A COMPARISON BETWEEN THE KINETIC THEORY OF
DILUTE GAS MIXTURES AND THE KINETIC THEORY OF
POLYMER MIXTURES
by
R. Byron Bird
Chemical and Biological Engineering Department
University of Wisconsin-Madison
Madison, Wisconsin 53706 U.S.A.
It may not be generally recognized that a phase-space kinetic theory
for polymer mixtures may be established in a fashion similar to the
kinetic theory of dilute gas mixtures. In what follows, we summarize
the theory for gases, and then we proceed to do the same for
polymers. Then we give some of the results for polymers that point
out future directions for research in this interesting and challenging
field.
Since this discussion will focus primarily on theory, it is
important to heed the statements made by two famous Dutch
scientists. First, we have the well-known quote from H. Kamerlingh
Onnes (possibly the most often misspelled name in science): Door
meten tot weten (through measurement to knowledge). And second,
there is the quote from I. M. Kolthoff: Die Theorie leitet, das Experiment
entscheidet (the theory guides, the experiment decides). The second
quote acknowledges the role of theory, but both men are stressing the
fact that experimental data are essential. We can only hope that the
approach to kinetic theory of polymers described below will
ultimately prove useful in organizing our thinking and in suggesting
useful experiments.
1. DILUTE GAS MIXTURES
Many attempts have been made to develop the nonequilibrium
statistical mechanics for gases with concentration, velocity, and
temperature gradients. The most satisfactory of these begin with an
equation for a distribution function in the phase-space for a gas
composed of N molecules. That is, we imagine a hyperspace of 6N
dimensions, with one axis each for the x, y, z coordinates of the N
molecules and one axis each for the x, y, z components of the
momenta. Then one point in this phase space describes the current
dynamical state of the system. The equations of motion then describe
all future states of the system.
1a. The Liouville equation
Next we imagine an ensemble of systems: a very large number
of identical containers of gas, each one represented by a point in the
phase space. These points move around and appear just like a
flowing fluid. Since no systems, and hence system points, are lost,
there will be an "equation of continuity" that describes the motion:
 

f

      i  r& i f   i  p& i f 
t
p

 i  r
(1.1)
where f is the distribution of points in the phase space and thus a
function of all r i ,p i . When Newton's laws of motion, F i  p& i and
p i  m r& i , for the ith molecule of species  with mass m , are
substituted into Eq. 1.1, we then get the Liouville equation for the gas
mixture:
 p i 
f

   
  i f  F i   i
t
p
 i  m r

f   Lf

(1.2)
where L is the Liouville operator. It was shown by Kirkwood [1] that
this equation could be converted into the Boltzmann equation for
species  :
f 
 
  

    r& f       g f    J 
 r
  &

t
r
(1.3)
where J  is a very complex term that contains information regarding
the dynamics of a binary encounter between two monatomic
molecules, and g is the force per unit mass acting on a molecules of
species  .
1b. The general equation of change
Equation 1.3 may be multiplied by a property B and integrated
over all momenta to get the general equation of change forB:
 B 

B  
B  
 

B f  d&
r      r& Bf  d&
r        r&     g     f  d&
r

 r

t
r  
&
r 
 t 
(1.4)
  BJ  d&
r
It can be shown that if B is conserved in a collision, then the last term
on the right side vanishes.
1c. Special equations of change
We may now write down the equations of change by
setting B successively equal to mass m , momentum  m r& , and
energy
 12 m r& 2
(keeping in mind that the only energy for
monatomic molecules is the kinetic energy—for diatomic molecules,
see BSL-2e, §0.3):
Species continuity equation:
Equation of motion:
Energy equation:

      v    j 
(1.5)
t 
----==

v     vv        g
t
----==
(1.6)

t
 v
1
2
2
   v
 Û    
1
2
2

 Û v
 

   q     v   v    g
==
--------------  j  g 

===
(1.7)
The single underlined terms are the convective fluxes of species mass,
momentum, and energy, respectively, and the doubly underlined
terms are the corresponding molecular fluxes. In the energy equation
(which can be thought of as a generalization of U  Q  W ) there is
a molecular heat flux q and a molecular work flux   v . The
internal energy for an ideal monatomic gas mixture Û  32 nkT in Eq.
1.7 is obtained from equilibrium statistical mechanics.
At the same time that one gets the above three conservation
equations, one also obtains the expressions for the molecular fluxes:


