Page 253 - Bird R.B. Transport phenomena

§8.2 Rheometry and Material Functions 237
functions are defined and measured. Information about the actual measurement equip-
ment and other material functions can be found elsewhere. 12 It is assumed throughout
this chapter that the polymeric liquids can be regarded as incompressible.

Steady Simple Shear Flow
We consider now the steady shear flow between a pair of parallel plates, where the ve-
locity profile is given by v = yy, the other velocity components being zero (see Fig. 8.2-
x
1). The quantity y, here taken to be positive, is called the "shear rate/' For a Newtonian
fluid the shear stress r is given by Eq. 1.1-2, and the normal stresses (т т , and T ) are
yx хх/ уу ZZ
all zero.
For incompressible non-Newtonian fluids, the normal stresses are nonzero and un-
equal. For these fluids it is conventional to define three material functions as follows:
dv
Tyx v d x
v
ii -r\ (8.2-2)
dv^ 2
T -T =-4 2 ЫЧ (8.2-3)
~dy
zz
yy
r
in which 7] is the non-Newtonian viscosity, 4 is the first normal stress coefficient, and
l
^ is the second normal stress coefficient. These three quantities—77, Ф ^ —are all
2 и 2
functions of the shear rate y. For many polymeric liquids 77 may decrease by a factor of
4
as much as 10 as the shear rate increases. Similarly, the normal stress coefficients may
7
decrease by a factor of as much as 10 over the usual range of shear rates. For polymeric
fluids made up of flexible macromolecules, the functions 77(7) and (y) have been found
^
r
experimentally to be positive, whereas 4 (y) is almost always negative. It can be shown
2
that for positive ^i(y) the fluid behaves as though it were under tension in the flow (or
x) direction, and that the negative 4? (y) means that the fluid is under tension in the
2
transverse (or z) direction. For the Newtonian fluid 77 = /x, ^ = 0, and ^ 2 = 0-
The strongly shear-rate-dependent non-Newtonian viscosity is connected with the
behavior given in Eqs. 8.1-1 to 3, as is shown in the next section. The positive 4^ is pri-
l
marily responsible for the Weissenberg rod-climbing effect. Because of the tangential
flow, there is a tension in the tangential direction, and this tension pulls the fluid toward
the rotating rod, overcoming the centrifugal force. The secondary flows in the disk-and-
cylinder experiment (Fig. 8.1-4) can also be explained qualitatively in terms of the posi-
tive 4f Also, the negative ^2 c a n be shown to explain the convex surface shape in the
v
tilted-trough experiment (Fig. 8.1-5).
Upper plate moves at a constant speed
J' • Fig. 8.2-1. Steady simple shear flow be-
^/ tween parallel plates, with shear rate y.
^ X Ф For Newtonian fluids in this flow, r =
xx
=
T
v -yy T yy 2z = 0, but for polymeric fluids the
x
normal stresses are in general nonzero
— — — — and unequal.
1 J. R. Van Wazer, J. W. Lyons, K. Y. Kim, and R. E. Colwell, Viscosity and Flow Measurement,
Interscience (Wiley), New York (1963).
2 K. Walters, Rheometry, Wiley, New York (1975).