Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-16T22:14:07.771Z Has data issue: false hasContentIssue false
  • English
  • Français

Inherent thermal convection in a granular gas inside a box under a gravity field

Published online by Cambridge University Press:  16 November 2018

Francisco Vega Reyes Open the ORCID record for Francisco Vega Reyes [Opens in a new window] ,
Andrea Puglisi ,
Giorgio Pontuale  and
Andrea Gnoli
Francisco Vega Reyes*
Affiliation:
Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, 06071 Badajoz, Spain
Andrea Puglisi
Affiliation:
Istituto dei Sistemi Complessi, CNR and Dipartimento di Fisica, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Rome, Italy
Giorgio Pontuale
Affiliation:
Istituto dei Sistemi Complessi, CNR and Dipartimento di Fisica, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Rome, Italy
Andrea Gnoli
Affiliation:
Istituto dei Sistemi Complessi, CNR and Dipartimento di Fisica, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Rome, Italy
*
†Email address for correspondence: fvega@unex.es
Get access Rights & Permissions [Opens in a new window]

Abstract

We theoretically prove the existence in granular fluids of a thermal convection that is inherent in the sense that it is always present and has no thermal gradient threshold (convection occurs for all finite values of the Rayleigh number). More specifically, we study a gas of inelastic smooth hard disks enclosed in a rectangular region under a constant gravity field. The vertical walls act as energy sinks (i.e. inelastic walls that are parallel to gravity), whereas the other two walls are perpendicular to gravity and act as energy sources. We show that this convection is due to the combined action of dissipative lateral walls and a volume force (in this case, gravitation). Hence, we call it dissipative lateral walls convection (DLWC). Our theory, which also describes the limit case of elastic collisions, shows that inelastic particle collisions enhance the DLWC. We perform our study via numerical solutions (volume-element method) of the corresponding hydrodynamic equations in an extended Boussinesq approximation. We show that our theory describes the essentials of the results for similar (but more complex) laboratory experiments.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection Granular media: Granular media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 859 , 25 January 2019 , pp. 160 - 173
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andreotti, B., Forterre, Y. & Pouliquen, O. 2013 Granular Media: Between Fluid and Solid. Cambridge University Press.Google Scholar
Ansari, I. H. & Alam, M. 2016 Pattern transition, microstructure, and dynamics in a two-dimensional vibrofluidized granular bed. Phys. Rev. E 93, 052901.Google Scholar
Bagnold, R. A. 1954 The Physics of Blown Sand and Desert Dunes. Dover Publications Inc. Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Batchelor, G. K. 1982 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Baxter, G. W. & Olafsen, J. S. 2007 Experimental evidence for molecular chaos in granular gases. Phys. Rev. Lett. 99, 028001.Google Scholar
Bénard, H. 1900 Les tourbillons cellulaires dans une nappe liquide. Rev. Gén. Sci. Pures Appl. 11, 1261–1271.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.Google Scholar
Bougie, J. 2010 Effects of thermal noise on pattern onset in continuum simulations of shaken granular layers. Phys. Rev. E 81, 032301.Google Scholar
Boussinesq, J. 1903 Théorie analytique de la chaleur, mise en harmonie avec la ther modynamique et avec la théorie mécanique de la lumiére, vol. 2. Gautier-Villars.Google Scholar
Brey, J. J. & Cubero, D. 2001 Hydrodynamic transport coefficients of granular gases. In Granular Gases (ed. Pöschel, T. & Luding, S.), Lectures Notes in Physics, vol. 564, pp. 59–78. Springer.Google Scholar
Brey, J. J., Dufty, J. W., Kim, C. S. & Santos, A. 1998 Hydrodynamics for granular flow at low density. Phys. Rev. E 58, 4638–4653.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover Publications Inc. Google Scholar
Chapman, C. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851–1112.Google Scholar
Daniels, P. G. 1977 Asymptotic sidewalls effects in rotating Bénard convection. Z. Angew. Math. Phys. J. Appl. Math. Phys. 358, 577–584.Google Scholar
Dufty, J. W. 2001 Kinetic theory and hydrodynamics for a low density granular gas. Adv. Complex Syst. 4, 397–406.Google Scholar
Egolf, D. A. 2000 Equilibrium regained: from nonequilibrium chaos to statistical mechanics. Science 287, 101–103.Google Scholar
Egolf, D. A., Melnikov, I. V., Pesch, W. & Ecke, R. E. 2000 Mechanisms of extensive spatiotemporal chaos in Rayleigh–Bénard convection. Nature 404, 733–736.Google Scholar
Eshuis, P., van der Meer, D., Alam, M., van Gerner, H. J., van der Weele, K. & Lohse, D. 2010 Onset of convection in strongly shaken granular matter. Phys. Rev. Lett. 104, 038001.Google Scholar
Eshuis, P., van der Weele, K., van der Meer, D. & Lohse, R. Bos D. 2007 Phase diagram of vertically shaken granular matter. Phys. Fluids 19, 123301.Google Scholar
