Onset of instability

Figure 8.13. Time dependence after onset of instability of the driving energy and the fragmentation energy. The energy eondition (see text) prohibits spall fragmentation for times prior to intersection of the energy curves. An insufficient flaw structure would delay spall fragmentation to yet later times.

Whirl from fluid trapped in the rotor. This type of whirl oeeurs when liquids are inadvertently trapped in an internal rotor eavity. The meehanism of this instability is shown in Figure 5-24. The fluid does not flow in a radial direetion but flows in a tangential direetion. The onset of instability oeeurs between the first and seeond eritieal speeds. Table 5-4 is a handy summary for both avoidanee and diagnosis of self-exeitation and instabilities in rotating shafts. 

The temperature dependences of the isothermal elastic moduli of aluminium are given in Figure 5.2 [10]. Here the dashed lines represent extrapolations for T> 7fus. Tallon and Wolfenden found that the shear modulus of A1 would vanish at T = 1.677fus and interpreted this as the upper limit for the onset of instability of metastable superheated aluminium [10]. Experimental observations of the extent of superheating typically give 1.1 Tfus as the maximum temperature where a crystalline metallic element can be retained as a metastable state [11], This is considerably lower than the instability limits predicted from the thermodynamic arguments above. 

Tangney et al.92 studied the friction between an inner and an outer carbon nanotube. Realistic potentials were used for the interactions within each nanotube and LJ potentials were employed to model the dispersive interactions between nanotubes. The intra-tube interaction potentials were varied and for some purposes even increased by a factor of 10 beyond realistic para-meterizations, thus artificially favoring the onset of instabilities and friction. Two geometries were studied, one in which inner and outer tubes were commensurate and one in which they were incommensurate. 

Note that the radial and vertical components are out of phase, and that the coefficient multiplying r is only half that multiplying z. Thus, the effect of the ac field is to exert an oscillatory force on the particle with an effective field strength in the vertical direction that is twice the radial field strength. As a result of the larger field strength in the z-direction, the onset of instability is governed by the z-component of the equation of motion, so we need examine only that component. 

The replacement of Mn by Cr in the Ti-48Al-2Cr-2Nb alloy (Fig. 10.68) causes the predicted onset of instability with respect to a Cr-rich B2 phase. This in good accord with experimental observations in alloys of this type (Fuchs 1995, Kelly and Austin 1996) and at low temperatures there is also a potential for the formation of... 

At low liquid rates, the onset of instability occurs at a constant value of the total superficial velocity, and is predictable from holdup and flooding data for wetted wall columns. As liquid flow rates increase, Nicklin and Davidson predict that unstable flow begins at lower values of the gas flow rate. For high liquid flow rates, however, the slug length becomes important, and the unstable flow will begin at higher values of gas flow rate. Therefore, a definite liquid flow rate exists at which an unstable flow pattern appears with a minimum gas flow rate. 

With this simplification of the two dimensional step flow problem, we can study the long time behavior of the step train well beyond the initial onset of instability. We start with an array of 40 steps with small perturbations from an initial uniform configuration. We discretize the y coordinate so that each step has 2000 segments. Periodic boundary conditions are used in x and y direction. The time evolution problem of Eqs. (15) using (16) is converted into a set of difference equations. We control the time step so... 

Numerical simulations illustrating the role of diffusion in the onset of instabilities have been carried out by Hannusse.5 They confirm the modification of macroscopic behavior by local fluctuations that introduce such phenomena as delays or metastable states. 

The conditions under which the above stationary-state solution loses its stability can be determined following the recipe of 2.6. Again we find that instability may arise, and hence oscillatory behaviour is possible, in this reversible case. The condition for the onset of instability can be expressed in terms of the reactant concentration p < p p, where... 

The progress of a given reaction depends on the temperature, pressure, flow rates, and residence times. Usually these variables are controlled directly, but since the major feature of a chemical reaction is composition change, the analysis of composition and the resetting of the other variables by its means is an often used means of control. The possible occurrence of multiple steady states and the onset of instabilities also are factors in deciding on the nature and precision of a control system. 

Shibuya (SlO) dealt with the case of the onset of instability in film flow on the outer surface of a vertical tube. By assuming a mixed disturbing velocity of the cosine-hyperbolic cosine type, it was found that the numerical value of the Reynolds number for instability was approximately... 

