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Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime

  • Peter P. Mitrano (a1), John R. Zenk (a1), Sofiane Benyahia (a2), Janine E. Galvin (a2), Steven R. Dahl (a1) and Christine M. Hrenya (a1)...

In this work we quantitatively assess, via instabilities, a Navier–Stokes-order (small-Knudsen-number) continuum model based on the kinetic theory analogy and applied to inelastic spheres in a homogeneous cooling system. Dissipative collisions are known to give rise to instabilities, namely velocity vortices and particle clusters, for sufficiently large domains. We compare predictions for the critical length scales required for particle clustering obtained from transient simulations using the continuum model with molecular dynamics (MD) simulations. The agreement between continuum simulations and MD simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers ( $K{n}_{u} $ ) at the time of cluster detection reveals $K{n}_{u} \gg 1$ due to the presence of large velocity gradients associated with vortices. Although kinetic-theory-based continuum models are based on a small- $Kn$ (i.e. small-gradient) assumption, our findings suggest that, similar to molecular gases, Navier–Stokes-order (small- $Kn$ ) theories are surprisingly accurate outside their expected range of validity.

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R. K. Agarwal , K. Y. Yun & R. Balakrishnan 2001 Beyond Navier–Stokes: Burnett equations for flows in the continuum-transition regime. Phys. Fluids 13, 30613085.

L. Almazán , J. A. Carrillo , V. Salueña , Garzó & T. Pöschel 2013 A numerical study of the Navier–Stokes transport coefficients for two-dimensional granular hydrodynamics. New J. Phys. 15, 043044.

T. Bartels-Rausch , V. Bergeron , J. H. E. Cartwright , R. Escribano , J. L. Finney , H. Grothe , P. J. Gutiérrez , J. Haapala , W. F. Kuhs , J. B. C. Pettersson , S. D. Price , C. I. Sainz-Díaz , D. J. Stokes , G. Strazzulla , E. S. Thomson , H. Trinks & N. Uras-Aytemiz 2012 Ice structures, patterns, and processes: A view across the icefields. Rev. Mod. Phys. 84, 885.

J. J. Brey , M. J. Ruiz-Montero & D. Cubero 1999 Origin of density clustering in a freely evolving granular gas. Phys. Rev. E 60, 3150.

J. J. Brey , M. J. Ruiz-Montero & F. Moreno 2001 Hydrodynamics of an open vibrated granular system. Phys. Rev. E 63, 061305.

N. Brilliantov , C. Saluena , T. Schwager & T. P. Pöschel 2004 Transient structures in a granular gas. Phys. Rev. Lett. 93, 134301.

R. Brito & M. Ernst 1998 Extension of Haff’s cooling law in granular flows. Europhys. Lett. 43, 497.

S. L. Conway & B. J. Glasser 2004 Density waves and coherent structures in granular Couette flows. Phys. Fluids 16, 509.

J. E. Galvin , C. M. Hrenya & R. D. Wildman 2007 On the role of the Knudsen layer in rapid granular flows. J. Fluid Mech. 585, 73.

V. Garzó 2005 Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, 021106.

V. Garzó & J. W. Dufty 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895.

B. J. Glasser , S. Sundaresan & I. G. Kevrekidis 1998 From bubbles to clusters in fluidized beds. Phys. Rev. Lett. 81, 1849.

I. Goldhirsch 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267.

I. Goldhirsch , M. L. Tan & G. Zanetti 1993 A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. J. Sci. Comput. 8, 1.

I. Goldhirsch & G. Zanetti 1993 Clustering instability in dissipative gases. Phys. Rev. Lett. 70, 1619.

P. Goldreich & S. Tremaine 1982 The dynamics of planetary rings. Annu. Rev. Astron. Astrophys. 20, 249.

A. Herman 2011 Molecular-dynamics simulation of clustering processes in sea-ice floes. Phys. Rev. E 84, 056104.

M. A. Hopkins & M. Y. Louge 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 47.

C. M. Hrenya 2011 Kinetic theory for granular materials: polydispersity. In Computational Gas-Solids Flows and Reacting Systems: Theory, Methods and Practice (ed. S. Pannala , M. Syamlal & T. O’Brien ), pp. 102127. IGI Global.

C. M. Hrenya , J. E. Galvin & R. D. Wildman 2008 Evidence of higher-order effects in thermally driven rapid granular flows. J. Fluid Mech. 598, 429.

J. Li & J. A. M. Kuipers 2003 Gas-particle interactions in dense gas-fluidized beds. Chem. Engng Sci. 58, 711.

J. J. Lissauer 1993 Planet formation. Annu. Rev. Astron. Astrophys. 31, 129.

S. Luding & H. J. Herrmann 1999 Cluster-growth in freely cooling granular media. Chaos 9, 673.

T. W. Martin , J. M. Huntley & R. D. Wildman 2005 Hydrodynamic model for a vibrofluidized granular bed. J. Fluid Mech. 535, 325.

P. P. Mitrano , S. R. Dahl , D. J. Cromer , M. S. Pacella & C. M. Hrenya 2011 Instabilities in the homogeneous cooling of a granular gas: a quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, 093303.

P. P. Mitrano , S. R. Dahl , A. M. Hilger , C. J. Ewasko & C. M. Hrenya 2013 Dual role of friction in granular flows: attenuation versus enhancement of instabilities. J. Fluid Mech. 729, 484.

P. P. Mitrano , V. Garzó , A. M. Hilger , C. J. Ewasko & C. M. Hrenya 2012 Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases. Phys. Rev. E 85, 041303.

E. Rericha , C. Bizon , M. Shattuck & H. Swinney 2001 Shocks in supersonic sand. Phys. Rev. Lett. 88, 14302.

R. B. Rice & C. M. Hrenya 2009 Characterization of clusters in rapid granular flows. Phys. Rev. E 79 (2), 021304.

N. Sela & I. Goldhirsch 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41.

J. Schmidt , H. Salo , F. Spahn & O. Petzschmann 2001 Viscous overstability in Saturn’s B-ring: II. Hydrodynamic theory and comparison to simulations. Icarus 153, 316.

R. Soto , M. Mareschal & M. M. Mansour 2000 Nonlinear analysis of the shearing instability in granular gases. Phys. Rev. E 62, 3836.

S. Sundaresan 2000 Modelling the hydrodynamics of multiphase flow reactors: current status and challenges. AIChE J. 46, 11021105.

S. Sundaresan 2003 Instabilities in fluidized beds. Annu. Rev. Fluid Mech. 35, 63.

C. R. Wassgren , J. A. Cordova , R. Zenit & A. Karion 2003 Dilute granular flow around an immersed cylinder. Phys. Fluids 15, 3318.

T. Wildman , T. Martin , J. Huntley , J. Jenkins , H. Viswanathan , X. Fen & D. Parker 2008 Experimental investigation and kinetic-theory-based model of a rapid granular shear flow. J. Fluid Mech. 602, 63.

J. J. Wylie & D. L. Koch 2000 Particle clustering due to hydrodynamic interactions. Phys. Fluids 12, 964.

H. Xu , M. Louge & Reeves 2003 Solutions of the kinetic theory for bounded collisional granular flows. Contin. Mech. Thermodyn. 15, 321.

X. Yin , J. R. Zenk , Mitrano & C. M. Hrenya 2013 Impact of collisional versus viscous dissipation on flow instabilities in gas–solid systems. J. Fluid Mech. 727, R2.

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Journal of Fluid Mechanics
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