Ferziger, J. H. & Perić, M. 2002 Computational Methods for Fluid Dynamics. Springer.Google Scholar
Foerster, S. F., Louge, M. Y., Chang, H. & Allis, K. 1994 Measurements of the collision properties of small spheres. Phys. Fluids 6, 1108–1115.Google Scholar
Forterre, Y. & Pouliquen, O. 2003 Long-surface-wave instability in dense granular flows. J. Fluid. Mech. 486, 21–50.Google Scholar
Godfrey, D. A. 1990 The rotation period of saturn’s polar hexagon. Science 247 (4947), 1206–1208.Google Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293.Google Scholar
Gollup, J. P. 1995 Order and disorder in fluid motion. Proc. Natl. Acad. Sci. USA 92, 6705–6711.Google Scholar
Gray, D. D. & Giorgini, A. 1976 The validity of the Boussinesq approximation for liquids and gases. Intl J. Heat Mass Transfer 19, 545–551.Google Scholar
Guyer, J. E., Wheeler, D. & Warren, J. A. 2009 FiPy: Partial differential equations with Python. Comput. Sci. Engng 11 (3), 6–15.Google Scholar
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430.Google Scholar
Hall, P. & Walton, I. C. 1977 The transition to a convective régime in a two-dimensional box. Proc. R. Soc. Lond. A 358, 199–221.Google Scholar
Hilbert, D. 1912 Begründung der kinetischen gastheorie. Math. Ann. 72, 562–577; English translation of the original German text may be found in Kinetic theory, vol. 3 by S. G. Brush (Pergamon, 1972).Google Scholar
Hunt, J. C. R. & Durbin, P. A. 1999 Pertubed vortical layers and shear sheltering. Fluid Dyn. Res. 24, 375–404.Google Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202.Google Scholar
Johnson, P. C. & Jackson, R. 1987 Frictional–collisional constitutive relations for granular materials, with applications to plane shearing. J. Fluid Mech. 176, 67–93.Google Scholar
Kadanoff, L. P. 2001 Turbulent heat flow: structures and scaling. Phys. Today 54, 34–39.Google Scholar
Kanatani, K.-I. 1979 A micropolar continuum theory for the flow of granular materials. Intl J. Engng Sci. 17, 419–432.Google Scholar
Khain, E. & Meerson, B. 2003 Onset of thermal convection in a horizontal layer of granular gases. Phys. Rev. E 67, 021306.Google Scholar
Lord Rayleigh 1916 On convection currents in horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. Ser. 6 32 (192), 529–546.Google Scholar
Losert, W., Bocquet, L., Lubensky, T. C. & Gollub, J. P. 2000 Particle dynamics in sheared granular matter. Phys. Rev. Lett. 85, 1428–1431.Google Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256.Google Scholar
Mutabazi, I., Wesfreid, J. E. & Guyon, E. 2006 Dynamics of Spatio-Temporal Cellular Structures: Henri Bénard Centenary Review, Springer Tracts on Modern Physics, vol. 207. Springer.Google Scholar
Nott, P. R., Alam, M., Agrawal, K., Jackson, R. & Sundaresan, S. 1999 The effect of boundarieson the plane Couette flow of granular materials: a bifurcation analysis. J. Fluid Mech. 397, 203–229.Google Scholar
Olafsen, J. S. & Urbach, J. S. 1998 Clustering, order, and collapse in a driven granular monolayer. Phys. Rev. Lett. 81, 4369–4372.Google Scholar
Pontuale, G., Gnoli, A., Vega Reyes, F. & Puglisi, A. 2016 Thermal convection in granular gases with dissipative lateral walls. Phys. Rev. Lett. 117, 098006.Google Scholar
Prevost, A., Egolf, D. A. & Urbach, J. S. 2002 Forcing and velocity correlations in a vibrated granular monolayer. Phys. Rev. Lett. 89, 084301.Google Scholar
Puglisi, A. 2015 Transport and Fluctuations in Granular Fluids. Springer.Google Scholar
Puglisi, A., Gnoli, A., Gradenigo, G., Sarracino, A. & Villamaina, D. 2012 Structure factors in granular experiments with homogeneous fluidization. J. Chem. Phys. 136, 014704.Google Scholar
Reynolds, O. 1885 On the dilatancy of media composed of rigid particles in contact. with experimental illustrations. Phil. Mag. 20 (5), 432–437.Google Scholar
Risso, D., Soto, R., Godoy, S. & Cordero, P. 2005 Friction and convection in a vertically vibrated granular system. Phys. Rev. E 72, 011305.Google Scholar
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41–74.Google Scholar
Vega Reyes, F. & Lasanta, A. 2017 Hydrodynamics of a granular gas in a heterogeneous environment. Entropy 19, 536.Google Scholar
Vega Reyes, F., Santos, A. & Garzó, V. 2010 Non-Newtonian granular hydrodynamics. What do the inelastic simple shear flow and the elastic fourier flow have in common? Phys. Rev. Lett. 104, 028001.Google Scholar
Vega Reyes, F. & Urbach, J. S. 2009 Steady base states for Navier–Stokes granular hydrodynamics with boundary heating and shear. J. Fluid Mech. 636, 279–283.Google Scholar
Wildman, R. D., Huntley, J. M. & Parker, D. J. 2001a Convection in highly fluidized three-dimensional granular beds. Phys. Rev. Lett. 86, 3304–3307.Google Scholar
Wildman, R. D., Huntley, J. M. & Parker, D. J. 2001b Granular temperature profiles in three-dimensional vibrofluidized granular beds. Phys. Rev. E 63, 061311.Google Scholar
Windows-Yule, C. R. K., Rivas, N. & Parker, D. J. 2013 Thermal convection and temperature inhomogeneity in a vibrofluidized granular bed: the influence of sidewall dissipation. Phys. Rev. Lett. 111, 038001.Google Scholar