Instability secondary

In what follows, we will explore the first of the two suggested scenarios. Starting with a multilamellar cylinder configuration, we will study its stability under shear flow (Fig. 17). Our aim is to check whether we can find an instability of the cylinder-a secondary instability-that would be responsible for the break-up into onions. 

Cavities withPr 0.7 and W/H a 8. For HIL > 5 and 0 < 0 < 60°, direct application of the horizontal scaling law (Eq. 4.104) introduces significant errors when Pr = 0.7 and Ra = 104 [144] these errors have been shown [66, 235] to result from a secondary instability that appears at a Rayleigh number only slightly greater than that for the primary instability discussed in the section on horizontal rectangular parallelepiped and circular cylinder cavities. A modified scaling relation, taken from Hollands et al. [144], is recommended for 0 < 0 < 60° ... 

First we discuss conhgurations which can be described by the standard model where patterns appear either directly or as a secondary instability (section 2.1). Then we discuss briefly EC phenomena not covered by the standard model (section 2.2.). 

In the following, we first discuss the situations where EC occurs as a primary forward bifurcation and where the standard model is directly apphcable (cases A and B). Then we discuss configurations where EC sets in as a secondary instability upon an already distorted Freedericksz ground state and compare it with experiments (cases C and D). Note that in this case the linear analysis based on the standard model already becomes numerically demanding. Finally, we address those combinations of parameters where a direct transition to EC is not very robust, since it is confined to a narrow Ca range around zero. For cases E and H this range may be accessible experimentally while for cases F and G it is rather a theoretical curiosity only. 

For a/e > 0.0057 EC occurs superimposed onto the Freedericksz state (secondary instability) at a higher voltage Uc > Upi- The standard model can also be applied here by carrying out numerical linear stability analysis of the Freedericksz distorted state, and one is faced with similar modifications and difficulties as mentioned in case C before. The ea-dependent Uc and Qc, presented by the dotted hnes in Fig. lO(a-b), have been calculated numerically. It should be noted that the convection rolls are now oriented parallel to the initial director alignment, contrary to the normal rolls in case A or C. 

The analysis of an alternation of flexodomains and EC patterns at low AC frequencies discussed in Section 4.4 is much more complicated. For frequencies / /t, where the threshold voltages of both patterns are near to each other, one expects them to flash up independently at the onset. With increasing voltage the flexodomains and rolls will start to interact, apart from the fact that each pattern type might develop its own secondary instabilities. To disentangle these processes is certainly a very demanding task both in theory and in experiment. 

As far as our understanding of the stability is concerned, minimum-average-B is fairly well established. The calculations, even for finite p, or the critical P for multipoles have been done. The value of/3 turns out to be rather low, and this may also be one of its disadvantages. Secondary instabilities are trapped particle instabilities or instabilities associated with atoroidal field added toa multipole. I would say that this subject has not been sufficiently studied, although the primary stability has been well explored. 

W. Zhang and J. Vinals, Secondary instabilities and spatiotemporal chaos in parametric surface waves, Phys. Rev. Lett, 74, 690, 1995. 

The final state of the instability sometimes manifests itself as a string of large drops and small ones ( satellites ). The satellites originate from a secondary instability of the liquid sheath as two adjacent drops are in the process of forming. When we deal with instabilities of fluid streams (particularly with viscous liquids), it is sometimes possible to observe a whole hierarchy of satellites, the appearance of which is considered a nuisance in many applications (for example, in ink jet printers, where the phenomenon has a disturbing tendency to degrade the resolution of the printed text). 

Cheng, M., and Chang, H.-C. (1995). Competition between subharmonic and sideband secondary instabilities on a falling film. Phys. Fluids 7(l) 34-54. 

The terms occurring are those allowed by symmetry. Due to the anisotropy more terms appear than in isotropic systems [94]. The clue for the characteristic appearance of the zigzag instabihty as a secondary bifurcation is that 4 is typically negative in nematics [23, 24] leading, in contrast to isotropic fluids, to amplification of transverse modulations of roll patterns. Model calculations that include this feature [25, 26] were quite successful in describing quahtatively the secondary instability and the behaviour beyond. 

Remark 7.6. Because of the special stmcture of the inertia matrix of the linear system (7.8), in addition to the flutter boundary, (7.49), there exist a secondary instability boundary defined by (7.48). This additional boundary corresponds to the... 

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