Primary instability

At the inflow and at the top of the computational domain one calculates the flow variables, as induced by the freestream vortex via Biot-Savart interaction rule. At the outflow, fully developed condition is applied for the wall-normal component of the velocity ( = 0) and using the same in SFE, one can obtain the vorticity boundary condition at the outflow from Equation (3.4.2). At the top frame of Fig. 3.8, one sees incipient unsteady separation on the wall. In subsequent frames, one notices secondary and tertiary events induced by the primary instability. In these computed cases, one does not notice TS waves and the vortices formed on the wall are essentially due to unsteady separation that is initiated by the freestream convecting vortex. These ensemble of events have been noted as the vortex-induced instability or bypass transition in Sengupta et al. (2001, 2003), Sengupta Dey (2004) and in Sengupta Dipankar (2005). 

The discussion following Eqn. (5.1.8) imply a single Hopf bifurcation when Reynolds number increases beyond Rccr It is interesting to note that Landau (1944) talked about further instabilities following the nonlinear saturation of the primary instability mode. This is akin to Floquet analysis of the resulting time periodic system (Bender Orszag (1978)). The possibility of multiple bifurcation was also mentioned in Drazin Reid (1981) who stated that in more complete models of hydrodynamic stability we shall see that there may he further bifurcations from the solution A = 0, e.g. where the next least stable mode of the basic flow becomes unstable, and from the solution A = Ae- To the knowledge of the present authors, no theoretical analysis exist that showed multiple bifurcation before for this flow. Here,... 

The buoyancy force of an initial density instability is a function of its volume. In the absence of extension, a relatively strong overburden may thus be able to resist the growth of all but the very largest of initial instabilities. This could explain why subvolcanic granitoid complexes, which provide high-amplitude primary instabilities, may be favoured to grow into final domes (e.g. Hickman 1984 Collins 1989). 

The existence of low-frequency combustion oscillations superimposed on the primary instability has been reported by De Zilwa et al. [9] for premixed gaseous systems. As noted earlier, the present pressure and CH-photo-diode measurements also reveal low-frequency oscillations at around 12 Hz. This low-frequency mode is due to the cavity between the acoustically-closed upstream inlet end and the constricted exit nozzle of the air-delivery inlet that leads to a bulk mode oscillation in the flame. The existence of these oscillations can be clearly seen in Figs. 15.2, 15.5, and 15.6, where the time-variations in the peak pressure and... 

Cavities withPr 0.7 and W/H a 8. For HIL > 5 and 0 < 0 < 60°, direct application of the horizontal scaling law (Eq. 4.104) introduces significant errors when Pr = 0.7 and Ra = 104 [144] these errors have been shown [66, 235] to result from a secondary instability that appears at a Rayleigh number only slightly greater than that for the primary instability discussed in the section on horizontal rectangular parallelepiped and circular cylinder cavities. A modified scaling relation, taken from Hollands et al. [144], is recommended for 0 < 0 < 60° ... 

Figure 5. Bifurcation diagram on the plane of the two control parameters p and a. The solid lines 1 and 2 mark the primary instability, where the homogeneous homeotropic orientation becomes unstable. At 1, the bifurcation is a stationary (pitchfork) bifurcation, and a Hopf one at 2. The two lines connect in the Takens-Bogdanov (TB) point. The solid lines 3 and 4 mark the first gluing bifurcation and the second gluing bifurcation respectively. The dashed lines 2b and 3b mark the lines of the primary Hopf bifurcation and the first gluing bifurcation when calculated without the inclusion of flow in the equations.

Note that the third stability condition corresponds to Aj > 0. According to (1.38), the wavenumber zero mode of the activator-inhibitor-substrate system undergoes a Hopf bifurcation if A2 goes through 0. For mass-action kinetics, (12.20) implies that G or Po- Consequently, the third stability condition can fail, if the total substrate is too low. Then the Turing bifurcation ceases to be the primary instability, and a uniform Hopf bifurcation occurs first in the fiiU system. 

Since T < 0, an oscillatory instability of the unifor m steady state of Q occurs only for networks with inhibitory coupling, i.e., and F have opposite signs. Interestingly, for kinetic schemes that like the Oregonator model and activator-inhibitor schemes in general have J22 < 0, a stationary instability also occurs only for networks with inhibitory coupling. If r > A//22 > the stationary instability is the primary instability otherwise, the oscillatory instability occurs first. 

Park, H., S. D. Heister A numerical study of primary instability on viscous high-speed jets, Comput. Huids 35, 1033-1045 (2006). 

Black-eye pattern A more complex structure appears well beyond the primary instability of a hexagonal array through increase in the malonic acid concentration [52]. 

Following the primary instability, a liquid sheath remains, but it too is unstable and the final image is often an alternation of large and small droplets. However, the period is imposed by Am-... 

For hydrophilic proteins and peptides the double emulsion/solvent evaporation method is considered as the most convenient because of its encapsulation efficiency and biologically active stability [43]. However, the double emulsion/solvent evaporation method posesses several liabilities such as the required use of toxic solvents that are not easy to fully eliminate (hydrophobic and halogenated solvent) and the inadequate protein release profile. [44-46]. Fxuthermore, the production of acid degradation products, particularly lactic acid, is the primary instability source for the encapsulated acid-labile protein or peptide [6,8],... 

The region where the system is Turing unstable is represented in Figure 1. Other equivalent representations can be found in the literature [12,17]. Fully developed Turing structures may exist outside this domain, sometimes called the Turing space , only sets the limits of the primary instability. 

Far beyond the primary instability, the patterns become time dependent with only short range order in both space and time [10]. This chemical turbu-... 

Under certain conditions in experiments with a polyacrylamide gel, the width of the first hexagonal region decreased to zero within the experimental resolution, and the primary instability then led to stripes rather than hexagons, as Figure 3c illustrates [43]. 

The theory of pattern formation predicts other instabilities beyond the primary instability. One instability leads to the zigzag patterns discussed in Section 4. Another instability that has been well studied, particularly in convective systems, is the Eckhaus instability, which produces a long wavelength pattern. The Eckhaus instability has not been observed in reaction-diffusion systems but should exist for some parameter conditions. 

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