Oscillatory instabilities

W, g potential functions, k 1, has been discussed in various papers (see, for example, [6, 11, 9, 16, 3]). It has been pointed out that, for step-sizes /j > e = 1/ /k, the midpoint method can become unstable due to resonances [9, 16], i.e., for specific values of k. However, generic instabilities arise if the step-size k is chosen such that is not small [3, 6, 18], For systems with a rotational symmetry this has been shown rigorously in [6j. This effect is generic for highly oscillatory Hamiltonian systems, as argued for in [3] in terms of decoupling transformations and proved for a linear time varying system without symmetry. 

A mechanical system, typified by a pendulum, can oscillate around a position of final equilibrium. Chemical systems cannot do so, because of the fundamental law of thermodynamics that at all times AG > 0 when the system is not at equilibrium. There is nonetheless the occasional chemical system in which intermediates oscillate in concentration during the course of the reaction. Products, too, are formed at oscillating rates. This striking phenomenon of oscillatory behavior can be shown to occur when there are dual sets of solutions to the steady-state equations. The full mathematical treatment of this phenomenon and of instability will not be given, but a simplified version will be presented. With two sets of steady-state concentrations for the intermediates, no sooner is one set established than the consequent other changes cause the system to pass quickly to the other set, and vice versa. In effect, this establishes a chemical feedback loop. 

Ozawa M, Akagawa K, Sakaguchi T, Tsukahara T, Fuji T (1979) Oscillatory flow instabilities in air-water two-phase flow systems. Report. Pressure drop oscillation. Bull JSME 22 1763-1770 Qu W, Yoon S-M, Mudawar 1 (2004) Two-phase flow and heat transfer in rectangular microchannels. J Electron Packag 126 288-300... 

Various factors affecting the nonpremixed edge speed, such as flame stretch, preferential diffusion, and heat loss, have also been investigated, including cellular and oscillatory instabilities of edge flames [1,39 3]. 

Electrochemical oscillation during the Cu-Sn alloy electrodeposition reaction was first reported by Survila et al. [33]. They found the oscillation in the course of studies of the electrochemical formation of Cu-Sn alloy from an acidic solution containing a hydrosoluble polymer (Laprol 2402C) as a brightening agent, though the mechanism of the oscillatory instability was not studied. We also studied the oscillation system and revealed that a layered nanostructure is formed in synchronization with the oscillation in a self-organizational manner [25, 26]. 

A simple model was proposed on the basis of the experimental findings, in which the solution pH plays a key role for the oscillatory instability [51]. The theoretical... 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... 

Integral action degrades the dynamic response of a control loop. We will demonstrate this quantitatively in Chap. 10. It makes the control loop more oscillatory and moves it toward instability. But integral action is usually needed if it is desirable to have zero offset. This is another example of an engineering trade-off that must be made between dynamic performance and steadystate performance. 

Note that the radial and vertical components are out of phase, and that the coefficient multiplying r is only half that multiplying z. Thus, the effect of the ac field is to exert an oscillatory force on the particle with an effective field strength in the vertical direction that is twice the radial field strength. As a result of the larger field strength in the z-direction, the onset of instability is governed by the z-component of the equation of motion, so we need examine only that component. 

Interactions between the flame and the surrounding wall (in a combustion chamber) could influence the contaminant production. This is examined by Dionisios Vlachos and his group at the University of Delaware (formerly at the University of Massachusetts at Amherst) using numerical bifurcation techniques (Chapter 26). For the first time, oscillatory instabilities have been found and control methodologies have been proposed to reduce flame temperatures and NO2 emissions. 

THE ROLE OF FLAME-WALL THERMAL INTERACTIONS IN OSCILLATORY INSTABILITIES... 

To explain the role of transport, simulations have been also performed in an isothermal PSR. Oscillatory instabilities were again found [8]. These facts indicate that oscillations are radical induced. However, without the heat of reactions, no self-sustained oscillations are found for these conditions. The heat of reactions is a prerequisite at these conditions to pull the HB point outside the multiplic-... 

When an energetic material burns in a combustion chamber fitted with an exhaust nozzle for the combustion gas, oscillatory combustion occurs. The observed frequency of this oscillation varies widely from low frequencies below 10 Hz to high frequencies above 10 kHz. The frequency is dependent not only on the physical and chemical properties of the energetic material, but also on its size and shape. There have been numerous theoretical and experimental studies on the combustion instability of rocket motors. Experimental methods for measuring the nature of combustion instability have been developed and verified. However, the nature of combustion instability has not yet been fully understood because of the complex interactions between the combustion wave of propellant burning and the mode of acoustic waves. 

Combustion of a propellant in a rocket motor accompanied by high-frequency pressure oscillation is one of the most harmful phenomena in rocket motor operation. There have been numerous theoretical and experimental studies on the acoustic mode of oscillation, concerning both the medium-frequency range of 100 Hz-1 kHz and the high-frequency range of 1 kHz-30 kHz. The nature of oscillatory combustion instability is dependent on various physicochemical parameters, such... 

