Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-20T08:50:53.423Z Has data issue: false hasContentIssue false
  • English
  • Français

Simple shear flows of dilute gas–solid suspensions

Published online by Cambridge University Press:  26 April 2006

Heng-Kwong Tsao  and
Donald L. Koch
Heng-Kwong Tsao
Affiliation:
School of Chemical Engineering, Cornell University, Ithaca, NY 14853, USA
Donald L. Koch
Affiliation:
School of Chemical Engineering, Cornell University, Ithaca, NY 14853, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Kinetic theory and numerical simulations are used to explore the dynamics of a dilute gas–solid suspension subject to simple shear flow. The particles experience a Stokes drag force and undergo solid-body interparticle collisions. Two qualitatively different steady-state behaviours are possible: an ignited state, in which the variance of the particle velocity is very large; and a quenched state, in which most of the particles follow the local fluid velocity. Theoretical results for the ignited state are obtained by perturbing from a Maxwell distribution, while predictions for the quenched state result from consideration of the collision of particles that initially move with the fluid. A composite theory, which includes effects of collisions driven by both the mean shear and the velocity fluctuations, predicts the existence of multiple steady states. Dynamic simulations and calculations using the direct-simulation Monte Carlo method confirm the result that, for certain volume fractions and shear rates, either the quenched or ignited state can be achieved depending on the initial velocity variance.

Simulations are also performed for particles experiencing a nonlinear drag force. Both the theory of rapid granular flow, which neglects drag, and our theory for the ignited state with linear drag predict that the particle velocity variance can grow without bound as ϕ → 0, where ϕ is the volume fraction. The nonlinear drag force eliminates the divergence and leads to a particle velocity variance that will always decrease with decreasing volume fraction in the limit ϕ → 0.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 296 , 10 August 1995 , pp. 211 - 245
Copyright
© 1995 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

Alfonso, M., Gañán-Calvo, A. M. & Lasheras, J. C. 1991 The dynamics and mixing of small spherical particles in a plane, free shear layer. Phys. Fluids A 3, 1207–1217.Google Scholar
Bagnnold, R. A. 1954 Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 49–63.Google Scholar
Bird, G. A. 1970 Direct simulation and the Boltzmann equation. Phys. Fluids 13, 2676–2681.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1977 Dynamics of Polymeric Liquids. John Wiley & Sons.
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34.Google Scholar
Campbell, C. S. 1989 The stress tensor for simple shear flows of a granular material. J. Fluid Mech. 203, 449–473.Google Scholar
Campbell, C. S. 1990 Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57–92.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematic Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press.
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles. Academic.
Einstein, A. 1906 Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 19, 289–306 (see also 34, 591–592, 1911).Google Scholar
Elghobashi, S. E. & Abou-Arab, T. W. 1983 A two-equation turbulence model for two-phase flow. Phys. Fluids 26, 931–938.Google Scholar
Goddard, J. D. 1986 Dissipative materials as constitutive models for granular media. Acta Mechanica 63, 3–13.Google Scholar
Goldreich, P. & Tremaine, S. 1978 The velocity dispersion in Saturn's rings. Icarus 34, 227–239.Google Scholar
Herdegen, N. & Hess, S. 1982 Nonlinear flow behavior of the Boltzmann gas. Physica 115A, 281–299.Google Scholar
Hopkins, M. A. & Shen, H. H. 1992 A Monte Carlo solution for rapidly shearing granular flows based on the kinetic theory of dense gases. J. Fluid Mech. 244, 477–491.Google Scholar
Jenkins, J. T. & Richman, M. W. 1988 Plane simple shear of smooth inelastic circular disks: the anisotropy of the second moment in the dilute and dense limits. J. Fluid Mech. 192, 313–328.Google Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 187–202.Google Scholar
Koch, D. L. 1990 Kinetic theory for a monodisperse gas—solid suspension. Phys. Fluid A 2, 1711–1723.Google Scholar
Kumaran, V., Tsao, H.-K. & Koch, D. L. 1993 Velocity distribution functions for a bidisperse, sedimenting particle—gas suspension. Intl J. Multiphase Flow 19, 665–681.Google Scholar
Louge, M., Mastorakos, E. & Jenkins, J. T. 1991 The role of particle collisions in pneumatic transport. J. Fluid Mech. 231, 345–359.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.Google Scholar
Maxey, M. R. 1987 The motion of small spherical particles in a cellular flow field. Phys. Fluids 30, 1915–1928.Google Scholar
Mei, R. & Adrian, R. J. 1992 Flow past a sphere with an oscillation in the free-stream velocity, and unsteady drag at finite Reynolds number. J. Fluid Mech. 237, 323–341.Google Scholar
Lawrence, C. J. & Mei, R. 1995 Long-time behaviour of the drag on a body in impulsive motion. J. Fluid Mech. 283, 307–327.Google Scholar
Richman, M. R. 1989 The source of second moments in dilute granular flows of highly inelastic spheres. J. Rheol. 33, 1293–1306.Google Scholar
Sangani, A. S., Mo, G.-B., Tsao, H.-K. & Koch, D. L. 1995 Rapidly sheared, dense gas—solid suspensions. J. Fluid Mech. (submitted).Google Scholar
Savage, S. B. & Jeffrey, D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255–272.Google Scholar
Sinclair, J. L. & Jackson, R. 1989 Gas-particle flow in a vertical pipe with particle—particle interactions. AIChE J. 35, 1473–1486.Google Scholar
Squires, K. & Eaton, J. K. 1990 Particle response and turbulence modification in isotropic turbulence. Phys. Fluids A 2, 1191–1203.Google Scholar
Tsao, H.-K., Koch, D. L., Mo, G.-B. & Sangani, A. S. 1993 Inertial effects in suspensions. In Proc. 5th NSF-DOE Workshop on Fluid Solids Transports, Ithaca, NY, pp. 121–130.
Zwanzig, R. 1979 Nonlinear shear viscosity of a gas. J. Chem. Phys. 71, 4416–4420.Google Scholar