Rayleigh unstable wavelength

As Rayleigh understood very well a long time ago, the particular wavelength A = 27t/ wins out for kinetic reasons. Of all the possible unstable wavelengths, it is the one that grows the fastest and overshadows all the others. That wavelength is given by... 

The growth rate R(q), not to be confused with the Rayleigh factor R(q), Is always negative for metastable mixtures, as expected. For unstable mixtures, R(q) becomes positive for fluctuations of a sufficiently long wavelength that q < q, exhibits a maximum at... 

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... 

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... 

This simple analysis indicates only what size jet is unstable and how the surface forces cause the breakup of the jets. It does not indicate how fast such a breakup proceeds. Rayleigh s analysis of the most favored wavelength for breakup was based on the assumption that the viscosity of the fluid was negligible. For thick jets of liquids such as water, the experimental data [10] indicate that the breakup occurs most frequently at lengths corresponding lOrQ to 12 rQ rather than the 9ff, predicted by Rayleigh, j... 

While the Rayleigh instability of a cylindrical surface is driven by the surface energy, the flat surface of a thin film can become unstable under the influence of van der Waals forces (see Chapter 5). In this case, the surface tension provides a restoring force which stabilizes short-wavelength undulations. Again, the critical wavelength is determined by a balance of the free energies (see the problems at the end of this chapter). 

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. 

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