Page 276 - Bird R.B. Transport phenomena

260 Chapter 8 Polymeric Liquids

The integration by parts allows the integration to be done more easily. Show that the final
result is

Verify that, when the yield stress goes to zero, this result simplifies to the Newtonian fluid
result in Problem 2B.3.
8B.6 Derivation of the Buckingham-Reiner equation. 2 Rework Example 8.3-1 for the Bingham
model. First find the velocity distribution. Then show that the mass rate of flow is given by

(8B.6-1)

in which r R = (^ 0 - ty^R/lL is the shear stress at the tube wall. This expression is valid only
when T > т .
R 0
8B.7 The complex-viscosity components for the Jeffreys fluid.
(a) Work Example 8.4-1 for the Jeffreys model of Eq. 8.4-4, and show that the results are Eqs.
8.5-12 and 13. How are these results related to Eqs. (G) and (H) of Table 8.5-1?
(b) Obtain the complex-viscosity components for the Jeffreys model by using the superposi-
tion suggested in fn. 3 of §8.4.
8B.8 Stress relaxation after cessation of shear flow. A viscoelastic fluid is in steady-state flow be-
tween a pair of parallel plates, with v x = yy. If the flow is suddenly stopped (i.e., у becomes
zero), the stresses do not go to zero as would be the case for a Newtonian fluid. Explore this stress
relaxation phenomenon using a 3-constant Oldroyd model (Eq. 8.5-3 with Л = /л = /л\ = /x = 0).
0
2
2
(a) Show that in steady-state flow
(8B.8-1)
To what extent does this expression agree with the experimental data in Fig. 8.2-4?
(b) By using Example 8.5-1 (part a) show that, if the flow is stopped at t = 0, the shear stress
for t > 0 will be
)
Ъ-' / Л 1 (8B 8 2
-"

This shows why X } is called the "relaxation time/' This relaxation of stresses after the fluid
motion has stopped is characteristic of viscoelastic materials.
(c) What is the normal stress r xx during steady shear flow and after cessation of the flow?
8B.9 Draining of a tank with an exit pipe (Fig. 7B.9). Rework Problem 7B.9(a) for the power law
fluid.
8B.10 The Giesekus model.
(a) Use the results in Table 8.5-1 to get the limiting values for the non-Newtonian viscosity
and the normal stress differences as the shear rate goes to zero.
(b) Find the limiting expressions for the non-Newtonian viscosity and the two normal-stress
coefficients in the limit as the shear rate becomes infinitely large.
(c) What is the steady-state elongational viscosity in the limit that the elongation rate tends to
zero? Show that the elongational viscosity has a finite limit as the elongation rate goes to infinity.

2
E. Buckingham, Proc. ASTM, 21,1154-1161 (1921); M. Reiner, Deformation and Flow, Lewis, London
(1949).