The jump of the viscosity between these extreme, Δη = η max À η min , is given by
Δη ¼
α 1
4
À
α 5 À α 6
ð
Þ
α 3 À α 2
α 2 þ α 3
4
(24)
Equations 22, 23, and 24 have been used to assess the degree of the accuracy of
the planar transversal isotropic fluid, and of the nonplanar Leslie-Ericksen equations,
by comparing them with experimental measurements (Ericksen 1969; Rey 1993a, b;
Han and Rey 1994a, b; 1995a, b).
1
a
b
0
–1
Scales Stress,
Ω
0
5
1 0
1 5
20
Er=500
Er=800
Er=1000
Strain
–1
–2
–3
–4
–5
1
0
–3
–2
–1
0
Tilt Angle, θ
Twist Angle,
θ
1
2
3
Fig. 11 (a) Gray scale visualization of the shear stress surface as a function of the twist and tilt
angles for N D . Lighter regions correspond to local stress maxima and darker to minima. U and S
denote the unstable and stable orientation, respectively. Path 1: evolution from velocity direction to
flow-alignment produces 1 overshoot, path 2: from velocity gradient to flow alignment produces
monotonic change, path 3; from vorticity direction to flow alignment produces 1 undershoot. (b)
Scaled dimensionless shear stress during flow start-up for three Ericksen numbers, with initial
alignment at 60
to flow direction. Overshoots, undershoots, and scaling is predicted. (Adapted
from Grecov and Rey 2003c)
296
A. D. Rey et al.