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On the viscosity of suspensions of solid spheres

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Abstract

A cell theory is used to derive the dependence of the zero-shear-rate viscosity on volume concentration for a suspension of uniform, solid, neutrally buoyant spheres. This result reduces to Einstein's solution at infinite dilution and to Frankel and Acrivos's expression in the limit as the concentration approaches its maximum value. Good agreement is found between the solution and the available data for the entire concentration range, provided that the maximum concentration is determined from the viscosity data themselves.

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Abbreviations

a :

radius of sphere

d :

the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers

E :

energy dissipation rate in one-half the liquid volume separating the spheres

E cell :

total energy dissipation rate in the cell

E homogeneous :

energy dissipation rate in the cell of a hypothetical one phase fluid

E interaction :

energy dissipation rate in the cell due to sphere interactions

E sphere :

energy dissipation rate in the cell due to the sphere at the cell center

F :

force on one sphere

h :

minimum separation distance between two spheres

J :

(1/2)d = one-half the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers

p :

pressure

W :

velocity of one sphere in squeezing flow between two spheres relative to the midpoint of the line connecting the sphere centers

δ i :

unit vectors in thei-th direction

κ :

elongation rate

μ :

viscosity of the suspending fluid

μ r :

μ*/μ = relative viscosity

μ*:

viscosity of the suspension

π :

the total stress tensor

τ :

the part of the total stress tensor that vanishes at equilibrium

φ :

volume fraction of spheres

References

  1. Einstein A (1906) Ann Phys 19, 289; (1911) Ibid, 34, 591

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  2. Frankel NA and Acrivos A (1967) Chem Engng Sci 22, 847

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  3. Jeffrey DJ and Acrivos A (1967) AIChEJ 22, 417

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  4. Thomas DG (1965) J Colloid Sci 20, 267

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  5. Quemada D (1977) Rheol Acta 16, 82

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  6. Ackermann NL and Shen HT (1979) AIChEJ 25, 327

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  7. Rutgers R (1962) Rheol Acta 16, 202, 305

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  8. Gadala-Maria F and Acrivos A (1980) J Rheol 24, 799

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Author notes

  1. Alan L. Graham

    Present address: Los Alamos National Laboratory, P.O. Box 1663, 87545, Los Alamos, NM, USA

Authors and Affiliations

  1. Department of Chemical Engineering and Rheology Research Center, University of Wisconsin, 53706, Madison, WI, USA

    Alan L. Graham

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  1. Alan L. Graham
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Graham, A.L. On the viscosity of suspensions of solid spheres. Appl. Sci. Res. 37, 275–286 (1981). https://doi.org/10.1007/BF00951252

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  • DOI: https://doi.org/10.1007/BF00951252

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