Perturbation waves growth rate

A commonly used primary atomization model for liquid jets has been developed by Huh et al. [1], The model considers the effects of both infinitesimal wave growth on the jet surface and jet turbulence including cavitation dynamics. Initial perturbations on the jet surface are induced by the turbulent fluctuations in the jet, originating from the shear stress along the nozzle wall and possible cavitation effects. This approach overcomes the inherent difficulty of wave growth models, where the exponential wave growth rate becomes zero at zero perturbation amplitude. 

The above discussion identifies the growth-rate gradient of short waves, dajde, evaluated at Smb, as a further measure of fluidization quality. It provides the necessary additional dimension to the quantification in terms of the minimum bubbling void fraction, distinguishing between systems having the same Smb but different perturbation-amplitude growth rate characteristics. This gradient may be readily evaluated from eqn (10.4). 

Flat Sheets. Generally, the interface between a liquid sheet and air can be perturbed by aerodynamic, turbulent, inertial, surface tension, viscous, acoustic, or electrical forces. The stability of the sheet and the growth rate of unstable disturbances are determined by the relative magnitude of these forces. Theoretical and experimental studies 255112561 on disintegration mechanisms of flat sheets showed that the instability and wave formation at the interface between the continuous and discontinuous phases are the maj or factors leading to... 

An instability analysis of a multiphase flow is usually based on the continuum assumption and the equations of motion of the phases. Two different approaches are common. One approach is to study the wave propagation speed from the equations of motion using an analogy to the surface wave situation. The other approach is to study the growth rate of the perturbation wave when a small perturbation wave is introduced to the system. The criterion for stability can then be derived. A perturbation wave can be expressed by... 

Fig. 47. Schematic of the growth rate X(n) (as defined in Eqs. (48) and (49)) of a perturbation of the homogeneous steady state versus the wave number of the perturbation in the case of negative global coupling.

Clearly, a perturbation with a wave number n will decay, if the growth rate Xn is negative and for positive growth rates, the perturbations will grow. The qualitative dependence of the growth rate as a function of n is plotted in Fig. 47. It has a mo-notonically decreasing characteristic for n > 0, but exhibits a jump toward smaller... 

Fig. 68. Typical dispersion relations, displaying the growth rate of perturbations, A(n), vs. their wave number, n, of an S-NDR system for three different homogeneous steady states [33]. The lowest curve depicts the case of a stable homogeneous state, the middle one is close to a Turing-type bifurcation in which a stationary structure with the integer wave number closest to the maximum of the curve is born. The up-most curve shows a situation for which the homogeneous state is unstable with respect to perturbations lying within the wavelength range n, for which X(n) > 0.

This equation is only valid for small-amplitude disturbances, when the approximation C/a 1 holds. When this expression for p is equated to (1.23) at r = a and simplified in the linear approximation, the amplitude. A, disappears, and we are left with a characteristic relationship between the growth rate and the wave number determining the perturbation spectrum. Solving for the growth rate, we have ... 

Fig. 1.4 Nondimensional growth rate of capillary axisymmetric perturbations for an inviscid jet in terms of the wave number [18, Fig. 7], The symbols represent the experimentally measured growth rates for low viscosity jets (Courtesy of Cambridge University Press)...

Figure 1.5 shows the growth rate of the capillary instability for different liquid viscosities. Viscosity dampens the instability with a damping coefficient of hpl lp and shifts the fastest growing perturbations toward longer waves. For p = 0, Rayleigh solution is obtained, whereas for very viscous jets with (3pk /2p) al2pa , a> = a/ pd) — k a ). The breakup length for a viscous jet is found as ...

Fig. 1.5 Growth rate of small capillary perturbations of viscous jets instability in terms of the wave number...

For a sufficiently high voltage only the electrostatic interactions need to be considered. In a stability analysis, a small perturbation of the interface with wave number q, the growth rate and amplitude u is considered h x, t) = Hq + Mexp[(i x + t)/f. The modulation of h gives rise to the lateral pressure gradient inside the film inducing a Poiselle flow f. 

Note that in the expression for A, the wave number k and flow velocity v always appear as a product kv. This means that all observations concerning the dispersion relations between A or Re A and fc at a fixed v can be viewed as relations between A or Re A and u at a fixed k. We can state thus, that for any perturbation which is a Fourier harmonic with a finite k, the growth rate monotonically rises from Re A(0) < 0 to an > 0 as the velocity v grows from 0 to oo. This can be interpreted in terms of disengaging activator and inhibitor as the velocity of the differential flow grows, the separation of the activator and inhibitor becomes more and more effective until the growth rate of the unstable modes reaches the rate of autocatalysis, an. Since the notion of autocatalytic growth is associated with chemistry we call this kind of instability differential flow induced chemical instability . 

The wave number k is also real when both the numerator and denominator in eqn (8.33) are positive in this case we have mk > v > md- Under these conditions eqn (8.32) reveals the perturbation growth rate a to be positive, indicating unstable, heterogeneous fluidization, and hence the... 

The above speculations, linking fluidization quality to perturbation-amplitude growth and decay rates, are now examined. Although bubble-related phenomena clearly imply conditions outside the linear response limit of the system, initial growth rates, obtainable from the linearized relations, can be so large in these cases that they could be expected to play a major role in subsequent developments. The linearized particle bed model delivers explicit relations for perturbation-wave velocity and amplitude growth rate, thereby enabling the above considerations to be... 

In Chapter 10, expressions for the velocity v and amplitude growth rate a of a perturbation wave satisfying eqn (13.19) were presented as functions... 

When one fluid overlays a less dense fluid, perturbations at the interface tend to grow by Rayleigh-Taylor instability (LI, T4). Surface tension tends to stabilize the interface while viscous forces slow the rate of growth of unstable surface waves (B2). The leading surface of a drop or bubble may therefore become unstable if the wavelength of a disturbance at the surface exceeds a critical value... 

The front is inherently unstable, however, and this is often studied by a linear stability analysis. Infinitesimal perturbations are applied to all of the variables to simulate reservoir heterogeneities, density fluctuations, and other effects. Just as in the Buckley-Leverett solution, the perturbed variables are governed by force and mass balance equations, and they can be solved for a perturbation of any given wave number. These solutions show whether the perturbation dies out or if it grows with time. Any parameter for which the perturbation grows indicates an instability. For water flooding, the rate of growth, B, obeys the proportionality... 

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