Marginal stability analysis

The fact that an infinity of front velocities occurs for pulled fronts gives rise to the problem of velocity selection. In this section we present two methods to tackle this problem. The first method employs the Hamilton-Jacobi theory to analyze the dynamics of the front position. It is equivalent to the marginal stability analysis (MSA) [448] and applies only to pulled fronts propagating into unstable states. However, in contrast to the MSA method, the Hamilton-Jacobi approach can also deal with pulled fronts propagating in heterogeneous media, see Chap. 6. The second method is a variational principle that works both for pulled and pushed fronts propagating into unstable states as well as for those propagating into metastable states. This principle can deal with the problem of velocity selection, if it is possible to find the proper trial function. Otherwise, it provides only lower and upper bounds for the front velocity. 

W. van Saarloos, M. van Hecke and R. Holyst, Front propagation into unstable and metastable states in smectic-C liquid crystals Linear and nonlinear marginal-stability analysis, Phys. Rev. E, 52, 1773-1777 (1995). 

Develop final Perform structural design of analysis of component acceptable accuracy Determine structural response—stresses, support reactions, deflections, and stability—based on a structural analysis of acceptable accuracy. Determine acceptable accuracy based on economic value of component, consequences of failure, state-of-the-art capability in stress and stability analysis, margin of safety, knowledge about loads and materials properties, conservatism of loads, provisions for further evaluation by prototype testing... 

Also, Schellman s work is pertinent (1809). From studies on heats of dilution of urea in water he concludes that the N—H 0=C bond has an enthalpy of 1.5 kcal/mole in aqueous solution, and he carries this value over to proteins and polypeptides. Among these complicated materials he is forced to approximate—but he deduces relations which show the stability of helices and sheets in terms of H bond enthalpy and configurational entropy. From this he draws the important conclusion that H bonds, taken by themselves, give a marginal stability to ordered structures which may be enhanced or disrupted by the interactions of the side chains. Schellman ends his papers with a discussion of experimental tests needed to eliminate some of the assumptions in his theoretical analysis. 

We finally mention that the stability analysis of front solutions mentioned before (van Saarloos, 2003 Ebert and van Saarloos, 2000) reveals that the selected velocity (c or v for the pulled or the pushed situation respectively) arising from well localized initial conditions is the marginally stable one in the sense that it is the smallest velocity... 

Quality windows are also delineated by set values of the damping coefficients and attenuation factors that are computed in stability and frequency analysis. These too can be traced out efficiently in parameter space by augmented continuation schemes. The same is true of the turning points and bifurcation points in parameter space, points of marginal stability. These are the guides to situations in which there is more than one stable operating state. When such situations may arise, it becomes desirable to solve repeatedly the full equation system of flow for transient behavior in order to know how different start-up procedures and upsets select among the multiple stable states. 

The analysis is very similar to that of Section 3 except, of course, flow and transport in both phases must be considered. When a 1 (i.e., when both fluids are of great depth), and when solnte convection along the interface can be ignored, the marginal stability condition is that obtained by Stemling and Scriven (1959) ... 

According to linear stability analysis the trivial stationary point is an unstable saddle point, while the nontrivial stationary point is a marginally stable centre. 

From this numerical analysis we find, that within numerical precision ( 1%) the tip radius R(0,t) performs very small oscillations near the value R(0,t) r, i.e. just at the width of the dendrite, where the speed of the propagating sidebranch-front equals the axial velocity of the tip. Moreover, the result is practically independent of the parameter e( 1..2) which governs the side-branch amplitude. This behavior is also found in experiments [2]. Our explanation for this is summarized as the marginal stability principle, to be discussed in the next section. 

An alternative way of understanding the marginal stability mechanism is the philosophy, that the fastest mode of the set (8) dominates the systems behavior. This seems to be true in cases with periodic patterns [7,9]. In the simpler system [10], however, the mode marginally stable against perturbations (of finite support) is the slowest one. And again, the full nonlinear analysis there shows, that it is ultimately being selected [10]. 

We conclude, therefore, that a satisfactory criterion for mode selection requires a nonlinear analysis as sketched in the flow diagram fig. 5). For cases analyzed sofar, however, the marginal stability principle appears to play a dominant role. 

In previous chapters, Laplace transform techniques were used to calculate transient responses from transfer functions. This chapter focuses on an alternative way to analyze dynamic systems by using frequency response analysis. Frequency response concepts and techniques play an important role in stability analysis, control system design, and robustness analysis. Historically, frequency response techniques provided the conceptual framework for early control theory and important applications in the field of communications (MacFarlane, 1979). We introduce a simplified procedure to calculate the frequency response characteristics from the transfer function of any linear process. Two concepts, the Bode and Nyquist stability criteria, are generally applicable for feedback control systems and stability analysis. Next we introduce two useful metrics for relative stability, namely gain and phase margins. These metrics indicate how close to instability a control system is. A related issue is robustness, which addresses the sensitivity of... 

A linear stability analysis of the steady state xq of this system implies stability whenever the real normal mode frequency is negative, and instability otherwise, where xu = -1 + nx - nx. Points of marginal stability correspond to u = 0 and, according to the quadratic expression for oi, there may be 0, 1, or 2 points. It is easy to see that there is no instability for n < nc = 4. 

Time-Delay Compensation Time delays are a common occurrence in the process industries because of the presence of recycle loops, fluid-flow distance lags, and dead time in composition measurements resulting from use of chromatographic analysis. The presence of a time delay in a process severely hmits the performance of a conventional PID control system, reducing the stability margin of the closed-loop control system. Consequently, the controller gain must be reduced below that which could be used for a process without delay. Thus, the response of the closed-loop system will be sluggish compared to that of the system with no time delay. 

In this statement, we have used "polar plot of G0l" to replace a mouthful of words. We have added G0L-plane in the wording to emphasize that we are using an analysis based on Eq. (7-2a). The real question lies in what safety margin we should impose on a given system. This question leads to the definitions of gain and phase margins, which constitute the basis of the general relative stability criteria for closed-loop systems. 

Are the equilibrium constants for the important reactions in the thermodynamic dataset sufficiently accurate The collection of thermodynamic data is subject to error in the experiment, chemical analysis, and interpretation of the experimental results. Error margins, however, are seldom reported and never seem to appear in data compilations. Compiled data, furthermore, have generally been extrapolated from the temperature of measurement to that of interest (e.g., Helgeson, 1969). The stabilities of many aqueous species have been determined only at room temperature, for example, and mineral solubilities many times are measured at high temperatures where reactions approach equilibrium most rapidly. Evaluating the stabilities and sometimes even the stoichiometries of complex species is especially difficult and prone to inaccuracy. 

The wide margins of error associated with estimating ATS production make trend analysis difficult. It is clear, however, that following a strong increase of ATS production in the 1990s, the situation appears to have stabilized in the last few years. 

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