Fig. 13.21 shows another example of oscillatory burning of an RDX-AP composite propellant containing 0.40% A1 particles. The combustion pressure chosen for the burning was 4.5 MPa. The DC component trace indicates that the onset of the instability is 0.31 s after ignition, and that the instability lasts for 0.67 s. The pressure instability then suddenly ceases and the pressure returns to the designed pressure of 4.5 MPa. Close examination of the anomalous bandpass-filtered pressure traces reveals that the excited frequencies in the circular port are between 10 kHz and 30 kHz. The AC components below 10 kHz and above 30 kHz are not excited, as shown in Fig. 13.21. The frequency spectrum of the observed combustion instability is shown in Fig. 13.22. Here, the calculated frequency of the standing waves in the rocket motor is shown as a function of the inner diameter of the port and frequency. The sonic speed is assumed to be 1000 m s and I = 0.25 m. The most excited frequency is 25 kHz, followed by 18 kHz and 32 kHz. When the observed frequencies are compared with the calculated acoustic frequencies shown in Fig. 13.23, the dominant frequency is seen to be that of the first radial mode, with possible inclusion of the second and third tangential modes. The increased DC pressure between 0.31 s and 0.67 s is considered to be caused by a velocity-coupled oscillatory combustion. Such a velocity-coupled oscillation tends to induce erosive burning along the port surface. The maximum amplitude of the AC component pressure is 3.67 MPa between 20 kHz and 30 kHz. - ...

D.W. Rice, CombstnFlame 8(1), 21-8 (1964) CA 60, 14325 (1964) (Effect of compositional variables upon oscillatory combustion of solid rocket propellants) N ) R.W. Hart F.T. McClure, "Theory of Acoustic Instability in Solid Propellant Rocket Combustion , lOthSympCombstn (1965), pp 1047-65 N2) E.W. Price, "Experimental Solid Rocket Combustion Instability , Ibid, pp 1067-82 Qi) R.S. Levine, "Experimental Status of High Frequency Liquid Rocket Combustion Instability , Ibid, pp 1083-99 O2) L. Crocco,... 

Let us return to Fig. 1.12(c), where there are multiple intersections of the reaction rate and flow curves R and L. The details are shown on a larger scale in Fig. 1.15. Can we make any comments about the stability of each of the stationary states corresponding to the different intersections What, indeed, do we mean by stability in this case We have already seen one sort of instability in 1.6, where the pseudo-steady-state evolution gave way to oscillatory behaviour. Here we ask a slightly different question (although the possibility of transition to oscillatory states will also arise as we elaborate on the model). If the system is sitting at a particular stationary state, what will be the effect of a very small perturbation Will the perturbation die away, so the system returns to the same stationary state, or will it grow, so the system moves to a different stationary state If the former situation holds, the stationary state is stable in the latter case it would be unstable. 

Can we understand this instability and, if so, can we predict when it will occur We would like to map out the experimental conditions under which oscillatory behaviour can be expected in terms of the rate constants k0 etc. and the concentration of the reactant. In fact we can do even better than this. We will see that much can be said about the details of the oscillations when they start, how they grow in period and amplitude, how long they will last, how and when they die out, and how many we can expect to see. Some typical results are given in Table 2.2. 

During the period of instability, the system will move spontaneously away from the stationary state. For the present model there is only ever one stationary state, so there is no other resting state for the system to move to. The concentrations of A and B must vary continuously in time. They eventually tend to a periodic oscillatory motion around the unstable state. We thus see oscillations over the range of conditions described by (2.20). 

We can characterize the oscillations in terms of their size (amplitude) and the period between successive peaks. It is particularly useful to establish how the amplitude and period vary with the reactant concentration. One way of doing this is artificially to hold p constant and then integrate the rate equations until a(t) and b(t) settle down to a steady oscillation. Figure 2.5 shows the stable oscillatory response obtained from eqns (2.2) and (2.3) with the reactant concentration held constant at the value p = 0.01 mol dm-3, inside the range of instability. The concentration of species A varies between a maximum value of 1.36 x 10 4 mol dm- 3 and a minimum of 2.77 x 10 7 mol dm-3. The difference between the maximum and minimum gives the amplitude of the oscillation appropriate to this value of p (and to the particular values of the rate constants used from Table 2.1), 1.36 x 10 4 mol dm-3. The period can easily be read off from the figure as the time between successive maxima tp = 19.0s. Similarly, b t) has a maximum of 1.235 x 10 4 mol dm 3 and a minimum of 6.48 x 10 7 mol dm 3, so the oscillatory amplitude is 1.229 x 10 4moldm 3. 

Close to the upper end of the range of instability, the oscillations have small amplitude and a short period near p, the waveform is close to sinusoidal. As p is decreased the excursions increase in amplitude, quite quickly, attaining a maximum at p x 0.015 mol dm- 3 with the particular values of the rate constants used here. The period is now longer and the waveform less symmetric. At yet lower reactant concentrations, the amplitude decreases slightly the period continues to increase smoothly as p decreases over most of the oscillatory range, and the oscillations become more and more sawtooth in form. Finally, extremely close to p, the oscillatory amplitude and the period decrease rapidly again. The amplitude tends to zero, although the period remains finite. 

Fig. 2.6. Variation of oscillatory waveform with precursor reactant concentration (a) stationary-state locus for intermediate A, a(p), showing region of instability for p < p < p indicating the magnitude of the oscillations (b) oscillations in a and b and the corresponding limit cycle for p = 0.005 (c) oscillations in a and h and the corresponding limit cycle for p = 0.010 (d) oscillations in a and b and the corresponding limit cycle for p = 0.0195.

Fig. 2.7. Variation in (a) absolute amplitude and (b) oscillatory period across region of instability for the pool chemical model with rate data from Table 2.1. (The qualitative form is appropriate for all combinations of rate constants giving oscillatory behaviour with this model.)...

The general philosophy of our test for oscillatory instability can be summarized as follows. At the pseudo-steady state, the net rates of change of the intermediate concentrations are zero da/dt = db/dt = 0. If we make a small... 

Instability, and hence potential oscillatory behaviour, also sets in exactly at this point. Taking the limit of eqn (2.25) also gives... 

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