Wave-like instability

Fig. 9.12 Undulation or wave-like instability in the smectic A layer subjected to a dilatation-compression distortion...

In both the cases considered, an optical contrast of the patterns observed in isotropic liquids is very small. Certainly, the anisotropy of Uquid crystals brings new features in. For instance, the anisotropy of (helectric or diamagnetic susceptibility causes the Fredericks transition in nematics and wave like instabilities in cholesterics (see next Section), and the flexoelectric polarizaticm results in the field-controllable domain patterns. In turn, the anisotropy of electric conductivity is responsible for instability in the form of rolls to be discussed below. All these instabilities are not observed in the isotropic liquids and have an electric field threshold controlled by the corresponding parameters of anisotropy. In addition, due to the optical anisotropy, the contrast of the patterns that are driven by isotropic mechanisms , i.e. only indirectly dependent on anisotropy parameters, increases dramatically. Thanks to this, one can easily study specific features and mechanisms of different instability modes, both isotropic and anisotropic. The characteristic pattern formation is a special branch of physics dealing with a nonlinear response of dissipative media to external fields, and liquid crystals are suitable model objects for investigation of the relevant phenomena [39]. 

FIGURE 6.30. Homeotropic texture of a smectic A in an electric field Ae > 0. (a) Initial geometry (b) Parodi texture transition [109] (c) wave-like instability [7], experimentally observed patterns corresponding to a Parodi transition [109] (d) from a homeotropic orientation and (e) from a planar orientation. 

The field-induced wave-like instability discussed theoretically [113] is still to be observed in experiment. 

Then the mixture with droplets is quenched into the spinodal instability region to some T < Ta (Concentration c(r) within droplets starts to evolve towards the value C(,(T) > C(,(T ), but the evolution type depends crucially on the value Act = cj(T) — Ch(Ta). At small Act we have a usual diffusion with smooth changes of composition in space and time. But when Act is not mall (for our simulations Act O.2), evolution is realised via peculiar wave-like patterning shown in Figs. 8-10. 

The irregularity in the color of the outer isosurface appears to indicate wave-like fluctuations in the near-wall pressure, i.e. near-wall turbulent boundary layer structures, consistent with what has been stated about the instability of the near-wall flow in Sect. 3.1.1. The low pressure in the inner part of the vortex and in the vortex finder is clearly to be seen. A very low pressure can be seen also to exist in the apex of the cone. Its location is consistent with what is seen experimentally when the vortex extends to the bottom of the cyclone, not terminating upon a vortex stabilizer or due to the natural... 

The theories of elastic and viscoelastic materials can be obtained as particular cases of the theory of materials with memory. This theory enables the description of many important mechanical phenomena, such as elastic instability and phenomena accompanying wave propagation. The applicability of the methods of the third approach is, on the other hand, limited to linear problems. It does not seem likely that further generalization to nonlinear problems is possible within the framework of the assumptions of this approach. The results obtained concern problems of linear viscoelasticity. 

As a final comment, it is interesting to note that this FS(K) study of the hydrogen molecule offers a new and simple illustration of the behavior of sophisticated Hartree-Fock schemes like UHF, PHF and EHF. Furthermore, it provides a very efficient numerical example of instabilities in the standard Hartree-Fock method. It is important to see that the UHF, PHF and EHF schemes all correct the wrong RHF behavior and lead to the correct dissociation limit. However, the UHF and PHF schemes only correct the wave function for large enough interatomic distances and the effect of projection in the PHF scheme even results in a spurious minimum. The EHF scheme is thus the only one which shows a lowering of the energy with respect to RHF for all interatomic distances. 

The design of a die for an appiication is a very complex process. The design must be such that regions with long residence times do not exist, and it must provide uniform flow at the exit. Moreover, coextrusion dies must be designed with iayer uniformity and without process instabilities known as the wave instability and the zig-zag instability [i2, i3]. These instabilities are beyond the scope of this book and thus are not discussed further. Like transfer lines, the velocities in a die shouid be at least i.5 cm/s for thermally stable resins (with antioxidant systems) and at least 3 cm/s for resins that are not thermally stable. 

Convecting free-stream disturbances within this speed range are likely to trigger strong sustained instability, because of the high amplification rate that such modes would experience. Also for c > 0.4f/oo, the convecting disturbances would create damped wave packets. 

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