16.4 Viscosity and Alignment in Nematics 333
pμν =− 2 η∇μvν− 2 η ̃ 1 nμnλ ∇λvν − 2 η ̃ 3 nμnνnκnλ∇λvκ− 2 η ̃ 2(
−εμλκωλnκnν)
−ζ nμnν∇λvλ, (16.101)pμ=−γ 1 (ωμ−nμnνων)+γ 2 εμνλnνnκ∇κvλ,p ̃=−ηV∇λvλ−ζ nμnν ∇μvν.The shear viscosityη,thetwist viscositycoefficientγ 1 and the bulk viscosityηVare
positive, all other coefficients may have either sign. The coefficientζcharacterizes
the coupling between the irreducible tensors of rank 0 and 2, the Onsager symmetry
is already taken into account. The coupling between the irreducible tensors of rank
1 and 2 is specified byγ 2 andη ̃ 2. These coefficients obey the Onsager relation
2 η ̃ 2 =γ 2. (16.102)The coefficientsγ 1 andγ 2 occurring in the equation for the axial vector associ-
ated with the antisymmetric part of the pressure tensor are calledLeslie viscosity
coefficients.
The Miesowicz viscosity coefficientsηi,i= 1 , 2 ,3, [144], see also Sect.16.3.3,
are defined for a plane Couette flow with the velocity in thex-direction and its
gradient in they-direction, are inferred from
pyx=−ηi∂vx
∂y, (16.103)
where the casesi= 1 , 2 ,3 correspond to the direction of the field and thusnparallel
to thex-,y- andz-axes, respectively. Theηiare related to the viscosities defined in
(16.101)by
η 1 =η+1
6
η ̃ 1 +1
2
η ̃ 2 +1
4
γ 1 +1
4
γ 2 ,η 2 =η+1
6
η ̃ 1 −1
2
η ̃ 2 +1
4
γ 1 −1
4
γ 2 ,η 3 =η−1
3
η ̃ 1. (16.104)Notice thatη=(η 1 +η 2 +η 3 )/3 is the average of theηi. Furthermore, one has
η 1 −η 2 = ̃η 2 +1
2
γ 2 =γ 2. (16.105)The second equality follows from the Onsager symmetry relation (16.102). The
experimentallyobserveddifferencebetweenη 1 andη 2 isanevidencefortheexistence
of an antisymmetric part of the pressure tensor.