Sedimentation of Suspensions of Non-Brownian Spheres
Some recent papers...
Long-Range
Correlations in Sedimentation
Nonuniversal
Velocity Fluctuations of Sedimenting Particles
P.N. Segrč, E. Herbolzheimer, and P.M. Chaikin, Phys. Rev. Lett. 79, 2574 (1997).
(Related and Interesting Web Sites Related and Interesting Papers)
The problem is to understand the flow dynamics of spheres as they settle. It's a very simple problem conceptually, but one that has produced a variety of conflicting experimental findings which have confounded theorists for quite some time.
In our experiments the samples consisted of small, 15 mm diameter, polystyrene spheres settling in water. The particles are large enough that Brownian motion is negligable, and they settle at such slow rates, V~1 inch/hour, that the Reynolds number is <<1 so that the linear equations of creeping flow should apply. As a single particle settles, it sets up a flow field around it which varies as U~1/r, i.e. a power law decay that lacks any characteristic decay length, i.e. it is of infinite range. Consequently in real samples at finite concentrations calculating the flow fields present in the fluid as the particles settle is a complex many-body problem. To give some insight into the phenomena think of an initially dilute suspension of spheres as shown below. Since the spheres are heavier than water, there is a buoyant force driving them downward to the final state also shown below. The individual velocities of the spheres, the quantity of interest in the experiments, are not all the same though because:
(1) When individual spheres come close together, they fall faster than a single sphere would alone. This inherently links local density with local velocity so that fluctuations in density lead to varying settling rates.
(2) As the particles settle, they displace the
background liquid which must work its way up through the matrix of
particles, hindering their settling motions.
The Big question is whether there is a characteristic correlation
length for the hydrodynamic interactions that develop??
Shown below is a copy of Fig. 1 from our paper. Fig.s (a) and (d) show velocity vector maps from two different concentration samples of settling spheres in water. Fig's (b,c) and (e,f) show the behavior of the fluctuations in the settling velocities. It is immediately apparent that the velocity fluctuations organize themselves into 'swirls' of some finite size, involving thousands of particles, i.e. there is a correlation length. For more information on this see the paper that we published.
Related and Interesting Web Sites
Dan Joseph (Minnesota) and 4 other groups. DIRECT SIMULATION OF THE MOTION OF PARTICLES IN FLOWING LIQUIDS
|
Related and Interesting Papers
(Paper 0) Analogies between Colloidal Sedimentation and Turbulent Convection at High Prandtl Numbers P. Tong and B. J. Ackerson Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078 PHYSICAL REVIEW E DECEMBER 1998 VOLUME 58, NUMBER 6.
(Paper 1) On Chaos at Zero Reynolds Number
These Chaotic patterns were found by Imre Janosi et al. (PRE, 1997) from the slow settling of three particles.
(Paper 2) Simulations have also seen swirl-like patterns during Sedimentation.
These are from -- S. Schwarzer, PRE 52, 6461 (1995).
(Paper 3) Intrinsic Convection in Concentrated Settling Suspensions
During Sedimentation at low Re, and f=20%, the particle mean velocities are greater in the center of the cell, than at the wall (a), but so too are the relative fluctuation velocities in (b), so that the normalized fluctuation amplitudes are fairly constant accross the cell width (c).
(Paper 4) Sedimentation in Dilute Fiber Suspensions -- B. Herzhaft et al. (Paper 5) Screened and Unscreened Phases in Sedimentation -- A. Levine et al.
