we have: in two-dimensional hyperbolic flow Tr¼2; in uniaxial axisym-
metric flow Tr¼3; in biaxial axisymmetric flow Tr¼6.
Any flow exerts a frictional orviscous stressZC[see Eq. (5.1)] onto the
wall of the vessel in which the liquid is flowing or onto particles in the liquid.
This has several consequences, the simplest one being that particles move
with the liquid; some others are illustrated in Figure 5.3, which applies to
simple shear flow. A solid sphere rotates, as mentioned. Solid anisometric
particles also rotate, but not at a completely constant rate. An elongated
particle will rotate slower when it is oriented in the direction of flow than
when in a perpendicular direction. This means that such particles showon
averagea certain preference for orientation in the direction of flow, although
they keep rotating.
Note This can give rise to so-called flow birefringence; see Section
9.1, under Optical anisotropy.A liquid sphere can become elongated, if some conditions are fulfilled
(see Section 11.3.2). As depicted, it obtains an orientation of about 45to
the direction of flow, and the liquidinsidethe particle rotates. Much the
same holds for a (random) polymer coil: it is also elongated, and its
rotational motion now implies that the coil is compressed and extended (as
indicated in the figure) periodically. It may be noted that this is a good
example of the difference between the time scale of an event and the time
that an experiment lasts. The time scale of compression/extension is simply
2 =C(i.e., 1 ms if the shear rate is 2000 s^1 ), however long the shearing lasts.
FIGURE5.3 Motion of particles in simple shear flow. The arrows indicate the
direction of flow relative to the particles. In the central plane the flow velocity is zero
or, in other words, the coordinate system moves with the geometric center of the
particle. See text.