Page 265 - Bird R.B. Transport phenomena

§8.5 The Corotational Derivatives and the Nonlinear Viscoelastic Models 249

Note that there is an important difference between the problems in the last two ex-
amples. In Example 8.4-1 the velocity profile is prescribed, and we have derived an ex-
pression for the shear stress required to maintain the motion; the equation of motion was
not used. In Example 8.4-2 no assumption was made about the velocity distribution, and
we derived the velocity distribution by using the equation of motion.

.5 THE COROTATIONAL DERIVATIVES AND THE
NONLINEAR VISCOELASTIC MODELS
In the previous section it was shown that the inclusion of time derivatives (or time inte-
grals) in the stress tensor expression allows for the description of elastic effects. The lin-
ear viscoelastic models can describe the complex viscosity and the transmission of
small-amplitude shearing waves. It can also be shown that the linear models can de-
scribe elastic recoil, although the results are restricted to flows with negligible displace-
ment gradients (and hence of little practical interest).
In this section we introduce the hypothesis ' 1 2 that the relation between the stress ten-
sor and the kinematic tensors at a fluid particle should be independent of the instanta-
neous orientation of that particle in space. This seems like a reasonable hypothesis; if
you measure the stress-strain relation in a rubber band, it should not matter whether
you are stretching the rubber band in the north-south direction or the east-west direc-
tion, or even rotating as you take data (provided, of course, that you do not rotate so
rapidly that centrifugal forces interfere with the measurements).
One way to implement the above hypothesis is to introduce at each fluid particle a
corotating coordinate frame. This orthogonal frame rotates with the local instantaneous
angular velocity as it moves along with the fluid particle through space (see Fig. 8.5-1).
In the corotating coordinate system we can now write down some kind of relation

8 =Л
У
у
у
У у . £
У
У
У \ "'
\
Fluid particle
7 Fluid particle at time t
at timef Fluid particle
trajectory
О х
Fig. 8.5-1. Fixed coordinate framewith origin at O, and the coro-
^
tating frame with unit vectors 8 8 / з t n a t move with a fluid par-
2
lr
ticle and rotate with the local, instantaneous angular velocity
£[V X v] of the fluid.

1
G. Jaumann, Grundlagen der Bewegungslehre, Leipzig (1905); Sitzungsberichte Akad. Wiss. Wien, Ha, 120,
385-530 (1911); S. Zaremba, Bull. Int. Acad. Set, Cracooie, 594-614,614-621 (1903). Gustaf Andreas Johannes
Jaumann (1863-1924) (pronounced "Yow-mahn") who taught at the German university in Briinn (now
Brno), for whom the "Jaumann derivative" is named, was an important contributor to the field of
continuum mechanics at the beginning of the twentieth century; he was the first to give the equation of
change for entropy, including the "entropy flux" and the "rate of entropy production" (see §24.1).
2
J. G. Oldroyd, Proc. Roy. Soc, A245, 278-297 (1958). For an extension of the corotational idea, see
L. E. Wedgewood, Rheol. Ada, 38, 91-99 (1999).