A transition between screened and unscreened phases occurs for critical values of the Peclet number. (Paper 6) Crystalline Fluidized Beds - M. Rutgers et al., PRE, May 1995. (Paper 7) Diffusion, Dispersion, and Settling of Hard Spheres (DWS!) -
J.Z. Xue et al., PRL, 14 Sep. 1992. (Paper 9) Sedimentation of homogeneous suspensions of non-Brownian spheres -AJC Ladd, Physics of Fluids, 3/97. (Paper
10) Hydrodynamic screening in
Sedimentating Suspensions of Non-Brownian Spheres -AJC Ladd, PRL
2/96. Additional References:
1. Effect of the vessel size on the hydrodynamic diffusion of sedimenting spheres Helene Nicolai, Elisabeth Guazzelli Physics of Fluids January 1995 2. Particle velocity fluctuations and hydrodynamic self-diffusion of sedimenting... H. Nicolai, B. Herzhaft, E. J. Hinch, L. Oger... Physics of Fluids January 1995 3. A study of the sedimentation of noncolloidal bidisperse, concentrated suspensions M. Hoyos, J. C. Bacri, J. Martin, D. Salin Physics of Fluids December 1994 4. Enhanced sedimentation of suspensions in porous media Abraham Marmur, Sigalit Bar Physics of Fluids September 1994
5. Hydrodynamic diffusion in a suspension of sedimenting point particles with periodic... Donald L. Koch Physics of Fluids September 1994 6. Deformation and orientation of an elastic slender body sedimenting in a viscous... Xianghua Xu, Ali Nadim Physics of Fluids September 1994 7. Hydrodynamic dispersion broadening of a sedimentation front J. Martin, N. Rakotomalala, D. Salin Physics of Fluids October 1994 8. The nature of particle contacts in sedimentation Shulin Zeng, Edward T. Kerns, Robert H. Davis Physics of Fluids June 1996 9. Velocity fluctuations of a heavy sphere falling through a sedimenting suspension Helene Nicolai, Yannick Peysson, Elisabeth Guazzelli Physics of Fluids April 1996 10. Magnetic resonance imaging study of sedimenting suspensions of noncolloidal spheres Michael A. Turney, Man Ken Cheung, Michael J. McCarthy... Physics of Fluids May 1995 11. Accurate determination of the sedimentation flux of concentrated suspensions J. Martin, N. Rakotomalala, D. Salin Physics of FluidsOctober 1995 12. Nuclear magnetic resonance imaging investigation of sedimentation of concentrated suspensions in non-Newtonian fluids Bobroff S, Phillips RJ JOURNAL OF RHEOLOGY 42: (6) 1419-1436 NOV-DEC 1998. 13. A pulsed field gradient NMR technique for the determination of the structure of suspensions of non-Brownian particles with application to packings of spheres Talini L, Leblond J, Feuillebois F JOURNAL OF MAGNETIC RESONANCE 132: (2) 287-297 JUN 1998 14. Particle settling in oil-in-water emulsions Beydoun D, Guang D, Chhabra RP, Raper JA POWDER TECHNOLOGY 97: (1) 72-76 JUN 1998 15. Mean velocities in polydisperse suspensions Tory EM, Kamel MT POWDER TECHNOLOGY 93: (3) 199-207 OCT 15 1997 16. Non-colloidal sedimentation compared with Kynch theory Chang D, Lee T, Jang Y, Kim M, Lee S POWDER TECHNOLOGY 92: (1) 81-87 JUN 1997 17. Unexpected phenomena observed in particle settling in non-Newtonian media Gheissary G, vandenBrule BHAA JOURNAL OF NON-NEWTONIAN FLUID MECHANICS 67: 1-18 NOV 1996 18. Wall effects on the sedimentation velocity of suspensions in viscous flow DiFelice R, Parodi E AICHE JOURNAL 42: (4) 927-931 APR 1996 19. Sedimentation of bidisperse, uncharged colloidal sphere suspensions: Influence of viscosity and irregular surfaces ThiesWeesie DME, Philipse AP, Lekkerkerker HNW JOURNAL OF COLLOID AND INTERFACE SCIENCE 177: (2) 427-438 FEB 10 1996 20. Sedimentation of bidisperse, uncharged colloidal sphere suspensions: Influence of viscosity and irregular surfaces ThiesWeesie DME, Philipse AP, Lekkerkerker HNW JOURNAL OF COLLOID AND INTERFACE SCIENCE 177: (2) 427-438 FEB 10 1996
|
