Page 274 - Bird R.B. Transport phenomena

258 Chapter 8 Polymeric Liquids

QUESTIONS FOR DISCUSSION
1. Compare the behavior of Newtonian liquids and polymeric liquids in the various experi-
ments discussed in §§8.1 and 8.2.
2. Why do we deal only with differences in normal stresses for incompressible liquids (see Eqs.
8.2-2 and 3)?
3. In Fig. 8.2-2 the postulated velocity profile is linear in y. What would you expect the velocity
distribution to look like if the gap between the plates were not small and the fluid had a very
low viscosity?
4. How is the parameter n in Eq. 8.3-3 related to the parameter n in Eq. 8.3-4? How is it related to
the slope of the non-Newtonian velocity curve from the dumbbell kinetic theory model in
§8.6?
5. What limitations have to be placed on use of the generalized Newtonian models and the lin-
ear viscoelastic models?
6. Compare and contrast Examples 8.4-1 and 2 regarding the geometry of the flow system and
the assumptions regarding the velocity profiles.
7. To what extent does the Oldroyd model in Eq. 8.5-3 include a generalized Newtonian model
and a linear viscoelastic model? Can the Oldroyd model describe effects that are not de-
scribed by these other models?
8. Why is it necessary to put restrictions on the parameters in the Oldroyd model? What is the
relation between these restrictions and the subject of rheometry?
9. What advantages do molecular expressions for the stress tensor have over the empirical ex-
pressions?
10. For what kinds of industrial problems would you use the various kinds of models described
in this chapter?
11. Why may the power law model be unsatisfactory for describing the axial flow in an annulus?

PROBLEMS 8A.1 Flow of a polyisoprene solution in a pipe. A 13.5% (by weight) solution of polyisoprene in
n
3
isopentane has the following power law parameters at 323 K: n = 0.2 and m = 5 X 10 Pa • s .
It is being pumped (in laminar flow) through a horizontal pipe that has a length of 10.2 m and
an internal diameter of 1.3 cm. It is desired to use another pipe with a length of 30.6 m with
the same mass flow rate and the same pressure drop. What should the pipe radius be?
8A.2 Pumping of a polyethylene oxide solution. A 1 % aqueous solution of polyethylene oxide at
333 К has power law parameters n = 0.6 and m = 0.50 Pa • s". The solution is being pumped
between two tanks, with the first tank at pressure p and the second at pressure p . The pipe
2
A
carrying the solution has a length of 14.7m and an internal diameter of 0.27 m.
It has been decided to replace the single pipe by a pair of pipes of the same length, but
with smaller diameter. What diameter should these pipes have so that the mass flow rate will
be the same as in the single pipe?
8B.1 Flow of a polymeric film. Work the problem in §2.2 for the power law fluid. Show that the
result simplifies properly to the Newtonian result.
8B.2 Power law flow in a narrow slit. In Example 8.3-2 show how to derive the velocity distribu-
tion for the region -B < x < 0. Is it possible to combine this result with that in Eq. 8.3-13 into
one equation?
8B.3 Non-Newtonian flow in an annulus. Rework Problem 2B.7 for the annular flow of a power
law fluid with the flow being driven by the axial motion of the inner cylinder.
(a) Show that the velocity distribution for the fluid is

wr j:,./-,! (8B3 1)
"