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Evolution of clusters of sedimenting low-Reynolds-number particles with Oseen interactions

Published online by Cambridge University Press:  30 April 2008

G. SUBRAMANIAN  and
DONALD L. KOCH
G. SUBRAMANIAN
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, 560064, India
DONALD L. KOCH
Affiliation:
School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
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Abstract

A theoretical framework is developed to describe, in the limit of small but finite Re, the evolution of dilute clusters of sedimenting particles. Here, Re =aU/ν is the particle Reynolds number, where a is the radius of the spherical particle, U its settling velocity, and ν the kinematic viscosity of the suspending fluid. The theory assumes the disturbance velocity field at sufficiently large distances from a sedimenting particle, even at small Re, to possess the familiar source--sink character; that is, the momentum defect brought in via a narrow wake behind the particle is convected radially outwards in the remaining directions. It is then argued that for spherical clusters with sufficiently many particles, specifically with N much greater than O(R0U/ν), the initial evolution is strongly influenced by wake-mediated interactions; here, N is the total number of particles, and R0 is the initial cluster radius. As a result, the cluster first evolves into a nearly planar configuration with an asymptotically small aspect ratio of O(R0U/N ν), the plane of the cluster being perpendicular to the direction of gravity; subsequent expansion occurs with an unchanged aspect ratio. For relatively sparse clusters with N smaller than O(R0U/ν), the probability of wake interactions remains negligible, and the cluster expands while retaining its spherical shape. The long-time expansion in the former case, and that for all times in the latter case, is driven by disturbance velocity fields produced by the particles outside their wakes. The resulting interactions between particles are therefore mutually repulsive with forces that obey an inverse-square law. The analysis presented describes cluster evolution in this regime. A continuum representation is adopted with the clusters being characterized by a number density field (n(r, t)), and a corresponding induced velocity field (u (r, t)) arising on account of interactions. For both planar axisymmetric clusters and spherical clusters with radial symmetry, the evolution equation admits a similarity solution; either cluster expands self-similarly for long times. The number density profiles at different times are functions of a similarity variable η = (r/t1/3), r being the radial distance away from the cluster centre, and t the time. The radius of the expanding cluster is found to be of the form Rcl (t) = A (ν a)1/3N1/3t1/3, where the constant of proportionality, A, is determined from an analytical solution of the evolution equation; one finds A = 1.743 and 1.651 for planar and spherical clusters, respectively. The number density profile in a planar axisymmetric cluster is also obtained numerically as a solution of the initial value problem for a canonical (Gaussian) initial condition. The numerical results compare well with theoretical predictions, and demonstrate the asymptotic stability of the similarity solution in two dimensions for long times, at least for axisymmetric initial conditions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 603 , 25 May 2008 , pp. 63 - 100
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Batchelor, G. K. 1967 Introduction to Fluid Dynamics Cambridge University Press, p. 348.Google Scholar
Batchelor, G. K. 1974 Low-Reynolds-number bubbles in fluidized beds. Arch. Mech. 26, 339.Google Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian Dynamics. Annu. Rev. Fluid Mech. 20, 111.CrossRefGoogle Scholar
Bretherton, F. P. 1964 Inertial effects on clusters of spheres falling in a viscous fluid. J. Fluid Mech. 20, 401.CrossRefGoogle Scholar
Byrd, P. F. & Friedman, M. D. 1971 Handbook of Elliptic Integrals for Engineers and Scientists. Springer.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.Google Scholar
Ekiel-Jezewska, M. L., Metzger, B. & Guazzelli, E. 2006 Spherical cloud of point particles falling in a viscous fluid. Phys. Fluids 18, 038104.CrossRefGoogle Scholar
Feng, J., Hu, H. H. & Joseph, D. D. 1994 Direct simulation of initial-value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation J. Fluid Mech. 261, 95.CrossRefGoogle Scholar
Fortes, A. F., Joseph, D. D. & Lundgren, T. S. 1987 Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177, 467.CrossRefGoogle Scholar
Gadala-Maria, F. & Acrivos, A. 1980 Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol 24, 799.CrossRefGoogle Scholar
Gakhov, F. D. 1966 Boundary Value Problems Pergamon.CrossRefGoogle Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1965 Table of Integrals, Series and Products. Academic.Google Scholar
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Noordhoff.Google Scholar
Hocking, L. M. 1964 The behavior of clusters of spheres falling in a viscous fluid. Part 2. Slow motion theory. J. Fluid Mech. 20, 129.CrossRefGoogle Scholar
Hogg, A. J., Hallworth, M. A. & Huppert, H. E. 2005 On gravity currents driven by constant fluxes of saline and particle-laden fluid in the presence of a uniform flow. J. Fluid Mech. 539, 349.CrossRefGoogle Scholar
Hogg, A. J., Ungarish, M. & Huppert, H. E. 2001 Effects of particle sedimentation and rotation on axisymmetric gravity currents. Phys. Fluids 13, 3687.CrossRefGoogle Scholar
Hu, H. H., Joseph, D. D. & Crochet, M. J. 1992 Direct simulation of fluid particle motions. Theor. Comput. Fluid Dyn. 3, 285.CrossRefGoogle Scholar
Jayaweera, K. O. L. F., Mason, B. J. & Slack, G. W. 1964 The behavior of clusters of spheres falling in a viscous fluid. Part 1. Experiment. J. Fluid Mech. 20, 121.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
Koch, D. L. 1993 Hydrodynamic diffusion in dilute sedimenting suspensions at moderate Reynolds numbers. Phys. Fluids A 5, 1141.CrossRefGoogle Scholar
Koch, D. L. & Hill, R. J. 2001 Inertial effects in suspension and porous-media flows. Annu. Rev. Fluid Mech. 33, 619.CrossRefGoogle Scholar
Kojima, M., Hinch, E. J. & Acrivos, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27, 19.CrossRefGoogle Scholar
Leshansky, A. M., Lavrenteva, O. M. & Nir, A. 2003 The weakly inertial settling of particles in a viscous fluid. Proc. R. Soc. Lond. A 459, 3079.CrossRefGoogle Scholar
Machu, G., Walter, M., Nitsche, L. C. & Schaflinger, U. W. E. 2001 Coalescence, torus formation and breakup of sedimenting drops: experiments and computer simulations. J. Fluid Mech. 447, 299.CrossRefGoogle Scholar
Metzger, B., Nicolas, M. & Guazzelli, E. 2007 Falling clouds of particles in viscous fluids. J. Fluid Mech. 580, 283.CrossRefGoogle Scholar
Muskhelishvili, N. I. 1992 Singular Integral equations. Dover.Google Scholar
Nicolas, M. 2002 Experimental study of gravity-driven dense suspension jets. Phys. Fluids 14, 3570.CrossRefGoogle Scholar
Nitsche, J. M. & Batchelor, G. K. 1997 Break-up of a falling drop containing dispersed particles. J. Fluid Mech. 340, 161.CrossRefGoogle Scholar
Vasseur, P. & Cox, R. G. 1977 The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech 80, 561.CrossRefGoogle Scholar
Yin, X. & Koch, D. L. 2007 Hindered settling velocity and microstructure in suspensions of spheres with moderate Reynolds number. Phys. Fluids 19, 093302.CrossRefGoogle Scholar