Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-03T11:07:31.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Sedimentation in a dispersion with vertical inhomogeneities

Published online by Cambridge University Press:  20 April 2006

Fran¸ois Feuillebois
Fran¸ois Feuillebois
Affiliation:
Laboratoire d'Aérothermique du Centre National de la Recherche Scientifique, F 92190 Meudon, France
Get access Rights & Permissions [Opens in a new window]

Abstract

Statistical studies of hydrodynamic interactions between many particles in a dilute dispersion raise a problem of divergent integrals. This problem arises in particular when calculating the average velocity of sedimentation of solid spheres in a viscous fluid. The solution to this problem was given by Batchelor (1972) for monodisperse suspensions of spheres, on the basis of an assumption of homogeneity. This assumption is removed here. The problem of divergent integrals is reconsidered. The solution treats as successive steps:

  1. the average flow due to random statistically independent point forces;

  2. the average flow due to random statistically independent solid spheres, without hydrodynamic interactions;

  3. the average sedimentation velocity of random, pairwise-dependent solid spheres with hydrodynamic interactions, in a dilute suspension.

Considering the case of identical spheres, and assuming homogeneity in any horizontal plane, an expression is obtained for the average sedimentation velocity of a sphere in an otherwise inhomogeneous dispersion. The formula is written in terms of integrals involving probability distributions. It reduces, when the suspension is homogeneous, to a formula obtained by Batchelor.

The probability distributions are not calculated in this paper. In order to evaluate numerically the average velocity of sedimentation, a simple expression for the pair distribution function is assumed, and two different concentration profiles are considered, viz. a sinusoidal variation and a step function. In the case of sinusoidal concentration wave, it is found that the contribution of the inhomogeneity is, for small wavelengths, comparable in magnitude to that calculated for a homogeneous dispersion by Batchelor (1972), i.e. –6.55c.

The difference in velocity between the crest and trough of the wave is an increasing function of the wavelength. For a step function in concentration, particles at the top of the cloud start to fall faster, this effect being limited to a top layer about 10 radii thick.

For future study of the long-term behaviour of a sedimenting cloud, the evolution of the pair distribution function should be added to the present theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 139 , February 1984 , pp. 145 - 171
Copyright
© 1984 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

Acrivos, A. & Herbolzheimer, E. 1979 Enhanced sedimentation in settling tanks with inclined walls J. Fluid Mech. 92, 435–457.Google Scholar
Adler, P. M. 1981 Interaction of unequal spheres. I. Hydrodynamic interaction. Colloidal forces J. Coll. Interface Sci. 84, 461–474.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres J. Fluid Mech. 52, 245–268.Google Scholar
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction J. Fluid Mech. 74, 1–29.Google Scholar
Batchelor, G. K. 1982 Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory J. Fluid Mech. 119, 379–408.Google Scholar
Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56, 401–428.Google Scholar
Batchelor, G. K. & Wen, C. S. 1982 Sedimentation in a dilute polydisperse system of interacting spheres. Part 2. Numerical results J. Fluid Mech. 124, 495–528.Google Scholar
Chwang, A. T. & Wu, T. Y. T. 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flow J. Fluid Mech. 67, 787–815.Google Scholar
Felderhof, B. U. 1976 Rheology of polymer suspensions Physica A82, 596–622.Google Scholar
Feuillebois, F. 1980 Certains problèmes d’écoulements mixtes fluide—particules solides. Thèse de Doctorat d'Etat. Univ. Paris VI.
Feuillebois, F. 1984 Singularities in Stokes flow in terms of the theory of distributions (to be published).
Friedlander, F. G. 1982 Introduction to the theory of distributions. Lecture course, Cambridge University.
Gel'Fand, I. M. & Vilenkin, N. YA. 1964 Generalized Functions, vol. 4. Academic.
Haar, D. Ter 1961 Elements of Statistical Mechanics. Holt, Rinhart & Winston.
Haber, S. & Hetsroni, G. 1981 Sedimentation in a dilute dispersion of small drops of various sizes J. Coll. Interface Sci. 79, 56–75.Google Scholar
Hearn, A. C. 1973 REDUCE 2. User's Manual, 2nd edn. University of Utah, Salt Lake City.
Herczynski, R. & Pienkowska, I. 1980 Towards a statistical theory of suspension Ann. Rev. Fluid Mech. 12, 237–269.Google Scholar
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension J. Fluid Mech. 83, 695–720.Google Scholar
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond A 335, 355–367.Google Scholar
Jeffrey, D. J. 1974 Group expansions for the bulk properties of a statistically homogeneous, random suspension. Proc. R. Soc. Lond A 338, 503–516.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow J. Fluid Mech. 139, 261–290.Google Scholar
Saffman, P. G. 1973 On the settling speed of free and fixed suspensions Stud. Appl. Maths 52, 115–127.Google Scholar
Schwartz, L. 1966 Théorie des Distributions. Hermann.
Schwartz, L. 1979 Méthodes Mathématiques pour les Sciences Physiques. Hermann. [English transl. of the 1st edn: Mathematics for the Physical Sciences. Hermann/Addison-Wesley, 1966.]