Tensors for Physics

16.4 Viscosity and Alignment in Nematics 345

vector associated with the antisymmetric part of the pressure tensor, cf. Sect.16.3.5.
This contribution vanishes, when the average angular velocity is equal to the vortic-
ity, as already assumed above. In the presence of an external field, which exerts a
torqueTμ, the entropy production contains an additional contribution proportional to
wμ(Tμ+ 2 kBTεμνλaνκΦκλ). The entropy production, however, should not depend
explicitly on the vorticity, sinceωμ=0 can also be achieved by a solid body like
rotation. This implies thatTμ=− 2 kBTεμνλaνκΦκλhas to hold true. When fur-
thermore, the relaxation of the internal angular momentumJis fast, compared with
the orientational relaxation, one has effectivelydJ/dt=0 andpμmatches the
torque density, viz.pμ =(ρ/m)Tμ. Thus the pseudo-vector associated with the
antisymmetric part of the pressure is related to the alignment by

pμ=− 2

ρ

m

kBTεμνλaνκΦκλ. (16.137)

As expected, both the symmetric traceless and the antisymmetric parts of the pressure
tensor associated with the alignment vanish in thermal equilibrium where one has
Φμν=0.
Multiplication of (16.135)byτaελκνaκμyields

τaMλ+ελκνaκμΦμν=−ελκνaκμ

(

τap

√

2 ∇μvν− 2 κτa ∇μvσaσν

)

,

with

Mλ=ελκνaκμ

(

daμν

dt

− 2 εμαβωαaβν

)

. (16.138)

Then (16.137) is equal to

pλ= 2

ρ

m

kBTMλ+ 2

ρ

m

kBTελκνaκμ

(

τap

√

2 ∇μvν− 2 κτa ∇μvσaσν

)

.

(16.139)

16.4.6 Limiting Cases: Isotropic Phase, Weak Flow

in the Nematic Phase

For a plane Couette flow, a symmetry adapted ansatz for the alignment and for
the pressure tensors can be made in analogy to (16.90). Then each of the tensorial
equations reduces to three coupled equations for the relevant 3 components. In detail,
these can be inferred from the more general case of all 5 components as presented
in Sect.17.3. Here just some results are stated for the nonlinear viscous behavior in
the isotropic phase, where terms nonlinear in the alignment tensor are disregarded.