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232 Chapter 8 Polymeric Liquids

only in flows with exceedingly small displacement gradients. Next in §8.5 we give several
nonlinear viscoelastic models, and these are intended to be applicable in all flow situations.
As we go from elementary to more complicated models, we enlarge the set of observed
phenomena that we can describe (but also the mathematical difficulties). Finally in §8.6
there is a brief discussion about the kinetic theory approach to polymer fluid dynamics.
Polymeric liquids are encountered in the fabrication of plastic objects, and as addi-
tives to lubricants, foodstuffs, and inks. They represent a vast and important class of liq-
uids, and many scientists and engineers must deal with them. Polymer fluid d3mamics,
heat transfer, and diffusion form a rapidly growing part of the subject of transport phe-
nomena, and there are many textbooks, 1 treatises, 2 and journals devoted to the subject.
The subject has also been approached from the kinetic theory standpoint, and molecular
theories of the subject have contributed much to our understanding of the mechanical,
thermal, and diffusional behavior of these fluids. 3 Finally, for those interested in the his-
tory of the subject, the reader is referred to the book by Tanner and Walters. 4

§8.1 EXAMPLES OF THE BEHAVIOR OF POLYMERIC LIQUIDS
In this section we discuss several experiments that contrast the flow behavior of New-
tonian and polymeric fluids. 1

Steady-State Laminar Flow in Circular Tubes

Even for the steady-state, axial, laminar flow in circular tubes, there is an important dif-
ference between the behavior of Newtonian liquids and that of polymeric liquids. For
Newtonian liquids the velocity distribution, average velocity, and pressure drop are
given by Eqs. 2.3-18, 2.3-20, and 2.3-21, respectively.
For polymeric liquids, experimental data suggest that the following equations are
reasonable:
v f \a/n)+\ / \ (i/ ) + i
7 r v w
77 * 1 ~ h? a n d 7) ~ n i \ • о (8.1-1, 2)
V У
\RJ (1/п) + 3
where n is a positive parameter characterizing the fluid, usually with a value less than
unity. That is, the velocity profile is more blunt than it is for the Newtonian fluid, for
which n = 1. It is further found experimentally that
<3> - <3> ~ w n (8.1-3)
0 L
The pressure drop thus increases much less rapidly with the mass flow rate than for
Newtonian fluids, for which the relation is linear.

1
A. S. Lodge, Elastic Liquids, Academic Press, New York (1964); R. B. Bird, R. C. Armstrong, and
O. Hassager, Dynamics of Polymeric Liquids, Vol. 1., Fluid Mechanics, Wiley-Interscience, New York, 2nd
edition (1987); R. I. Tanner, Engineering Rheology, Clarendon Press, Oxford (1985).
2
H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology, Elsevier, Amsterdam (1989);
H. Giesekus, Phanomenologische Rheologie: Eine Einfuhrung, Springer Verlag, Berlin (1994). Books
emphasizing the engineering aspects of the subject include Z. Tadmor and C. G. Gogos, Principles of
Polymer Processing, Wiley, New York (1979), D. G. Baird and D. I. Collias, Polymer Processing: Principles
and Design, Butterworth-Heinemann, Boston (1995), J. Dealy and K. Wissbrun, Melt Rheology and its Role
in Plastics Processing, Van Nostrand Reinhold, New York (1990).
3
R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2,
Kinetic Theory, Wiley-Interscience, New York, 2nd edition (1987); C. F. Curtiss and R. B. Bird, Adv.
I
Polymer Sci, 125,1-101 (1996) and /. Chem. Phys. l l , 10362-10370 (1999).
R. I. Tanner and K. Walters, Rheology: An Historical Perspective, Elsevier, Amsterdam (1998).
4
1
More details about these and other experiments can be found in R. B. Bird, R. C. Armstrong, and
O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Fluid Dynamics, Wiley-Interscience, New York, 2nd edition
(1987), Chapter 2. See also A. S. Lodge, Elastic Liquids, Academic Press, New York (1964), Chapter 10.