Mass flux vector:
j  m  r&  v f  d&
r
Momentum flux tensor:
   m r&  v r&  v f  d&
r
Heat flux vector:
q   12 m  r&  v

(1.8)



 r&  vf
2


(1.9)
d&
r
(1.10)
Notice that the molecular fluxes have the same general structure as
the convective fluxes. Each of these molecular-flux expressions
contains the distribution function, which must be obtained by solving
the Boltzmann equation.
1d. Solution of the Boltzmann equation
When a gas mixture is at rest, the distribution function is given
by the Maxwell-Boltzmann distribution function, obtainable from
equilibrium statistical mechanics. In the presence of gradients,
however, we have to multiply the equilibrium solution by a
correction factor to account for the influence of these gradients:
f

 m 
r& ,r,t  n 

 2 kT 





3 2

exp  m r&  v


2

2kT  1     L


(1.11)

in which  r& ,r,t is the first "correction term," which is taken to be
of the form:

 



 r& ,r,t   A  lnT  B :v  n C d

(1.12)
The vectors A and C and the second-order tensors B are all
functions of r& ,r,t , and are given as solutions of integrodifferential
equations [2]. The general diffusional driving forces, d  , include the
concentration (or activity) driving force, the pressure gradient
driving force, and the external driving force (BSL, p. 766):
cRTd  c RTlna     p   g      g 
(1.13)
(although for ideal gases, a somewat simpler expression can be used:
BSL, p. 860).
1e. The fluxes in terms of the transport properties
When the Boltzmann equation has been solved, then one can
express the fluxes in terms of the transport properties:
Species mass flux:
j r,t       D d   DT lnT
Momentum flux:
†
 r,t   p   v  v   


(1.14)
      v 
2
3
(1.15)
Heat flux:
q r,t    T  
H
j
M 
cRTx x DT
   

D
 j
j 
 

   
(1.16)
Here, the transport properties are:
 = viscosity
 = dilatational viscosity (zero for dilute monatomic gases)
 = thermal conductivity
DT = thermal diffusion coefficients
D = generalized Fick multicomponent diffusion coefficients
D = generalized Maxwell-Stefan multicomponent diffusion
coefficients
An equation relating the D and the D was given for the first time
by Curtiss and Bird [3], based on the method suggested by Merk [4],
who studied the connection as a graduate student while at the
Technical Hogeschool in Delft.
1f. The transport properties in terms of molecular parameters
All that remains is to get the transport properties in terms of the
constants appearing in the intermolecular force law. Such
calculations have been done for several force laws (see MTGL,
Chapter 8). For the simplest properties, namely, self-diffusivity,
viscosity, and thermal conductivity of pure monatomic gases, we
have the following formulas:
3  mkT 1
8  2 D 
Self diffusivity
D =
(1.17)
Viscosity

5  mkT
16  2  
(1.18)
Thermal conductivity

25  mkT
Ĉ
32  2 k V
(1.19)
Here m is the molecular mass,  is the mass density, and the omegas
are functions of kT  , and  and  are parameters in the intermolecular force expression. If the omegas are set equal to 1, then the
results above are those for a gas of rigid spheres of diameter  .
2. POLYMER MIXTURES
This discussion is a summary of a series of publications by
Curtiss and Bird during the period 1996 to 1999, in which we tried, as
much as possible, to parallel the above discussion for dilute gas
mixtures [5]. We visualize a polymer molecule as a collection of
mass-points ("beads") connected by some kind of interbead forces
("springs"). The springs may be chosen in several different ways:
c
Hookean spring:
(2.1)
F   HQ
This is a simple linear spring, which can be stretched
indefinitely. It is easy to handle analytically, but gives
generally poor results for describing rheological behavior.
H is a spring constant.
Warner or "FENE" spring:
F  
c
HQ

