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The Rheology of Two-Phase Flows

  • Chapter
Rheological Measurement

Abstract

There is renewed interest in multiphase flows. This has arisen from the need for information by diverse industries, as well as from intellectual curiosity within the scientific community, now armed with new mathematical and physical tools. In the polymer industry, future developments in the area of polymer alloys, blends and composites (PABC) will depend, to a great degree, on the ability of suppliers and processors to control the morphology of the finished product. In the petroleum industry the behavior of drilling mud and high pressure flow of liquids through porous media is crucial for secondary and tertiary oil recovery. There is a growing trend in developing technology for slurry transporation of coal, ore and other large volume particulate materials. Here again the role of multiphase flow is important.

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Abbreviations

A i :

equation constants

A 0, A :

initial and final flow channel cross-section in con-vergent flow eqn (15.84)

a i :

semiaxis in eqns (15.22), (15.23) parameters

a c, a T :

concentration and temperature shift factor

B, B 0 :

extrudate swell and its value for Newtonian liquid, Respectively

C ij :

elastic constants in eqn (15.86)

D :

droplet deformability defined in eqn (15.101) characteristic diameter defined in eqn (15.118)

D c, D e :

capillary and extrudate diameter, respectively

D r :

rotational Brownian diffusion coefficient, eqn (15.18)

d, d i :

diameter, diameter of ith generation of particles in poly disperse suspensions eqns (15.15) and (15.16)

d v/s :

volume-to-surface average particle diameter, eqn (15.120)

d̄ :

average value of d

E :

elasticity of interphase in eqn (15.114)

E:

(subscript) uniaxial extension (superscript) excess Value

E sep :

Esep energy required to separate two particles eqn (15.36)

e :

shape factor in eqn (15.56)

F, Fy, Fm :

function, its yield value and that of a matrix, eqn (15.43)

Fij :

interparticle interaction force in eqn (15.65)

F~max :

dimensionless stress at yield per drop in eqn (15.116)

f :

free volume fraction

f i :

orientation function in eqns (15.86)—(15.95)

f(x) :

function of a parameter x

G, G′, G″:

shear modulus, and storage and loss dynamic shear moduli, respectively

G y′ , G y″ :

yield values for G′ and G″

Ğ:

dimensionless shear rate defined in eqn (15.111)

g :

acceleration due to gravity

gi :

orientation function in eqns (15.86)—(15.94)

H :

enthalpy

H̄G :

reduced gross relaxation spectrum in eqns (15.156) and (15.157)

J, J0, J e0 :

creep compliance, its value at t = 0 and steady state J eqn (15.48)

k i ≃ 1/ømi :

constants in eqn (15.14)

k i :

equation constants in eqn (15.42)

k H, kM :

Huggins, Martin constants

L :

length of a dispersed particle

L(τ):

retardation spectrum

M, M w, Mn :

molecular weight and its weight and number averages, respectively

M :

parameter in eqn (15.35)

M e :

entanglement molecular weight

m i :

parameters of eqns (15.33) and (15.46)

N 1, N 2 :

first and second normal stress difference

N +1 N +2 :

normal stress growth functions

n :

power law exponent eqn (15.74)

n i :

number of particles

ne :

entanglement degree of polymerization eqn (15.147)

P :

pressure

Pe :

Peclet number eqn (15.18)

P e :

entrance-exit pressure drop in capillary flow

p, p * :

aspect ratio and its generalized value

r ij :

center-to-center distance between particles eqn (15.65)

r, r c :

radial position of a particle and its critical value

R * :

gas constant

R :

hard spheres diameter ratio eqn (15.10)

R c :

particle diameter-to-capillary diameter ratio eqn (15.27)

R e :

Reynolds number eqn (15.31)

R 0 :

distance from center of a particle

R T, R 0T :

Trouton ratio and its limit at small deformation rates

rh:

parameter in eqn (15.158)

∹S〉:

elastic stress component of spherical suspension eqn (15.65)

S i :

initial slope of stress growth function in uniaxial Extension

s:

subscript indicating suspension

s i :

specific surface area of z’th particle

T :

temperature

T m, T g :

melting point glass transition temperature

T s :

separation temperature eqn (15.146)

t, t p :

time, period of rotation for anisometric particles eqn (15.17) and (15.30)

U :

particle velocity eqn (15.31)

Ui:

functions defined in eqns (15.75)—(15.79)

V, V i :

volume, volume of /th fraction of particles eqn (15.13)

w i :

weight fraction of specimen /

X ij :

energetic interaction parameter eqn (15.125)

x :

variable

x i :

mole fraction

Y :

packing function in eqn (15.2)

α, αi :

equation parameters eqns (15.4), (15.22), (15.23) etc.

