Capillary wave instability

Recently, a study[265] has been conducted on the formation of droplets by the capillary wave instability of a spherical liquid shell with pulsating cavity. 

Brochard and Dalliant have indicated that liquid films on non-wettable substrates are spinodally unstable against thickness fluctuation if the thickness e is thinner than lOOnm (7). Their model of linearized capillary wave instability predicts that the thickness fluctuation with a characteristic wave vector qu is amplified most rapidly to nucleate initial dry spots separated by 2it/qtt fix>m each odiw. Their result is... 

Dm =(4 3c7/pi 02)1/3 Dm = 0.42 3 for largg pL(Oqc 2o Derived on the basis of capillary wave theory in terms of Taylor instability, for thick films Peskin Raco [483]... 

The theoretical framework, within which the existence of surface instabilities created by capillary waves can be predicted is the linear stability analysis [23, 24]. This model assumes a spectrum of capillary waves with wave vectors q and time constant r (Fig. 1.8a). 

The electric field experiment shown here can be considered as a test case for the quantitative nature of capillary instability experiments. It shows the precision, with which the capillary wave pattern reflects the underlying destabilizing force. In the case of electric fields, this force is well understood. Therefore, the good fit in Fig. 1.10b demonstrates the use of film instability experiments as a quantitative tool to measure interfacial forces. The application of this technique to forces that are much less well understood is described in the following section. 

In the last two sections, we considered mass transfer from the film toward the droplets and the reverse, from droplets toward the film. In both cases, the diffusion fluxes lead to stabilization of the film. Here we consider the third possible case corresponding to mass transfer from the first droplet toward the second one across the film between them. In contrast with the former two cases, in the last case the mass transfer is found to destabilize the films. Experimentally, the diffusion transfer of alcohols, acetic acid, and acetone was studied. - The observed destabilization of the films can be attributed to the appearance of Marangoni instability, which manifests itself through the growth of capillary waves at the interfaces, which eventually can lead to film rupture. 

Problem 12-2. Capillary Instability of a Varicose Sheet. We have seen that a cylinder of fluid surrounded by air is unstable because of growing capillary waves. An equivalent problem is the stability of capillary waves on a round laminar jet when the velocity profile across the jet is uniform. We now wish to consider whether the same type of instability is relevant to a fluid sheet of finite width <7 and infinite lateral extent (the sheet has an interface above and below). Equivalently, we could also ask whether a planar (2D) laminar jet is unstable to capillary wave growth. Assume that the sheet is subject to an infinitesimal ID wavelike disturbance that is symmetric about the center plane and corresponds to an initial sheet thickness h = ho( + e sin foe). Prove whether the sheet is linearly stable or unstable to this type of disturbance. 

In 1950 Taylor studied the formation of sinusoidal waves on the free surface of a fluid in air undergoing acceleration perpendicular to the free surface and in the direction of the force of gravity, and almost a century prior, Rayleigh considered the formation of similar waves if the denser fluid was placed above the free surface both illustrate a similar mechanism that has come to be described by the Rayleigh-Taylor instability. Curiously, the instability will appear between the interfaces of other phases of media, including even solid-solid interfaces, and the form of the equations that describe the instability will remain essentially the same [11]. The formation of droplets from capillary waves may be described by the Rayleigh-Taylor instability. 

A more elaborated theoretical model based on interfacial Taylor instability triggering the surface wave was developed by Peskin and Raco [39]. A thin layer of a liquid, wetting the surface of a solid resonator wbch vibrates to its plane, forms a chessboard-like pattern of stationary capillary waves. This phenomenon occurs when the vibration amplibde exceeds a threshold value. Further on, ligament breakup of the liquid occurs and droplets are hurled from the crests of the capillary waves. Together with the wavelength, they introduced wave amplibde and the sheet thickness as parameters to determine the droplet size [39]. 

Another class of methods for achieving patterning of polymer surfaces on a length scale of micrometers or less relies on a physical instability with an intrinsic length scale. Examples of such processes, which will not be further discussed in this review, include dewetting [86,87], buckling produced by stresses arising from dispersion forces [88,89] or residual mechanical stress [65]. Further approaches exploit the amplification of capillary waves by various means [90,91]. 

Irrespective of the precise origin of an instability, the amplification of the capillary waves leads to the deformation of the free surface (film-air interface) and results in localized flow of liquid from the thinner parts to the thicker parts of the film [38, 56-168]. This phenomenon eventually results in the rupture of the film with the formation of dry patches or holes, when the growing amplitude of the capillary wave spectrum equals the film thickness [65]. As the film ruptures, the two distinct interfaces (film-air and film-substrate) merge and a three phase contact line (film-air-substrate) is formed. Depending on the thermodynamics of the system. 

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