Page 271 - Bird R.B. Transport phenomena

§8.6 Molecular Theories for Polymeric Liquids 255

molecular theory has thus resulted in a model with four adjustable constants: 7} , A , n,
s
H
and b, which can be determined from rheometric experiments. Thus the molecular the-
ory suggests the form of the stress tensor expression, and the rheometric data are used to
determine the values of the parameters. The model described by Eqs. 8.6-2, 3, and 4 is
called the FENE-P model (finitely extensible nonlinear elastic model, in the Peterlin ap-
2
2
proximation) in which (Q/Q ) in Eq. 8.6-1 is replaced by (Q )/Qo-
0
This model is more difficult to work with than the Oldroyd 6-constant model, be-
cause it is nonlinear in the stresses. However, it gives better shapes for some of the mate-
rial functions. Also, since we are dealing here with a molecular model, we can get
information about the molecular stretching and orientation after a flow problem has
been solved. For example, it can be shown that the average molecular stretching is given
2
1
by (Q )/Qo = 1 — Z" where the angular brackets indicate a statistical average.
The following examples illustrate how one obtains the material functions for the
model and compares the results with experimental data. If the model is acceptable, then
it must be combined with the equations of continuity and motion to solve interesting
flow problems. This requires large-scale computing.

EXAMPLE 8.6-1 Obtain the material functions for the steady-state shear flow and the steady-state elongational
flow of a polymer described by the FENE-P model.
Material Functions for
the FENE-P Model SOLUTION
(a) For steady-state shear flow the model gives the following equations for the nonvanishing
components of the polymer contribution to the stress tensor:

S (8.6-5)
P Р/у
Zr pAJX = -пкТХ у (8.6-6)
н
Here the quantity Z is given by
Z = 1 + (3/«[l - (т ^/ЗикТ)] (8.6-7)
р
These equations can be combined to give a cubic equation for the dimensionless shear stress
contribution T = т /ЗпкТ
yx рух
T] + 3pT + 2q = 0 (8.6-8)
x yx
in which p = (b/54) + (1/18) and q = (fr/108)A y. This cubic equation may be solved to give 6
H
T yx = -2p V2 sinh(| arcsinh qp~ ) (8.6-9)
3/2
The non-Newtonian viscosity based on this function is shown in Fig. 8.6-3 along with some
experimental data for some polymethyl-methacrylate solutions. From Eq. 8.6-9 we find for
the limiting values of the viscosity

6
For у = 0: V-Vs = шТХ "{]^) ( 8 - " 1 0 )
(h Л \ 1 / 3
For у -> oo: - r) ~ HKTA (j ф-J (8.6-11)
v s H
Hence, at high shear rates one obtains a power law behavior (Eq. 8.3-3) with n = \. This can be
taken as a molecular justification for use of the power law model.

s
K. Rektorys, Survey of Applicable Mathematics, MIT Press, Cambridge, MA (1969), pp. 78-79.