Instability differential flow

Note that in the expression for A, the wave number k and flow velocity v always appear as a product kv. This means that all observations concerning the dispersion relations between A or Re A and fc at a fixed v can be viewed as relations between A or Re A and u at a fixed k. We can state thus, that for any perturbation which is a Fourier harmonic with a finite k, the growth rate monotonically rises from Re A(0) < 0 to an > 0 as the velocity v grows from 0 to oo. This can be interpreted in terms of disengaging activator and inhibitor as the velocity of the differential flow grows, the separation of the activator and inhibitor becomes more and more effective until the growth rate of the unstable modes reaches the rate of autocatalysis, an. Since the notion of autocatalytic growth is associated with chemistry we call this kind of instability differential flow induced chemical instability . 

A special stability problem was studied by Yakhnin et al. [12,13] for cross-flow moving-bed reactors. The instability here is caused by the differential flow between heat acting as autocatalyst and the reacting matter at elevated Lewis numbers. The cross flow removes the system from equilibrium all along the reactor and thereby furthers the occurrence of instabilities. The study is purely theoretical without any experimental verification for the present. 

V.Z. Yakhnin, A.B. Rovinsky, and M. Menzinger. Differential flow instability of the exothermic standard reaction in a tubular cross-flow reactor, Chem. Eng. Sci. 49 3257 (1994). 

Figure A3.14.15. The differential flow-induced chemical instability (DIFICI) in the BZ reaction. (Reprinted with permission from [44], The American Physical Society.)... 

We distinguish (Figure 1) two types of differential flow induced instabilities (although the classification may be tentative as are their names) the... 

While studying dynamic models with three variables we realized, however, that differential flow induced instability may occur in systems of > 3 dynamical dimensions even in the absence of an activator (or unstable subsystem). Such systems cannot exhibit the TI and the origin of the destabilization... 

It was clear from thp outset that the above instabilities would affect not only chemical systems but also a large class of physical, biological, ecological and engineering systems that share its essential mathematical structure. Hence it is more appropriate to refer to the entire class of differential flow instabilities as DM. 

This paper is organized as follows in Section 2, after a review of the relevant aspects of the Turing instability, we consider different aspects of the DIFICI. The differential flow instability in systems without an activator, DIFIRI, is described in Section 3, and Section 4 deals with some applications of the differential flow induced instabilities. 

The Differential Flow Induced Chemical Instability (DIFICI) of Activator-Inhibitor Systems... 

Chemical Instability Induced by the Differential Flow of Activator AND Inhibitor... 

The Differential Flow Induced Resonance Instability (DIFIRI) in Dynamical Systems without an Activator... 

The necessary condition for the differential transport induced instabilities - the DIFICI and Turing instability (TI) - is the existence of an unstable subsystem, an activator [2,3,4,7,8]. The physical cause of these instabilities is the following the differential transport of activator and inhibitor, be it through a differential flow (DIFICI) or through differential diffusivity (TI), spatially decouples the counteracting species, and thereby releases locally the natural tendency of the activator to grow. 

We demonstrate here that a differential flow may lead to an instability of an otherwise stable homogeneous steady state even if the system does not contain such an unstable subsystem as long as it consists of at least three dynamical variables. The previous analysis showed that a two-variable system may be destabilized by the differential flow only if one of the species is an activator. It is clear that the mechanism of this new instability must be fundamentally different from that of DIFICI and TI. The results imply that the class of systems that may be destabilized by the differential flow of their components to form inhomogeneous patterns is much wider than anticipated earlier. 

It is important to see the difference between the differential flow instabilities in systems with and without an activator that becomes apparent from... 

Including diffusion of the activator in addition to the convective term, apparently brings about the short-wavelength cut-off in the dispersion relations and the appearance of a critical velocity (below which no instability is possible), similar to the case considered above (Sections 2.2.1 and 2.2.3). For this reason it may be difficult to use the dispersion relations to distinguish the two differential flow instabilities. 

It is clear that once the presence of an activator is not required in systems of three variables, it is not necessarily needed in systems of a larger number of variables one could imagine the three variable system embedded into a larger system with no or weak feedback from the embedding system. Intuitively, it seems also reasonable that the statements about the destabilization by the differential flow between an activator and inhibitor subsystems and the necessary requirement for the diffusive instability also hold true in systems of arbitrary number of variables. However, a rigorous proof of this conjecture is still needed. 

The mechanism of the differential flow instability without an activator can be easily interpreted in the case when the two-variable subsystem (or one of the subsystems with more than one variable in a system with > 3 variables) is a damped oscillator (its steady state is a stable focus). Then we can think about a resonance between the frequency of the damped oscillator and the Doppler frequency u = kv of a Fourier harmonic with which the other, moving subsystem was perturbed. Although each of the separate modes tends to decay, their resonant coupling may cause them to grow. 

It is interesting, however, that the instability of a system without activator may take place even if neither of its subsystems is a damped oscillator all the steady states are stable nodes. We argue that the mechanism of the instability is still of a resonant type. The heuristic arguments are as follows. When a system of at least three variables is split by a differential flow, one of the subsystems involves at least two variables. We assume that the steady state of this subsystem is a stable node. The response of such a system to a perturbation is a linear combination of at least two exponential functions [i.e. exp(—Aif) + 0 exp(—A2f) (-1-...) Aj > 0]. Although this response asymptotically decays, for certain a it may initially grow. Then its Fourier spectrum will have a maximum at a finite frequency. Therefore, the subsystem is most sensitive to a perturbation at this frequency. This can be interpreted as a resonance (although with a small quality factor). The instability is thus caused by the resonance which is induced by the differential flow. For this reason we call the instability of a system without an activator differential flow induced resonance instability . It becomes clear now why only modes with wavenumbers within a finite range [Equation (41)] may be unstable all other modes are out of resonance . 

Although the illustration of the differential flow instability without an activator refers to a chemical system, similar examples can be found in physics. Apparently, the multi-mode instability in a laser without a saturable absorber and with bad cavity is of the same nature [5]. Beam instabilities in plasmas can probably also be treated in the same way, although the common approach involves analysis of dielectric permittivity rather than dynamic equations [6]. However, this kind of instability has previously not been considered in chemical or biological systems where it may be no less important. 

Again, our enquiry centers around the instability of the homogeneous reference state, which in this case is the state of continuous wave (CW) laser operation. The movement of the photons relative to the stationary laser medium and absorber is intrinsic, hence it provides a required differential flow. The saturable absorber, whose absorption coefficient + hFhi) (the symbols are defined below) depends on the photon flux density F, gives rise to an unstable subsystem as follows. The absorber causes a localized fluctuation of the light field to grow, since the transmission of the partly saturated absorber increases (decreases) in response to an increasing (decreasing) photon flux. In the three-variable description of the laser, in terms of photon density and the populations of absorber and laser medium, the unstable (or activator) subsystem is therefore formed by the photon density and the population of the absorber. As shown below, the three-variable description reduces to the classical two-variable case [10]. 

The following considerations suggest that many biological systems may be affected by differential flow instabilities DIFI. First, differential flows occur naturally, for instance in the circulatory, digestive and lymphatic systems where a flow containing one key species may interact with a counteracting species that is bound to the wall of the vessel. In vitro, cells or enzymes may be readily immobilized on appropriate supports for experiments in a flow system similar to the one shown in Figure 8. 

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