Instability Rayleigh-Taylor

We shall now discuss a suspended film and its instability (known as Rayleigh-Taylor instability). We shall first find what wavelengths turn out to be unstable and then go on to examine the dynamics of the phenomenon. 

FIGURE 5.7. Drops hanging under a horizontal panel. The drops are formed either by condensation or by destabilization of a liquid film. Eiach drop is approximately 1 cm in size (courtesy Marc Fermigier). 

We retain the notation introduced earlier in equation 5.9. The film thickness is modulated with an amplitude 6e about the median value cq 6e Co) and a wavevcctor q. The wavelength X = 2Trfq is assumed to be large compared to Cq. Wc also assume for simplicity that the deformation is a function of the coordinate x only (the corrugated panel model). 

The surface term can be evaluated by calculating the curved length of the interface. In the limit of small deformations 5e g[ A), the quantity ds can be expanded in the form ds w da [l -I- (l/2)(de/dx) ]. Equation (5.13) can then be readily integrated to yield 

Equation (5.4) gives the flow rate of a liquid that is due to a volumetric force Vp. In the present case, the force is determined both by the weight of the liquid, which is pgdejdx per unit volume, and by the Laplace force - yd e/dx induced by the surface curvature. The force, and hence the flow rate, can thus easily be deduced from the interface profile [equation (5.9)]. Invoking the flow rate equation (5.6) leads to an equation describing the evolution of the interface  

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As described previously, in the atomization sub-model, 232 droplet parcels are injected with a size equal to the nozzle exit diameter. The subsequent breakups of the parcels and the resultant droplets are calculated with a breakup model that assumes that droplet breakup times and sizes are proportional to wave growth rates and wavelengths obtained from the liquid jet stability analysis. Other breakup mechanisms considered in the sub-model include the Kelvin-Helmholtz instability, Rayleigh-Taylor instability, 206 and boundary layer stripping mechanisms. The TAB model 310 is also included for modeling liquid breakup. 

A horizontal interface between two fluids such that the lower fluid is the less dense tends to deform by the process known as Rayleigh-Taylor instability (see Section UFA). Spikes of the denser fluid penetrate downwards, until the interface is broken up and one fluid is dispersed into the other. This is observed, for example, in formation of drops from a wet ceiling, and of bubbles in film boiling. For low-viscosity fluids, the equivalent diameter of the particle formed is of order Ja/gAp. 

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... 

Physical Model We propose a symplified model for the non-linear-like phenomena in X-ray sources, considering the properties of Rayleigh-Taylor instability on the magnetopause which is formed by the accreting matter to the neutron star, and then obtain the motion of this magnetopause. ... 

However numerical simulations of early supernova-driven winds fail to find any evidence for substantial gas ejection from luminous ( L ) galaxies. One can ask what is wrong with the hydrodynamic simulations Certainly, the simulations lack adequate resolution. Rayleigh-Taylor instabilities enhance wind porosity and Kelvin-Helmholtz instabilities enhance wind loading of the cold interstellar medium. Both effects are certain to occur and will enhance the wind efficacity. Yet another omission is that one cannot yet resolve the motions of massive stars before they explode. This means that energy quenching is problematic and the current results are inconclusive for typical massive galaxies. 

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)). 

Dzwinel W, Yuen DA (2001) Mixing driven by Rayleigh-Taylor instability in the mesoscale modeled with dissipative particle dynamics. Int l J Modem Phys C 12 91-118. 

Schmeling, H. 1987. On the relation between initial conditions and late stages of Rayleigh-Taylor instabilities. Tectonophysics, 133, 65-80. 

Figure 11 Phenomena connected with drop breakup (a) deformation (splitting) of droplets, (b) deformation into lenticular shape, (c) development and separation of the boundary layer, (d) velocity distribution near the separation point of the boundary layer, (e) Kelvin Helmholtz instability, and (f) Rayleigh Taylor instability, pi < P2-...

Next, we turn to the classic problem of Rayleigh-Taylor instability for the gravitationally driven overturning of a pair of immiscible superposed fluids in which the upper fluid has a higher density than the lower fluid. This is another example of a problem in which the viscosity of the fluid is not an essential factor in its instability. 

Clearly, Gk > 0 for R > 0. Instability corresponding to R > 0 is the analog of Rayleigh Taylor instability. For a flat interface, we could have instability only for R > 0. Here, however, we may still have instability even if R <0, provided R/R is sufficiently large. To determine the condition for G > 0 in terms of controllable parameters, we can substitute for R/R in (4-315) by using the inviscid form of the Rayleigh Plesset equation, (4-204), with y = 0, that is,... 

This type of instability is known as Rayleigh-Taylor instability. It is discussed in Chap. 12. 

The generalization of the analysis of the Rayleigh-Taylor instability to the acceleration of a spherical interface was independently reported by two authors M. S. Plesset, On the stability of fluid flows with spherical symmetry, J. Appl. Phys. 25, 96-8 (1954) G. Birkhoff, Note on Taylor instability, Q. Appl. Math. 12, 306-9 (1954) Stability of spherical bubbles, Q. Appl. Math. 13, 451-3 (1956). 

However, the inverse problem, with the film on the underside of the solid substrate, now features a competition between the stabilizing (leveling) effect of capillary forces against the destabilizing effect of gravity. This corresponds to a well-known stability problem, called Rayleigh-Taylor instability, applied to the thin film. In this case,... 

In a film of infinite lateral extent, k can range from 0 to oo, so a necessary condition for instability is that AH > 2npgh. Since all wave numbers are available in a film of infinite extent, we see that this analysis predicts that the thin film will always be unstable, even with the stabilizing influence of surface tension, to disturbances of sufficiently large wavelength when van der Waals forces are present. Similarly, the Rayleigh Taylor instability that occurs when the film is on the underside of the solid surface will always appear in a film of infinite extent. In reality, of course, the thin film will always be bounded, as by the walls of a container or by the finite extent of the solid substrate. Hence the maximum wavelength of the perturbation of shape is limited to the lateral width, say W, of the film. This corresponds to a minimum possible wave number... 

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