Boundaries of stability

The onset of flow instability in a heated capillary with vaporizing meniscus is considered in Chap 11. The behavior of a vapor/liquid system undergoing small perturbations is analyzed by linear approximation, in the frame work of a onedimensional model of capillary flow with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes on the flow stability is studied. A scenario of a possible process at small and moderate Peclet number is considered. The boundaries of stability separating the domains of stable and unstable flow are outlined and the values of the geometrical and operating parameters corresponding to the transition are estimated. 

Peaking and Non-isothermal Polymerizations. Biesenberger a (3) have studied the theory of "thermal ignition" applied to chain addition polymerization and worked out computational and experimental cases for batch styrene polymerization with various catalysts. They define thermal ignition as the condition where the reaction temperature increases rapidly with time and the rate of increase in temperature also increases with time (concave upward curve). Their theory, computations, and experiments were for well stirred batch reactors with constant heat transfer coefficients. Their work is of interest for understanding the boundaries of stability for abnormal situations like catalyst mischarge or control malfunctions. In practice, however, the criterion for stability in low conversion... 

The combustion regime at the limit (including the limit which depends on the heat transfer) is located at the boundary of stability of the steady regime. The boundary of feasibility of the steady regime (see [11]) is not attained. 

Fig. 76. Diagrams of mineral equilibria in silicate iron-formations in the absence of carbon dioxide (isothermal sections) / = actual boundaries of stability fields of minerals 2 = boundaries unrealistic under the given conditions S = isobars of fluid pressure (P, = jO + kbar) 4 = isobars of log...

The experimental results for the unmodified seesaw appear in Figs. 5.7, 5.8, and 5.9 for pole locations A, = 1, 2 and 3, respectively. For A. = 1, the system was on the boundary of stability and continually oscillated. For A. = 2 and 3, the system was stable and set point changes were possible. The quite large discrepancy between simulation and experiment can be attributed to the inevitable peak in sensitivity function coupled with the modelling approximations involved. Higher values of A, gave violent instability. 

To answer the question raised in the preceding paragraph, we must first itemize the main types of boundaries of stability regions of equilibrium states and periodic orbits. To do this we must undertake a systematic classification of the information we have presented in all previous chapters. We will pay special attention to the features that distinguish each type of boundaries. 

One more codimension-one boundary of stability of periodic trajectories which corresponds to the blue sky catastrophe [152]. It may occur in n-dimensional systems where n > 3. 

Dynamically definite and indefinite boundaries of stability regions... 

The answer is obvious when we deal with principal safe boundaries of stability regions the representative point tends to a new stable regime which,... 

Bautin, N. N. [1949] Behavior of Dynamical Systems near the Boundaries of Stability Regions (OGIZ GOSTEHIZDAT Leningrad). 

Bautin, N. N. and Shilnikov, L. P. [1980] Suplement I Safe and dangerous boundaries of stability regions, The Hopf Bifurcation and Its Applications Russian translation of the book by Marsden, J. E. and McCracken, M. (Mir Moscow). 

In general, phase behavior of three-component systems is compUcated and includes one-, two-, and three-phase regions. The analysis of the whole phase behavior of a system described by Eq. (7.1), then, is fairly complicated. Ginzburg [84], and He, Ginzburg, and Balazs [85] considered a simpler problem of detertniriing the spinodal boundaries, that is, the boundaries of stability of homogeneous (one-phase) systems. The condition of stabUity of the homogeneous phase could be written as ... 

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