362 17 Tensor Dynamics
17.3 Alignment Tensor Subjected to a Shear Flow
17.3.1 Dynamic Equations for the Components
In the presence of a shear flow the equation (16.149), viz.
daμν
dt− 2 εμλκΩλaκν− 2 κΓμκaκν+Φμν=√
3
2
λKΓμν,is governing the dynamics of the alignment tensor. For a plane Couette flow with
the velocity inx-direction and its gradient iny-direction and with the imposed shear
rateΓ, this equation is equivalent to 5 coupled equations for theai:
∂
∂ta 0 +1
3
√
3 κΓa 2 +Φ 0 = 0 , (17.36)
∂
∂ta 1 −Γa 2 +Φ 1 = 0 ,
∂
∂ta 2 +Γa 1 +1
3
√
3 κΓa 0 +Φ 2 =1
2
√
3 λKΓ,
∂
∂ta 3 −1
2
Γ( 1 +κ)a 4 +Φ 3 = 0 ,
∂
∂ta 4 +1
2
Γ( 1 −κ)a 3 +Φ 4 = 0 ,whereΦi=(θ+a^2 )ai+Qi,i= 0 , ..,4. ForQisee (17.29).
Stationary solutions of these equations correspond to problems discussed in
Sects.16.3.6and16.4.6. Another application is the study of the effect of a shear
flow on the phase transition isotropic-nematic, as first presented in [169] and inde-
pendently treated in [170]. The solutions found for equation (17.36), however, are
much richer, cf. [183].
17.3.2 Types of Dynamic States
For a stationary imposed shear rate, not only stationary solutions exist. Also periodic
and even chaotic behavior is found for the alignment tensor, subjected to a plane
Couette flow. In the following, the namemain directoris used for the direction of
the principal axis associated with the largest eigenvalue of the tensor. The various
types of dynamic states are
- Symmetry adapted states witha 3 =a 4 =0:
A Aligning: stationary in-plane flow alignment witha 0 <0. Furthermore, one
may distinguish statesA+andA−pertaining to positive and negative values