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

Published online by Cambridge University Press:  10 January 2009

HAITAO XU ,
ROLF VERBERG ,
DONALD L. KOCH  and
MICHEL Y. LOUGE
HAITAO XU
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
ROLF VERBERG
Affiliation:
School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
DONALD L. KOCH*
Affiliation:
School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
MICHEL Y. LOUGE
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
*
‡Email address for correspondence: [email protected]
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Abstract

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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 618 , 10 January 2009 , pp. 181 - 208
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Acrivos, A. & Chang, E. 1986 A model for estimating transport quantities in two-phase materials. Phys. Fluids 29, 3–4.Google Scholar
Anderson, T. B. & Jackson, R. 1967 A fluid mechanical description of fluidized beds. Ind. Engng Chem. Fundamentals 6, 527–539.CrossRefGoogle Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Bizon, C., Shattuck, M. D., Swift, J. B. & Swinney, H. L. 1999 Transport coefficients for granular media from molecular dynamics simulations. Phys. Rev. E 60, 4340–4351.Google Scholar
Bolio, E. J., Yasuna, J. A. & Sinclair, J. L. 1995 Dilute, turbulent gas–solid flow in risers with particle–particle interactions. AIChE J. 41, 1375–1388.Google Scholar
Brinkman, H. C. 1949 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34.CrossRefGoogle Scholar
Carman, P. C. 1937 The determination of the specific surface area of powder I. J. Soc. Chem. Ind. 57, 225–236.Google Scholar
Carnahan, N. F. & Starling, K. E. 1969 Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51, 635–636.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles. Academic.Google Scholar
Dasgupta, S., Jackson, R. & Sundaresan, S. 1994 Turbulent gas–particle flow in vertical risers. AIChE J. 40, 215–228.Google Scholar
Ergun, S. 1952 Pressure drop through granular beds. Chem. Engng Prog. 48, 84–88.Google Scholar
Garzó, V. & Dufty, J. W. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895–5911.Google ScholarPubMed
Goldhirsch, I., Noskowicz, S. H. & Bar-Lev, O. 2005 Nearly smooth granular gases. Phys. Rev. Lett. 95, 068002.Google Scholar
Gopinath, A., Chen, S. B. & Koch, D. L. 1997 Lubrication flows between spherical particles colliding in a compressible non-continuum gas. J. Fluid Mech. 344, 245–269.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001 a The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 213–241.CrossRefGoogle Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001 b Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243–278.CrossRefGoogle Scholar
Hopkins, M. A. & Louge, M. Y. 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 47–57.CrossRefGoogle Scholar
Jenkins, J. T. 2001 Boundary conditions for collisional grain flows at bumpy, frictional walls. In Granular Gases (ed. Pöschel, T. & Luding, S.), pp. 125–139. Springer.CrossRefGoogle Scholar
Jenkins, J. T. & Richman, M. W. 1985 Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355–377.CrossRefGoogle Scholar
Jenkins, J. T. & Richman, M. W. 1986 Boundary conditions for plane flows of smooth, nearly elastic, circular disks. J. Fluid Mech. 171, 313–328.CrossRefGoogle Scholar
Koch, D. L. & Sangani, A. S. 1999 Particle pressure and marginal stability limits for a homogeneous monodisperse gas fluidized bed: kinetic theory and numerical simulations. J. Fluid Mech. 400, 229–263.Google Scholar
Kumaran, V. 2006 The constitutive relation for the granular flow of rough particles, and its application to the flow down an inclined plane. J. Fluid Mech. 561, 1–42.CrossRefGoogle Scholar
Liss, E. D., Conway, S. L. & Glasser, B. J. 2002 Density waves in gravity-driven granular flow through a channel. Phys. Fluids 14, 3309–3326.Google Scholar
Louge, M. Y., Jenkins, J. T. & Hopkins, M. A. 1990 Computer simulations of rapid granular shear flows between parallel bumpy boundaries. Phys. Fluids A 2, 1042–1044.Google Scholar
Louge, M. Y., Mastorakos, M. & Jenkins, J. T. 1991 The role of particle collisions in pneumatic transport. J. Fluid Mech. 231, 345–359.Google Scholar
Louge, M. Y., Jenkins, J. T. & Hopkins, M. A. 1993 The relaxation of the second moments in rapid shear flows of smooth disks. Mech. Mat. 16, 199–203.Google Scholar
Louge, M. Y., Jenkins, J. T., Reeves, A. & Keast, S. 2000 Microgravity segregation in collisional granular shearing flows. In Proc. IUTAM Symp. on Segregation in Granular Materials (ed. Rosato, A. D. & Blackmore, D. L.), pp. 103–112. Kluver.Google Scholar
Louge, M. Y., Jenkins, J. T., Xu, H. & Arnarson, B. Ö. 2001 Granular segregation in collisional shearing flows. In Mechanics for a New Millennium (ed. Aref, H. & Phillips, J.), pp. 239–252. Kluver.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 flow field. J. Fluid Mech. 140, 223–256.CrossRefGoogle Scholar
Maxwell, J. C. 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
Mitarai, N. & Nakanishi, H. 2007 Velocity correlations in dense granular shear flows: effects of energy dissipation and normal stress. Phys. Rev. E 75, 031305.Google Scholar
Mueth, D. M., Debregeas, G. F., Karczmar, G. S., Eng, P. J., Nagel, S. R. & Jaeger, H. 2000 Signatures of granular microstructure in dense shear flows. Nature 406, 385–389.Google Scholar
Onoda, G. Y. & Liniger, E. G. 1990 Random loose packing of uniform spheres and the dilatancy onset. Phys. Rev. Lett. 64, 2727–2730.CrossRefGoogle ScholarPubMed
Richman, M. W. 1988 Boundary conditions based upon a modified Maxwellian velocity distribution function for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227–240.CrossRefGoogle Scholar
Richman, M. W. & Chou, C. S. 1988 Boundary effects on granular flows of smooth disks. Z. Angew. Mech. Phys. 39, 885–901.CrossRefGoogle Scholar
Sangani, A. S. & Behl, S. 1989 The planar singular solutions of Stokes and Laplace equations and their application to transport processes near porous surfaces. Phys. Fluids A 1, 21–37.Google Scholar
Sangani, A. S., Mo, G., Tsao, H.-K. & Koch, D. L. 1996 Simple shear flows of dense gas–solid suspensions at finite Stokes numbers. J. Fluid Mech. 313, 309–341.CrossRefGoogle 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
Sinclair, J. L. & Jackson, R. 1989 Gas–particle flow in a vertical pipe with particle–particle interactions. AIChE J. 39, 1473–1486.Google Scholar
Smart, J. R. & Leighton, D. T. Jr., 1989 Measurement of the hydrodynamic surface roughness of noncolloidal spheres. Phys. Fluids A 1, 52–60.CrossRefGoogle Scholar
Sundaram, S. & Collins, L. R. 1997 Collision statistics in an isotropic particle-laden turbulent suspension. I. Direct numerical simulations. J. Fluid Mech. 335, 75–109.Google Scholar
Sundararajakumar, R. R. & Koch, D. L. 1996 Non-continuum lubrication flows between particles colliding in a gas. J. Fluid Mech. 313, 283–308.Google Scholar
Torquato, S. 1995 Nearest-neighbor statistics for packings of hard spheres and disks. Phys. Rev. E 51, 3170–3182.Google Scholar
Verberg, R. & Koch, D. L. 2006 Rheology of particle suspensions with low to moderate fluid inertia at finite particle inertia. Phys. Fluids 18, 083303.CrossRefGoogle Scholar
Wylie, J. J., Koch, D. L. & Ladd, A. J. C. 2003 Rheology of suspensions with high particle inertia and moderate fluid inertia. J. Fluid Mech. 480, 95–118.CrossRefGoogle Scholar
Xu, H. 2003 Collisional granular flows with and without gas interactions in microgravity. PhD dissertation, Cornell University.Google Scholar
Xu, H., Louge, M. & Reeves, A. 2003 Solution of the kinetic theory for bounded collisional granular flows. Continuum Mech. Thermodyn. 15 (4), 321–349.CrossRefGoogle Scholar
Zarraga, I. S., Hill, D. A. & Leighton Jr, D. T. 2000 The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J. Rheol. 44, 185–220.CrossRefGoogle Scholar