Page 264 - Bird R.B. Transport phenomena

248 Chapter 8 Polymeric Liquids

EXAMPLE 8.4-2 Extend Example 4.1-3 to viscoelastic fluids, using the Maxwell model, and obtain the attenua-
tion and phase shift in the "periodic steady state/'
Unsteady Viscoelastic
Flow Near an SOLUTION
Oscillating Plate
For the postulated shearing flow, the equation of motion, written in terms of the stress tensor
component gives

The Maxwell model in integral form is like Eq. 8.4-9, but with a single exponential:

УА
П
?еМ-(*-П/^У !' *Г (8.4-17)
Combining these two equations, we get

As in Example 4.1-3 we postulate a solution of the form
1ш1
v (y, t) = Щу\у)е } (8.4-19)
x
where v°(y) is complex. Substituting this into Eq. 8.4-19, we get

dy z Jo Ai
-"
V l l (8 4 20)
dy 2
Removing the real operator then gives an equation for v°(y)

= о (8.4-21)
dy 2
Then if the complex quantity in the brackets [ ] is set equal to (a + /j3), the solution to the dif-
2
ferential equation is
+
JP = e-(° Wy (8.4-22)
Vo
Multiplying this by e and taking the real part gives
la)t
axj
v (y, t) = v$~ cosM - J3y) (8.4-23)
x
This result has the same form as that in Eq. 4.1-57, but the quantities a and /3 depend on fre-
quency:
(8.4-24)

= №- [Vl + (A ) - \M~ m (8.4-25)
2
lW
2i
\ 7o
That is, with increasing frequency, a decreases and /3 increases, because of the fluid elasticity.
This result shows how elasticity affects the transmission of shear waves near an oscillating
surface.