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Viscosity of suspensions modeled with a shear-dependent maximum packing fraction


A large amount of data from the literature on viscosity of concentrated suspensions of rigid spherical particles are analyzed to support the new concept that the maximum packing fraction (ϕ M ) is shear-dependent. Incorporation of this behavior in a rheological model for viscosity (η) as a function of particle volume fraction (ϕ) succeeds in describing virtually all non-Newtonian effects over the entire concentration range and also accounts for a yield stress. The most successful model is one proposed by Krieger and Dougherty for Newtonian viscosities,η (ϕ, ϕ M ), but withϕ M varying from a low-shear limitϕ M0 to a high-shear limitϕ M. Microstructural interpretations of this behavior are advanced, with arguments suggesting that similar rheological models should apply to suspensions of nonspherical and irregular particles.

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a :

particle size scale (for spheres, the diameter)

A :

lumped kinetic parameter in eqs. (23) and (24)



C :

coefficient in Arrhenius model, eq. (2)

D :

coefficient in Mooney model, eq. (3)

e i :

parameter representing one of the three electroviscous effects (i = 1, 2, or 3)

f :

fraction of total particulates that exist in the dispersed phase, eq. (22)

h :

solution factor, in Arrhenius model, eq. (2)

k :

crowding factor, in Mooney model, eq. (3)

k D ,k F :

kinetic rate coefficient for producing particles of dispersed or flocculated type, respectively

K :

Einstein coefficient for particles of any shape, eq. (1); equal to [η]


Krieger-Dougherty model, eq. (6)

m :

exponent to characterize shear-dependence in viscosity models of Cross, eq. (10), and eq. (23), and also in yield stress prediction eq. (24)

N :

number of monodisperse components in a blend of spheres with different diameters


polydispersity (in size) parameter

S :

generalized shape parameter

T :


V c :

volume of “chamber” in figure 6, representing the entire volume of the sample

V P :

total volume of particles in the sample

V D ,V F :

sample volumes in which dispersed particles or flocculated particles, respectively, prevail; volumes of the “dispersed phase” or “flocculated phase”, containing both particles and carrier fluid

V PD ,V PF :

particle volume within the phase volumeV D orV F , respectively

α :

coefficient in definition ofτ c in eq. (8); of order unity

β :

coefficient regulating\(\dot \gamma \)-sensitivity in eq. (10)

\(\dot \gamma \) :

shear rate,dv 1/dx 2 in simple shear

η :

shear viscosity of the suspension

η 0,η :

low-shear and high-shear limiting values ofη

η s :

viscosity of the suspending fluid


intrinsic viscosity,\(\mathop {\lim }\limits_{\phi \to 0} (\eta - \eta _s )/\phi \eta _s \)

η r :

reduced viscosity,η/η s


Boltzmann's constant; inτ c

τ :

shear stress

τ c :

parameter characterizing sensitivity of viscosity to stress, in eq. (8)

τ B :

dynamic yield stress in the floc model

τ y :

yield stress

ϕ :

volume fraction occupied by solids in a suspension

ϕ M :

maximum value ofϕ attainable by a given collection of particles under given conditions of flow

ϕ M0,ϕ M :

limiting values ofϕ M at the low-τ and high-τ conditions, respectively


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Correspondence to M. C. Williams.

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Wildemuth, C.R., Williams, M.C. Viscosity of suspensions modeled with a shear-dependent maximum packing fraction. Rheol Acta 23, 627–635 (1984).

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Key words

  • Suspension viscosity
  • maximum packing fraction
  • spherical particle
  • concentrated suspension