Here dv/dxis the extensional strain rate, andeis the extensional viscosity. The
correlation between the ideal extensional viscosity and the shear viscosity for a
Newtonian fluid can be described by theTrouton’s ratio(Trouton 1906 ), as given by
e
¼ 3 (7.5)
7.1.4 Non-Newtonian Fluids
In most practical applications, polymer fluids do not behave like ideal Newtonian
fluids. The occurrence of non-ideal viscoelastic behaviors of shear flow is often
associated with a dimensionless number, called the Weissenberg number We
(Weissenberg 1947 ; Dealy 2010 ),
Wetg^0 (7.6)which reflects the ratio between the relaxation time of the fluidtand the driven time
of the external fieldg^0 ^1. When the value ofWeis relatively small, the relaxation
time of the fluid is short, and the fluid can relax itself immediately, exhibiting the
characteristics of a Newtonian fluid. When the value ofWeis relatively large, the
fluid cannot relax itself in a short time, and the flow becomes unstable, deviating
from the ideal Newtonian fluid.
According to the relationship between the shear stresssand the shear rateg^0 ,
non-Newtonian fluids can be classified into the following conventional types, as
illustrated in Fig.7.4. Curve d represents thedilatant fluid, whose viscosity(i.e.
the slope of the curve) increases with the increase of shear rateg^0. Such a
Fig. 7.3 Illustration of the profile of flow velocity in the extensional flow
130 7 Polymer Flow