Stability neutral

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. 

In between the two extremes of stability and instability there are numerous near-neutral stability situations, resulting in varying degrees of vertical mixing. In this regime, the mixing depends on such factors as shear-induced turbulence and vertical mixing in and by clouds. 

Figure 4. Neutral stability diagrams from linear analysis for values of the thermophysical properties considered in this paper, (a) Diagram for properties given as Case I in Table I as a function of the segregation coefficient k. Continued on next page.

Figure 6. Expanded view of neutral stability diagram for System I with k = 0.865. Critical values for bifurcation of families of cells in a 2Xc sample size are marked.

Neutral stabilized collisional-radiative recombination is also quite slow under typical afterglow conditions. In his review, Flannery4 gives the simple formula,... 

In the introduction to this chapter a combustion wave was considered to be propagating in a tube. When the cold premixed gases flow in a direction opposite to the wave propagation and travel at a velocity equal to the propagation velocity (i.e., the laminar flame speed), the wave (flame) becomes stationary with respect to the containing tube. Such a flame would possess only neutral stability, and its actual position would drift [1], If the velocity of the unbumed mixture is increased, the flame will leave the tube and, in most cases, fix itself... 

Theoretical considerations and observations for neutral stability indicate for small x that, where o-J is determined for averaging times... 

The quantity in brackets is With respect to Z, observations for neutral stability and small x indicate that Hilst (1957) made observations of cr under very stable conditions, so that vertical meandering was almost completely suppressed and [Pg.273]

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.

Moreover, the periodic orbits are here of neutral stability. 

Statement 2. Substitution into the concentration dynamics (equations (8.2.12) and (8.2.13)) of the reaction rate K — K(Na, Nb), dependent on the current concentrations, changes the nature of the singular point. In particular, a centre (neutral stability) could be replaced by stable or unstable focus. This conclusion comes easily from the topological analysis its illustrations are well-developed in biophysics (see, e.g., a book by Bazikin [30]). 

Benjamin (B5) has given a detailed treatment of the onset of two-dimensional instability in film flow, taking capillary effects into account. The expression for neutral stability found in this work can be given as... 

In a simplified treatment for longer waves, Benjamin derived an amplification factor a, defined as the amplification experienced by the wave of maximum instability in traveling a distance of 10 cm. If neutral stability exists, a = 1, and if a > 1, wavy flow can be expected. For a vertical wall,... 

Substituting the condition a = 1 for neutral stability, it is seen that... 

Hanratty and Hershman (H2) have given an interesting treatment of the stability problem. For the case of free film flow (i.e., film flow without a bounding gas flow), the condition for neutral stability becomes... 

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.

Anionic surfaces flat adsorption (DADMAC with low or moderate molar mass) partial adsorption (copolymers with low charge density, homo-and copolymers with high molar mass, branched structures) recharge of surfaces, neutralization, stabilization... 

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