Backflow in the Bend Geometry

We now turn our attention to the bend homeotropic to planar Preedericksz transition geometry of Fig. 3.8 on page 78. First note that if flow is considered to be negligible then when H is near He an approximation for Ton is given analogously to (5.427) by 

However, it is known that flow plays a significant role in this geometry and we shall show that the switch-on time is substantially reduced by the presence of backflow. Motivated by the geometry and notation of Fig. 3.8, consider 

By reasoning similar to that used in equations (5.429) to (5.450) to obtain the governing dynamic equations previously given in equations (5.451) to (5.457), it can be shown that there are analogous equations for this present case. We omit the details because of the identical arguments used and proceed by stating the results [220]. The dynamic equations analogous to (5.452) and (5.453) are 

The boundary conditions remain as those given in equation (5.451). Under similar suppositions as before, it is easily shown that the analogues of the inequalities in (5.455) axe 

The switch-on time is given by the result for r in equation (5.464) using the relevant parameters given by the definitions (4.169), (5.502)2 and (5.504). It is found that 

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Section 5.9 introduces the dynamics of the Preedericksz transition in the classical geometries described in Chapter 3 in Section 3.4.1. The switch-on and switch-off times will be defined when flow is considered to be negligible in the usual twist geometry, as detailed in Section 5.9.1. In some instances, however, flow turns out to be quite influential and leads to the phenomena of backflow and kickback y as to be discussed in detail in the case of the splay geometry in Section 5.9.2. Backflow in the bend geometry is discussed in Section 5.9.3. 

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