Capillary wave theory

Analytical and empirical correlations for droplet sizes generated by ultrasonic atomization are listed in Table 4.14 for an overview. In these correlations, Dm is the median droplet diameter, X is the wavelength of capillary waves, co0 is the operating frequency, a is the amplitude, UL0 is the liquid velocity at the nozzle exit in USWA, /Jmax is the maximum sound pressure, and Us is the speed of sound in gas. Most of the analytical correlations are derived on the basis of the capillary wave theory. Experimental observations revealed that the mean droplet size generated from thin liquid films on... 

Dm o.34i = o.u ma pLa>l)in Derived on the basis of capillary wave theory (semi-empirical) Lang [127]... 

Fig. 1 illustrates the two mechanisms proposed for the processes of liquid disintegration and aerosol generation within ultrasonic nebulizers. The capillary-wave theory relates to the production of capillary waves in the bulk liquid. These waves constructively interfere to form peaks and a central geyser. When the amplitude of the applied energy is sufficiently high, the crests of the capillary waves break off, and droplets are formed. The rate of generation of capillary waves is dependent on both the physicochemical properties of the nebulized fluid and the intensity of the ultrasonic vibration. Mercer used Eq. (1) to calculate the threshold amplitude for the generation of capillary waves ... 

It may be of interest to note that Mandelstamm s analysis forms the basis of the capillary wave theory of surface tension. ... 

To complete the mathematical problem a relationship r(c), a so-called adsorption isotherm, is needed. For the simple case of bubble or drop oscillations (with the surfactant only outside the drop) a solution was derived in Ref. 189 in analogy to the capillary wave theory (183, 184). 

Stiffness Coefficient The temperature and the electric field are the main external system parameters that determine the step fluctuations. The range over which the fluctuations can be described in the framework of the capillary wave theory is limited by the fact that a lattice model is used. Depending on the metal, the temperature range and the applied electric field, the fluctuations can be too weak or too intense. If they are too weak, no reasonable statistics can be obtained if they are too strong, the fluctuations are too large to be described by capillary wave theory. 

Simulations for a field of 5 X 10 ° Vm were also performed, but the fluctuations were too large to be described by capillary wave theory. These findings are in line with the observations that two-dimensional metal structures on metal surfaces become more mobile with increasing electrode potential. 

The Marcus theory is difficult to apply directly in a practical situation, owing to lack of knowledge of the probability of the formation of protrusions. One way to overcome this problem could be to employ capillary wave theory for the interface between two immiscible electrolyte solutions. Recently, theoretical efforts have been made to describe capillary waves at soft electrified interfaces [83]. It may be possible to use such theories to quantify the value of P(h)Ah. One of the major complications is related to the fact that the surface tension is dependent on the Gal-vani potential difference between the two... 

Some recent results in the theory of the interface between continuous fluid phases at equilibrium are described. Emphasis is given to the role played by the external field in determining the microscopic structure of the interface, its anomalous effect on the critical behavior of fluid interfaces in two dimensions, the success of capillary wave theory and the failure of traditional van der Waals theory to describe not only transverse but also longitudinal interfacial correlations, as well as to account for the optical reflectivity of the interface of simple fluids and binary mixtures near the critical point. 

A different approach to the problem of fluid interfacial structure advocates capillary waves, which have no analogue in the bulk, as the proper modes of the interfacial fluctuations. While originally designed for low temperatures, capillary wave theory has been successfully extrapolated into the critical region of three-dimensional fluids. ... 

In the two-dimensional version of capillary wave theory, we have an infinitely sharp fluctuating line separating two two-dimensional fluid phases. The probability of a single-valued distortion z x) is proportional to exp(-W(z)/AjjT), with ... 

We finally observe that for fluids in four dimensions, capillary wave theory predicts ... 

In view of the striking disagreement between the predictions of capillary wave and van der Waals theories in two dimensions, it is useful to reexamine critically the basic assumptions of capillary wave theory. 

As seen from Section II. 1, these results are in full agreement with the prediction p = (d )(o of ordinary capillary wave theory for the case of a gravitational field and an intrinsic profile of zero thickness, as well as with the identical prediction of the generalized van der Waals theory. The critical behavior in d > 3 is thus insensitive to the choices of the intrinsic profile and external field. 

The few available experimental data had not enabled one to discriminate between the vastly diverging points of view advocated in the van der Waals and capillary wave theories. Although no reflectivity measurements have been performed very close to the critical point since the pioneering studies of Webb and his co-workers,new measurements and analyses of the critical parameters (bulk correlation length and surface tension) of the systems originally studied by Webb et al.. have been performed recently and very accurate values for both quantities, which differ appreciably from those used in earlier work, have recently become available. 

Capillary wave theory, on the other hand, yields, in three dimensions ... 

Capillary wave theory considers the density variation at the interface to be the result of the superposition of thermally excited density fluctuations on a bare intrinsic profile. Mathematically, the instantaneous local density at a... 

Figure 3 A test of capillary wave theory for a Lennard-Jones liquid/vapor interface. The circles represent independent simulations of the interface width at different temperatures. r is the surface tension. (Data adapted from Ref. 131. Copyright 2012 American Instimte of Physics.)...

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