Couple/Torque surface

The physical mechanism of the instability is related to several coupled phenomena discussed by Helfrich [42]. His elegant calculation of the instability threshold is reproduced here for the simplest steady state one-dimensional model shown in Fig. 11.33a. A homogeneously aligned nematic liquid crystal layer of thickness d is stabilised by the rubbed surfaces of the limiting glasses. The dielectric torque is considered negligible (s = 0). At first, a small director fluctuation 9(x) with a period dh postulated ... 

The shear stress cr is given by the force per unit area on the inner cylinder. If the torque on the inner cylinder is T (otherwise known as the couple or moment, with units of N.m), the turning force F on the inner cylinder is given by T/ai. Since the surface area over which the force F acts is 2miH, where H is the cylinder height, the shear stress cris given by... 

If we assume the polar field comes from the temperature gradient between z=d and z=0, i.e., T] - To> d that the director is unbound at the surfaces (i.e., weak anchoring), then the surface torques or couple stresses Zii- ijk i ik zero, which leads to... 

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