Flow alignment angle

X of a flow-aligning nematic with flow-alignment angle 9a in a steady shearing flow, with x the flow direction and... 

In ordinary nematics p and p are both negative and the flow alignment angle 0 is usually small. This equilibrium orientation of the director is... 

Gahwiller discovered that nematics that undergo a transformation to the smectic phase at lower temperatures exhibit an unusual type of instability as the temperature approaches the transition point. The limiting value of the flow alignment angle 0 defined as... 

It has been suggested that in the phase, the disc-like shape of the molecule may have a significant effect on p and 3. The stable orientation of the director under planar shear will now be as shown in fig. 6.5.2(0). Thus it can be argued that both p and p should be positive, and the flow alignment angle 0o should lie between —45 and —90°. It then follows that when p < 0, the director tumbles and the flow becomes... 

Flow alignment angle. Consider a shear as shown in Figure 5.20. The shear rate dv/dx = constant and the liquid crystal director is uniformly oriented in the xz plane. Show that when Y lY2 < 1, the lilt angle is given by cos 20 = - Y yz in the steady state. 

Figure 7, The shear stress (left) and viscosity as functions of the shear rate for Xk = T25 and = 0, in the nematic phase, at the temperature = 0.8, where the flow alignment angle, at small shear rates, is about 10 degrees. The large gray dots stem from calculations with constant shear rates, with a maximum shear deformation 150. The smaller black dots are for imposed shear stress.

The first normal stress difference Ni/2 = Si (upper left), the zz-stress difference (upper right), i.e., the quantity So <7zz - [<7xx + cTj,j,)/2, cf. (23), the average flow alignment angle (lower left), and the magnitude of the in-plane alignment, i.e., a + (lower right) are... 

Figure 17 shows the first and the zz normal stress differences (top row), the flow alignment angle and the in-plane alignment (bottom row) as functions of the shear rate for Ak = 1.05 and k = 0.4, at the temperature = 0. Comparison should be made with Fig. 14. The main qualitative difference is in the plots for the zz normal stress difference So which is non-zero for k 0 and rather similar to the in-plane alignment. As a consequence, the magnitude of the second normal stress difference becomes smaller than 0.5 lVi. ...

P (7 (that is, 0 at the plates) is taken from the numerical solution. Figure 9 shows the viscosity ratio 772/77 as a function of 1 /Z)= k/T p for (77 -772)7772=8 and different flow alignment angles, l/t] gives an asymptotically linear dependence on l/D for large D-values. 

Figure 9. Viscosity ratio 772 77 as a function of ilD = k/T Pj for different flow alignment angles.

The flow alignment angle depends on the ratio of the axes of the molecules according to Eq. (36)... 

The flow alignment angle for discotics should therefore be approximately 90°. A stability analysis shows that the angle above 90° is the stable one [44]. Thus for both types of nematic liquid crystals the configuration with the large dimension nearly parallel to the flow direction is the stable alignment. Figure 12 demonstrates this phenomenon. 

Correspondingly, the flow alignment angles around all axes are... 

The nine coefficients can, therefore, be determined from the seven viscosity coefficients, the rotational viscosity coefficient 2A5 and the relaxation time for the flow alignment for y/=0. Instead of the two last determinations it is also possible to use two flow alignment angles at different V values. 

To determine rj2 direction of the magnetic field and plate movement must coincide. This causes many mechanical problems. Sometimes no magnetic field is applied, in which case the viscosity is determined under flow alignment. The procedure cannot be used for larger flow alignment angles. 

A stable angle obviously requires that I- = IAI> 1. Indeed, in most nematics, I Al> 1, and there is an equilibrium with flow alignment angles in the range... 

Illustration of the flow alignment angle in simple shear flow. 

This is analogous to the flow alignment angle in nematics (see (4.22)), which again shows that the behavior of the c-director within the layers is similar to the meaning of the director in the nematic phase. 

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