Instability criteria

The thrustor was considered to consist of two sections 1) where the mixture is formed and 2) where combustion takes place and the pressure is generated. The principal mechanism involved in the combustion process was assumed to be successive ignition, but other mechanisms such as turbulent frontal combustion were also considered. The analysis yielded two instability criteria, expressed in terms of the Mach number in zone 1, the velocity ratio in zones 1 and 2, the isentropic exponent in zone 2, the activation energy, the temperature of the cold gas, the pressure upstream of the combustion zone, and the pressure drop due to the combustion... 

Escoffier (E5), 1961 Analysis of onset of instability in open channel flow and origin of waves of instability. Discussion of earlier instability criteria. 

It is worthwhile illustrating the proof of the principle of exchange of stabilities for the Rayleigh-Benard problem. Not only will this allow us to discuss the derivation of instability criteria for the case of no-slip boundaries, but the approach to proving this principle can also be applied to other problems. 

However, the full instability criteria (8.5.1)-(8.5.3) still cannot distinguish stable states from metastable states but then, no differential test can make this distinction. To distinguish stable states from metastable states, we must apply an appropriate equilibrium criteria. For example, if T and P have been specified for a proposed state, then the stable state is the one that minimizes the Gibbs energy. Using this as a basis, we showed how to identify the stable state for pure fluids and for binary mixtures. 

Finally, we consider in some detail a selection of prominent fracture mechanisms in polymers and finish with a table of the measured initial fracture-instability criteria. 

When the mean zonal flow is more realistic, with both latitudinal and vertical shear [u y, z)], the basic-state potential vorticity P replaces the absolute vorticity in determining instability criteria. A necessary condition for inertial instability is that fP<0 somewhere in the fluid. This criterion is easily related to the Richardson number if the basic-state flow is assumed geostrophic. From Section I.G.4, the potential vorticity can be written... 

Here, the intrinsic rate is represented by rc = kiT)fiC), and the subscript b denotes the bulk fluid conditions. The exponent 9 is a constant depending on pellet geometry q — 0, 1, and 2 for a slab-like, cylindrical, and spherical pellets, respectively. It is rather difficult to establish stability for the general problem as posed since this requires an analysis of a very wide class of perturbations. It is much easier to establish instability conditions. As a result, most of the studies available are concerned with instability criteria. 

As mentioned in Chap. 3, for depfli/diameter ratios less than 0.5, such as in large cylindrical LPG and LNG tanks, flie boundary layer flow across the base may be broken by the creation of vertical fliermals spaced horizontally at intervals approximating to the liquid depth according to Rayleigh s instability criteria for natural convection. In aU cases, the heat in-flow is carried by boundary layer flows, and thermals, to the liquid surface. 

Analysis of flow excursion The threshold of flow excursion can be predicted by evaluating the Ledinegg instability criterion in a flow system or a loop, Eq. (6-1),... 

Novozhilov (Ref 9) noted other instances where the instability criterion of Zel dovich is not satisfied. He also noted ZeTdovich s assertion that the form of the stability criterion may change if the variation in the surface temperature and the inertia of the reaction layer of the condensed phase are taken into account, and stability criteria obtained under the assumption that the chemical reaction zone in the condensed phase and all of the processes in the gas phase are without inertia. Novozhilov used a more general consideration of the problem to show that the stability region is determined by only two parameters Zel dovich s k and the partial derivative r of the surface temperature with respect to the initial temperature at constant pressure t=(dTi/(fro)p. Combustion is always stable if k 1, combustion is stable only when r >(k — l) /(k +1)... 

Kekule s instability criterion failed completely in the case of many coordination compounds, which were classified as molecular compounds by sheer dint of necessity although they were extremely resistant to heat and chemical reagents. For example, although hexaamminecobalt(III) chloride contains ammonia, it neither evolves this ammonia on mild heating nor does it react with acids to form ammonium salts. Also, addition of a base to its aqueous solution fails to precipitate hydrated cobalt(III) hydroxide. 

The differential stability criteria were derived by finding conditions that maximize the total entropy in an isolated system. Those conditions constrain how the system responds to thermal, mechanical, and diffusional fluctuations. In the derivations, those constraints are conveniently posed as stability criteria they show us that a stable substance must always obey the thermal criterion (8.1.23), the mechanical criterion (8.1.31), and the diffusional criterion (8.3.14). But the converses of those statements are not always true for example, a mechanically stable fluid always has Kj > 0, but a fluid having Kj > 0 is not necessarily stable— it might be metastable. Therefore, in using these differential criteria (as opposed to merely deriving them), many ambiguities can be avoided if we repose each constraint in the form of an instability criterion such criteria identify those thermodynamic states at which a pure substance or mixture is differentially unstable. 

The first instability criterion is that a thermally unstable substance always has... 

The forms (8.5.1)-(8.5.3) show that these differential criteria are inclusive a mixture that is diffusionally stable is also mechanically stable, and a mechanically stable substance is also thermally stable. Inversely, a thermally unstable fluid is also mechanically unstable, and a mechanically unstable mixture is also diffusionally unstable. In addition, use of the diffusional instability criterion (8.5.3), may remind us that a binary mixture can be diffusionally unstable because Kj < 0 even when Gji > 0. 

