322 16 Constitutive Relations
direction such that〈Gxx〉is larger than〈Gyy〉, then the average rotational velocity is
smaller than the vorticity. Non-Equilibrium Molecular Dynamics (NEMD) computer
simulations presented in [119] show that the relation (16.66) is obeyed rather well.
Simple models for the test of (16.66) are studied in [120, 121].
16.3.6 Flow Birefringence
A fluid composed of non-spherical, i.e. optically anisotropic particles, has optical
isotropic properties in its isotropic phase. A viscous flow, however, causes a partial
orientation of the particles. As a consequence, the symmetric traceless part of the
dielectric tensor becomes non-zero. The resulting double refraction or birefringence
is calledflow birefringenceorstreaming double refraction. This effect was looked
for, first observed and described by Maxwell [122], it is also referred to asMaxwell
effect. The phenomenological ansatz is
εμν = 2 Mεisoη∇μvν=− 2 β∇μvν. (16.67)HereMis theMaxwell coefficient,εisoandηare the orientationally averaged dielec-
tric coefficient and the viscosity. The flow birefringence coefficientβis related to the
Maxwell coefficient byβ=−Mεisoη. For a plane Couette flow, with the velocity in
x-direction and its gradient iny-direction, two of the principal axes of the dielectric
tensor are parallel to the unit vectorse(^1 ,^2 )=(ex±ey)/
√
2, the third axis is parallel
toez.
The relation (16.67) holds true in the absence of additional external fields and for
small shear rates. In general, flow birefringence is described by a fourth rank tensor,
analogous to the shear viscosity tensor (16.43). The constitutive relation for flow
birefringence is
εμν =− 2 βμνμ′ν′∇μ′vν′. (16.68)The simple case (16.67) corresponds toβμνμ′ν′=βΔμν,μ′ν′.
The symmetric traceless part of the dielectric tensor of molecular liquids and
colloidal dispersions is proportional to the alignment tensor, viz.εμν =εaaμν,for
εasee (12.20). Here flow birefringence is due to the shear flow induced alignment
which results from a coupling between the alignment tensor and the friction pressure
tensor. Point of departure for a treatment within the framework of irreversible ther-
modynamics is an expression for the contribution of the alignment to the entropy.
In lowest order, this contribution is proportional to−aμνaμν, see Sect.12.2.6.The
time change of this expression is proportional to−aμνdaμν/dt. For the time change
of the alignment tensor the educated guess
daμν
dt− 2 εμλκωλaκν=(
δaμν
δt)
irrev