1  Q 2 Q02

(2.2)
This spring has a maximum length of Q0 . It can describe
many nonlinear rheological properties, but is difficult to
handle analytically. (FENE = finitely extensible nonlinear
elastic)
HQ
c
F  
"FENE-P" spring:
(2.3)
2
2
1  Q Q0
Here the ratio in the denominator is averaged at the local
conditions. "P" stands for Anton Peterlin who used a similar
approximation.
Fraenkel spring:
F    H Q  L 
c
(2.4)
When the spring constant is allowed to go to infinity, the
Fraenkel spring becomes a rigid rod of length L.
Whatever model is chosen, the beads are presumed to be acted on by
a Stokes law type of drag force, with a drag coefficient  .
2a. The Liouville equation for general bead-spring models
(i.e., models with any connectivity and complexity)
In setting up the Liouville equation, we have to take into
account the fact that the ith molecule of species  will have "beads"
which will be indicated by an additional index  , and that these
beads will be spread out around the center of mass of the molecule.
Therefore, the complete phase space will consist of all the bead
positions, r i , and the bead momenta, p i . We are then concerned
with a distribution function that is a function of all the bead positions
and momenta; this function will be described by an "equation of
continuity" in the complete phase space:
 

f

     i  r& i f   i  p& i f 
t
p
 i  r

(2.5)
When this equation is combined with Newton's laws of motion for
the beads, F i  p& i and p i  m r& i , we then get the Liouville equation
for the polymer mixture:
 p i 

f

   i      i f  F i   i  f   Lf
t
p
 m r

where L is the Liouville operator.
2b. The general equation of change
(2.6)
When the Liouville equation is multiplied by some property B
and then integrated over the entire phase space, we get the general
equation of change:

B  LB
t
B   Bfdx
where
(2.7; 2.8)
in which x is shorthand for all the phase space coordinates.
2c. Special cases for the equations of change
We now make special choices for B so that we will get the
familiar equations of change:
B
B at position r
i  m  r i  r 
(2.9) 

  i  p i r i  r 
(2.10) 
v
(2.11) 
1
2
(2.12) 
r  v 
 p i  p i

i
i

U
  i  
   r  r

 2m


  i  r i  p i   r i  r 

 v 2  Û
To get the mass density  at position r, we have to require the
masses of the various beads to be located at r, and this is
accomplished by means of the delta function in Eq. 2.9. Similar
comments may be made for the computation of the momentum
density, the energy density, and the angular momentum density.
By the first choice of B above (in Eq. 2.9) we get the equation of
continuity for species  :

      v    j 
t 
(2.13)
in which the mass flux is given by:
i
¨   dQ
j   m  ©
r

v
™
°Æ 
´ 
(2.14)

in which
indicates integration of the enclosed quantity over the
momentum space for a molecule of species  (the only contribution
in the case of a dilute monatomic gas), and  L  dQ indicates an
integration over all the internal coordinates of a molecule of species
 (needed because of the extension of the polymer molecule in
space). The Q  Q1,Q 2,Q3, L is the set of appropriately chosen


"connector vectors" (often coincidental with the springs).
When we make the choice of B in Eq. 2.10, we then get the
equation of motion:

v     vv        g
t
(2.15)
in which the momentum flux is given by                  :
k



e

k
¨   dQ


&
&
      m  ©
r

v
r

v
™
°Æ 

´ 
(2.16)
      R F   dQ
(2.17)
e
e
  
     R F   dQ
d
d 
% dR dQ dQ
    12          R
 F



1
2
d
(2.18)
(2.19)
The "kinetic contribution"   is due solely to the molecular motion;
this is the only contribution for the ideal monatomic gas. The
k
"external force contribution"    takes into account the different
forces acting on the various beads, because of electric charges. The
e
"intramolecular force contribution"    accounts for the tensions in
the springs that cross a plane in the fluid. Finally, the "intermolecular

force contribution"    accounts for the bead-bead interactions for
beads on two different (d) molecules. The R's appearing in these
expressions are:
R = position vector for bead  with respect to the center of
mass of molecule 

R = vector from bead  to bead 
d
R
 = vector from bead  to bead 
R = vector from the center of mass of molecule  to the
center of mass of molecule 
In addition  is the two-molecule configurational distribution
function. Now the contribution    does not seem to be taken into
account in the so-called "tube" models for undiluted polymers. No
explanation is ever given for the omission of this contribution, and, to
date, no one has attempted to make an estimate of its size.
When we make the choice for B given in Eq. 2.11, we then get
the energy equation:
d

t
 v
1
2
2
   v
 Û    
1
2

2

 Û v
 

   q     v   v    g   j  g 
(2.20)
When this is done, the expression for the heat flux plus the work flux
is obtained: q  q   q   q   q 
k
k
q  
e

d
2
© 
¨

™
r&  v r&  v °°  dQ
 
™
´
Æ
 ©
¨   dQ
&
 12      
r

v
™
°Æ 

´
d  ©r&  v ¨  


 12          
° % dR dQ dQ
´™
Æ
1
2
m
 



e
e

¨   dQ
&
q      R F   ©
r

v
°Æ 
´™
(2.21)
(2.22)

q  
1
2


¨
     R F   ©
™
´ r&  v °Æ  dQ
q  
1
2
d  

¨
% ©
         R
 F
 ™
´ r&  v °Æ

d
(2.23)

dR dQ dQ
(2.24)
The contribution q  is the kinetic transport of kinetic, intramolek
cular, an intermolecular energy. The contributions q  , q  , and q 
represent the work done against external, intramolecular, and intermolecular forces. The structure of these last three contributions is
closely related to the analogous contributions to the momentum flux
tensor.
e

d
2d. Solution to the Fokker-Planck equation
To get the one-molecule phase-space distribution function
needed for describing the behavior of polymer solutions, one can
derive and solve an equation of the Fokker-Planck type. This
equation is:
 1



f


    p   f     F   f 
t
r
p 
 m




 

b
h
     F   F  f 
 p

(2.25)
which contains F , the mechanical forces on bead  , the sum of the
intramolecular, intermolecular, and external forces:
F  F   F   F 

d
e
(2.26)
and the stochastic forces on bead  :
b
h
F   F  = the sum of the Brownian and drag forces
(2.27)
For any bead-spring model the solution of Eq. 2.25 is:
3


n
n
f r ,p ,t  f 0  1    g 1 e n h   O  2 


n1


 
(2.28)
n
Here   tcoll thydr is an expansion parameter, the h  are tensorial
n
Hermite polynomials of order n, and the g  1 are nth order tensors
that contain the gradients of velocity, temperature, and concentration. Furthermore e n indicates an nth order dot product.
The pair distribution function, which appears in the
expressions above, has not been obtained in an analogous fashion.
All we have at present is an approximate procedure for estimating
the pair distribution function. This is discussed in the Curtiss-Bird
paper in J. Chem. Phys. (1999) dealing with diffusion.
3. SOME RESULTS FOR POLYMERIC FLUIDS
Next we will discuss some of the phenomena that we have been
able to describe with the phase-space kinetic theory. From these
results it is also possible to focus on problems that need to be solved
as well as experiments that are needed.
3a. Momentum transport
The main driving force for studying kinetic theory of polymers
was the need to develop a rational basis for developing constitutive
equations for solving fluid dynamics problems for polymer solutions
and undiluted polymers (i.e., polymer melts). The first textbook
devoted to this topic was that of Bird, Curtiss, Armstrong &
Hassager. Earlier research monographs were those of Kirkwood [7]
and Yamakawa [8]. An introduction to the theories based on the
"tube" models may be found in the treatise of Doi and Edwards [9];
the tube models are not discussed in the present article. None of these
references discuss the problems of heat and mass transport or the
coupling of these with momentum transport.
The simplest model considered for the connection between
polymer structure and rheology was the elastic (Hookean) dumbbell.
For a dilute solution of such dumbbells, it was shown [10] many
years ago that a convected Maxwell equation for the polymer
contribution to the "extra stress tensor"  is the appropriate
constitutive equation; that is,
  p   s   p
where
 p   p1  nkT H &
(3.1)
Here  1 is a convected time derivative of  ,  s is the solvent
contribution to the stress tensor, & v  v  is the rate-of-strain
tensor, and H   4 H is a time constant (  being the hydrodynamic
drag coefficient for a bead moving through the solvent, and H is the
spring constant). This simple result is of limited value, because it
cannot describe the observed non-Newtonian viscosity or the normal
stress coefficient in steady-state shear flow. In linear viscoelasticity, it
cannot describe the observed spectrum of relaxation times.
For the FENE-P model, described at the beginning of §2, we
find [11]
†
Z p   p1  H   p  1   bnkT 
DlnZ
  1   bnkT H &
Dt
(3.2)
where
tr p 
3
Z  1   1   b
b
3nkT 
(3.3)
and b  HQ02 kT is a dimensionless quantity—about 50—that is a
measure of the extent to which the molecule can be stressed; the
parameter  is given by   2  b b  2 . This model gives a viscosity
that goes as & 2 3 and a first normal stress coefficient as & 4 3 , both of
which are in fair agreement with experiment. It also gives an
elongational viscosity that goes to 2nkT H b as the elongation rate
goes to   , and to

