Wavenumber instability

Here, for convenience, we have used values of the wavenumber that are scaled by the undisturbed film width, = ho W 1 and kc = hokc. One way to interpret this condition is that the film height must be less than a critical value for instability... 

If the wavenumber of the roll solution satisfies the condition K < 0 (i.e. A < 1), a transverse modulational instability takes place with respect to disturbances with k] = 0, ky 0. It is also called the zigzag instability. 

Figure 12. Intervals of the modulation wavenumber K corresponding to Eckhans and zigzag instabilities.

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. 

When wetting interactions are present (g > 0) the dispersion relation changes to cr = —gk" - -k —k, see Fig.2b. One can see that wetting interactions suppress the elastically-driven instabihty. With wetting interactions, the instability occurs for < gc = 1/4 at a wavenumber kc = Thus, wetting inter-... 

Typical dispersion curves defined by (39) are qualitatively the same as shown in Fig.2b. Note that the critical wavenumber at the threshold does not depend on the wetting potential and is determined only by the surface stiffness and the energy of edges and corners. For the parameter values typical of semiconductors like Si or Ge, with the surface energy 7 2.0 Jm , surface stiffness a 0.2 Jm , the lattice spacing ao 0.5 nm and the regularization parameter u Oq 5.0 X 10 J, the wavelength of the structure at the onset of instability is 14.0 nm. 

For a spatially unbounded system, the onset of instability (i.e., the Turing bifurcation) occurs at 8 = 8 when H(fe ) = 0, because the wavenumbers are dense over the H(k ) curve. For a spatially finite system with no-flux boundary conditions, however, only particular wavenumbers satisfy the boundary conditions because the functional form of the solution must be cos kr). Thus, in a bounded one-dimensional system, perturbations will grow with wavenumbers given by... 

Point B is much farther from the Turing bifurcation, and we anticipate the pattern formation to be more complex. Following the same procedure as described above, we find the following range of wavenumbers over which the diffusion-induced instability occurs... 

The dispersive spectrometers suffer from greater wavenumber errors, of a less predictable form, owing to their general mechanical and thermal instability and can also be affected by non-uniform illumination across the monochromator entrance slit [26]. FT-spectrometers typically use a He-Ne laser as a reference beam to monitor the displacement of the moving optical element, so providing an active internal absolute wavelength calibration... 

Theorem 10.4 The critical wavenumber kj of the Turing instability near a doublezero point is given by... 

The uniform steady state of a DIRW can undergo a spatial Hopf or wave bifurcation only if the right-hand side of (10.115) is positive. In other words, the rate of activation must exceed the loss rate of the activator in the steady state. If a spatial Hopf instability occurs, then all spatial modes with wavenumbers bigger than jj, the positive root of (10.107), are unstable. 

This curve has a single minimum, (A x, bj), which corresponds to the Turing instability of the uniform steady state. The Turing threshold bj and the critical wavenumber kj depend on q and read in parametric form... 

Note that the third stability condition corresponds to Aj > 0. According to (1.38), the wavenumber zero mode of the activator-inhibitor-substrate system undergoes a Hopf bifurcation if A2 goes through 0. For mass-action kinetics, (12.20) implies that G or Po- Consequently, the third stability condition can fail, if the total substrate is too low. Then the Turing bifurcation ceases to be the primary instability, and a uniform Hopf bifurcation occurs first in the fiiU system. 

Here k = 27rj/i > j = 1,2,..., s = Kkju is the non-dimensional wavenumber and fi = Koj/u is the non-dimensional frequency of oscillation. Instability occurs when a pair of complex conjugate eigenvalues crosses the imaginary axis... 

Using the physical values for the fluids used and layer thicknesses a = b = 50 pm, negative imaginary roots of co were found as a function of wavenumber and increasing applied field when the applied field was beyond the incipience of instability as shown in Fig. 4. The fastest growth wavenumber is the minimum of the negative imaginary roots. 

We expect disturbances with wavenumbers near to dominate the early stages of the instability where the linear theory applies. In practice, the dominant disturbance observed experimentally in many cases of instability frequently has a wavenumber near even though the linear analysis is not applicable for the rather large perturbations that are usually required to detect the instability. 

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