αi :

polarizabilities eqn (15.95)

2α:

convergence angle eqn (15.152)

β:

Onsager parameter in eqn (15.147)

β, β12 :

slip factor in Lin’s equations (15.154), (15.155)

Γ0 :

critical parameter for droplet breakup

γ,γ ̇:

strain, rate of strain in shearing

γ ̇c, γ ̇y :

critical value of γ ̇ for onset of dilatancy or yield

γ R:

recoverable shear strain

γr :

strain recovery

∆Eη :

activation energy of flow

∆G m, ∆G el :

Gibbs free energy of mixing and an elastic contribution

∆Hm :

heat of mixing

∆V:

overlapping volume eqn (15.36)

δ:

segmental density eqn (15.147)

ε ε ̇:

Hencky strain in extension and rate of straining, respectively

ε*:

Lennard-Jones characteristic constant

εmax :

maximum filament shrinkage eqn (15.156)

εb :

maximum strain at break

η, η0 η∞:

shear viscosity and its upper and lower Newtonian plateau value

η’, η”, η*:

dynamic, loss and complex viscosity, respectively

η E η +e :

elongational viscosity, stress growth function in ex- tension at ε ̇ = const

ηr, ηsp :

relative, specific viscosity

η 0r ηE. r:

zero-shear relative viscosity, relative viscosity in elongation

[η], [η]s, [η]a:

intrinsic viscosity, intrinsic viscosity of suspensions and that for anisometric particle suspensions

[η]e,[η]E.d:

emulsion and deformable droplet emulsion intrinsic viscosity, respectively

ηp, ηm :

viscosity of dispersed liquid and matrix liquid of emulsion eqn (15.96)

ηi, ηsi, ηei :

interface viscosity and its shear and extensional components respectively

ηr,i :

relative viscosity of /th generation of particles in polydisperse suspension

η+, η +e :

time dependent viscosity (or the stress growth function) in shear and uniaxial extension

ηapp :

apparent viscosity

θ:

anisometry factor eqns (15.84), (15.85)

θh :

parametereqn (15.158)

K:

parameter defined in eqn (15.100)

Δ, λ:

viscosity ratios defined in eqns (15.98) and (15.99), respectively

v, v0 :

dynamic interfacial tension coefficient and its equilibrium value

Vij :

kinematic viscosity coefficient eqn (15.131)

Ξ:

function defined in eqn (15.63)

ρ:

density

σ, σ*:

shape constant in eqns (15.20) and (15.21), and Lennard-Jones constant, respectively

σij :

i, j component of the stress tensor

σcrit :

(t) critical value of shear stress to break bonds in a thixotropic system eqn (15.38)

σy, σy,c, σy,E:

yield shear stress, its value in compression and extension, respectively

σ +ij :

stress growth function in shear (ij = 12) or elongation (ij = 11)

σc :

critical shear stress for droplet breakup

σm :

critical shear stress for melt fracture

τ:

retardation time eqn (15.49)

τ *:

relaxation time

τ 2 :

free volume contribution eqn (15.125)

Φ:

Farris volume fraction as defined in eqn (15.12)

Ø, Øm :

volume fraction maximum packing ø

Øm.N :

(f)m for N generations of spheres eqns (15.8), (15.9)

Øm0 Øm∞ :

maximum volume packing fraction at shear stress (σ = 0 and σ → ∞, respectively

ØF :

effective volume of a flock eqn (15.37)

Ø| :

angle of orientation of a spheroid eqn (15.30)

Xij :

thermodynamic interaction coefficient between species i and j eqn (15.123)

Ψ(t) :

retardation function eqns (15.48), (15.49)

Ψ Ψ0:

first normal stress difference coefficient and its value at γ ̇ → 0.

Ψ, Ψ0 :

fiber orientation angle in convergent flow eqns (15.84), (15.85)

ω:

frequency

~:

indicates reduced variables

*:

(superscript) reducing variables

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  1. Industrial Materials Research Institute, National Research Council Canada, Boucherville, Quebec, Canada

    L. A. Utracki

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  1. Department of Applied Physics, Sheffield City Polytechnic, Sheffield, UK

    A. A. Collyer

  2. Department of Metallurgy and Materials Science, University of Nottingham, Nottingham, UK

    D. W. Clegg

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Utracki, L.A. (1993). The Rheology of Two-Phase Flows. In: Collyer, A.A., Clegg, D.W. (eds) Rheological Measurement. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2898-0_15

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