Threshold hydrodynamic instabilities

If (5.20) is not valid other types of electrooptical effects take place and the modulated FVederiks transition cannot be observed in experiment. Figure 5.4 shows how doping liquid crystals with conducting and dielectric impurities can violate the inequality (5.20) and, consequently, the electro-hydrodynamic instabilities 1 and 5 (Table 5.1) are observed within the whole frequency range (curve B). Considerable change in the threshold voltage and inversion frequency also takes place for different values of the low-frequency dielectric anisotropy (curves A and C). 

In Chapter 3 the steady-state hydrodynamic aspects of two-phase flow were discussed and reference was made to their potential for instabilities. The instability of a system may be either static or dynamic. A flow is subject to a static instability if, when the flow conditions change by a small step from the original steady-state ones, another steady state is not possible in the vicinity of the original state. The cause of the phenomenon lies in the steady-state laws hence, the threshold of the instability can be predicted only by using steady-state laws. A static instability can lead either to a different steady-state condition or to a periodic behavior (Boure et al., 1973). A flow is subject to a dynamic instability when the inertia and other feedback effects have an essential part in the process. The system behaves like a servomechanism, and knowledge of the steady-state laws is not sufficient even for the threshold prediction. The steady-state may be a solution of the equations of the system, but is not the only solution. The above-mentioned fluctuations in a steady flow may be sufficient to start the instability. Three conditions are required for a system to possess a potential for oscillating instabilities ... 

Consequently, a parallel alignment of smectic layers is linearly stable against undulations even if the perpendicular alignment might be more preferable due to some thermodynamic considerations. As we have shown in Fig. 8, this rigorous result of standard smectic A hydrodynamics is weakened in our extended formulation of smectic A hydrodynamics. When the director can show independent dynamics, an appropriate anisotropy of the viscosity tensor can indeed reduce the threshold values of an undulation instability. 

Abstract A systematic overview of various electric-field induced pattern forming instabilities in nematic liquid crystals is given. Particular emphasis is laid on the characterization of the threshold voltage and the critical wavenumber of the resulting patterns. The standard hydrodynamic description of nematics predicts the occurrence of striped patterns (rolls) in five different wavenumber ranges, which depend on the anisotropies of the dielectric permittivity and of the electrical conductivity as well as on the initial director orientation (planar or homeotropic). Experiments have revealed two additional pattern types which are not captured by the standard model of electroconvection and which still need a theoretical explanation. 

Several cases of dielectric, hydrodynamic, and flexoelectric instabilities and domain structures have been observed and extensively studied in CLCs. Their appearance depends on the initial orientation of molecules, the physical parameters of the material, and the applied electric field. In CLCs with positive dielectric anisotropy Ae > 0, an electric field applied along the helix axis of a planar (Grandjean) texture can induce a two-dimensional spatially periodic deformation which has the form of a square grid [96], The period and threshold voltage of this field-induced instability depend on the elastie constants, the dieleetric anisotropy, and the sample thickness [97],... 

We now calculate the threshold of domain formation following the concepts of HelfrichJ We consider a thin planar cell with electrodes on the two surfaces. We assume that the dielectric anisotropy is negative (Ae = 6 — < 0) and that the liquid crystal is aligned uniformly parallel to the surface by suitable surface treatment. The geometry is that of Fig. 1 the director lies in the x-z plane and is parallel to the jc-axis at the surfaces. There is no variation in the y-direction and all variations in the jc-direction are periodic with period A. In this geometry, the electric field alone does not distort the liquid and any instability must be of hydrodynamic nature. 

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