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On viscous flow and effective viscosity of concentrated suspensions and emulsions

Effect of particle concentration and surfactant impurities

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Summary

Previous analysis byHappel (3) of viscous flow in concentrated solid suspensions has been extended to include concentrated emulsions of slightly deformable fluid particles in the presence or absence of surfactant impurities.

General expressions were obtained for viscous flow in multi-particle systems when arbitrary shear fields are imposed. Specific relations were then derived for uniform,Couette and hyperbolic fields. The behavior is found to be strongly dependent upon particle concentration and surfactant concentration. The theoretical expressions obtained for effective viscosity of emulsions compare favorably with experimental data ofNeogy andGhosh (18),Sibree (15),Sherman (17), andBroughton andWindebank (16). These results support other studies on ensemble velocities [(10), (12), and in particular (22)], which strongly indicate the practical value and factual reliability of cell models in predicting the behavior of suspensions and emulsions.

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Abbreviations

a :

radius of particle

A mn , Â mn , B mn ,B mn ,...., I mn ,Î mn :

general coefficients defined by [A 1] to [A 9]

c :

concentration of surfactants in continuous phase

C 1,C 2,C 3,C 4,C 5 :

coefficients defined by [A 14] to [A 18]

D s b :

diffusivity of surfactants in continuous phase

D s i :

surface diffusivity of surfactants

E :

rate of energy dissipation per unit volume

F :

frictional force

g :

acceleration due to gravity

G :

shear

h :

distance from center of particle to point of zero velocity

i, j, k :

Cartesian unit vectors

J n :

surface flux of surfactants

K :

constant defined by [27]

L m n , L mn , M mn , M mn , N mn , N mn :

parameters defined by [A 11] to [A 13]

m,n :

integers

p :

pressure

p n :

solid spherical harmonic of ordern

r :

radius vector

R 1,R 2 :

radii of curvature

s :

surface area

S n θ, ϕ :

surface spherical harmonic of ordern

t r :

unit vector in radial direction

T 0 :

torque about center of particle

U :

velocity of uniform imposed field

U s :

Stokes velocity of particle

v :

velocity vector

V m n , V mn , X mn , X mn , Y mn , Ŷ mn , Z mn , Z mn :

parameters defined by [66] to [69]

W 1,W 2,Y 1,Y 2 :

parameters defined by [52] to [55]

x, y, z :

Cartesian coordinates,z aligned with axis of particle

α * :

overall adsorption rate constant

β n :

viscosity parameter defined by [A 19]

γ :

ratio of radii of particle and cell

γ n :

interfacial retardation viscosity defined by [A 20]

Γ :

surface concentration of surfactants

Γ 0 :

equilibrium surface concentration of surfactants

Γ′ :

deviation ofΓ fromΓ 0

Γ m n , Γ mn :

coefficients defined by [31]

δ :

thickness ofNernst diffusional layer

Δ n :

parameter defined by [A 10]

ε :

parameter defined by [32]

ζ n :

general coefficient defined by [23]

η :

dimensionless radius,η =r/a

θ :

cone angle

λ :

viscosity ratio defined by [A21]

μ :

viscosity

μ eff :

effective viscosity

ξ n (θ, ϕ) :

deviation from sphericity function

ξ m n , ξ mn :

deviation parameters, defined by [61]

ϱ :

density

σ :

surface tension

τ r :

surface shear force

τ rr ,τ rθ ,τ rϕ :

components of stress tensor

ϕ :

polar angle

ϕ n :

solid spherical harmonic of ordern

ϰ n :

solid spherical harmonic of ordern

ψ :

parameter defined by [71]

ω r ,ω θ ,ω ϕ :

components of vorticity

α :

pertains to phase in general

c:

pertains to continuous phase

d :

pertains to dispersed phase

o :

pertains to imposed field

a :

at interface

o :

equilibrium value

r :

radial value

θ :

tangential value

Φ :

azimuthal value

〈 〉 :

expected value

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Author notes

  1. B. Gal-Or

    Present address: Department of Aeronautical Engineering, Technion - Israel Institute of Technology, Haifa, Israel

  2. I. Yaron

    Present address: Division of Fluid, Energy and Aerospace Sciences, Case Western Reserve University, Cleveland, USA

Authors and Affiliations

  1. Department of Aeronautical Engineering, Technion - Israel Institute of Technology, Haifa, Israel

    I. Yaron

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  1. I. Yaron
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  2. B. Gal-Or
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Yaron, I., Gal-Or, B. On viscous flow and effective viscosity of concentrated suspensions and emulsions. Rheol Acta 11, 241–252 (1972). https://doi.org/10.1007/BF01974767

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