Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-14T16:46:34.367Z Has data issue: false hasContentIssue false
  • English
  • Français

The effects of Brownian rotations in a dilute suspension of rigid particles of arbitrary shape

Published online by Cambridge University Press:  12 April 2006

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

Abstract

A set of constitutive equations is derived to describe the time-dependent flow of a dilute suspension of identical rigid particles of arbitrary shape which are influenced by Brownian couples. The form of the orientation probability distribution for small departures from isotropy is found. The case of weak flows with strong Brownian effects is studied in detail, and the viscoelastic approximation and second-order-fluid limit of the constitutive equation are derived for a general particle shape. A full numerical solution is given for ellipsoids. The general nearly spherical particle is also considered and constitutive equations for general flow strengths are obtained for this case.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 84 , Issue 2 , 30 January 1978 , pp. 237 - 263
Copyright
© 1978 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

Barthès-Biesel, D. & Acrivos, A. 1973 Deformation and burst of a liquid droplet freely suspended in a linear shear field. J. Fluid Mech. 61, 1.CrossRefGoogle Scholar
Batchelor, G. K. 1970a The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545.Google Scholar
Batchelor, G. K. 1970b Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419.Google Scholar
Brenner, H. 1964a The Stokes resistance of a slightly deformed sphere. Chem. Engng Sci. 19, 519.Google Scholar
Brenner, H. 1964b The Stokes resistance of an arbitrary particle. III. Chem. Engng Sci. 19, 631.Google Scholar
Brenner, H. 1967 Coupling between the translational and rotational Brownian motions of rigid particles of arbitrary shape. II. J. Colloid Interface Sci. 23, 407.Google Scholar
Brenner, H. & Condiff, D. W. 1972 Transport mechanics in systems of orientable particles. III. Arbitrary particles. J. Colloid Interface Sci. 41, 228.Google Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284.Google Scholar
Curtiss, C. F., Bird, R. B. & Hassager, O. 1976 Kinetic theory and rheology of macro-molecular solutions. Adv. Chem. Phys. 35, 31.Google Scholar
Fixman, M. & Evans, J. 1976 Dynamics of stiff polymer chains. IV. High frequency viscosity limit. J. Chem. Phys. 64, 3474.Google Scholar
Giesekus, H. 1962 Elasto-viskose Flussigkeiten, für die in stationaren Schichtstromungen samtliche Normalspannungskomponenten verschieden gross sind. Rheol. Acta 2, 50.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Hinch, E. J. 1972 Note on the symmetries of certain material tensors for a particle in Stokes flow. J. Fluid Mech. 54, 423.Google Scholar
Hinch, E. J. & Leal, L. G. 1972 The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles. J. Fluid Mech. 52, 683.Google Scholar
Hinch, E. J. & Leal, L. G. 1973 Time-dependent flows of a suspension of particles with weak Brownian rotations. J. Fluid Mech. 57, 753.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc. A 102, 161.Google Scholar
Leal, L. G. & Hinch, E. J. 1971 The effect of weak Brownian rotations on particles in shear flow. J. Fluid Mech. 46, 685.Google Scholar
Leal, L. G. & Hinch, E. J. 1972 The rheology of a suspension of nearly spherical particles subject to Brownian rotations. J. Fluid Mech. 55, 745.Google Scholar
Lodge, A. S. & Wu, Y. J. 1971 Constitutive equations for polymer solutions from the bead/spring model of Rouse and Zimm. Rheol. Acta 10, 539.Google Scholar
Nir, A., Weinberger, H. F. & Acrivos, A. 1975 Variational inequalities for a body in a viscous shearing flow. J. Fluid Mech. 68, 739.Google Scholar
Prager, S. 1957 Stress–strain relations in a suspension of dumbbells. Trans. Soc. Rheol. 1, 53.Google Scholar
Rallison, J. M. 1976 Fellowship dissertation, Trinity College, Cambridge.Google Scholar
Taylor, T. D. & Acrivos, A. 1964 The Stokes flow past an arbitrary particle – the slightly deformed sphere. Chem. Engng Sci. 19, 445.Google Scholar
Williams, M. C. 1975 Molecular rheology of polymer solutions: interpretation and utility. A.I.Ch.E. J. 21, 1.CrossRefGoogle Scholar
Workman, H. J. & Hollingsworth, C. A. 1969 Concerning the orientation distribution function of rigid particles in a suspension which is undergoing simple shear flow. J. Colloid Interface Sci. 29, 664.Google Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69, 377.Google Scholar