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Mathematical Modelling of Two-Phase Flows

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Rheological Measurement

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Abstract

When the size of the suspended particles is much smaller than the length scale of the flow, or equivalently the characteristic dimensions of the device in which the suspension is flowing, it is possible to treat the suspension as a homogeneous fluid having specific rheological properties, which can be investigated with classical experimental means. Owing to the number of parameters that may influence the rheology (particle concentration, size, geometry, deformability, type of flow, colloidal forces, etc.), it is usually extremely difficult, if not outright impossible, unambiguously to relate the few experimental rheological parameters (shear viscosity, normal stresses, elastic modulus, etc.) to the numerous intrinsic physical properties of the particulate fluid. It is thus the aim of suspension theory to establish a relationship between the microscopic parameters of the suspension and the measurable bulk properties.

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Abbreviations

∹ 〉:

bulk value of enclosed quantity

*:

superscript for bulk particle properties

s:

superscript for interfacial particle properties

—:

superscript bar denoting symmetric deviatoric part, defined in eqn (16.45)

∂/ ∂ t :

partial derivative with respect to time

𝔇)\ 𝔇)t:

corotational derivative with respect to time, defined in eqn (16.43)

d :

characteristic length scale for incompressible particles

n i :

outer unit vector normal to particle surface, Fig. 16.1

N :

number density of particles

∅ :

volume fraction of particles

d :

mean particle spacing

η :

viscosity of suspending medium

ρ :

density of suspending medium

γ:

shear rate

ω :

bulk vorticity intensity

P(C, t):

joint probability density function of finding particles in configuration C

x mi :

position of particle m between times t and t + at

f i :

constant body force per unit volume

δ ij :

Kronecker delta

∆V :

representative volume of suspension

∆L:

length scale of A F

eij :

SS local rate of strain tensor

σij :

local stress tensor

v i :

local velocity

v i :

deviation from mean velocity

p :

pressure, eqn (16.5)

P ij :

particle stress

α:

denotes a generic particle of AV, eqn (16.6)

V α , S α :

volume and surface area of a generic particle of AV, eqn (16.6).

f i :

inertia effect for particle motion, eqn (16.6)

Rep :

microscopic Reynolds number, eqn (16.7)

S αij :

deviatone dipole strength due to one particle, eqns (16.8) and (16.9)

F jα :

non-hydrodynamic resultant force on a particle at xff, eqn (16.10)

kT :

Boltzmann temperature, eqn (16.11)

∹ S ij 〉:

bulk particle stress tensor, eqn (16.12)

Pc(C∣x 0i ):

normalized conditional probability density, eqn (16.13)

C jj :

measure of particle anisotropy and orientation

P’ :

conditional probability of finding one particle at x*¡ with orientation Q

v xi :

undisturbed bulk velocity field, eqn (16.16)

Ωij :

vorticity tensor, eqn (16.16)

λ :

viscosity ratio of dispersed fluid to bulk fluid, eqn (16.21)

E *:

elastic bulk modulus of particles

η s, E s :

viscosity and elastic modulus of particle interface

X i , Xi :

position of given point in reference state at time t

x :

distance from particle surface to centre of mass

f(x i , t) :

equation of deformed particle surface

F si :

force exerted by interface, eqns (16.25) and (16.26)

λ s :

viscosity ratio between interface and bulk fluid

k :

capillary number, eqns (16.27) and (16.28)

p’ :

deviation from main pressure, eqn (16.30)

T ij :

symmetric and traceless coefficient, eqn (16.30)

η r :

relative viscosity of suspension

A ijkl :

expression dependent on instantaneous orientation of an ellipsoid, eqn (16.37)

PeB :

Brownian motion Peclet number, eqns (16.38), (16.39) and

D :

rotation diffusion coefficient, eqn (16.38)

α:

particle axis ratio, eqn (16.40)

ui :

unit vector oriented along axis of a spheroid, eqn (16.40)

P 0 (u i , t) :

orientation probability density

A, B, C, F :

shape factors, eqn (16.42)

A’, B :

shape factors, eqn (16.44)

[η] :

intrinsic viscosity

N 1, N 2 :

normal stress differences

ε:

measures deviation from sphericity, eqns (16.47)-(16.49)

w i :

local electric potential, eqns (16.50)—(16.52)

μ :

local charge density, eqn (16.50)

z :

1/471 (dielectric constant of suspending medium)

n :

ion velocity, eqn (16.51)

e :

ion mobility, eqn (16.51)

z :

ion valence, eqns (16.51) and (16.53)

n :

ion number density, eqn (16.51)

e :

electronic charge, eqn (16.51)

k :

reciprocal Debye length, eqn (16.53)

n∞ :

equilibrium ion number density, eqn (16.53)

C:

particle potential

PeE :

electrical Peclet number

H :

electrical Hartmann number

β :

first-order electroviscous correction to Einstein equation, eqn (16.54)

t:

suspension characteristic response time

D f :

capsule deformation, eqn (16.63)

τ1, τ2 :

relaxation times

Ą ij :

measure of in-plane interface deformation

L ij , M ij :

two linear combinations of deformation tensors C ij9 Ąij, eqns (16.67) and (16.68)

C ′ij :

measure of non-linearity of an elastic membrane constitutive law

K :

constant, eqn (16.71)

P 2 :

pair density, eqn (16.72)

r i :

distance of two spheres, eqn (16.72)

v ri :

relative velocity of two spheres, eqn (16.72)

D ij :

Brownian diffusion tensor, eqn (16.72)

F ρi :

electrostatic force between charged spheres, eqn (16.77)

U 2 :

electrostatic pair potential, eqn (16.77)

L :

characteristic particle separation, eqn (16.80)

b :

measures ratio of electrical to thermal forces, eqn (16.81)

P’ :

defined in eqn (16.85)

β1 :

coefficient, eqn (16.85)

∅ m :

maximum concentration, eqn (16.87)

Ć :

constant, eqn (16.87)

d’ :

interparticle distance

L c :

cluster size

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Authors and Affiliations

  1. Division de Biomécanique et Instrumentation Médicale, Université de Technologie de Compiègne, Compiègne, France

    D. Barthes-Biesel

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  1. D. Barthes-Biesel
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Editors and Affiliations

  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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© 1993 Springer Science+Business Media Dordrecht

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Barthes-Biesel, D. (1993). Mathematical Modelling 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_16

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  • DOI: https://doi.org/10.1007/978-94-017-2898-0_16

  • Publisher Name: Springer, Dordrecht

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