Neutral stability curves

A serious point is the neglect of surface tension and anisotropy in these derivations. In the experiments analyzed so far the relation VX const, seems to hold approximately, but what happens when the capillary anisotropy e goes to zero Numerically, tip-splitting occurs at lower velocities for smaller e. Most likely in a system with anisotropy e = 0 (and zero kinetic coefficient) the structures show seaweed patterns at velocities where the diffusion length is smaller than the short wavelength hmit of the neutral stability curve, as discussed in Sec. V B. 

Fig. 10.9. The neutral stability curve in the n n plane below the curve parametrized by tr(J) = 0 the uniform state is unstable to perturbations of appropriate spatial form...

We can think of the reactant concentration and some initial spatial distribution of the intermediate concentration and temperature profiles specifying a point on Fig. 10.9. If we choose a point above the neutral stability curve, then the first response of the system will be for spatial inhomogeneity to disappear. If the value of /r lies outside the range given by (10.79), then the system adjusts to a stable spatially uniform stationary state. If ji lies between H and n, we may find uniform oscillations. 

If, however, we start the system with a given non-uniform distribution, corresponding to n = 2 say, and a value for ji such that the initial point lies beneath the neutral stability curve, then the spatial amplitudes will not decay. Rather the positive real parts to the eigenvalues will ensure that the perturbation waveform grows. The system may move to a state which is varying both in time and position—a standing-wave solution. 

Fig. 10.10. The neutral stability curve for a system with k < 0.0279, showing curves parametrized by tr(J) = 0 and det(J) = 0. Within the latter, non-uniform profiles may be stable .

The neutral stability curve corresponding to the condition det(J) = 0 gives a closed region, within which we expect the appearance of stable (time-independent) spatially non-uniform profiles. 

As an example of how these curves should be interpreted, we consider a specific case. Let us take k = 0.02 and, for convenience, choose the size of the reaction zone such that y = 6n2 The dispersion, or neutral stability curve for this system, is shown separately in Fig. 10.12. The wave number n can only have integer values, so valid modes correspond to the horizontal lines with nn/y111 = l, etc. Only three of these horizontals intersect the... 

Fig. 10.11. The development of the neutral stability curve for stable pattern formation with the...

Fig. 10.12. Specific neutral stability curve for k = 0.02 and y = 16n2, showing the possibility of stabilizing patterns with n = 2, 3, or 4 over limit ranges of the precursor concentration fx.

Figure 13. Neutral stability curves computed by linear analysis for the succinonitrile-acetone system as a function of acetone concentration for fixed temperature gradient of G = 67°/cm.

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. 

This is a very complicated transcendental equation relating a and r = (Ra /a4)1, which must be solved numerically. We specify a and then use this equation to determine r, and then from r we calculate Ra. Because a = 0, the relationship of Ra versus a defines the neutral stability curve for the even solutions. A set of numerically calculated... 

Figure 12-8. Neutral stability curves from Eq. (12-300) for three different values of Bi. The critical Marangoni number for each Bi is the minimum value over the range of possible values for a.

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. 

Fig. 16. Neutral stability curves showing the effect of surfactants.

The stabilizing influence of the surface viscosity of a typical monolayer was found to be negligible, in contrast to that of the surface elasticity, which (even for a so-called gaseous monolayer) was sufficient to cause a dramatic stabilization. The shift in the Pearson neutral-stability curves brought about by the presence of a gaseous monolayer and also by a close-packed monolayer on a 1-mm deep water substrate is shown in Fig. 16. For this example, the gaseous monolayer having a surface pressure of 0.2 dyn/cm effected a 500-fold increase in the stability criterion, whereas, in... 

The neutral stabihty curve in the (s, z)-plane has a minimum at s = 0.5. The neutral stability curve is shown in Figure 8 (the lowest curve). The uniformly propagating wave is unstable in the region above the curve, i.e., for z > Zc, and it is stable below the curve. It can be easily checked that the parameter z is nothing else but the Zeldovich number Z and can be also written in the form... 

Figure 8. Neutral stability curves in the (s, z) plane for various values of P. Above respective curves, uniform propagation is unstable.

The neutral stability curve in the (s, z)-plane has a minimum at s = Sm > 0 for all P > 0. Neutral stability curves are shown in Figure 8 for physically meaningful values of P. We remark that the parameter P is related to the concentration of the unreacted monomer. Thus, it is not surprising that the dispersion relation for P = 0 coincides with the dispersion relation in the GC model (3.83). 

It appears that our measurement follows the Smith model. But it is noted that the Reynolds number of our data for low Gortler parameter is considerably small. Then, it seems that the Blasius flow no longer holds good for those data. In connection with this problem Ragab and Nayfeh computed the neutral stability curves taking into account of the effect of displacement thickness of boundary layer. According to their results the curves approach to that of the Smith model departing from that of modified Smith model at low wave number when R becomes small. 

In Figure 4, representative neutral stability curves are plotted in plane ((j), K), the stability regions being situated above the curves. These curves are distinguished by two parameters. 

Figure 4. Neutral stability curves for fluidized beds as foilow from the Carnahan-Starling model at Vi = 1 and different values of IgSc (figures at the curves) in the limiting regimes of constant and varying fiuctuation temperature (solid and dashed curves, respectively) dotted curves correspond to the osmotic pressure correction function calculated with the help of the Enskog model.

The result (7.9) was derived a few years ago by M. Takashima (1981a) and see, also, Regnier and Lebon (1995) paper, where the growth rate of disturbances for the non-zero mode is studied. For fixed values of Bi, Bo and Cr, relation (7.9) enables us to plot a neutral stability curve in the (k, Ma)-plane see, the Fig. 7.1... 

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) ... 

The typical neutral stability curve, Ma(k), represented in the Fig. 10.1 below, has two minima first Mai correspondingto k = 0, and describes the long-wave instability, and then Mas, related with kc 0, and indicates the threshold of short-scale convection. In this Fig. 10.1, the dashed line corresponds to layer with undeformable interface and in this case only one minimum exist. Mas. 

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