Tensors for Physics

328 16 Constitutive Relations

viscosity coefficients are given by

ηiso=ηNew

(

1 −

τap^2

τaτp

)

,ηNew=

ρ

m

kBTτp. (16.85)

Similarly, the pressure tensor is also given by

pνμ=− 2 ηNew∇νvμ+ pνμ

Gies

, (16.86)

pνμ

Gies

=−

√

2

ρ

m

kBTτpa

(

daμν

dt

− 2 εμλκωλaκν

)

.

The superscriptGiesrefers to Giesekus [138].
In analogy to (16.82), the real and imaginary parts of the complex viscosity coef-
ficient are now given by

η′(ω)=(ηNew−ηiso)

1

1 +(ωτa)^2

+ηiso,η′′(ω)=(ηNew−ηiso)

ωτa

1 +(ωτa)^2

,

(16.87)

with

ηNew−ηiso=ηNew

τap^2

τaτp

. (16.88)

Hereη′(ω)approaches the viscosityηisofor high frequencies whereωτa 1 applies.
Depending on the type of fluids, the relative viscosity difference(ηNew−ηiso)/ηNew
ranges from 10−^2 to 10^2 , or higher.
The expressions (16.82) and (16.87) show, and this is true in general, a fluid
can reveal its visco-elastic behavior only, when the frequencyωis not too small
compared with the reciprocal relaxation time. Depending on the type of fluid and on
the temperature, values for the relaxation time vary over many orders of magnitude.
Similarly, a non-linear flow behavior can be observed when the shear rate is not
too small compared with the reciprocal relaxation time. For this reason, typical
viscoelastic fluids also show nonlinear viscous behavior.

16.3.9 Nonlinear Viscosity.

The study of the viscoelastic and nonlinear viscous properties of complex fluids is
calledRheology[139, 140]. The nonlinear effects of the shear rate on the mater-
ial properties which are due to shear-induced distortions of the local structure or
the shear-induced partial orientation of particles, are also referred to asrheological
properties.