Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T13:26:28.589Z Has data issue: false hasContentIssue false
  • English
  • Français

The stress system in a suspension of force-free particles

Published online by Cambridge University Press:  29 March 2006

G. K. Batchelor
G. K. Batchelor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.

In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 41 , Issue 3 , 29 April 1970 , pp. 545 - 570
Copyright
© 1970 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

Cox, R. G. 1969 The deformation of a drop in a general time-dependent fluid flow. J. Fluid Mech. 37, 601–623.Google Scholar
Giesekus, H. 1962 Elasto-viskose Flüssigkeiten, für die in stationären Schichtströmungen sämtliche Normalspannungskomponenten verschieden gross sind. Rheol. Acta, 2, 50–62.Google Scholar
Goddard, J. D. & Miller, C. 1967 Nonlinear effects in the rheology of dilute suspensions. J. Fluid Mech. 28, 657–673.Google Scholar
Hand, G. L. 1962 A theory of anisotropic fluids. J. Fluid Mech. 13, 33–46.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Hashin, Z. 1964 Theory of mechanical behaviour of heterogeneous media. Appl. Mech. Rev. 17, 1–9.Google Scholar
Hill, R. 1963 Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids, 11, 357–372.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc. A 102, 161–179.Google Scholar
Jeffreys, H. & Jeffreys, B. J. 1966 Methods of Mathematical Physics. Cambridge University Press.
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Landau, L. D. & Lifschitz, E. M. 1959 Fluid Mechanics. Pergamon Press.
McTague, J. P. 1969 Magnetoviscosity of magnetic colloids. J. Chem. Phys. 51, 133–136.Google Scholar
Peterson, J. M. & Fixman, M. 1963 Viscosity of polymer solutions. J. Chem. Phys. 39, 2516–2523.Google Scholar
Roscoe, R. 1967 On the rheology of a suspension of viscoelastic spheres in a viscous liquid. J. Fluid Mech. 28, 273–293.Google Scholar
Rosenkilde, C. E. 1967 Surface-energy tensors. J. Math. Phys. 8, 84–88.Google Scholar
Takserman-Krozer, R. & Ziabicki, A. 1963 Behaviour of polymer solutions in a velocity field with parallel gradient, Parts I and II. J. Polymer Sci. A, 1, 491–515.Google Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. Roy. Soc. A 138, 41–48.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. Roy. Soc. A 146, 501–523.Google Scholar