Stability oscillations

The main features of the copper catalyzed autoxidation of ascorbic acid were summarized in detail in Section III. Recently, Strizhak and coworkers demonstrated that in a continuously stirred tank reactor (CSTR) as well as in a batch reactor, the reaction shows various non-linear phenomena, such as bi-stability, oscillations and stochastic resonance (161). The results from the batch experiments can be suitably illustrated with a two-dimensional parameter diagram shown in Pig. 5. 

In Fig. 6, separate regions of bi-stability, oscillations and single stable steady-states can be noticed. This cross-shaped phase diagram is common for many non-linear chemical systems containing autocatalytic steps, and this was used as an argument to suggest that the Cu(II) ion catalyzed autoxidation of the ascorbic acid is also autocatalytic. The... 

The transient process that follows after introducing the impulse perturbation to the feed is shown in Fig. 5. At the stable conditions stabilization of throughputs can be monotonous (curve 1 in Fig. 5a) and oscillating (curve 1 in Fig. 5b). The loss of stability for identical classifiers is always of fast stabilizing oscillation character but the oscillations occurs around the throughput values, which are different in comparison to the nominal ones. The reason for the appearance of these auto-oscillations is namely the similarity of... 

Halanay, A. [1966] Differential Equations Stability Oscillations (Academic Press New York) Series title Mathematics in Science and Engineering 23. 

EIOs), backward wave oscillators (BWOs) or magnetrons are available. Their spectral characteristics may be favourable however, they typically require highly stabilized high-voltage power supplies. Still higher frequencies may be obtained using far-infrared gas lasers pumped for example by a CO- laser [49]. 

Our understanding of the development of oscillations, multi-stability and chaos in well stirred chemical systems and pattern fonnation in spatially distributed systems has increased significantly since the early observations of these phenomena. Most of this development has taken place relatively recently, largely driven by development of experimental probes of the dynamics of such systems. In spite of this progress our knowledge of these systems is still rather limited, especially for spatially distributed systems. 

Galerkin method becomes unstable and useless. It can also be seen that these oscillations become more intensified as a becomes larger (note that the factor affecting the stability is the magnitude of a and oscillatory solutions will also result using large negative coefficients). 

The main limitation of the pilot-operated regulator is stability. When the gain in the pilot amplifier is raised too much, the loop can become unstable and oscillate or hunt. The two-path pilot regulator (see b) is also available. This regulator combines the effects of self-operated and the pilot-operated styles and mathematically produces the equivalent of proportional plus reset control of the process pressure. 

It is noteworthy that eq. (4.15a) is nothing but the linearized classical upside-down barrier equation of motion (8S/8x = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15a) describes how the trajectory escapes from the instanton solution, when it deviates from it. The parameter X, referred to as the stability angle [Gutzwil-ler 1967 Rajaraman 1975], generalizes the harmonic-oscillator phase co, which would appear in (4.15), if CO, were a constant. The fact that X is real indicates the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det( — -I- co, ) is a function of X only,... 

It should be noted that, whereas ferroelectrics are necessarily piezoelectrics, the converse need not apply. The necessary condition for a crystal to be piezoelectric is that it must lack a centre of inversion symmetry. Of the 32 point groups, 20 qualify for piezoelectricity on this criterion, but for ferroelectric behaviour a further criterion is required (the possession of a single non-equivalent direction) and only 10 space groups meet this additional requirement. An example of a crystal that is piezoelectric but not ferroelectric is quartz, and ind this is a particularly important example since the use of quartz for oscillator stabilization has permitted the development of extremely accurate clocks (I in 10 ) and has also made possible the whole of modern radio and television broadcasting including mobile radio communications with aircraft and ground vehicles. 

Torsional vibrations are due to the stick-slip effect of the stabilizers in deviated boreholes. They can be seen at surface as large torque oscillations with a period of 3 to 10 s. Figure 4-308 shows a near-bit stabilizer in a deviated borehole. The stick-slip effect increases with WOB and RPM. 

The pressure spike introduces a disruption in the flow. Depending on the local conditions, the excess pressure inside the bubble may overcome the inertia of the incoming liquid and the pressure in the inlet manifold, and cause a reverse flow of varying intensity depending on the local conditions. There are two ways to reduce the flow instabilities reduce the local liquid superheat at the ONB and introduce a pressure drop element at the entrance of each channel, Kandlikar (2006). Kakac and Bon (2008) reported that density-wave oscillations were observed also in conventional size channels. Introduction of additional pressure drop at the inlet (small diameter orifices were employed for this purpose) stabilized the system. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

Chapter 11 consists of following Sect. 11.2 deals with the pattern of capillary flow in a heated micro-channel with phase change at the meniscus. The perturbed equations and conditions on the interface are presented in Sect. 11.3. Section 11.4 contains the results of the investigation on the stability of capillary flow at a very small Peclet number. The effect of capillary pressure and heat flux oscillations on the stability of the flow is considered in Sect. 11.5. Section 11.6 deals with the study of capillary flow at a moderate Peclet number. 

In this section the influence of the pressure in the capillary and the heat flux fluctuations on the stability of laminar flow in a heated capillary tube is analyzed. All the estimations performed in the framework of the general approach and developed in the previous section are kept also in the present cases. Below we will assume that the single cause for capillary pressure oscillations is fluctuations of the contact angle due to motion of the meniscus, whereas heat flux oscillations are the result of fluid temperature fluctuations only. 

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