Rayleigh disturbances

If the capillary number exceeds its critical value the droplets undergo deformation and the so-formed long slender bodies become unstable due to interfacial tension driven disturbances, the so-called Rayleigh disturbances. Locally the radii become small enough that the interfacial tension exceeds the deformation stresses and the drops break up (Fig. 5.3). This will continue until the radius of the drop is such that the critical capillary number is reached ... 

The interfacial tension will want to reduce the interface between the two phases, minimizing the surface-to-volume ratio. The smallest surface-to-volume ratio is achieved in a sphere, S/V = 3/R. Thus, an extended liquid thread will tend to break up due to the interfacial tension. The breakup is initiated by small disturbances at the interface, so-called Rayleigh disturbances. These disturbances grow due to the interfacial tension and eventually breakup can occur. The progress of the breakup process is illustrated in Fig. 7.149. 

The first theoretical analysis of the breakup of a Newtonian thread in a quiescent Newtonian matrix was performed over a century ago by Rayleigh [284]. The disturbances that initiate the breakup process are often referred to as Rayleigh disturbances. Rayleigh analyzed only the effect of surface tension, neglecting the viscosities of the two phases. This work was extended by Tomotika [285] by including the effect of viscosity. The analysis considers a sinusoidal liquid cylinder the radius as a function of axial distance z is ... 

Once the flow has stopped, the interfacial tension drives two competing processes. One is the relaxation back to the original sphere the other is the development of capillary waves. Capillary waves are like Rayleigh disturbances on a liquid thread. The relaxation of the drop is caused by the pressure difference over the interface of the drop ... 

The affine deformation of the drops causes the drops to extend into long thin threads, which is referred to as fibrillation. This process continues until the local radii become so small that the Weber (Capillary) number starts to approach the critical Weber number. At this point the threads become unstable and disintegrate as a result of interfacial tension-driven processes the interfaces are now active. The most important mechanisms are the growth of Rayleigh disturbances in the midpart of the thread, end-pinching, retraction, and necking in the case of relatively short dumbbell-shaped threads. 

If the tube is not infinitesimal in radius, the calculation becomes more difficult, because we shall have to consider not only the form of the meniscus (iri order to calculate its volume) but also the direction in which that form tends to change under the disturbance imagined. The general problem has been solved by Rayleigh (Proo. Boy. 800. A, xcil. 184, 1915) for sufficiently small tubes with the result... 

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... 

Lord Rayleigh (31) was the first to investigate the stability of an infinitely long, liquid cylinder embedded in an immiscible liquid matrix driven by surface tension, taking into account inertia. Weber (32) considered stresses in the thread, and Tomotika (33) included the viscosity of the matrix as well. The analysis follows the evolution in time of small Rayleigh sinusoidal disturbance in diameter (Fig. 7.19) ... 

Fig. 7.19 A liquid thread of radius Rq with a Rayleigh sinusoidal disturbance.

Above Eth. the field is sufficiently strong to cause limited translation of the domain wall without disturbing to any significant extent the overall domain structure. This process is described as reversible (more correctly as nearly reversible ) to distinguish it from the very hysteretic and clearly irreversible process evidenced by the hysteresis loop (Fig. 2.46). In this regime the P-E characteristic is a narrow loop, the Rayleigh loop referred to above (c.f. Fig. 6.9). 

Using the small disturbance approach gives a value of 1708 for the Rayleigh number at which disturbances start to grow, i.e., at which fluid motion will develop in the enclosure. This value will be independent of Prandtl number because the fluid motion for Rayleigh numbers near the critical value is very weak and the effect of the convective terms in the momentum equation is then negligible and the governing equations, i.e., Eqs. (8.137) to (8.139) then are ... 

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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