Developing a quantitative theory of the conditions for shear localization is the subject of ongoing numerical (finite element-based) research. Key earlier papers are (Recht 1964) in which the instability criterion dx/dy = 0 was first applied, (Semiatin and Rao 1983) in which it was argued that dx/dy needed to be substantially negative and (Hou and Komanduri 1997) in which the complexities of temperature distributions in shear localized chips were examined in more detail than in previous work. Adiabatic shearing has been the subject of a number of general reviews, for example, Walley (2007), and books, for example, Bai and Dodd (1992). These mention but do not have a main focus on machining. Walley (2007) mentions nine earlier reviews. 

In the case of horizontal shear only, //, is proportional to /(/ + f) the result of the previous derivation is obtained. For the case of vertical shear only, //is proportional to Ri -1 and the instability criterion becomes Ri< 1. 

Though the details of the stability calculations cannot be given here, the nature of the results can be seen from the example of the linear multipole. For flute modes the instability criterion 6W < 0 can be written ... 

But Kekuld s stability criterion, or to be more accurate, instability criterion failed completely in the case of many coordination compounds, especially the metal-ammines, which were classified as molecular compounds by sheer dint of necessity even though they were extremely resistant to heat and chemical reagents. For example, look at Figure 1. Although hexaamminecobalt(III) chloride contains ammonia, it neither evolves this ammonia on mild heating nor does it react with acids to form ammonium salts. Also, despite its cobalt content, addition of a base to its aqueous solution fails to precipitate hydrated cobalt(III) hydroxide. It remained for Alfred Werner to explain successfully the constitution of such compounds, but the time was not yet ripe. Before considering Werner s coordination theory, we must examine one more theory of coordination compounds, perhaps the most successful of the pre-Wemer theories, namely, the Blomstrand-J0rgensen chain theory. 

We note that the instability criterion in the form of Equations 6.79-6.81 can be applied only for the K-plane of special composition Nr. This is the additional condition, similar to the sign of the second-order derivative determining the type of extremum (maximum, minimum, or inflection point) for some function (the first derivative being equal to zero). The condition Equation 6.76 can be used for finding the prospective unstable candidates. ... 

Again, we encounter our previous instability criterion. A perturbation will grow dSefdt > 0) provided that 6 < 1 [that is the same as condition (5.22) with eo time constant r( ), which varies as -the inverse of (1 — This time constant is... 

What has been shown here is that energy considerations for a viscoelastic medium in the non-inertial approximation give no more than a Griffith instability criterion similar to that for an elastic medium. One cannot therefore hope to obtain a condition determining crack velocity from a non-inertial energy equation. The status of conditions which emerge by using approximate solutions [Christensen (1979), Christensen and Wu (1981)] has been discussed by Christensen and McCartney (1983). 

For this problem, follow the steps given below to obtain an instability criterion. 

The lattice-vibration instability model [40, 55-59] extends Lindemann s vibrational-lattice instability criterion [60]. The melting behavior of a nanosolid is related to the ratio (/ ) of the root-mean-square displacement (RMSD, of an atom at the surface to the RMSD of an atom inside a spherical dot. p is a size-independent parameter ... 

Equations [84] and [85] suggest that the droplet size in the steady state is essentially determined by competition of the interfacial tension and the viscous shear stress, as considered in the classical Taylor theory " for an instability criterion of a single droplet. However, the behavior of the normal stress difference (eqn [83b]) is not fully understood from this theory. Concerning this point, Doi and Ohta proposed a phenomenological model for the interface anisotropy and spedfic interfacial area in blends having the characteristic length determined only by the shear. The predictions of the Doi-Ohta model are consistent with the experimental observation (eqns [83]-[85]) as well as the scaling behavior observed for... 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

For purposes of this survey an important distinction must be made between a commonly used criterion of instability and the sense in which the term is used here. A stable dispersion is not necessarily one which will not settle, because the only disturbance involved in settling is motion of translation of the parti-... 

Below, the procedure for the determination of dominant campaigns in a version that was proposed by Lazaro et al. (1989) is outlined. Their methodology includes enumeration of feasible production sequences, selection of dominant production lines, task sequencing, and search for an optimum with constraints. All possible production variants are generated by an enumeration procedure that takes into account the possibility of available equipment working in parallel, initial and final task overlapping, and instability of intermediate products. Non-feasible sequences are eliminated so that only favourable candidates are subjected to full evaluation. Dominant production lines are selected by maximizing the criterion ... 

Most substances which appear in the examples of this chapter are analysed In Part Two and their enthalpy of decomposition determined experimentally. This is because most of them are considered hardly stable. This is one of the reasons for assigning no Tow risk in the suggested classifications. But it is also indisputable that criterion Cf overestimates the instability risk. It is the case for all aromatic compounds that are generally very stable. In the examples above, N-methylaniline, dichlorobenzene... 

Criterion C2 takes into account criterion C as well as the enthalpy of combustion of the compound. This criterion is the result of the observation that compounds of simiiar enthalpies of decomposition and combustion are the only ones that present a risk bonded to instability. 

When analysing the resulte it is noticed this time that criterion C4 is the least severe of the four CHETAH criteria. It emphasises the unstable property of nitroaniline but under mates the instability of ammonium nitrate and ammonium dichromate for which there is no indication of danger whatsoever. 

It is to be noticed that three criteria depend on this criterion which seems somehow redundant. It is particularly true in the case of C4 and C, which from a thermodynamic point of view are exactly the same. Knowing the dangerous nature of these compounds, it would seem that C4 is more representative of instability risk than C. ... 

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