336 16 Constitutive Relations
The friction pressurepνμA, in the affine space, obeys the constitutive law
−pAνμ= 2 ηAΓνμA+ηAV∇AλvλAδμν,which is standard for an isotropic fluid with the shear viscosityηAand the bulk
viscosityηAV, cf. Sect.16.3. Notice that∇λAvAλ =∇λvλ. The symmetric traceless
deformation rate tensorΓνμAin affine space is related to the velocity gradient in real
space by
ΓμνA =1
2
(
A−μλ^1 /^2 A^1 νκ/^2 +A−νλ^1 /^2 A^1 μκ/^2)
∇λvκ−1
3
∇λvλδμν.The friction pressure tensor, in real space, and for∇·v=0,
pνμ=−ηA(
A−νλ^1 Aμκ∇λvκ+∇νvμ)
, (16.112)
contains symmetric traceless and antisymmetric parts.
For ellipsoids of revolution, with their symmetry axis parallel to the unit vector
u, which is identical with the directornof the perfectly oriented fluid, the volume
conserving transformation is governed by
Aμν=Q^2 /^3[
δμν+(Q−^2 − 1 )nμnν]
, A−μν^1 =Q−^2 /^3[
δμν+(Q^2 − 1 )nμnν]
,
cf. (5.58). HereQ=a/bis the axes ratio of an ellipsoid with the semi-axesaand
b=c.
Comparison of the resulting friction pressure in real space with the ansatz for the
anisotropic viscosity made in the previous section leads to
η=[
1 +
1
6
(Q−Q−^1 )^2
]
ηA, η ̃ 1 =1
2
(Q−Q−^1 )^2 ηA,η ̃ 2 =1
2
(Q−^2 −Q^2 )ηA, η ̃ 3 =−1
2
(Q−Q−^1 )^2 ηA,and
γ 1 =(Q−Q−^1 )^2 ηA,γ 2 =(Q−^2 −Q^2 )ηA. (16.113)
The pertaining Miesowicz and the Helfrich viscosity coefficients are
η 1 =Q−^2 ηA,η 2 =Q^2 ηA,η 3 =ηA,η 12 =−(Q−Q−^1 )^2 ηA. (16.114)The Onsager-Parodi relationη 1 −η 2 =γ 2 is fulfilled. Furthermore, one hasηV=ηVA.
Thus all the viscosity coefficients of this perfectly ordered nematic liquid are related
to the shear and volume viscosities of the reference liquid with the same density and
to the axes ratioQof the ellipsoids. Clearly. for prolate particles withQ>1, the