Rev. E 58, 4629 (1998)ADSCrossRefGoogle Scholar
  50. 50.
    Bizon, C., Shattuck, M.D., Swift, J.B., McCormick, W.D., Swinney, H.L.: Patterns in 3d vertically oscillated granular layers: simulation and experiment. Phys. Rev. Lett. 80, 57 (1998)ADSCrossRefGoogle Scholar
  51. 51.
    Ramírez, R., Risso, D., Cordero, P.: Thermal convection in fluidized granular systems. Phys. Rev. Lett. 85, 1230 (2000)ADSCrossRefGoogle Scholar
  52. 52.
    Hsiau, S.S., Chen, C.H.: Granular convection cells in a vertical shaker. Powder Technol. 111, 210 (2000)CrossRefGoogle Scholar
  53. 53.
    Wildman, R.D., Huntley, J.M., Parker, D.J.: Convection in highly fluidized three-dimensional granular beds. Phys. Rev. Lett. 86, 3304 (2001)ADSCrossRefGoogle Scholar
  54. 54.
    Sunthar, P., Kumaran, V.: Characterization of the stationary states of a dilute vibrofluidized granular bed. Phys. Rev. E 64, 041303 (2001)ADSCrossRefGoogle Scholar
  55. 55.
    He, X., Meerson, B., Doolen, G.: Hydrodynamics of thermal granular convection. Phys. Rev. E 65, 030301 (2002)ADSCrossRefGoogle Scholar
  56. 56.
    Garcimartin, A., Maza, D., Ilquimiche, J.L., Zuriguel, I.: Convective motion in a vibrated granular layer. Phys. Rev. E 65, 031303 (2002)ADSCrossRefGoogle Scholar
  57. 57.
    Talbot, J., Viot, P.: Wall-enhanced convection in vibrofluidized granular systems. Phys. Rev. Lett. 89, 064301 (2002)ADSCrossRefGoogle Scholar
  58. 58.
    Hsiau, S.S., Wang, P.C., Tai, C.H.: Convection cells and segregation in a vibrated granular bed. AIChE J. 48, 1430 (2002)CrossRefGoogle Scholar
  59. 59.
    Ohtsuki, T., Ohsawa, T.: Hydrodynamics for convection in vibrating beds of cohesionless granular materials. J. Phys. Soc. Jpn. 72, 1963 (2003)ADSCrossRefMATHGoogle Scholar
  60. 60.
    Cordero, P., Ramirez, R., Risso, D.: Buoyancy driven convection and hysteresis in granular gases: numerical solution. Physica A 327, 82 (2003)ADSCrossRefMATHGoogle Scholar
  61. 61.
    Miao, G., Huang, K., Yun, Y., Wei, R.: Active thermal convection in vibrofluidized granular systems. Eur. Phys. J. B 40, 301 (2004)ADSCrossRefGoogle Scholar
  62. 62.
    Tai, C.H., Hsiau, S.S.: Dynamics behaviors of powders in a vibrating bed. Powder Technol. 139, 221 (2004)CrossRefGoogle Scholar
  63. 63.
    Risso, D., Soto, R., Godoy, S., Cordero, P.: Friction and convection in a vertically vibrated granular system. Phys. Rev. E 72, 011305 (2005)ADSCrossRefGoogle Scholar
  64. 64.
    Isobe, M.: Bifurcations of a driven granular system under gravity. Phys. Rev. E 64, 031304 (2001)Google Scholar
  65. 65.
    Khain, E., Meerson, B.: Onset of thermal convection in a horizontal layer of granular gas. Phys. Rev. E 67, 021306 (2003)Google Scholar
  66. 66.
    Paolotti, D., Barrat, A., Marconi, U.M.B., Puglisi, A.: Thermal convection in monodisperse and bidisperse granular gases: a simulations study. Phys. Rev. E 69, 061304 (2004) Google Scholar
  67. 67.
    Pak, H.K., Behringer, R.P.: Surface waves in vertically vibrated granular materials. Phys. Rev. Lett. 71, 1832 (1993)ADSCrossRefGoogle Scholar
  68. 68.
    van der Hoef, M.A., Ye, M., van Sint Annaland, M., Andrews IV, A.T., Sundaresan, S., Kuipers, J.A.M.: Multi-scale modeling of gas-fluidized beds. Adv. Chem. Eng. 31, 65 (2006)CrossRefGoogle Scholar
  69. 69.
    Deen, N.G., van Sint Annaland, M., van der Hoef, M.A., Kuipers, J.A.M.: Review of discrete particle modeling of fluidized beds. Chem. Eng. Sc. 62, 28 (2007)CrossRefGoogle Scholar
  70. 70.
    Grossman, E.L., Zhou, T., Ben-Naim, E.: Towards granular hydrodynamics in two-dimensions. Phys. Rev. E 55, 4200 (1997)ADSCrossRefGoogle Scholar
  71. 71.
    Meerson, B., Pöschel, T., Bromberg, Y.: Close-packed floating clusters: granular hydrodynamics beyond the freezing point? Phys. Rev. Lett. 91, 024301 (2003)ADSCrossRefGoogle Scholar
  72. 72.
    Brey, J.J., Ruiz-Montero, M.J., Moreno, F.: Hydrodynamics of an open vibrated granular system. Phys. Rev. E 63, 061305 (2001)ADSCrossRefGoogle Scholar
  73. 73.
    Garcia-Rojo, R., Luding, S., Brey, J.J.: Transport coefficients for dense hard-disk systems. Phys. Rev. E 74, 061305 (2006)ADSCrossRefGoogle Scholar
  74. 74.
    Khain, E.: Hydrodynamics of fluid-solid coexistence in dense shear granular flow. Phys. Rev. E 75, 051310 (2007)ADSCrossRefGoogle Scholar
  75. 75.
    Orszag, S.A.: Accurate solution of the orr-sommerfeld stability equation. J. Fluid Mech. 50, 689 (1971)ADSCrossRefMATHGoogle Scholar
  76. 76.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, New York (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Eshuis
    • 1
  • Ko van der Weele
    • 2
  • Meheboob Alam
    • 3
  • Henk Jan van Gerner
    • 1
  • Martin van der Hoef
    • 1
  • Hans Kuipers
    • 4
  • Stefan Luding
    • 5
  • Devaraj van der Meer
    • 1
  • Detlef Lohse
    • 1
  1. 1.Physics of Fluids & MESA + Research Institute University of TwenteEnschedeThe Netherlands
  2. 2.Mathematics DepartmentUniversity of PatrasPatrasGreece
  3. 3.Engineering Mechanics UnitJawaharlal Nehru Center for Advanced Scientific ResearchBangaloreIndia
  4. 4.Fundamentals of Chemical Reaction EngineeringTechnical University of EindhovenEindhovenThe Netherlands
  5. 5.Multi Scale MechanicsUniversity of TwenteEnschedeThe Netherlands

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