316 16 Constitutive Relations
Its interrelation with the viscosity coefficients occurring in (16.47) can be inferred
with the help of (14.62). The result is
ηCouette(h)= 3 h^2 xh^2 yη(^0 )+[
h^2 x+h^2 y− 4 h^2 xh^2 y]
η(^1 +)+[
1 +h^2 xh^2 y−h^2 x−h^2 y]
η(^2 +).
(16.51)
The Couette viscosities forhparallel to the flow velocity, to its gradient and its
vorticity, viz. the casesh=ex,ey,ezare denoted byη 1 ,η 2 , andη 3 , respectively.
From (16.51) follows
η 1 =η(^1 +),η 2 =η(^1 +),η 3 =η(^2 +). (16.52)The effective viscosity for the field parallel to the bisector between thex- andy-
direction, viz. forh^2 x=h^2 y= 1 /2 is denoted byη 45 , referring to the 45◦direction.
Here (16.51) implies
η 45 =(
3 η(^0 )+η(^2 +))
/ 4.
In the liquid crystal literature, the coefficientsη 1 ,η 2 ,η 3 are calledMiesowicz vis-
cositiesand 4 times the difference between the viscosityη 45 and one half of the sum
ofη 1 andη 2 ,viz.
η 12 ≡ 4 η 45 − 2 (η 1 +η 2 )= 3 η(^0 )+η(^2 +)− 4 η(^1 +). (16.53)is calledHelfrich viscosity. The four effective viscosity coefficients linked with the
Couette flow geometry,η 1 ,η 2 ,η 3 andη 12 suffice to characterize the anisotropy of
the shear viscosity.
The anisotropic viscosity tensor also gives rise tonormal pressure differences,
e.g.pxx−pyy, as well as to transverse components likepyz. The meaning of these
terms is elucidated for aplane Poiseuille flow.
Consider a plane Poiseuille flow inx-direction between two fixed flat plates which
are perpendicular to they-direction. The geometry is akin to that of the plane Couette,
in as much as∇μvν =γeyμexν. The shear rateγ = ∂v∂yxis now, however, not
constant, but a linear function ofy, such that∂γ∂y=∂
(^2) vx
∂y^2 =const.
For a stationary flow and in the absence of external accelerating forces, the linear
momentum balance (7.52) implies
∇νpνμ=∇μδP+kμ= 0 , kμ=∇νpfricνμ. (16.54)
HereδPis the flow-induced change of the hydrodynamic pressure andkμis the
force density associated with the friction pressure tensorpfricνμ= pνμ+···, where
the dots stand for term involving the scalar partp ̃and the antisymmetric part of the
tensor. When the two latter terms are zero or can be neglected, cf. (7.53), one has