Instability Thresholds

The th mode is stable if k min(iCst, Since the steady state of an isolated reactor is stable, i.e., T Q and A 0, see (13.159) and (13.160), it follows from (13.175) that the USS of the network can undergo a stationary bifurcation only if 

Similarly, it follows from (13.176) that the USS can undergo a spatial Hopf bifurcation only if 

The type of coupling that gives rise to either instability, i.e., inhibitory or activa-tory coupling, depends on the sign and value of the rate functions /j and /2 at the uniform steady state. 

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Figure 5.1.7a shows a side view of a lean propane flame, 10 cm in diameter, propagating downward in a top-hat flow. The flame speed is 9cm/s, below the stability threshold, and the flame is stable at all wavelengths. Figure 5.1.7b shows a near stoichiometric flame in the same burner. The flame is seen at an angle from underneath. The mixture is diluted with nitrogen gas to reduce to flame speed to the instability threshold (10.1 cm/s), so that the cells are linear in nature. The cell size here is 1.9 cm. Figure 5.1.7c shows a flame far above the instability threshold, the cell shape becomes cusped, and the cells move chaotically. 

Thus, the r and components of the velocity gradient are completely disregarded and Vq1 depends on those coordinates only through the r and dependence of the (radial) force density and the ground state shear velocity. The quality of this model increases with increasing kR. It is expected that the velocity perturbation is overestimated in this model and thus the hypothetical instability threshold is lowered, which makes the model appealing at least as a first attempt. 

Estimate the temperature of laser torch plasma corresponding to the threshold of capillary instability of liquid 100 nm copper drop. This estimate can be done using the instability threshold condition (1) rewritten in the form ... 

The pearls are just at the Rayleigh instability threshold, their density is that of a collapsed globule and their size is obtained from the Rayleigh charge. It is the so-called electrostatic blob size... 

Equation (17) has been suggested by Swift and Hohenberg [37] as a model for the description of convective pattern in a fluid layer heated from below. Here 4> is the order parameter proportional to the vertical fluid velocity, and is proportional to the difference between the actual temperature drop across the layer and its critical value corresponding to the instability threshold. Numerical simulations show that pattern formation in the framework of equation (17) is fully similar to that described by equation (16) [36]. The growth rate of the disturbance with the wavevector k = kx, ky) is determined by... 

The coefficient a characterizing the non-Boussinesq properties of the fluid can have either sign. Also, we shall consider the system both above the linear instability threshold (7 > 0) and below that threshold (7 < 0). 

The Newell-Whitehead-Segel equations are valid only near the instability threshold, large-scale modulations. For their description, another approach can be applied [53],... 

A typical feature of a non-potential systems is the non-stationary oscillatory behavior that usually manifests itself in the propagation of waves. We have shown that the nonlinear evolution of waves near the instability threshold is described by the complex Ginzburg-Landau (CGL) equation. This equation is capable of describing various kinds of instabilities of wave patterns, like the Benjamin-Feir instability. In two dimensions, the CGL equation describes the formation of spiral waves that are observed in many biological and chemical systems characterized by the interplay of diffusion and chemical reactions at nano-scales. 

Now we consider a strongly nonlinear evolution of ID arrays of islands farther from the instability threshold studied by means of numerical simulations of eq.(5) using a pseudospectral method with periodic boundary conditions. For the parameter values corresponding to region 1 in Fig.3, near the instability threshold, one observes the formation of a sinusoidal surface profile. With the increase of the supercriticality (i.e. with the decrease of g from Qc = 1/4),... 

For the values of the parameters p and q from region 2 in Fig.3, where periodic arrays of islands near the instability threshold are unstable due to the presence of the zero mode, one observes the formation of localized (or strongly modulated) patches of islands shown in Fig.5. Depending on the sign of the parameter p, these can be either patches of cones or caps . 

Localized solutions shown in Fig.5 are found only near the instability threshold in the region 2 in Fig. 3. With the increase of the supercriticality, one observes either the formation of periodic arrays of cones or caps , or the blow-up. The latter can be of either island -type, shown in Fig.6a, or a pit -type, shown in Fig.6b. Spontaneous formation of nano-pits has been recently observed in experiments [17]. 

