330 16 Constitutive Relations
the shear viscosity and the viscometric functions approach constant values at small
shear rates.
A generic model for a non-linear viscous behavior is a Maxwell model, cf. (16.81),
with a co-rotational time derivative and an additional deformational contribution, viz.
∂
∂tpμν− 2 εμλκωλpκν − 2 κ∇μvλ pλν +τ−M^1 pμν =− 2 G∇μvν.(16.94)The pertaining Newtonian viscosity isηNew=η=GτM. The coefficientκchar-
acterizes the influence of the symmetric traceless part of the velocity gradient on
the dynamics of the friction pressure tensor. The caseκ=0 is referred to as the
co-rotational Maxwell modelor asJaumann-Maxwell model. For a plane Couette
flow, this special model yieldsη 0 =0, andη+,η−are given by the expressions
(16.82) for the real and imaginary parts of the complex viscosity coefficientη(ω),
with the frequencyωreplaced by the shear rateγ. The resulting decrease of the shear
viscosity with increasing shear rate is termedshear thinning.
Due toη 0 =0, one hasΨ 2 =− 0. 5 Ψ 1 , in this case. For many polymeric liquids,
however, typicallyΨ 2 ≈− 0. 1 Ψ 1 is observed in the small shear rate limit. As will
be pointed out in Sect.16.4.6, this behavior follows fromκ≈ 0 .4. In general, the
parameterκis non-zero, as it can e.g. be inferred from microscopic approaches based
on kinetic equations [41, 141]. The phenomenological Maxwell model equation
(16.94) withκ=0 is also referred to asJohnson-Segalman model[142, 143], the
casesκ=±1 are calledco-deformationalorconvectedMaxwell model.
By analogy, the nonlinear viscous properties associated with the alignment are
essentially described by (16.87). Assuming that the dynamics of the alignment is
governed by the co-rotational time derivative,η+andη−are given by the expressions
forη′andη′′withωreplaced by the shear rateγ, andη 0 =0. Again, the nonlinear
viscosity shows a shear thinning behavior, however, it approaches a finite value, viz.
ηisowhich is also calledsecond Newtonian viscosity.
More general cases withκ=0 and where terms nonlinear in the alignment and
in the friction pressure tensor are included in the dynamic equations, are treated in
Sects.16.4.6and17.4.
16.3.10 Vorticity Free Flow.
A flow field for which the velocity gradient tensor has no antisymmetric part is
referred to asvorticity free. Two main types are distinguished: uniaxial and planar
biaxial flow fields.
(i) The uniaxial extensional or compressional flow is considered with the special
geometryvz= 2 εzandvx=−εx,vy=−εy.Hereε=^12 ∂vz/∂z=−∂vx/∂x=
−∂vy/∂yis the extension or compression rate. The symmetry of the flow field is that
of the uniaxial squeeze-stretch field as sketched in Fig.7.3. The velocity gradient