Tensors for Physics

346 16 Constitutive Relations

The non-Newtonian viscosity coefficientη+, cf., Sect.16.3.9, is found to be

η+=ηNewH+(Γ ), H+(Γ )= 1 +

τap^2

τaτp

[

1 +Γ^2 ( 1 +κ^2 / 3 )

( 1 +Γ^2 ( 1 −κ^2 / 3 )^2

− 1

]

. (16.140)

The dimensionless viscosity coefficientH+(Γ )is a function of the dimensionless
shear rateΓ, here defined by

Γ=τγ=A−^1 τa

∂vx

∂y

. (16.141)

Notice that

τap^2
τaτp =(ηNew−ηiso)/ηNew <1, cf. (16.85). For|κ|<1, (16.140)
describes a shear thinning behavior, for 1<|κ|<

√

3, shear thickening, i.e. an
increase of the shear viscosity with increasing shear rate results for smaller values of
Γ, followed by a shear thinning at higher shear rates. In any case,ηisois approached
forΓ→∞. Similarly, the viscosity coefficientsη−andη 0 are given by

η−=(η+(Γ )−ηiso)Γ, η 0 =−κ(ηNew−ηiso)

[

1 +Γ^2 ( 1 −κ^2 / 3 )

]− 1

Γ.

(16.142)

Bothη−andη 0 approach 0 forΓ1 andΓ 1. The normal pressure differences
and the pertaining viscometric functions, as defined in Sect.16.3.9, can be inferred
from (16.142). In particular, the ratio between the first and the second viscometric
function is found to be

−Ψ 2

Ψ 1

=

1

2

−κ

[

1 +Γ^2 ( 1 −κ^2 / 3 )

]− 1 (

1 +Γ^2 ( 1 +κ^2 / 3 )

)

. (16.143)

The special caseκ=0, corresponding to a pure co-rotational time derivative of the
alignment tensor, impliesη 0 =0 and the small shear rate limitΨ 2 /Ψ 1 =− 0 .5. The
valueκ≈ 0 .4, suggested by the Fokker-Planck approach, yieldsΨ 2 /Ψ 1 ≈− 0 .1,
which is typical for many polymeric liquids.
For a weak flow in the nematic phase, the alignment tensor maintains its uniaxial
formaμν=aμνeq=

√

3 / 2 aeqnμnν, where the directorn, in general, depends on the

time and the position. The equilibrium order parameteraeq=

√

5 S, whereS=S 2
is recalled as the Maier-Saupe order parameter, is assumed not to be affected by the
flow. In this caseMreduces to

Mλ=

3

2

aeq^2 ελκνnκNν, Nν=

dnν

dt

−ενλκωλnκ. (16.144)

The vectorNis the co-rotational time derivative of the directorn,cf.(16.108). For
dn/dt=0, as considered in (16.136), the pseudo-vectorMis proportional to the
component of the vorticity, which is perpendicular ton,viz.Mλ=−^32 a^2 eq(ωλ−
nλnκωκ). In the weak flow limit, (16.139) reduces to