324 16 Constitutive Relations
To study the effect of the vorticity on the flow birefringence, a plane Couette
geometry is considered, with the flow velocity inx-direction and its gradient in
y-direction, cf. (7.28), Fig.7.6, and Sect.12.4.6. In this case, the shear rate tensor
and the vorticity are given byγμν= ∇νvμ =γexνeyμ andωλ=−^12 γezλ, where
γ=∂vx/∂y is the shear rate andei,i=x,y,zare the unit vectors parallel to the
coordinate axes. Insertion of the symmetry adapted ansatz
aμν=a 1 (exνeyμ−exνeyμ)/√
2 +a 2√
2 exνeyμinto the inhomogeneous relaxation equation (16.74) leads to coupled equations for
the coefficientsa 1 anda 2. These equations are
a ̇ 1 −γa 2 +τa−^1 a 1 = 0 , (16.77)
a ̇ 2 +γa 1 +τa−^1 a 2 =−τapτa−^1 γ.For a stationary situation, one obtainsa 1 =γτaanda 2 =−γτap( 1 +γ^2 τa^2 )−^1 .In
the small shear rate limit whereγτa1 applies, this result fora 2 corresponds to
(16.75), for the geometry considered here. Due toa 1 =acos 2φ,a 2 =asin 2φ,
wherea^2 =a 12 +a 22 , the stationary solutions are equivalent to
a=|τap|γ
√
1 +γ^2 τa^2, tan 2φ=1
γτa. (16.78)
For large shear rates, the magnitudeaof the alignment saturates at the value|τap|τa−^1.
The angleφ, in the present context referred to as flow angle, indicates the directions
of the principal axes of the alignment tensor, within thexy-plane. More specifically,
one of these axes encloses the angleφwith thex-direction, the other one the angle
φ+ 90 ◦, the third principal direction is parallel to thez-axis. In the small shear rate
limit, one hasφ= 45 ◦. At higher shear rates, this principal axis approaches the
flow direction.
The results (16.77) and (16.78) pertain to (16.74) where terms nonlinear in the
shear rate enter only via the co-rotational time derivative. In general, other nonlin-
earities occur in the dynamic equation for the alignment tensor. These are, e.g. an
additional term proportional toγμκaκνand terms nonlinear in the alignment tensor, as
encountered in connection with the phase transition isotropic-nematic. The relevant
equations and their consequences are discussed in Sects.16.4.4,16.4.5, and17.3.
Forgases of rotating molecules,cf.(13.65), one hasεμν =εTaaμνT, whereaTμνis
the tensor polarization associated with the rotational angular momenta. In this case
not only a theoretical treatment analogous to that one above is possible, but a kinetic
theory approach based on theWaldmann-Snider equation, a generalized quantum
mechanical Boltzmann equation. The resulting inhomogeneous relaxation equations
(16.59) are similar to the ansatz (16.71), just with the role of forces and fluxes