Skip to main content
    • Aa
    • Aa

Dense, bounded shear flows of agitated solid spheres in a gas at intermediate Stokes and finite Reynolds numbers


We consider moderately dense bounded shear flows of agitated homogeneous inelastic frictionless solid spheres colliding in a gas between two parallel bumpy walls at finite particle Reynolds numbers, volume fractions between 0.1 and 0.4, and Stokes numbers large enough for collisions to determine the velocity distribution of the spheres. We adopt a continuum model in which constitutive relations and boundary conditions for the granular phase are derived from kinetic theory, and in which the gas contributes a viscous dissipation term to the fluctuation energy of the grains. We compare its predictions to recent lattice-Boltzmann (LB) simulations. The theory underscores the role played by the walls in the balances of momentum and fluctuation energy. When particle inertia is large, the solid volume fraction is nearly uniform, thus allowing us to treat the rheology using unbounded flow theory with an effective shear rate, while predicting slip velocities at the walls. When particle inertia decreases or fluid inertia increases, the solid volume fraction becomes increasingly heterogeneous. In this case, the theory captures the profiles of volume fraction, mean and fluctuation velocities between the walls. Comparisons with LB simulations allow us to delimit the range of parameters within which the theory is applicable.

Corresponding author
Email address for correspondence:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

A. Acrivos & E. Chang 1986 A model for estimating transport quantities in two-phase materials. Phys. Fluids 29, 34.

T. B. Anderson & R. Jackson 1967 A fluid mechanical description of fluidized beds. Ind. Engng Chem. Fundamentals 6, 527539.

E. J. Bolio , J. A. Yasuna & J. L. Sinclair 1995 Dilute, turbulent gas–solid flow in risers with particle–particle interactions. AIChE J. 41, 13751388.

H. C. Brinkman 1949 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 2734.

N. F. Carnahan & K. E. Starling 1969 Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51, 635636.

S. Dasgupta , R. Jackson & S. Sundaresan 1994 Turbulent gas–particle flow in vertical risers. AIChE J. 40, 215228.

I. Goldhirsch , S. H. Noskowicz & O. Bar-Lev 2005 Nearly smooth granular gases. Phys. Rev. Lett. 95, 068002.

R. J. Hill , D. L. Koch & A. J. C. Ladd 2001 aThe first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 213241.

R. J. Hill , D. L. Koch & A. J. C. Ladd 2001 bModerate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243278.

M. A. Hopkins & M. Y. Louge 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 4757.

J. T. Jenkins 2001 Boundary conditions for collisional grain flows at bumpy, frictional walls. In Granular Gases (ed. T. Pöschel & S. Luding ), pp. 125139. Springer.

J. T. Jenkins & M. W. Richman 1985 Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355377.

E. D. Liss , S. L. Conway & B. J. Glasser 2002 Density waves in gravity-driven granular flow through a channel. Phys. Fluids 14, 33093326.

M. Y. Louge , J. T. Jenkins & M. A. Hopkins 1990 Computer simulations of rapid granular shear flows between parallel bumpy boundaries. Phys. Fluids A 2, 10421044.

M. Y. Louge , J. T. Jenkins & M. A. Hopkins 1993 The relaxation of the second moments in rapid shear flows of smooth disks. Mech. Mat. 16, 199203.

M. Y. Louge , J. T. Jenkins , A. Reeves & S. Keast 2000 Microgravity segregation in collisional granular shearing flows. In Proc. IUTAM Symp. on Segregation in Granular Materials (ed. A. D. Rosato & D. L. Blackmore ), pp. 103112. Kluver.

J. C. Maxwell 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231256.

D. M. Mueth , G. F. Debregeas , G. S. Karczmar , P. J. Eng , S. R. Nagel & H. Jaeger 2000 Signatures of granular microstructure in dense shear flows. Nature 406, 385389.

G. Y. Onoda & E. G. Liniger 1990 Random loose packing of uniform spheres and the dilatancy onset. Phys. Rev. Lett. 64, 27272730.

M. W. Richman 1988 Boundary conditions based upon a modified Maxwellian velocity distribution function for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227240.

M. W. Richman & C. S. Chou 1988 Boundary effects on granular flows of smooth disks. Z. Angew. Mech. Phys. 39, 885901.

A. S. Sangani & S. Behl 1989 The planar singular solutions of Stokes and Laplace equations and their application to transport processes near porous surfaces. Phys. Fluids A 1, 2137.

J. R. Smart & D. T. Leighton Jr., 1989 Measurement of the hydrodynamic surface roughness of noncolloidal spheres. Phys. Fluids A 1, 5260.

R. Verberg & D. L. Koch 2006 Rheology of particle suspensions with low to moderate fluid inertia at finite particle inertia. Phys. Fluids 18, 083303.

H. Xu , M. Louge & A. Reeves 2003 Solution of the kinetic theory for bounded collisional granular flows. Continuum Mech. Thermodyn. 15 (4), 321349.

I. S. Zarraga , D. A. Hill & D. T. Leighton Jr 2000 The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J. Rheol. 44, 185220.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 14 *
Loading metrics...

Abstract views

Total abstract views: 66 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 27th May 2017. This data will be updated every 24 hours.