Tensors for Physics

364 17 Tensor Dynamics

0.50.25 0 0.250.50. 75 1

a 2

0

0.2

0.4

0.6

0.8

1

1.2

a 1

0.250.20.150.10.05 0 0.050.1

a 3

0.1

0.05

0

0.05

0.1

0.15

0.2

a 4

Fig. 17.3Kayaking wagging orbits in the 1–2- and 3–4-planes of the alignment

0.2 0 0.20.4 0.60.8 1

a 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

a 1

1 0.5 0 0.5 1

a 3

0.4

0.2

0

0.2

0.4

a 4

Fig. 17.4Chaotic orbits in the 1–2- and 3–4-planes of the alignment

The shear rates for the KT and KW solutions areΓ= 2 .0 and 1.75, for the chaotic
solution it is 3.75.
The kayaking type of solutions, also referred to as “out of plane solutions”, were
first discussed in [190]. For a discussion of the complex dynamics of polymeric liquid
crystals and of related computer simulation studies see also [165, 191]. Observations
of the complex orientational dynamics in solutions of rod-like viruses are reported
in [192].
The scenarios for the route to chaos in nonlinear dynamics [194, 195], e.g. tran-
sitions via period doubling and via intermittent states do occur for the equations
considered here which govern the dynamics of the alignment tensor in the presence
of a Couette flow, cf. [183–187]. Equation(17.36) can be supplemented by an equa-
tion for the shear rate in order to control the shear stress, cf. [193]. Then it is possible
to stabilize stationary or periodic solutions for parameters where a constant shear
rate leads to chaotic behavior. For a survey of chaos control in other areas see [196].