1
nkT H b
2
as the elongation rate tends toward
  . This seems to be qualitatively in agreement with the limited
experimental data available.
For bead-spring chains, the Rouse model, gives a result that is
just a superposition of Hookean dumbbells, with a spectrum of
relaxation times. Here again, however, non-Newtonian viscosity,
normal stresses, and elongational viscosity are not described by the
chain model.
For a dilute solution of elastic dumbbells in which there is a
concentration gradient, the following constitutive equation is
obtained [5a]:
 p   p1  Dtr H 2  p  nkTH &
(3.4)
This equation had been obtained earlier by El Kareh and Leal [12]. A
summary of the effects of diffusion on the constitutive equation has
been given by Beris and Mavrantzas [13].
In addition, the effect of temperature gradients on the stress
tensor has been considered [5a], and the behavior of a charged Rouse
chain in an electric field has also be considered [5a].
3b. Heat transport
First we discuss the relation between the thermal conductivity
and the type of spring used in modeling polymer molecules as
dumbbells. What we find is that the thermal conductivity is
extremely sensitive to the nature of the springs. For example, if we
compare Hookean dumbbells with Fraenkel dumbbells, we have:
Hooke:
41 nk 2T

12 
Fraenkel:
1 nk 2T

c
3 
(3.5; 3.6)
For the Fraenkel dumbbell c  HL2 2kT , where L is the length of the
rigid dumbbell when H   . For the solution of Fraenkel dumbbells,
the thermal conductivity may become arbitrarily large, when the
spring is "tightened up." It is found [5f] that the major contribution to
the thermal conductivity in dilute solutions is that of q  . We can
make a similar comparison for the zero-shear-rate rheological
properties:

Hooke:
  nkTH
Fraenkel:
  32 nkTH c
Hooke:
1  2nkTH2
Fraenkel:
1  158 nkT H2 c 2 (3.9; 3.10)

(3.7; 3.8)

Inasmuch as the product H c   4H  HL2 2kT is independent of
H, it is apparent that "tightening up" the springs in the Fraenkel
model will have no effect on the rheological properties of the
solution.
The thermal conductivity of the dilute solution of Rouse chains
has also been worked out (not a simple problem) and one finally gets
for a chain of N beads in a solvent at rest:
nk 2T
  3.6539N  5.0525N  2.3337
N

2

(3.11)
The analogous problem for a chain with Fraenkel springs has not
been worked out.
The energy equation can be written in terms of temperature,
and this is a simple exercise for Newtonian fluids (BSL, p. 337). For a
solution of Rouse chains, however, the problem is more difficult, and
one gets [5g]:
ĈV
DT
D
   q :v 
Dt
Dt
 tr 
1
2
p
(3.11)
The term involving the trace of the polymer contribution to the stress
tensor arises in the equation of change for temperature for polymeric
fluids, whereas there is no such term for Newtonian liquids.
Another difference between Newtonian fluids and polymeric
liquids is in the heat conduction equation for a stationary fluid.
According to the continuum mechanics for linear thermoviscoelasticity [14]
T0  m t  t  
t
T t  
t 
dt     2T
(3.12)
Here T0 is the temperature at t   , and m t  t  is a timedependent thermal property corresponding to the heat capacity per
unit volume. The general phase space kinetic theory for this situation
gives [5h] Eq. 3.12 with:
m t  t   
ĈV ,eq
T0
 t  t  
3nk t t  H
e
2 H

t  t 
1


H 

(3.13)
Thus there is a contribution to the heat capacity that has a "fading
memory." Insofar as we know, this effect has yet to be measured.
3c. Mass transport
For solutions it is known that the mass flux depends on the
concentration gradient. However, according to the phase-space
kinetic theory, there may also be a dependence on the velocity
gradient as well (which cannot be allowed in the thermodynamics of
irreversible processes [5a]).
j       
(3.14)
That is, we now have to deal with second-order diffusion tensor,  ,
that includes velocity gradients as well as the concentration gradient.
For a steady shear flow, vx  &y , and a dilute solution of Rouse
chains, the diffusion tensor is given by:
 1  8 N 4 & 4H 2
45
kT  1 2

