Instability condition

The fact that symmetry breaking occurs at shorter interatomic distances for multiple bonds than for single bonds may be understood within two different languages. One refers to the instability conditions of the symmetry-adapted solution. In multiple bonds some bonding... 

Each of the transitions shown in Figure 3.3 has been interpreted in the light of force balances. Thus, the transition between wavy (SW) flow and annular (AD) (or intermittent, I) flow is caused by the occurrence of instability conditions for the waves when the gaseous phase accelerates and produces a reduction of the local... 

Now we can readily deduce the stability/instability conditions from Figure 6. For < J the attraction is too weak to overcome the particle entropy loss and the dispersion is stable. 

Details of the instability conditions of the debond process and Zmax are discussed in Section 4.3.4. Further, the solution for the initial frictional pull-out stress, (T, upon complete debonding is determined when the debond length, , reaches the embedded length, L, and the crack tip debond stress, ai, is zero ... 

The instability condition requires that the derivative of the partial debond stress with respect to the remaining bond length z = L — j k equal to or less than zero, i.e., dtr /dz O (Kim et al., 1991). Therefore, the fiber debond process becomes unstable if L - t) is smaller than a critical bond length, Zmax. where the slopes of the curves become zero in Figs. 4.23 and 4.24. At these bond lengths, the partial debond stress, maximum debond stress, crj. The Zmax value is determined from Eq. (4.102) as... 

Eq. (4.106) is essentially similar to the solution of the debond stress derived earlier by Takaku and Arridge (1973). The above instability condition for the partial debond stress of Eq. (4.105) gives a rather simple equation for z ax as... 

It is always very useful to be able to predict at what level of external stress and in which directions the macroscopic yielding will occur under different loading geometry. Mathematically, the aim is to find functions of all stress components which reach their critical values equal to some material properties for all different test geometries. This is mathematically equivalent to derivation of some plastic instability conditions commonly termed as the yield criterion. Historically, the yield criteria derived for metals were appHed to polymers and, later, these criteria have been modified as the knowledge of the differences in deformation behavior of polymers compared to metals has been acquired [20,25,114,115]. 

Therefore, the energy decreases continuously with time if the Rayleigh instability condition is satisfied,... 

At the beginning of this chapter, we mentioned that Ivanova [167,168], by using the Clark algorithm [235], had recently found sufficiently general conditions to ensure steady state uniqueness. Additionally, she formulated the instability conditions for this steady state. 

The particle switches if the equilibrium condition and the instability condition... 

An alternative instability condition of - 0 is now apparent since ddjd -> °° when ... 

Problem 12-15. Stability of a Fluid Layer in the Presence of Both Marangoni and Buoyancy Effects. A fluid layer is heated from below. It has a rigid, isothermal boundary at the bottom, but its upper surface is a nondeforming fluid interface. There are now two potential mechanisms for instability when the fluid is heated from below buoyancy-driven and surface-tension-gradient-driven convection. Determine the eigenvalue problem (i.e., the ODE or equations and boundary conditions) that you would need to solve to predict the linear instability conditions. Is the principle of exchange of stabilities valid Discuss how you would approach the solution of this eigenvalue problem. 

Symmetry and stability analysis. The semi-empirical unrestricted Hartree-Fock (UHF) method was used for symmetry and stability analysis of chemical reactions at early stage of our theoretical studies.1,2 The BS MOs for CT diradicals are also expanded in terms of composite donor and acceptor MOs to obtain the Mulliken CT theoretical explanations of their electronic structures. Instability in chemical bonds followed by the BS ab initio calculations is one of the useful approaches for elucidating electronic structures of active reaction intermediates and transition structures.2 The concept is also useful to characterize chemical reaction mechanisms in combination with the Woodward-Hoffman (WH) orbital symmetry criterion,3 as illustrated in Figure 1. According to the Woodward-Hoffmann rule,3 there are two types of organic reactions orbital-symmetry allowed and forbidden. On the other hand, the orbital instability condition is the other criterion for distinguishing between nonradical and diradical cases.2 The combination of the two criteria provides four different cases (i) allowed nonradical (AN), (ii) allowed radical (AR), (iii) forbidden nonradical (FN), and (iv) forbidden radical (FR). The charge and spin density populations obtained by the ab initio BS MO calculations are responsible for the above classifications as shown in Fig. 1. 

It is possible to relate the instability condition (2.20) obtained from linear stability analysis to the form of the two nullclines and, more specifically, to the position of the steady state on the product nullcline... 

Taking into account relations (2.15), we can rewrite the instability condition (2.20), obtained from linear stabihty analysis, in the form ... 

Given the definition A = qaik, the comparison of relations (2.23) and (2.25) allows us to express the instability condition in the phase plane as a function of the slope (da/dy) at steady state, in the compact form of inequality (2.26) ... 

The linear stability analysis of these equations, coupled to phase plane analysis, again leads to the instability condition (2.26) obtained for the model in the absence of product recycling. 

INSTABILITY CONDITIONS LEADING TO INSTABILITY CALCULABLE VARIABLES RELEVENT TO INSTABILITY... 

When D < 0, There is no equilibrium state, the system is not stable, we get specific instability conditions... 

Sun M G, et al. 2005. The mechanism of water inrush of based on the layered rock mass seepage instability conditions. Journal of China University of Mining and Technology, 34(3) 284-288. 

K fixed. Then these equations determine the instability threshold for each structural mode, fXc(Pi)- The explicit form of the instability conditions depends on the particular internal kinetics, F. The threshold of the Turing instability is given by fxj = min /Uc(/32), , if the USS becomes unstable as fx increases, or... 

If the coupling occurs via the activator only, = 0, then the instability conditions read... 

In the limit of short wavelength perturbations, k oo, and for p, p, the following relation, which is similar to the Kelvin-Helmholtz instability condition [27, 28] is obtained ... 

Table 5.11 Values of the Statistics Subject to Various Instability Conditions... 

In this Chapter, an electronic solution for the instability conditions of the organic devices is provided. The circuit is an example of the voltage programming technique for the compensation of the threshold voltage variation and mobility degradation of the OTFT and the OLED luminance degradation. The circuit is also simulated using PSPICE (circuit simulation tool) to verify the results. 

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