Tensors for Physics

17.3 Alignment Tensor Subjected to a Shear Flow 363

for the flow alignment angleχ. For nematics composed of rod-like particles the
first case occurs for small, the latter one for very large shear rates.
T Tumbling: in-plane tumbling of the alignment tensor, the main director is in the
flow plane and rotates about the vorticity axis.
W Wagging: in-plane wagging or librational motion of the main director about the
flow direction.
L Log-rolling: stationary alignment witha 1 =a 2 =0 anda 0 >0. This out-of-
plane solution is instable, in most cases.

• Symmetry breaking states witha 3 = 0 ,a 4 =0:

SB Stationary symmetry breaking states, which occur in pairs ofa 3 ,a 4 and−a 3 ,
−a 4.
KT Kayaking-tumbling: the projection of the main director onto the flow plane
describes a tumbling motion.
KW Kayaking-wagging: a periodic orbit where the projection of the main director
onto the flow plane describes a wagging motion.
C Complex: complicated motion of the alignment tensor. This includes periodic
orbits composedof sequences of KTandKW motionwithmultipleperiodicity
as well as aperiodic, erratic orbits. The largest Lyapunov exponent for the
latter orbits is positive, i.e., these orbits arechaotic.

For a given choice of parameters, in general, only a subset of these solutions are found
by increasing the shear rateΓ. The T and W states can be distinguished in a plot of
a 1 versusa 2. The point(a 1 ,a 2 )=( 0 , 0 )is included in the cycle for tumbling and
excluded for wagging. Similarly, in a plot ofa 3 versusa 4 , the point(a 3 ,a 4 )=( 0 , 0 )
is included in the cycle for the KT orbits and excluded for the KW orbits. ‘Phase
portraits’ of this kind are also useful to recognize more complicated periodic and
also irregular orbits. Examples for orbits pertaining tokayaking tumbling,kayaking
wagging, andchaoticsolutions are shown in Figs.17.2,17.3and17.4. All curves
are computed forθ=0, whereaeq= 3 /2, forλK= 1 .25, andκ=0, the tumbling
parameter isλ= 5 / 6 ≈ 0 .833. The initial state has small, but finite valuesa 0 , ..,a 4.

0.50.25 0 0.250.50. 75 1

a 2

0.25

0

0.25

0.5

0. 75

1

1.25

a 1

1 0.5 0 0.5 1

a 3

0.5

0

0.5

1

a 4

Fig. 17.2Kayaking tumbling orbits in the 1–2- and 3–4-planes of the alignment