Page 262 - Bird R.B. Transport phenomena

246 Chapter 8 Polymeric Liquids

3
This Jeffreys model contains three constants: the zero shear rate viscosity and two time
constants (the constant A is called the retardation time).
2
One could clearly add terms containing second, third, and higher derivatives of the
stress and rate-of-strain tensors with appropriate multiplicative constants, to get a still
more general linear relation among the stress and rate-of-strain tensors. This gives
greater flexibility in fitting experimental data.

The Generalized Maxwell Model

Another way of generalizing Maxwell's original idea is to "superpose" equations of the
form of Eq. 8.4-3 and write the generalized Maxwell model as

T(0 = E т (0 where ч к + кЛ* = ~ту (8.4-5,6)
к
к
fc=i d t
in which there are many relaxation times k k (with A > A ^ A . . . ) and many constants
1
3
2
7] with dimensions of viscosity. Much is known about the constants in this model from
k
polymer molecular theories and the extensive experiments that have been done on poly-
meric liquids. 4
The total number of parameters can be reduced to three by using the following em-
pirical expressions: 5
% = % ^ and A = £ (8.4-7,8)
t
in which r/ 0 is the zero shear rate viscosity, A is a time constant, and a is a dimensionless
constant (usually between 1.5 and 4).
Since Eq. 8.4-6 is a linear differential equation, it can be integrated analytically, with
the condition that the fluid is at rest at t = — oo. Then when the various т are summed
к
according to Eq. 8.4-5, we get the integral form of the generalized Maxwell model:
G(t-t')y(t')dt f (8.4-9)

In this form, the "fading memory" idea is clearly present: the stress at time t depends on
the velocity gradients at all past times V, but, because of the exponentials in the inte-
grand, greatest weight is given to times V that are near t; that is, the fluid "memory" is
better for recent times than for more remote times in the past. The quantity within braces
{ } is called the relaxation modulus of the fluid and is denoted by G(t - V). The integral ex-

3
This model was suggested by H. Jeffreys, The Earth, Cambridge University Press, 1st edition
(1924), and 2nd edition (1929), p. 265, to describe the propagation of waves in the earth's mantle. The
parameters in this model have been related to the structure of suspensions and emulsions by H. Frohlich
and R. Sack, Proc. Roy. Soc, A185,415-430 (1946) and by J. G. Oldroyd, Proc. Roy. Soc., A218,122-132
(1953), respectively. Another interpretation of Eq. 8.4-4 is to regard it as the sum of a Newtonian solvent
contribution (s) and a polymer contribution (p), the latter being described by a Maxwell model:

= " VsT, + A, т р = - t] p y (8.4-4a, b)
T S т р j t
so that т = T S + T p . Then if Eqs. 8.4-4a, 8.4-4b, and A : times the time derivative of Eq. 8.4-4a are added, we
get the Jeffreys model of Eq. 8.4-4, with rj 0 = r\ s + rj p and A 2 = {У] 5 /{Г) 5 + y] p ))k v
4
J. D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 3rd edition (1980). See also
N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior, Springer-Verlag, Berlin (1989);
and R. B. Bird, R. С Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics,
Wiley-Interscience, New York, 2nd edition (1987), Chapter 5.
5 T. W. Spriggs, Chem. Eng. Sci., 20, 931-940 (1965).