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Properties of a bidisperse particle–gas suspension Part 1. Collision time small compared with viscous relaxation time

Published online by Cambridge University Press:  26 April 2006

V. Kumaran  and
Donald L. Koch
V. Kumaran
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
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Abstract

The properties of a dilute bidisperse particle–gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit τcτv using a perturbation analysis in the small parameter v, which is proportional to the ratio of timescales τc/τv. Here, τc is the time between successive collisions of a particle, and tv is the viscous relaxation time. The leading-order distribution functions for the two species are isotropic Gaussian distributions, and are identical to the molecular velocity distributions in a two-component gas at equilibrium. Balance equations are written for the mean and mean-square velocities, using a distribution function that is a small perturbation from the isotropic Gaussian. The collisional terms are calculated by performing an ensemble average over the relative configurations of the colliding particles, and the mean velocity and velocity variances are calculated correct to O(v2) by solving the balance equations. The difference in the mean velocities of the two species is O(v) smaller than the mean velocity of the suspension, and the fluctuating velocity is O(v½) smaller than the mean velocity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 247 , February 1993 , pp. 623 - 641
Copyright
© 1993 Cambridge University Press

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References

Batchelor, G. K. 1988 A new theory of the instability of a fluidized bed. J. Fluid Mech. 193, 75– 110.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.
Didwania, A. K. & Homsy, G. M. 1982 Resonant sideband instabilities in wave propogation in fluidized beds. J. Fluid Mech. 122, 433– 438.Google Scholar
Jackson, R. 1963 The mechanics of fluidised beds. Parts 1 and 2. Trans. Inst. Chem. Engrs 41, 13– 28.Google Scholar
Jenkins, J. T. 1987 Balance laws and Constitutive relations for rapid flows of granular materials. In Constitutive Models of Deformation (ed. J. Chandra & R. Srivastava). SIAM.
Jenkins, J. T. & Mancini, F. 1989 Kinetic theory for binary mixtures of smooth, nearly elastic spheres. Phys. Fluids A 1 2050– 2057.Google Scholar
Jenkins, J. T. & Richman, M. W. 1985 Kinetic theory for plane flows of a dense gas of identical, rough, inelastic circular disks. Phys. Fluids A 28, 3485– 3494.Google Scholar
Koch, D. L. 1990 Kinetic theory for monodisperse gas–solid suspension. Phys. Fluids A 2, 1711– 1723.Google Scholar
Kumaran, V. & Koch, D. L. 1993a Properties of a bidisperse particle–gas suspension. Part 2. Viscous relaxation time small compared to collision time. J. Fluid Mech. 247, 643– 660.Google Scholar
Kumaran, V. & Koch, D. L. 1993b Approximate velocity distribution functions for a bidisperse particle–gas suspension. Intl J. Multiphase Flow (submitted).Google Scholar
Tham, M. K. & Gubbins, K. E. 1971 Kinetic theory of multicomponent dense fluid mixtures of rigid spheres. J. Chem. Phys. 55, 268– 279.Google Scholar
Yen, S. M. 1984 Numerical solution of the nonlinear Boltzmann equation. Ann. Rev. Fluid Mech. 16, 67– 97.Google Scholar