Tensors for Physics

16.4 Viscosity and Alignment in Nematics 349

The model parameters occurring in the scaled equations are the reduced temper-
atureθ, the dimensionless shear rateΓ, the tumbling parameterλK, andκ. Notice
thatλKserves as a measure for the coupling between the alignment and the flow. In
some applications of the equations, as presented in Sect.17.3, it suffices to treat the
special caseκ=0.

16.4.8 Spatially Inhomogeneous Alignment

In a spatially inhomogeneous situation, the equation governing the dynamics of the
alignment tensor contains terms linked with spatial derivatives. There are two sources

for terms of this kind: firstly, the divergence of the flux tensorbλμν∼〈cλuμuν〉,
wherecis the peculiar velocity of a particle, and secondly, the terms characterizing
the elasticity in the free energy and its derivative with respect to the alignment, see
e.g. (15.38). More specifically, the relaxation equation for the alignment tensor is

daμν

dt

− 2 εμλκωλaκν−...+∇λbλμν+τa−^1

(

Φμν−ξ 02 Δaμν

)

+...= 0 ,

where the dots...indicate the terms involving the deformation rate tensor∇μvν,

as in (16.135). It is understood thatdadμνt stands for the substantial derivative, i.e.
daμν
dt =

∂aμν
∂t +vλ∇λaμν. As before,Φμνis the derivative of the potentialΦ, with
respect toaμν, e.g. the Landau de Gennes expression (15.13). The term involving
ξ 02 Δaμνcorresponds to the simple case of an isotropic elasticity. The lengthξ 0 is
linked with the quantities occurring in (15.35)viaξ 02 =kBε^0 Tξref^2 σ 2 , whereξrefis
the reference length which, in (15.35), was denoted byξ 0. Equations for the three
irreducible parts of the tensorbλμν, which are tensors of ranks 1, 2 ,3, can be derived
by kinetic theory or by irreversible thermodynamics. When the relaxation times for
these three parts are practically equal to a single relaxation timeτb, the approximation

bλμν=−Da∇λ

(

Φμν−ξ 02 Δaμν

)

, Da=

kBT

m

τb,

is obtained. With thediffusion lengthadefined by^2 a=Daτa, the generalization of
(16.135) to a spatially inhomogeneous fluid becomes, [172],

d

dt

aμν− 2 εμλκωλaκν− 2 κ ∇μvκaκν (16.153)

+τa−^1

[

1 −^2 aΔ

](

Φμν−ξ 02 Δaμν

)

=−τa−^1 τap

√

2 ∇νvμ.

Second and fourth order spatial derivatives occur. The second order terms involve
diffusional and elastic contributions proportional to^2 aandξ 02 , respectively. In the