Instability, temporal

A potential pitfall with stop-time experiments comes with temporal instability of responses. When a steady-state sustained response is observed with time, then a linear portion of the production of reporter can be found (see Figure 5.15b). However, if there is desensitization or any other process that makes the temporal responsiveness of the system change the area under the curve will not assume the linear character seen with sustained equilibrium reactions. For example, Figure 5.16 shows a case where the production of cyclic AMP with time is transient. Under these circumstances, the area under the curve does not assume linearity. Moreover, if the desensitization is linked to the strength of signal (i.e., becomes more prominent at higher stimulations) the dose-response relationship may be lost. Figure 5.16 shows a stop-time reaction dose-response curve to a temporally stable system and a temporally unstable system where the desensitization is linked to the... 

Section 5 contains a summary of the basic mechanisms giving rise to spatio-temporal instabilities in electrochemical systems and discusses perspectives and challenges in future research. 

A second type of eigenstates, illustrated in Fig. 3.10, is the band of states formed by the exciton-contaminated photon continuum. Far from the critical area cK co0, this band presents a lorentzian resonance (cf. Fig. 3.9), whose temporal instability (cf. Fig. 3.8) is described by an exponential decay. Thus, the exact solution leads back to that of second-order perturbation theory, obtained in Section III.A.2.b above. 

Thus, the streaming velocities Ui and U2 do not affect the response of the system. If in addition, p > 1, i.e. a heavier liquid is over a lighter liquid, then the buoyancy force causes temporal instability (if / is considered real) - as is the case for Rayleigh-Taylor instability (see Chandrasekhar (I960)). 

Presence of the imaginary part with negative sign implies temporal instability for all wave lengths. Also, to be noted that since the group velocity and phase speed in y-direction is identically zero, therefore the Kelvin-Helmholtz instability for pure shear always will lead to two-dimensional instability. 

Above inviscid mechanism of instability is often encountered in free shear layers and jets. A fundamental difference between flows having an inflection point (such as in free shear layer, jets and wakes and the cross flow component of some three-dimensional boundary layers) and flows without inflection points (as in wall bounded flows in channel or in boundary layers) exists. Flows with inflection points are susceptible to temporal instabilities for very low Reynolds numbers. One can find detailed accounts of invis-... 

For different systems, we have different signs of the real and imaginary part of Landau coefficient /. Here, we will keep our attention focused to flow past a circular cylinder, that works as a prototypical model for bluff-body flow instability. This instability begins as a linear temporal instability and its first appearance with respect to the Reynolds number is referred to as Hopf bifurcation. Thus, the Reynolds number at which the first bifurcation occurs is given by Rccr- Thus, above Rccr the value of <7 > 0 signifies linear instability. One of the most important aspect of this linear instability is the subsequent non-linear saturation that can be adequately explained by the Landau s equation, if only R is positive. We will focus upon this type of flow only in the next. 

Vortex shedding behind a circular cylinder is explained theoretically as a Hopf bifurcation which is a consequence of linear temporal instability of the flow. In this point of view, the above temporal instability is moderated by nonlinearity of the system, that is quite adequately explained by Landau equation, as given in Landau (1944) and Drazin Reid (1981). Earlier numerical investigations by Zebib (1987), Jackson (1987) and Morzynski Thiele (1993) have identified the onset of vortex shedding to be at a critical Reynolds number (Rccr) between 45 and 46. 

Moresco, P. and Healey, J.J. (2000). Spatio-temporal instability in mixed convection boundary layers, J. Fluid Mech. 402 89-107. 

The other canonical flow geometry considered in the first part consists of bluff-body flow instability dealt in chapter 5. This introduces the flow past a cylinder that actually suffers linear temporal instability moderated by nonlinear stabilization. This flow is different from that is discussed primarily in chapters 2 to 4, where the linear instability is via spatial growth. Also, for such flows nonlinearity leads to further destabilization, whereas for the flow past a cylinder, the nonlinearity stabilizes the linear instability and takes the flow to another equilibrium flow. In chapter 6, the effects of heat transfer via the restrictive condition of Boussinesq approximation for the canonical flow past flat plates is studied. This problem has been solved... 

