17.3 Alignment Tensor Subjected to a Shear Flow 365
17.3.3 Flow Properties
The type of orientational behavior strongly affects the rheological behavior of the
fluid, due to the coupling between alignment and flow. The expansion with respect
to the basis tensors and the component notation can also be used for the symmetric
traceless part of the pressure tensor or the stress tensor. From (16.151)to(16.152)
one deduces expressions for the (dimensionless) shear stressσxy, and the normal
stress differencesN 1 =σxx−σyyandN 2 =σyy−σzzin terms of the dimensionless
tensor componentsΣi≡ΣμνalTμνi. These relations are
σxy=ηisoΓ+Σ 2 , N 1 = 2 Σ 1 , N 2 =−√
3 Σ 0 −Σ 1. (17.37)
Hereηisostands for the scaled second Newtonian viscosity and one has
Σ 2 =
2
√
3
λ−K^1[
φ 2 − ̃κ(
a 2 φ 0 +a 0 φ 2 −√
3
2
(a 4 φ 3 +a 3 φ 4 ))]
,
Σ 1 =
2
√
3
λ−K^1[
φ 1 − ̃κ(
a 1 φ 0 +a 0 φ 1 −√
3
2
(a 3 φ 3 −a 4 φ 4 ))]
, (17.38)
Σ 0 =
2
√
3
λ−K^1[
φ 0 − ̃κ(
a 0 φ 0 −a 1 φ 1 −a 2 φ 2 +1
2
(a 3 φ 3 +a 4 φ 4 ))]
,
withκ ̃= 2 κ/( 3 λK).
Examples for the rheological properties like the shear stress, the non-newtonian
viscosity and the normal stress differences as functions of the shear rate for a few
selected values of the temperature and for the other model parametersλKandκare
e.g. found in [185–188].Rheochaos, a term coined by Cates [197], is found for those
parameter ranges, where the dynamics of the alignment tensor is chaotic.
Solutions of the coupled equations for the velocity and the alignment tensor, for
a boundary driven plane Couette flow, show pulsed jets in the velocity field, [189].
The coupled dynamics of the alignment and the electric polarization was studied
in [97]. An extension of the theory to active materials involving swimmers was
introduced in [198], see also [199].
17.4 Nonlinear Maxwell Model
The Maxwell model equation, cf. (16.81) and (16.94), governing the dynamics of the
friction pressure tensor contains a linear relaxation term. The model can be extended
to include damping terms nonlinear in the pressure tensor [200]. Here the notation
follows [201].