Nonuniform perturbations

VI. On the Formation of DS in Response to Nonuniform Perturbations of Finite Amplitude... 

VI. ON THE FORMATION OF DS IN RESPONSE TO NONUNIFORM PERTURBATIONS OF FINITE AMPLITUDE... 

We focus on the case, most relevant in applications, that the period T of Dy t) is much longer than the characteristic time scale of the kinetics, which implies that P T 1. Analysis of (11.25), (11.26), and (11.27) leads to the conclusion that there exists a /r > 1, i.e., the uniform steady state is unstable to nonuniform perturbations if either (i) P > r or (ii) [F -I- (P ) - (P ) ] < 4r (P ). Somewhat lengthy further calculations show that, to leading order for large T, the uniform steady state will be driven unstable by diffusion if and only if either... 

To answer this question, consider the change in system free energy corresponding to the spatially nonuniform perturbed composition... 

Calculations for nniforin systems showed that the perturbative treatment usually overestimates the fluctuative contribution . Thus more refined, cluster inetods have been developed, such as the ( VM and CFM". They can be extended to nonuniform systems. In particular, the pair cluster (PCA) expression for F c, can be written out analytically" ... 

Recognizing that the information transmission is often both nonuniform (since the left and right fronts do not always propagate outward from the perturbation with a single well-defined velocity), and dependent on choice of initial states, one can instead define maximal and minimal average propagation speeds, and A " " ... 

Therefore, a computational paradigm could be formulated as follows If C becomes negative under cr-control, study the stability of this state towards charge density perturbations. It is very possible that a phase transition to a nonuniform state appears. ... 

Typically, the insertion induces sharp variation of the membrane profile at the distances 0.5-1.0nm from the membrane-peptide interface [79-82]. The steepness of this perturbation indicates that the short-A, behavior of membrane moduli must be important in the estimates of the elastic energy. In addition, a peptide inserted in a membrane almost certainly perturbs the membrane s elastic moduli in the immediate vicinity of the inclusion. Both these effects, membrane nonlocality and nonuniform modification of elastic properties by insertions, might play an important role in resolving the contradiction between the local calculations [80] and the experimental data for the mean lifetime of a gramicidin channel [81,109,110]. ... 

Across real surfaces and interfaces, the dielectric response varies smoothly with location. For a planar interface normal to a direction z, we can speak of a continuously changing s(z). More pertinent to the interaction of bodies in solutions, solutes will distribute nonuniformly in the vicinity of a material interface. If that interface is charged and the medium is a salt solution, then positive and negative ions will be pushed and pulled into the different distributions of an electrostatic double layer. We know that solutes visibly change the index of refraction that determines the optical-frequency contribution to the charge-fluctuation force. The nonuniform distribution of solutes thereby creates a non-uniform e(z) near the interfaces of a solution with suspended colloids or macromolecules. Conversely, the distribution of solutes can be expected to be perturbed by the very charge-fluctuation forces that they perturb through an e(z).5... 

The signal intensities, S, are theoretically proportional to the number of spins promoted from the lower to the upper spin state at resonance, but in pulsed FT NMR these intensities are perturbed in a nonuniform way by several factors that need to be suppressed for quantitative analysis ... 

In what follows both cases will be examined qualitatively from the point of view of the stability theory. To do this let us slightly perturb the free surface of the film to a wave shape (Fig, 11) Such perturbations arise naturally In any system because of thermal or mechanical perturbations. In the system considered hern there are additional chemical causes, for instance the nonuniform surface reduction. These perturbations can be either amplified in time or they can decay. In the former case the film ruptures, while in the latter the film is stable. The following thermodynamic considerations provide some insight regarding the two types of behavior. First, one may note that the... 

The averaging technique characteristic of the second approach may apply to the case of a tubular reactor where the ratio of the characteristic catalyst particle size to the diameter of a single tube is close to unity, but it is invalid, as will be shown, in the general case of fixed-bed reactors. This approach keeps out of a researcher s field of vision the problem of the reactor stability to local perturbations. At the same time, the technologist is often faced with hot spots in the catalyst bed of a fixed-bed reactor, which make its operation imperfect and even lead to an emergency situation in a number of cases, Until recently, nonuniformity of the fields of external parameters (e.g., nonuniform packing of the catalyst bed or nonuniformity of reactant stream velocity ) was considered the only cause of these phenomena. The question naturally arises whether the provision for uniformity of external conditions guarantees the uniformity of temperature and concentration profiles at the reactor cross-section. The present paper seeks to answer this question, which, as a matter of fact, has not yet been posed in such a form in the theory of chemical reactors. 

To conclude our discussion of the conditions for the origination of DS in catalytic systems, we consider the peculiarities of the process in the region of multiplicity of USS. It is known that in a model with a single equation of thermal conductivity (i.e., for e 1), a perturbation of certain amplitude will switch the system over from one USS to another (for details of the conditions and the dynamics of such a transition, see ). What will then happen in the model considered (e < 1) where nonuniform steady states exist along with uniform ones To which of them will the system transit On which parameters do the dynamics of such transitions depend This complicated, versatile problem will be treated here only at the definition level. 

---