  N & 4H 
N  3
 0
 13 N 2 & 4H 
1
0
0

0

1

(3.15)
As a result of the tensorial nature of  , the diffusion flux is not in the
same direction as the concentration gradient.
Another result that can be obtained from the phase-space
kinetic theory is the general expression for the diffusion fluxes in a
multicomponent mixture of polymers. For this situation, we have
found that there is a relation, similar to the Maxwell-Stefan relations:
(
 Z

 j
j 

        G         G
   
(3.16)
where G is the external force acting on species  , and G is the
(
external force acting on the fluid; the second order tensors Z suggest
that the fluxes are not necessarily aligned with the concentration
gradients. There will be additional thermal diffusion terms in Eq. 3.16
if higher terms in the fluxes are accounted for.
Still another type of diffusion can be described by the phasespace kinetic theory, namely the diffusion in the presence of
nonhomogeneous velocity gradients. For a dilute solution of Rouse
chains with N beads, the following generalization of Fick's second
law has been obtained [5a]:
D
kT

Dt
N




 2   : N  1   m n          

  m

eq


(3-17)
This equation allows for the description of the Uhlenhopp effect,
which states that in a coaxial rotating viscometer, the solvent
molecules will tend to diffuse toward the inner cylinder.
The kinetic theory of polymers suggests the existence of various
cross effects:
d
v
T
x
j

q
x
The shaded areas on the diagram represent the main flux-force
relations. Those marked with an "x" are the flux-force relations
permitted in the classical thermodynamics of irreversible processes.
The unmarked areas are coupling relations that are given by the
kinetic theory. Very few of these coupling relations have been
studied experimentally.
To calculate the properties of polymer liquid mixtures, one
needs to have an expression for the pair distribution function. Very
little is known about this quantity. It is needed in order to develop
the stress-tensor and heat-flux expressions for mixtures. These are
tough problems but to progress they must be attacked and solved.
Now polymers are really a mess.
It's been so for decades, I guess.
But let's not be fearful
Instead let's be cheerful:
'Tis better to "opt" than to "pess."
-o-o-o-o-oREFERENCES:
MTGL: Molecular Theory of Gases and Liquids, by J. O. Hirschfelder, C.
F. Curtiss and R. B. Bird, John Wiley and Sons, New York (1954,
1964).
BSL: Transport Phenomena, by R. B. Bird, W. E. Stewart, and E. N.
Lightfoot, John Wiley and Sons, New York, 2nd Revised Edition
(2007); corrigenda are posted on RBB's web page at U. W.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
J. G. Kirkwood, J. Chem. Phys., 15, 72 (1947); MTGL, §7.1c.
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H. J. Merk, Appl. Sci. Res., A8, 73-99 (1959).
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a.
C. F. Curtiss & R. B. Bird, Adv. Polymer Sci., 125, 1-101
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Diffusion:
b.
C. F. Curtiss & R. B. Bird, Proc. Nat. Acad. Sci. 93, 74407445 (1996).
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d.
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e.
R. B. Bird & C. F. Curtiss, Rheol. Acta., 35, 103-100 (1996).
f.
R. B. Bird, C. F. Curtiss & K. J. Beers, Rheol. Acta., 36,
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R. B. Bird & C. F. Curtiss, , J. Non-Newtonian Fluid Mech.,
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Fokker-Planck Equation:
i.
C. F. Curtiss & R. B. Bird, J. Chem. Phys., 106, 9899-9921
(1997).
R. B. Bird, C. F. Curtiss, R. C. Armstrong & O. Hassager,
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J. G. Kirkwood, Macromolecules, Gordon & Breach, New York
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[9]
[10]
[11]
[12]
[13]
[14]
M. Doi & S. F. Edwards, The Theory of Polymer Dynamics, Oxford
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H. Giesekus, Rheologica Acta, 1, 2-20 (1966).
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A. W. El Kareh and L. G. Leal, J. Non-Newtonian Fluid Mech., 33,
257-287 (1989).
A. N. Beris and V. G. Mavrantzas, J. Rheology, 38, 1235-1250
(1994).
R. M. Christensen, Theory of Viscoelasticity, 2nd Edition,
Academic Press, New York (1982), p. 114.