338 16 Constitutive Relations
and its gradient. In particular, for the geometry used above, e.g. in Sect.12.4.6, one
hasnx=cosχ,ny=sinχ,nz=0, then (16.115) implies
γ 1(
∂χ
∂t+Γ/ 2
)
+(Γ / 2 )γ 2 cos 2χ= 0 ,where the shear rate∂v∂yxhas been denoted byΓ. In a stationary situation, where
∂χ
∂t =0, theflow alignment angleχis determined by
cos 2χ=−γ 1
γ 2. (16.116)
This result is independent of the shear rate. For (16.116) to be applicable, thetumbling
parameter
λ≡−γ 2
γ 1(16.117)
has to obey the condition|λ|>1. The viscosity for this “free” flow in a flow aligned
state is
ηfree=1
2
(η 1 +η 2 −γ 1 )+η 12 ( 1 −γ 12 /γ 22 ). (16.118)In many nematics, the flow alignment angle is small, typically around 10◦, such
thatηfreeis not much larger than the smallest Miesowicz viscosityη 1. On the other
hand, close to the isotropic-nematic transition temperatureTni, the average viscosity
η=(η 1 +η 3 +η 3 )/ 3 >ηfreein the nematic phase is approximately equal to the
viscosity in the isotropic phase. This explains the surprising result that the viscosity
below the phase transition is smaller than the viscosity aboveTni.
For|λ|<1, no stationary solution of (16.115) is possible, even when the imposed
shear rate is constant. Then the director undergoes a tumbling motion, similar to that
one described by Jeffrey [161] for an ellipsoid in a streaming fluid. More specifically,
the tumbling period is related to the Ericksen-Leslie tumbling parameterλbyPJ=
4 π/(γ
√
1 −λ^2 ), for a full rotation of the director.16.4.4 Fokker-Planck Equation Applied to Flow Alignment
The equation governing the orientational dynamics of liquid crystals, both in the
isotropic and nematic phases, can be derived from a generalized Fokker-Planck equa-
tion [157–159]. To indicate the physics underlying this approach, the Langevin type
equation of a single non-spherical particle, immersed in streaming fluid, is consid-
ered first. Let the orientation of the particle be specified by the unit vectoru, its
angular velocity is written asω+Ω, whereωis the vorticity^12 ∇×v. Then the time
change ofuis given byu ̇μ=εμλνωλuν+εμλνΩλuν, andΩobeys the equation: