Wave number, “critical

From Eqs. (79a) to (79c), it is easily understood that only the fluctuation components having a positive amplitude coefficient p can grow unstably with time. According to Eq. (80), the critical wave number is obtained under the condition p = 0 as follows,... 

Nishi, et al. [ ] then find the value of B at which R(B) becomes zero, defined as the critical wave number, B ... 

For an approximate quantitative comparison of our theoretical results with the experiments on lyotropic liquid crystals we make a number of assumptions about the material parameters. As we have shown in Sect. 3.2 the different approaches cause only small variations in the critical wave number. For this estimate it suffices to use the critical wave number obtained in our earlier work [42], For lyotropics it is known [56, 57], that the elastic constants can be expressed as... 

From Eq. (29) one can easily extract a critical wave number qc... 

The so-called critical wave number kc, equal to the square root of the Bond number, thus separates the long-wavelength disturbances that are unstable from the shorter-wavelength disturbances that are stable because of the influence of capillary effects. We may note that the condition (6 102) shows that the thin-film analysis is valid provided Bo 0(1), as this is the condition for the wavelength of the disturbance to be large compared with the film thickness. 

Now, following the assumption of Taylor that the principle of exchange of stabilities is satisfied, so that a = 0 at the neutral stability point, the objective of the stability analysis is to obtain nontrivial solutions of (12-147) and (12-148) with a = 0 for various values of a and then determine the minimum value of T as a function of a. This minimum of T is the critical value of the Taylor number for transition to instability. The corresponding value of a is known as the critical wave number and represents the (dimensionless) wave... 

Figure 12-5. Stability criteria for the Rayleigh-Benard problem. The two curves shown are the neutral stability curves for the modes n = 1 and n = 2. The region above the curve for n = I is unstable, whereas that below is stable. The critical Rayleigh number is 657.511 at a critical wave number of 2.221...

For a given wave number, the least stable mode corresponds to n = 1 (this yields the smallest value of Ra ). If we plot Ra versus a lor n = 1, as in Fig. 12-5, we obtain the so-called neutral stability curve. For a given a, any value of Ra that exceeds Ra (a) corresponds to an unstable system, whereas any smaller value is stable. The critical Rayleigh number, Ra, for linear instability is the minimum value of Ra for all possible values of a, and the corresponding value of a = am, is known as the critical wave number. 

Now, for each value of Bi, we can plot the neutral stability curve, as shown in Fig. 12-8 for Bi = 0, 2, and 4. The critical Marangoni numbers for these three cases are approximately 80, 160, and 220. As noted earlier, the system is stabilized by increase of Bi because this leads toward an isothermal interface, and thus cuts the available Marangoni stress to drive convection. The critical wave numbers for these three cases are, respectively, 2.0, 2.3, and 2.5. 

Under the assumption of an optically thin cloud (every emitted photon can leave the cloud without being reabsorbed) a critical wave number similar to eq. (10.6) can be obtained ... 

Dimensionless Critical Wave Number for the Rayleigh Problem... 

The critical wave number is given by (10.42) and reads for the LE system... 

Flexoelectric patterns also exist for nematic layers with asjrmmetric boundary conditions, i.e. with homeotropic anchoring on one surface and planar anchoring on the other one hybrid-aligned nematics).The critical voltage and the critical wave number obtained with the one-elastic-constant approximation are in a good agreement with experimental re-sults. ... 

Note, that the problem (46) and (47) describes the case of tangentially immobile interfaces. One proves that the mobility of interfaces changes only slightly the values of her - it affects the value of the critical wave number, ker. 

The neutral stability curves, when Bi = 0 and Bo = 0.1, are shown in above Fig. 7.1 as a sample case. The values of Cr are given in the figure. Since the region below each curve represents stable state, the lowest point of each curve gives the critical Marangoni number Mac and the corresponding critical wave number kc. When Cr 0 (i.e. in the case of a non-deformable free surface, and in a such case Fr2—> 0 also but Bo 0), the result of Pearson s (1958) ... 

It follows from this latter fact that the highest point of each curve gives the critical Marangoni number Mac, for the onset of overstability and the corresponding critical wave number kc. When, according to (2.23), Ma = y d P/p (Vq) < 0, and upside of the layer is a free surface (Bo > 0), then it is necessary that (see (1.3) ) ... 

Finally, the information about the wave number of the most amplified disturbance at the breakup point, which is called the critical wave number A z,r,crit> allows the calculation of the drop size. It is assumed that the drop volume is equal to the volume of a cylindrical element with known dimensions. The radius of the cylinder is the radius of the time steady flow at the breakup point. Its length is the wave length of the most amplified disturbance Acrit = 2 rro/ z,r,crit- Finally, the drop diameter is given by ... 

In order to understand better the nature of the instabilities possible at these two points, consider the "critical" wave number me for disturbances which will produce instability at the smallest value of B. Minimizing the expressions in Eqs. (16a), (16b) with respect to m yields... 

It is clear from Eqs. 16 and 17 that the possible appearance of finite spatial inhonrogeneities involves a "cooperation" between chemistry and transport. Indeed, whenever the rates of diffusion and chemical reactions are "comparable," short wavelength disturbances may grow. For an instability at for example, the system amplifies a fluctuation of "critical" wave number me and ultimately reaches a stable nonuniform steady state (see Figure 10). 

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