Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-08T15:40:41.208Z Has data issue: false hasContentIssue false
  • English
  • Français

Stress in a suspension near rigid boundaries

Published online by Cambridge University Press:  12 April 2006

Aydin TÖZeren  and
Richard Skalak
Aydin TÖZeren
Affiliation:
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York 10027
Richard Skalak
Affiliation:
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York 10027
Get access Rights & Permissions [Opens in a new window]

Abstract

The stress system near a rigid boundary in a suspension of neutrally buoyant spheres is considered under the assumption of small Reynolds number. The suspending fluid is assumed to be Newtonian and incompressible. An ergodic principle is formulated for parallel mean flows, and the bulk stress is expressed as a surface average. Only dilute suspensions are considered and particle interactions are neglected. A uniform shear flow past a plane wall with a single spherical particle is studied first. A series solution is developed and the mean velocity and stress fields are computed for a force-free and couple-free sphere and also for spheres with couples applied by external means. The translational and angular velocities of the particle and the stress distribution on the surface of the particle are calculated. Properties of dilute suspensions of spheres are found by appropriate surface averages over the solutions for a single particle. The mean stress on a plane parallel to the wall is shown to reduce to the Einstein value when the distance from the boundary is sufficiently large. Mean velocity profiles of the suspension for Couette flow and Poiseuille flow are developed. It is shown that in an average sense particles rotate more slowly than the ambient fluid in a region approximately three sphere radii thick adjacent to the plane wall. But for the suspension as a whole, an apparent slip velocity always develops in this region. This results in an apparent viscosity which is less than the infinite-suspension value of Einstein.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 82 , Issue 2 , 7 September 1977 , pp. 289 - 307
Copyright
© 1977 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

Ariman, T., Turk, M. A. & Sylvester, N. D. 1973 Microcontinuum fluid mechanics — a review. Int. J. Engng Sci. 11, 905–930.Google Scholar
Ariman, T., Turk, M. A. & Sylvester, N. D. 1974 Applications of microcontinuum fluid mechanics. Int. J. Engng Sci. 12, 279–293.Google Scholar
Barbee, J. H. & Cokelet, G. R. 1971 The Fahraeus effect. Microvasc. Res. 3, 6–21.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Batchelor, G. K. 1971 The stress generated in a non-dilute suspension of elongated particles by pure straining motion. J. Fluid Mech. 46, 813–829.Google Scholar
Batchelor, G. K. 1973 Transport properties of two-phase materials with random structure. Lecture at 4th Can. Nat. Cong. Appl. Mech. Univ. Toronto.
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, 401–427.Google Scholar
Brenner, H. 1972a Suspension rheology. Prog. Heat Mass Transfer 5, 89–129.Google Scholar
Brenner, H. 1972b Dynamics of neutrally buoyant particles in low Reynolds number flows. Prog. Heat Mass Transfer 6, 509–574.Google Scholar
Brenner, H. & Weissman, M. H. 1972 Rheology of a dilute suspension of di-polar spherical particles in an external field. II. Effect of rotary Brownian motion. J. Colloid Interface Sci. 41, 499–531.Google Scholar
Bungay, P. M. & Brenner, H. 1969 Modeling of blood flow in the microcirculation-tube flow of rigid particle suspension. Proc. 7th Ann. SES meeting. St. Louis, Missouri.
Bungay, P. M. & Brenner, H. 1973 Pressure drop due to the motion of a sphere near the wall bounding a Poiseuille flow. J. Fluid Mech. 60, 81–96.Google Scholar
Cowin, S. C. 1974 The theory of polar fluids. Adv. in Appl. Mech. 14, 295–368.Google Scholar
Cox, R. G. & Brenner, H. 1971 The rheology of a suspension of particles in a Newtonian fluid. Chem. Engng Sci. 26, 65–93.Google Scholar
Dean, W. R. & O'Neill, M. E. 1963 A slow motion of viscous liquid caused by the rotation of a solid sphere. Mathematika 10, 13–24.Google Scholar
Einstein, A. 1906 Eine neue Bestimmung der Molekuldimensionen. Ann. Phys. 19, 289–306.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall — I, II. Chem. Engng Sci. 22, 637–660.Google Scholar
Guth, E. & Simha, R. 1936 Über die Viskostat von Kugelsuspensionen. Kolloid Z. 74, 266–275.Google Scholar
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics, 2nd edn. Noordhoff.
Hinch, E. J. 1972 Note on the symmetries of certain material tensors for a particle in Stokes flow. J. Fluid Mech. 54, 423–425.Google Scholar
Hinch, E. J. & Leal, L. G. 1972 The effects of Brownian motion on the rheological properties of a suspension of non-spherical particles. J. Fluid Mech. 52, 683–712.Google Scholar
Jeffery, G. B. 1915 Proc. Lond. Math. Soc. 14, 327–338.
Leal, L. G. & Hinch, E. J. 1972 The effect of weak Brownian rotations on particles in shear flow. J. Fluid Mech. 55, 745–765.Google Scholar
Maude, A. D. 1967 Theory of wall effect in the viscometry of suspensions. Brit. J. Appl. Phys. 18, 1193–1197.Google Scholar
Maude, A. D. & Whitmore, R. L. 1956 Wall effect and viscometry of suspensions. Brit. J. Appl. Phys. 7, 98–102.Google Scholar
O'Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving sphere. Mathematika 11, 67–74.Google Scholar
O'Neill, M. E. 1968 A sphere in contact with a plane wall in a slow linear shear flow. Chem. Engng Sci. 23, 1293–1298.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
Seshadri, V. & Sutera, S. P. 1970 Apparent viscosity of coarse, concentrated suspensions in tube flow. Trans. Soc. Rheol. 14, 351–373.Google Scholar
Tözeren, A. 1974 Stress in a suspension near rigid boundaries. Ph.D. dissertation, Columbia University, New York.
Whitmore, R. L. 1959 The viscous flow of disperse suspensions in tubes. In Rheology of Disperse Systems (ed. C. C. Mills), pp. 45–56. Pergamon.