Marginal stability condition

In this equation, the values of r as coefficients of A vanish and correspond to a transition point. Beyond this point, the real part Ar of the roots A] and A2 changes its sign and hence the system becomes unstable, and at the marginal state, we have A, I A2 0. Using Eq. (e), the marginal stability condition becomes... 

Or in terms of the critical affinity, we have the marginal stability condition... 

FIGURE 6.3 Marginal stability condition for thin layer heated from below. From Smith (1966) with permission. 

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) ... 

Thermodynamic equilibrium states correspond only to the stable or marginally stable states ( 0) of the mechanical analog. The first law of thermodynamics establishes the thermodynamic potential, while the second law of thermodynamics establishes the stability condition, as discussed in Chapter 5. 

It has been shown that the lowest order of the Q-expansion yields the macroscopic equation, and the next order the linear noise approximation, provided that the stability condition (X.3.4) holds. This condition is violated, albeit marginally, when a0() = 0. In this case the O-expansion takes an entirely different form its lowest approximation is a nonlinear Fokker-Planck equation. 

TDGL Formalism. Our object is to obtain a dynamical equation describing the onset of a pattern. The TDGL method has two main features it (l) identifies a few variables that specify the pattern and (2) develops relatively simple equations for these variables in the limit where the system is not too far from the point of marginal stability. We shall develop these ideas specifically in terms of the Fucus-type theory of the previous section and in particular near conditions wherein the X = 1 disturbance just becomes unstable. 

The differential solubilities exhibited by biomolecules thus should be appreciated as one of the most important aspects of the effects of water on living systems. Differential solubility is a critical principle in much of biochemical evolution, and it is a principle that is manifested in a number of contexts of adaptation to the environment. This is seen particularly clearly in the evolution of proteins in the face of different chemical and physical conditions. The amino acids selected to construct a particular protein reflect a finely tuned process that results in the generation of an appropriate three-dimensional structure and a correct balance between structural stability and flexibility—a balance termed marginal stability—that is essential for protein function. The marginal stability of the protein will be seen to be the consequence of complementary adaptations in the protein... 

If the reaction is five-centered (6 - 8), a silicon hydrotrioxide may be formed. Since there has been no previous report of a silicon hydro-trioxide, the stability of a species such as 8 under these conditions can only be estimated. Among carbon analogs, the reported dialkyltrioxides have only marginal stability at low temperatures (36, 37), and alkyl hydrotrioxides, proposed as intermediates in the ozonation of alcohols and ethers (38), decompose at ca. —10 °C. Admittedly, speculative extrapolation based on the comparative stabilities of other types of silicon and carbon analogs suggests that a silicon hydrotrioxide 8, should have been observable if it had been present, but this requirement is debatable. 

Figure 8.15 shows a possible form of the reaction rate along an adiabatic path. Points B and C ure such that the slope d rjd the curve exactly equals r/, and between these points the stability condition is violated. Since C undoubtedly lies to the left of E where d rfd vanishes, the region of unstable states will lie below the locus It is shown schematically as the cross-hatched region in Fig. 8.13. We notice that the design that minimizes 0 for given feed conditions (problem C, Sec. 8.2) corresponds to point C in Fig. 8.15. Such a reactor would be only marginally stable and the design would not be a good one. We also see from this figure that the problem was probably not particularly well specified. For 0 is the reciprocal of the slope of the line from the origin to a point on the curve. It therefore increases to a local maximum at B, decreases to a local minimum at C, and thereafter...

The choice of the thickness, AX, of the concrete layers does not appear as critical as that of At. Indeed, once the necessary stability condition (Equation A2.27) is satisfied with a certain margin, for example putting M 2 2N + 2), the transient is not very sensitive to the value of AX, specially after the first hours from the start of the accident. 

We have y= 72 mN/m, Pa = 1 g/cm, pg = pMIRT by the ideal gas law equals 0.00119 glcw . At marginal stability, the expression in brackets in Equation 5.139 vanishes. Inspection of Equation 5.139 shows that for a given value of V/ - Vgl, this condition occurs whenever the stabilizing quantity y a + (pa - 9 )glo. is minimized. Thus at marginal stability we have... 

Let us now suppose that the waves travel down the film with a velocity c and with no increase or decrease in amphtude. That is, we focus on the condition of marginal stability at the boundary between stable and unstable regions. Then the thickness h should have the form... 

At marginal stability, interfadal deflection remains constant so that the normal velocity atv = H is zero. Making use of Equations 6.21 and 6.22, we find from this condition that... 

Adapt the derivation of Section 3 to apply to the case of a solute with some surface activity which desorbs from a thin layer of liqitid into a gas phase. Derive the condition for marginal stability for the case of a flat interface = 0). Neglect surface diffusion effects. 

Another approach for analyzing the stability of the flow is based on wave-theory. In deriving the characteristics of kinematic and dynamic waves in two-component flow, Wallis has shown that the relations between the velocities of these two classes of waves govern the stability of the two stratified layers [74]. It has been shown that the condition of equal kinematic and dynamic waves velocities corresponds to marginal stability. Following this approach, Wu et al. determined the stratified/ nonstratified transition in horizontal gas-liquid flows [38]. The relations between the dispersion equation. Equation 16, and stability criteria Equation 33 on one hand, and the characteristics of kinematic and dynamic waves on the other hand, (for = 0), was shown in Brauner and Moalem Maron [45]and Crowley et al. [47]. 

Because of the thermodynamically marginal stability of the one structure relative to the other, the N D transition occurs in a relatively narrow trajectory of changing conditions. It is a highly cooperative process the disruption of any significant portion of the folded structure leads to the unfolding of all the rest. This cooperativity is illustrated in Figure 13.10. 

A comparison of observed and predicted hysteresis loops is shown in Fig. 7.10. Transition occurs before the marginal stability point is reached. Under the experimental conditions there is sufficient inhomogeneity to account for this effect. The temperature of the gas is probably not uniform throughout the sample so that observation is of a set of systems undergoing sequential transitions between the branches. 

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