First, let us discuss the evolution near the instability threshold by means of weakly nonlinear analysis. Consider wqi = 1/4 — 2e, e 1, introduce the long-scale coordinate X = ex and the slow time T = and expand... 

Figure 13. Parameter regions where a planar film surface is unstable (above the solid line) and where stable periodic structures can form near the instability threshold (near the solidline.

Eq.(61) has a special structure in that the linear operator is isotropic, while the nonlinear operator is anisotropic. The linear growth rate near the instability threshold, thus, does not depend on the wavevector orientation and the resulting dispersion relation is the same as in the 1+1 case, to = —woik — k, ... 

Since the linear operator of eq.(61) is isotropic and the nonlinear operator of eq.(61) has a quadratic nonfinearity that breaks h —s- —h symmetry, the preferred pattern near the instability threshold will have a hexagonal symmetry, caused by the quadratic resonant interaction between three different modes oriented at 120° with respect to one another and having the same Unear growth rate. The specific type of the pattern in this case is determined by the phase locking of the three resonant modes that depends on the quadratic resonant interaction coefficient. In order to compute this coefficient, take moi = 1/4 — 2 ye, < 1, introduce the slow time r = et, and use the expansions (45)-(47), as well as the expansion... 

The amplitude equations (65) cannot describe the nonlinear stabilization of a growing surface structure and cannot provide conditions for the formation of stable, spatially regular structures near the instability threshold. In order to obtain such conditions higher order (usually cubic) nonlinear terms in the amplitude equations need to be taken into account. However, in the presenee of the resonant quadratic interaction, the addition of cubic terms in the amplitude equations near the instability threshold is asymptotically rigorous only if the quadratic interaction coefficient is small, o e, which restricts the validity of... 

We have discussed certain aspects of self-assembly of quantum dots from thin solid films epitaxially grown on solid substrates. We have considered two principle mechanisms of instability of a planar film that lead to the formation of quantum dots the one associated with epitaxial stress and the one associated with the anisotropy of the film surface energy. We have focused on the case of particularly thin hlms when wethng interactions between the film and the substrate are important and derived nonlinear evolution equations for the him surface shape in the small-slope approximahon. We have shown that wetting interachons between the him and the substrate damp long-wave modes of instability and yield the short-wave instability spectrum that can result in the formahon of spahally-regular arrays of islands. We have discussed the nonlinear evoluhon of such arrays analyhcally, by means of weakly nonlinear analysis, and numerically, far from the instability threshold and have shown... 

Since near the instability threshold the decay rate of P2 is much larger than the growth rate of Pi, we can neglect the time derivative P2. Thus we obtain P2 = APf with A = 27t(p + 4) and arrive at ... 

Fig. 3.11.2. Experimental roll instability threshold curves for MBBA as functions of the applied voltage and frequency of shear for different values of the effective plate velocity d = 240 //m, = 3200 G. (From Pieranski and Guyon. )...

The stationary bifurcation and the Hopf bifurcation typically occur as one parameter is varied and are therefore known as codimension-one bifurcations. They represent the generic ways in which a steady state of a two-variable system can become unstable. It is sometimes possible to make the stationary and Hopf instability threshold coalesce by varying two parameters. Such an instability, where T = A = 0, is known as a Takens-Bogdanov bifurcation or a double-zero bifurcation, since Ai = A.2 = 0 at such a point [175], This bifurcation is a codimension-two bifurcation, since it requires the fine-tuning of two system parameters. 

K fixed. Then these equations determine the instability threshold for each structural mode, fXc(Pi)- The explicit form of the instability conditions depends on the particular internal kinetics, F. The threshold of the Turing instability is given by fxj = min /Uc(/32), , if the USS becomes unstable as fx increases, or... 

A strict upper bound for ic3 niax is given by c3,sup = 1- Fot given value of the coupling constant k, the instability thresholds of the two inhomogeneous structural modes coincide, i.e., the modes are degenerate, at... 

The eigenvalues of the three nonuniform structural modes, their instability thresholds, the maximum value of the coupling constant /Cmax its upper limit /Cjup as a 00 are given in Table 13.1. 

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