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Shear viscosity of slightly-elastic concentrated suspensions at low and high shear rates

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Abstract

A quasi-static asymptotic analysis is employed to investigate the elastic effects of fluids on the shear viscosity of highly concentrated suspensions at low and high shear rates. First a brief discussion is presented on the difference between a quasi-static analysis and the periodic-dynamic approach. The critical point is based on the different order-of-contact time between particles. By considering the motions between a particle withN near contact point particles in a two-dimensional “cell” structure and incorporating the concept of shear-dependent maximum packing fraction reveals the structural evolution of the suspension under shear and a newly asymptotic framework is devised. In order to separate the influence of different elastic mechanisms, the second-order Rivlin-Ericksen fluid assumption for describing normal-stress coefficients at low shear rates and Harnoy's constitutive equation for accounting for the stress relaxation mechanism at high shear rates are employed. The derived formulation shows that the relative shear viscosity is characterized by a recoverable shear strain,S R at low shear rates if the second normal-stress difference can be neglected, and Deborah number,De, at high shear rates. The predicted values of the viscosities increase withS R , but decrease withDe. The role ofS R in the matrix is more pronounced than that ofDe. These tendencies are significant when the maximum packing fraction is considered to be shear-dependent. The results are consistent with that of Frankel and Acrivos in the case of a Newtonian suspension, except for when the different divergent threshhold is given as [1 − (Φ/Φ m )1/2] − 1.

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Abbreviations

A (1) , A (2) :

Rivlin-Ericksen tensors, defined by eqs. (2) and (3)

D :

rate of deformation tensor, 1/2{(∇υ) + (∇υ)⊤} = 1/2(L +L ⊤)

De :

Deborah number (relation of the fluid/characteristic time of the flow)

D/D t :

Harnoy's objective time derivative, eq. (6)

e i :

base vector of coordinate

E :

rate of viscous dissipation of the suspension

E (0) :

rate of viscous dissipation of the suspending fluid

h :

thickness of the fluid layer in the near-contact region

h 0 :

particle spacing

L :

velocity gradient, (∇υ)⊤

n :

unit outer normal vector

N :

number of contact points

N 1 ,N 2 :

first and second normal-stress difference

p :

isotropic pressure

\(\bar p\) :

dimensionless isotropic pressure, defined by eq. (37)

P e :

Peclet number

r :

radius in cylindrical coordinates

R 0 :

particle radius

S 0 :

boundary of viscometric apparatus

dS :

increment of surface area

S R :

recoverable shear strain

U :

constant approach velocity of cells

υ i :

component of velocity vector

\(\bar v_i \) :

dimensionless velocity components, defined by eq. (37)

υ (0) :

velocity onS 0

V :

volume of liquid

α 1 ,α′ 1 :

Newtonian viscosity of the suspension medium

α * 1 ,α′ * 1 :

effective viscosity of the suspension

α 2 ,α 3 :

second-order coefficients of the Rivlin-Ericksen fluid

α * r ,α′ * r :

relative effective viscosity

\(\dot \gamma \) :

shear rate

\(\dot \gamma _0 \) :

macroscopic shear rate

ε :

dimensionless particle spacing

η,(0) :

dimensionless coordinates, defined by eq. (37)

λ :

relaxation time of the fluid

τ :

deviatoric stress tensor

τ ij :

component of the deviatoric stress tensor

Ω :

tensor of angular velocity of the principal axes of the rate-of-strain tensor

ø :

volume concentration of particles

ø 2d :

areal fraction obtained by 3/2ø

ø m :

maximum volume concentration of particles

ø 2d, m :

maximum areal fraction (= 3/2ø m ) of particles

ø m0 ,ø m∞ :

limiting values ofø m at the lowPe and highPe conditions

Ф v :

local rate of viscous dissipation per unit volume

l 0 :

effective radial dimension in the near-contact region

T :

total

⊤:

transpose

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Authors and Affiliations

  1. Department of Chemical Engineering, National Tsing Hua University, 30043, Hsinchu, Taiwan, ROC

    Maw-Ling Wang & Tei-Chih Cheau

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  1. Maw-Ling Wang
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  2. Tei-Chih Cheau
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Wang, ML., Cheau, TC. Shear viscosity of slightly-elastic concentrated suspensions at low and high shear rates. Rheol Acta 27, 596–607 (1988). https://doi.org/10.1007/BF01337455

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