Destabilizing terms

Dry State. The sidechain polar groups, neutral or charged, serve as the dipoles. The only additional energy term that needs to be added to Equation 4 is a destabilizing term on the right to account for the surface tension between the cluster and nonpolar phases. Equation 4 then takes the form,... 

The wave growth timescale is approximated by neglecting the surface tension and viscous effects and maintaining only the aerodynamic destabilizing term ... 

Equation 18 is essentially the K-H stability criterion. It is arranged in a form which clearly shows the relative contributions of the stabilizing and destabilizing effects (normalized with respect to gravity). The gravity and surface tension stabilizing terms constitute the rhs of Equation 18. The Ihs includes the destabilizing terms due to the two-phase inertia (J, J ). These vanish for particular combinations of wave celerity and (constant) shape factor related by = y ("f For... 

The relative contributions of the gas and liquid destabilizing terms along the stratified flow boundaries are to be carefully considered according to the particular physical system (e.g., horizontal gas-liquid, inclined gas-liquid, or liquid-liquid flows) and the range of operational conditions. In the extremes of 1 the... 

The destabilizing term 1, in Equation 25 requires knowledge of the memory coefficient, C, as defined in Equation 23. The coefficient, C, is to be extracted from experimental findings which reflect the dynamic interfacial interactions. Observations of stratified-smooth/stratified-wavy transitional boundaries from various laboratories reported in the literature bear a potential of a data-base for correlating C. These are summarized in Table 1. 

Figure 14 represents the liquid inertia destabilizing term, (normalized to gravity) along the stability boundary. Clearly, along the ZNS boundary, (k, J ) 0, and... 

Correspondingly, the liquid destabilizing effect as represented by certainly cannot be ignored. Moreover, in view of Figures 10-12 in the range which corresponds to stratified/intermittent transition, the liquid destabilizing term, in fact, even dominates,... 

Figure 16. Relative contribution of the destabilization terms aiong the ZNS boundary (J, JJ—effect of iiquid viscosity.

Table 2 indicates the various controlling destabilizing terms along the stratified-smooth/wavy transitional boundary in some limited physical situations. 

Jeffreys stability condition (Equation 43.1) with a constant value for s was applied by Taitel and Dukler [19] in attempting to predict the stratified-smooth/wavy transitional boundary for air-water flows in closed conduits. The value needed for the sheltering coefficient, in order to fit transitional data in 2.5 and 5.1cm tubes, was s = 0.01, in reasonable agreement with Figure 21. However, the omission of the inertia destabilizing terms, Ja, Jb in Taitel and Dukler [19], while employing Equation 43.1 for small conduits, should be carefully considered. 

Why is such a trend observed Actually, the reason that hard acids and bases prefer to interact with each other is different than the reason that soft acids and bases prefer to interact. To see this, let s examine some mathematics that is meant to model the interaction between Lewis acids and bases in an early stage of their interaction. The analysis derives from pertur-bational molecular orbital theory (PMOT), which was briefly introduced in Chapter 1, and is explored in more depth in Chapter 14. In essence, three forces are considered to mediate the energy of interaction (Ej) between the acid and base as they approach each other in space (Eq. 5.28). One is the electrostatic repulsion between the electron clouds of the two entities, referred to as Ecore/ a positive destabilizing term. The second and third factors are both attractive and stabilizing. An electrostatic attraction between an acid and base occurs due to opposite charges on the acid and base this is called E s- Lastly, a term called Eoveriap/ which is related to the net overlap of the nucleophilic and electrophilic orbitals, is found to lower the energy of the system as the nucleophilic electrons delocalize into the empty electrophilic orbital. 

The theory for the threshold of the instability in cells with thicknesses considerably exceeding the equilibrium pitch (1>Pq) has been considered by analogy with the case of dielectric instability [121, 266], but with allowance being made for the additional, destabilizing term in the free energy which is caused by the space charge. The frequency dependence of the threshold field for <0 has been shown to be similar to that caleulated for nematics. For a cholesteric liquid crystal with >0 the presence of electrical conductivity is revealed by a lowering of the threshold of the instability at low frequencies. 

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