The phenomenon of self organization occurs at nonstabHities of the sta tionary state and leads to the formation of temporal and spatio temporal dissipative structures. Remember that oscillating instabilities of stationary states of dynamic systems can be observed for the intermediate nonlinear stepwise reactions only, when no fewer than two intermediates are involved (see Section 3.5) and at least one of the elementary steps is kinet icaUy irreversible. The minimal sufficient requirements for the scheme of a process with temporal instabilities are not yet strictly formulated. However, in aU known examples of such reactions, the rate of the kineti caUy irreversible elementary reaction at one of the intermediate steps is at least in a quadratic dependence on the intermediate concentrations. Among these reactions are autocatalytic steps. 

The flow of the continuous phase is considered to be initiated by a balance between the interfacial particle-fluid coupling - and wall friction forces, whereas the fluid phase turbulence damps the macroscale dynamics of the bubble swarms smoothing the velocity - and volume fraction fields. Temporal instabilities induced by the fluid inertia terms create non-homogeneities in the force balances. Unfortunately, proper modeling of turbulence is still one of the main open questions in gas-liquid bubbly flows, and this flow property may significantly affect both the stresses and the bubble dispersion [141]. 

E. Scholl, A. Amann, M. Rudolf, and J. Unkelbach Transverse spatio-temporal instabilities in the double barrier resonant tunneling diode, Phys-ica B 314, 113 (2002). 

Significant production of sulfate has been detected and/or predicted in clouds and fogs in different environments (Hegg and Hobbs, 1987, 1988 Pandis and Seinfeld, 1989 Husain et al., 1991 Pandis et al., 1992 Swozdiak and Swozdiak, 1992 Develk, 1994 Liu et al., 1994). Detection of sulfate-producing reactions is often hindered by variability of cloud liquid water content and temporal instability and spatial variability in concentrations of reagents and product species (Kelly et al., 1989). 

Keywords Capillary instability of liquid jets Curvature Elongational rheology Free liquid jets Linear stability theory Nonlinear theory Quasi-one-dimensional equations Reynolds number Rheologically complex liquids (pseudoplastic, dilatant, and viscoelastic polymeric liquids) Satellite drops Small perturbations Spatial instability Surface tension Swirl Temporal instability Thermocapillarity Viscosity... 

With the limitations and the problems associated with both the perturbation analysis and the one-dimensional models, the full nonlinear equations of motion for the jet are solved numerically. One such solution is by Ashgriz and Mashayek [75]. They studied the temporal instability of an axisymmetric incompressible Newtonian liquid jet in vacuum and zero gravity. The variables are nondimensio-nalized by the radius of undisturbed jet, a, and a characteristic time (pa" jof. ... 

Fubiari, E. P. Temporal instability of viscous liquid microjets with spatially varying surface tension. J. Phys. A Math. Gen. 38, 263-276 (2005). 

There have been numerous studies on the temporal and spatial instability of liquid sheet [1-40]. This chapter is mainly on the temporal instability. Among these, Dombrowski and his coworkers [8-16] conducted extensive studies on the factors influencing the breakup of sheets and obtained information on the wave motions of high velocity sheets. More recent analyses are provided by Senecal et al. [20], and Rangel and Sirignano [21], This chapter provides only... 

Dombrowski and John [12] combined a linear model for temporal instability and a sheet breakup model for an inviscid liquid sheet in a quiescent inviscid gas, to predict the ligament and droplet sizes after breakup. The schematic of their wavy sheet is reproduced in Fig. 3.7. The equation of motion of the neutral axis mid-way... 

Droplet production by droplet stream generators takes place by pinch-off of liquid portions from jets. A trivial prerequisite for the application of this technique of drop production is, therefore, the formation of a laminar liquid jet from a round orifice or nozzle. The conditions of liquid flow through the orifice required to form a laminar jet are discussed in Sect. 26.3 below. Once the laminar jet is formed, its linear temporal instability against a disturbance with a non-dimensional wave number ka = 2nalX (with the wavelength X of the disturbance and the jet radius a) in a gaseous ambient medium under the action of surface tension, neglecting both the liquid viscosity and the dynamic interaction with the ambient gas, is described by the dispersion relation... 

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