Nb h Critical Reynolds number for onset of turbulence i Re, Reynolds number of gas stream N-Rei Reynolds number at onset of instability Nr Nusselt dimensionless film thickness parameter, defined by Eq. (97)... 

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

The Onset of Instability - The Critical Coagulation Concentration. Provided that the magnitude of the primary maximum is substantial, then the probability of the transition of the approaching particle into the primary minimum is small. However,... 

The phase behavior of mixtures of sterically stabilized dispersions and free polymer molecules is determined by the interparticle potential. In such systems, the interplay between the repulsive steric forces and the attractive forces due to the presence of free polymer molecules and van der Waals interaction between the particles determines the onset of instability. 

We show the reactor and jacket temperatures (T and T ) along with CA, the concentration of component A in the reactor. Initially, when the simulation started, the heat transfer area was sufficiently large to maintain open-loop static and dynamic stability. However, a few minutes into the simulation, we reduced UA by 20 percent. This creates dynamic instability with complex eigenvalues as in Eq. (4.25). The reactor temperature and composition start oscillating with a growing amplitude. However, the amplitude growth stops roughly 5 hours after the onset of instability and the reaction enters into a limit cycle of constant period and amplitude. 

The theoretical analysis indicated that asymmetric drainage was caused by the hydrodynamic instability being a result of surface tension driven flow. A criterion giving the conditions of the onset of instability that causes asymmetric drainage in foam films was proposed. This analysis showed as well that surface-tension-driven flow was stabilised by surface dilational viscosity, surface diffusivity and especially surface shear viscosity. 

The onset of instability is predicted at Rccr = 519 (based on displacement thickness as the length scale). It is important to realize that instability and transition are not synonymous. Actual process of transition begins with the onset of instability but the completion may depend upon multiple factors those form the basis for adjunct topics like secondary tertiary and... 

Following the path-breaking experiments of Schubauer Skramstad (1947), there have been sustained efforts to link the stability theory in predicting transition. Michel (1952) reported first that his compiled data showed the transition to be indicated when the total amplification of TS waves corresponded to A/Ao 10, where Aq is the disturbance amplitude at the onset of instability. This motivated Smith Gamberoni (1956) and van Ingen (1956) to use temporal theory results to show that at transition the total amplification is given by,... 

Fourier and Laplace transforms are linear transforms and are very often used for analyzing problems in various branches of science and engineering. Since receptivity is studied with respect to onset of instability, it is quite natural that these transform techniques will be the tool of choice for such studies. Fourier transform provides an approach wherein the differential equation of a time dependent system is solved in the transformed plane as. 

This equation can be used to describe the onset of instability, when a suitable mean flow is defined. We note that this equation is very generic for all incompressible flows (steady or unsteady flows), as it is based on full Navier-Stokes equation without making any assumptions. In Sengupta et al. (2006a) this equation has been used to explain the classical linear instability theory for parallel flows showing exactly identical TS waves obtained from Orr-Sommerfeld equation. In section 4.3, this is fully explained with the development of the actual equations and results. For the computational data, a mean flow was taken at t = 20 as representative undisturbed flow and the right hand side of (3.5.2) was calculated and plotted as shown in Fig. 3.9- at some representative times. 

Instability analyses do not provide good indications of fully developed transverse structures of detonations because these structures correspond to highly nonlinear phenomena. A great deal of nonlinear evolution would occur between onset of instability and attainment of a mature multidimensional detonation structure. Intersections of oblique shocks are known to constitute a central element in transverse structures of detonations [69], [72]. Oblique-shock relations are therefore relevant to the nonplanar structure. 

From the previous discussion it is apparent that provided the magnitude of is substantial the probability of transition into the primary minimum is small. However, as shown above when 1 = 1 = 0 the transition is facile (Fig. 11) and the particles coagulate. Therefore, we can define conditions for the onset of instability as (Verwey et oL, 1948)... 

The classical linear stability theory for a planar interface was formulated in 1964 by Mullins and Sekerka. The theory predicts, under what growth conditions a binary alloy solidifying unidirectionally at constant velocity may become morphologically unstable. Its basic result is a dispersion relation for those perturbation wave lengths that are able to grow, rendering a planar interface unstable. Two approximations of the theory are of practical relevance for the present work. In the thermal steady state, which is approached at large ratios of thermal to solutal diffusivity, and for concentrations close to the onset of instability the characteristic equation of the problem... 

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