plates. Some kind of elongational flow is always involved when a liquid is
accelerated or decelerated.
Axisymmetric simple shear flow occurs in a straight cylindrical tube;
this so-calledPoiseuille flowis depicted in Figure 5.4a, further on. The figure
also illustrates another point. The liquid velocityvequals zero at the wall of
the tube and is at maximum in the center, whereas the velocity gradientCis
zero in the center and is at maximum at the wall. This is more or less the case
in many kinds of flow. The flow velocity at the wall of a vessel always equals
zero, at least for a Newtonian liquid (explained below).
Long lasting simple shear flow of constant shear rate is often
approximated by Couette flow. The liquid is between two concentric
cylinders, one of which is rotating. If the ratio between the radii of the inner
and outer cylinders is close to unity,Cis nearly constant throughout the
gap.
Figure 5.2 illustrates thestrainsresulting from shear and elongational
flows. The elongational strain can be expressed in various ways and the so-
calledengineering strain, i.e.,ðLL 0 Þ=L 0 , is often used. The disadvantage is
that it gives the strain with respect to the original length, not with respect to
the length at the moment of measurement. The latter is to be preferred, and
this so-calledHencky strainor natural strain is given in Figure 5.2. During
flow, the strain alters, and we need to know thestrain rate, for instance
dg=dtor de=dt. These strain rates are both equal to the velocity gradientC.
(Unless stated otherwise, we will use the symbolCboth for the velocity
gradient and for the strain rate.)
During flow of a fluid, the relation holds
s¼ZC¼Zdg
dtð 5 : 1 Þ*where the part after the second equals sign only applies to simple shear flow.
The factorZis called theviscosityor more precisely the dynamic shear
viscosity; the S.I. unit is N?m^2 ?s¼Pa?s. Viscosity is a measure of the
extent to which a fluid resists flow (what Newton called the ‘‘lack of
slipperiness’’). For a so-called Newtonian liquid,Zis independent of the
velocity gradient. Pure liquids and solutions of small molecules virtually
always show Newtonian behavior, i.e., the velocity gradient is proportional
to the stress. The value of the viscosity depends, however, on the type of
flow, and its value for elongational flowðZelÞis always higher than that for
simple shear flow (Zssor simplyZ). The relation is given by
Zel¼Tr?Z ð 5 : 2 Þwhere Tr stands for the dimensionless Trouton ratio. For Newtonian liquids