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Kinetic theory and rheology of dumbbell suspensions with Brownian motion

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Fortschritte der Hochpolymeren-Forschung

Part of the book series: Advances in Polymer Science ((POLYMER,volume 8))

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Abbreviations

b :

dimensionless shear rate [Eq. (6.11)]

C :

cos θ

c :

cos φ

C m :

cos mφ

F :

probability density in position-velocity space

F i :

force on i th bead through the connector

F (c) :

tension in the connector of the dumbbell

H :

Hooke law constant for a spring

h :

function defined in Eq. (25.4)

i :

\(\sqrt { - 1}\)

j :

iλω

J :

normalization constant defined in Eq. (3.14)

J e :

equilibrium shear compliance [Eq. (13.3)]

k :

Boltzmann constant

L :

distance between bead centers in a rigid dumbbell

M :

molecular weight

m :

mass of a bead of the dumbbell

Ñ :

Avogadro's number

n :

normal unit vector

n 0 :

number density of dumbbell

P m n :

spherical harmonics [Eq. (5.5)]

p :

thermodynamic pressure

Q :

any function of R

r :

position vector for center of mass of dumbbell

r i :

position vector for i th bead (with coordinates x x , y i , z i )

R :

orientation vector with components X, Y, Z (= r 2 − r 1)

R :

separation between beads of dumbbell (= |R|)

R :

gas constant in Eq. (6.11)

S :

area of surface in Eqs. (4.11) and (4.12)

S :

sin θ

s :

sin φ

s m :

sin mφ

T :

absolute temperature

t :

time

v :

local fluid velocity

v′ :

perturbation velocity (§ 25)

x j :

mole fraction of j th species [Eq. (26.1)]

x i , y i , z i :

components of r i

X, Y, Z :

components of R

β :

secondary normal stress function [Eq. (6.3)]

β 0 :

secondary normal stress function at zero shear rate

Γ :

gamma function

\(\dot \gamma \) :

rate of deformation tensor [after Eq. (3.1)]

γ ∞ :

ultimate shear recovery [Eq. (12.1)]

γ :

infinitesimal strain

δ :

unit tensor

δ :

Dirac delta function

δ R :

unit vector in radial direction

δ i :

unit vector in i th Cartesian direction

ξ :

friction coefficient of a bead

η :

viscosity function in Eq. (6.1) (non-Newtonian viscosity)

η s :

solvent viscosity

η 0 :

zero shear rate viscosity

[η]:

intrinsic viscosity [Eq. (6.9)]

[η]0 :

zero-shear-rate intrinsic viscosity

η*:

complex viscosity (η′ − iη″)

\(\bar \eta\) :

elongational viscosity

θ :

angle down from z-axis in spherical coordinates

θ :

primary normal stress function [Eq. (6.2)]

θ 0 :

primary normal stress function at zero shear rate

κ :

tensor specifying the homogeneous velocity field [Eq. (3.1)]

κ :

shear rate in shear flow

\(\bar \kappa\) :

elongational rate in elongational flow

Λ :

operator in Eq. (5.1)

λ 1 :

time constant for Hookean dumbbell [Eq. (4.24)]

λ :

time constant for rigid dumbbell [Eq. (5.1)]

λ h :

time constant for rigid dumbbell with hydrodynamic interaction [before Eq. (25.15)]

λ j :

time constant defined after Eq. (26.2)

Ξ :

probability density in velocity space [Eq. (4.2)]

π :

pressure tensor (total stress tensor)

ϱ :

density of fluid

τ :

stress tensor (or extra stress tensor)

τ s :

solvent contribution to stress tensor [Eq. (4.1)]

τ p :

polymer contribution to stress tensor [Eq. (4.1)] ( = τ (b) p + τ (c) p )

Ψ :

geometrical ratio in § 21

φ :

angle about the z-axis in spherical coordinates

ψ :

probability density for two beads

ψ :

probability density for internal configuration of a dumbbell

ψ′ :

probability density for orientation of a rigid dumbbell [see Eq. (3.13 a)]

ψ j :

expansion functions in Eq. (6.4)

Ω :

Oseen tensor in § 25

Ω :

angular velocity in § 21

Ω :

operator in Eq. (5.1)

Ω e :

operator in Eq. (15.1)

ω :

frequency

ω :

vorticity tensor [Eq. (20.3)]

∇ :

nabla operator

\(\frac{D}{{Dt}}\) :

substantial derivative

\(\frac{\mathfrak{d}}{{\mathfrak{d}t}}\) :

convected derivative [Eq. (3.17)]

d R :

dXdYdZ

\(\frac{\partial }{{\partial R}}\) :

vector operator with components \(\frac{\partial }{{\partial X}},\frac{\partial }{{\partial Y}},\frac{\partial }{{\partial Z}}\)

ℛ e :

real part of

〈 〉:

expectation value defined in Eq. (3.13)

{A, B, C}:

special symbol defined in Eq. (12.13)

†:

transpose of a tensor

*:

complex viscometric functions (see §§7, 8, 9, 22)

·:

time differentiation

—:

expectation value in Eq. (4.25)

—:

complex conjugate in Eq. (7.9)

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  1. D. C. Evans

    Present address: Department of Applied Mathematics, University of Liverpool, Liverpool, England

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  1. Chemical Engineering Department and Rheology Research Center, University of Wisconsin, 53706, Madison, Wisconsin, USA

    R. B. Bird, H. R. Warner Jr. & D. C. Evans

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Bird, R.B., Warner, H.R., Evans, D.C. (1971). Kinetic theory and rheology of dumbbell suspensions with Brownian motion. In: Fortschritte der Hochpolymeren-Forschung. Advances in Polymer Science, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-05483-9_9

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