Uniform states stability

First, we ask whether it is possible that the diffusion of the intermediate A and the conduction of heat along the box might destabilize a stable uniform state. An important condition for this is that the diffusion and conduction rates should proceed at different rates (i.e. be characterized by different timescales). Secondly, if the well-stirred system is unstable, can diffusion stabilize the system into a time-independent spatially non-uniform state Here we find a qualified yes , although the resulting steady patterns may be particularly fragile to some disturbances. 

Thus, when the stirring stops, the uniform state remains a stationary solution of the system. Diffusion does not affect the existence of the uniform state, but it may influence its stability. In particular we are interested in determining whether this state can become unstable to spatially non-uniform perturbations. 

As a simple example, we might impose a perturbation with a cosine distribution as illustrated in Fig. 10.4. If the uniform state is stable to such a perturbation, the amplitude will decay to zero if the uniform state is unstable, the amplitude will grow. We could ask this question of stability with respect to any specific spatial pattern, but non-uniform solutions will also have to satisfy the boundary conditions. This latter requirement means that we should concentrate on perturbations composed of cosine terms, with different numbers of half-wavelengths between x = 0 and x = 1. 

Each component of the perturbations has been separated into two terms a time-dependent amplitude An and Tm, and a time-dependent spatial term cos (nnx). If the uniform state is stable, all the time-dependent coefficients will tend in time to zero. If the uniform state is temporally unstable even in the well-stirred case, but stable to spatial patterning, then the coefficients A0 and T0 will grow but the other amplitudes Ax-Ax and 7 1-7 0O will again tend to zero. If the uniform state becomes unstable to pattern formation, at least some of the higher coefficients will grow. This may all sound rather technical but is really only a generalization of the local stability analysis of chapter 3. 

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

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...

The analysis above refers to time-independent velocity fields and equal diffusion coefficients for all species, but Straube et al. (2004) have shown that similar behavior applies to mixing in time-dependent chaotic flows. It was shown numerically that A is a linear function of the reaction rate and the transition in the stability of the spatially uniform state to non-homogeneous perturbations takes place when the positive Lyapunov exponent of the local dynamics is equal to the exponent describing the decay rate of the dominant eigenmode. 

Before going further we note that result (3.264) for a special perturbed state with (3.243), (3.246) (i.e., (3.244) with zero velocity through the body) expresses the Gibbs stability of a uniform state with U°, V°, S° (3.258), (3.259), (3.264), deduced from the stability conditions (3.256), (3.257). 

At the spinodal, then, the work of nucleus formation vanishes, and the critical nucleus becomes arbitrarily large. The significance of these results is made explicit by considering the stability of an initially uniform state to infinitesimal isothermal density fluctuations. The Helmholtz energy change per unit volume due to such fluctuations is given by... 

For instance, for a two-component system (c = 2) the stability limit of the uniform state (n = 1), called spinodal, is a surface with / = 2. 

Several simulation runs follow for the two cases. The runs are represented by their initial conditions and a few selected profiles in order to show the progress of the dynamics. The spatial domain was divided into 100 segments so that A.x = 1 x 10. The dimensionless time increment for these runs was At = 1 x 10. At each time interval, the linearized response converged within 5 iterations to the nonlinear response. Figure 8.23 shows the dynamics of a startup from a cold reactor with a single steady state. For this run, the system exhibits uniform asymptotic stability. 

In Figure 8.24, the initial profile in temperature is linear and lies just above the steady state. The system indeed does not possess uniform asymptotic stability with respect to its steady state. The physical reason behind such nonuniform behavior is clear. Small upsets in the reactor feed are exaggerated by the effects of reaction, and the magnified upset must pass through the reactor before the system settles again. 

The polarity of the alignment layer surface does not have much influence on alignment phenomena for nematic liquid crj talline materials. However, in the case of FLC materials, the polarity of the alignment layer surface shows an important effect. This is because the interaction between the spontaneous polarization and the polarity of the surface becomes important. This matter has been approached theoretically [27]. The stable director orientation in the SSFLC device was determined by minimizing the total free energy of the surfaces and the bulk elastic distortion as functions of cell thickness, cone angle, helical pitch, elastic constant and surface interaction coefficient. Because of the tendency of the direction of the spontaneous polarization to point either into or out of the substrate surface due to polar surface interaction, the director of the molecules twists from the top to the bottom surface. Therefore, the uniform state can only be stabilized in the case of a small surface interaction coefficient. 

Excited-state quenching has, however, been strongly disputed as an important stabilizing mechanism, particularly on a practical basis where it is calculated that up to 10%w/w of uniformly distributed stabilizer would be required in order to produce effective quenching. 

In the second aspect, we again picture the catalytic material itself to have spatially invariant properties, but now we ask questions about the stability of a spatially uniform reaction state to spatial perturbations. This stability question is similar to that posed in studies of hydrodynamic stability and of the other reaction-diffusion problems considered by Turing [62], Prigogine [63,64], Nicolis [63], Othmer and Scriven [65,66] and their co-workers. Prigogine and his co-workers labeled this phenomena "symmetry-breaking" instabilities. The key idea is that since there is a finite rate of transport, the complex interactions between the rate of communication by diffusive transport and the rate of chemical change may make it dynamically impossible for a spatially uniform state to be sustained. 

There are a number of routes to self organization in reaction transport systems. The types of linear instability resulting in the monotonic growth of perturbations has been discussed [ 3 ] and are summarized in Fig. 1. Shown are four cases of the dependence of stability eigenvalues for perturbations, of wave vector k, from the uniform state. The first two cases were distinguished in [4] where the extrinsic type of instability was introduced. In the intrinsic case patterns arise at a well defined wave vector... 

This technique was introduced in [6] and applied there and in [7 ] to the study of the stability to the formation of textural patterns of the uniform state of a rock under stress. This is clearly a rather powerful technique for analyzing these phenomena in random elastic or other media and must be investigated in greater detail to completely unravel all the effects contained in the mechano-chemical coupling that can lead to pattern formation in stressed rocks. In the next two sections we investigate the application of these concepts to metamorphic layering and stylo-lization. 

The realization of this device geometry was first applied in 1980 in the surface-stabilized ferroelectric liquid crystal display and provided much faster switching times than the nematic devices of the time (<0.1 ms) however, the main drawback of the smectic device has been the stability of liquid crystal alignment within the pixels. Nematics are very fluid-like, and after a deformation, they rapidly revert to their previous uniform state of alignment (think about what happens when you press on your laptop screen). Smectics are much more viscous and unfortunately do not self-repair when deformed. 

If q y) is now positive (otherwise one has to go on with the procedure) the bifurcation saturates leading to the bifurcation diagram given in Figure Ic that now also involves a secondary saddle-node bifurcation [2] for /isn = -gj)/4g D- The stability of the various branches are then calculated in standard fashion. For < g < 0> one has bistability between the periodic structured state and the trivial uniform state with the possibility of observing localized structures (see Section 5). 

If 5v //v /coex is not small, the simple description Eq. (14) in terms of bulk and surface terms no longer holds. But one can find AF from Eq. (5) by looking for a marginally stable non-uniform spherically symmetric solution v /(p) which leads to an extremum of Eq. (5) and satisfies the boundary condition v /(p oo) = v(/ . Near the spinodal curve i = v /sp = Vcoex /a/3 (at this stability limit of the metastable states both and S(0) diverge) one finds "... 

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