Page 267 - Bird R.B. Transport phenomena

§8.5 The Corotational Derivatives and the Nonlinear Viscoelastic Models 251

Table 8.5-1 Material Functions for the Giesekus Model
Steady shear flow:

(A)
1 + (1 - 2a)f
(B)
2r/A a(l - / ) (Ay) 2
0
(C)
VoX (A y) :
where
2
16aq-a)(Ay) ] 1/2 -l

Small-amplitude oscillatory shear flow:

^o 1 + (Aw) 2
Steady elongational flow:

V 1 2 2
, , 3 + Л (Vl - 4(1 - 2a)k's + 4(Air) - V l + 2(1 - 2d)K's + (Агг)) (Н)
6a As

are not particularly simple. Superpositions of Giesekus models can be made to describe
the shapes of the measured material functions almost quantitatively. 6 The model has
been used widely for fluid dynamics calculations.

EXAMPLE 8.5-1 Obtain the material functions for steady shear flow, small amplitude oscillatory motion, and
steady uniaxial elongational flow. Make use of the fact that in shear flows, the stress tensor
Material Functions for components T and r yz are zero, and that in elongational flow, the off-diagonal elements of
XZ
the Oldroyd 6-Constant fa stress tensor are zero (these results are obtained by symmetry arguments ).
7
e
SOLUTION
(a) First we simplify Eq. 8.5-3 for unsteady shear flow, with the velocity distribution v (y, t) =
x
y(t)y. By writing out the components of the equation we get
1 + A, J^JT XX - (Ai + M-i)vy = +r/ (A + fi )y 2 (8.5-5)
2
2
0
X, j^jr + (X, - к)т у = -т)о(Л - /* )У 2 (8.5-6)
yy ух 2 2
1 + A, f)r = 0 (8.5-7)
zz
j
+ A ^ y (8.5-8)
2

6 W. R. Burghardt, J.-M. Li, B. Khomami, and B. Yang, /. RheoL, U7,149-165 (1999).
7
See, for example, R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids,
Vol. 1, Fluid Dynamics, Wiley-Interscience, New York, 2nd edition (1987), §3.2.