Transverse capillary waves

Fig. 3.23. Damping of transversal capillary waves, after van den Tempel van de Riet (1965)...

Among various relaxation spectrometry methods of liquid surface layers the transverse capillary waves has been used most frequently for micellar solutions [96 - 101]. The shape of the concentration dependence of the wavelength is the same for all investigated cationic, anionic and nonionic surfactants and resembles the corresponding dependence of surface tension. Figure 16 shows as an example the experimental results for solutions of SDS [96]. 

The influence of micellisation on the propagation of capillary waves has been discovered only for solutions of the nonionic surfactant - DePO. The determined values of Z2 are comparable with the results for solutions of DePB but they decrease monotonously with concentration. Therefore, the obtained results evidence that relations (5.284) and (5.285) describe the concentration dependence of the dynamic surface elasticity well. Hence, the method of transverse capillary waves can be used for studies of micellisation kinetics of surfactants with relatively low surface activity. For surfactants with higher surface activity where the formation and disintegration of micelles proceed slower the method of longitudinal surface waves can be used [102, 103]. The characteristics of longitudinal waves are more sensitive to the dynamic surface elasticity, and this allows one to study the micellisation kinetics under the condition... 

According to the capillary wave approximation , a rough interface is characterized by a width that diverges logarithmically with its transverse dimensions ... 

Fig. 6 The effect of transverse viscosity on the polar plot of Fig. 4. The damping coefficient, a, is plotted vs. the real capillary wave frequency, 0> for several different transverse viscosities (/x in the figure has units of 10 5 mNsm ). Only the isopleths for Sd = 0 and k = 0 are shown to give the outermost loop of Fig. 4. The plot was generated using the same condition as in Fig. 4, k = 32 431 m, ad = 71.97mN nr1, p = 997.0 kg nr3, r) = 0.894 mPa s and g = 9.80 m s 2...

Henceforth we shall use the term capillary waves, or capillary ripples for waves that are so small that interfacial tension contributes significantly to their properties. Two types of such waves can be distinguished spontaneous, or thermal waves and those externally applied. The former type is always present they are caused by spontaneous fluctuations cind have a stochastic nature. In secs. 1.10 and 1.15 it was shown how from these fluctuations interfacial tensions and bending moduli could be obtained. Now the second type will be considered. Transverse or longitudinal perturbations can be applied to the interface, for example by bringing in a mechanically driven oscillator (see sec. 3.7). Such waves are damped, meaning that the amplitude Is attenuated. Damping takes place by viscous friction in the... 

The surface motion of the sea takes place on a variety of scales from mm capillary waves to mesoscale eddies. At low wind conditions it seems that there are some scales that dominate with regard to slick-formation. They give rise to long filamentary structures observed on optical- as well as SAR-images of the sea surface, where they have a transverse dimension of the order of 100 m and longitudinal coherence for several kilometres (Scully-Powers 1986). 

For transverse waves the amplitudes of potential flow exceed by a few orders of magnitude the amplitudes of the vortical flow A20 Bo2 and A21 B12. This means that the inhomogeneities of surface tension mainly scatter the capillary waves, and the inhomogeneities of the dynamic surface elasticity play a minor role. 

In the general case transverse surface waves have a vortical component and Eq. (8) is assumed to be an approximation only. This means that Eq. (13) does not allow to calculate the damping coefficient. However, the real part of the complex wave number K for slightly damped capillary waves (ReK > > ImK) can be estimated. 

One of the most remarkable and most frequently studied effects of surface rheology is the damping of surface waves. Surface waves, transversal and longitudinal waves, are described by dispersion equations. Maim (1984) gave recently an overview on modem aspects of dynamic surface tension and capillary waves. 

Capillary Ripples Surface or interfacial waves caused by perturbations of an interface. When the perturbations are caused by mechanical means (e.g., barrier motion), the transverse waves are known as capillary ripples or Laplace waves, and the longitudinal waves are known as Marangoni waves. The characteristics of these waves depend on the surface tension and the surface elasticity. This property forms the basis for the capillary wave method of determining surface or interfacial tension. 

Lucassen-Reynders and Lucassen (Lucassen 1968, Lucassen-Reynders and Lucassen 1969) have derived the dispersion relation for a liquid surface in the presence of a surface film. They showed that periodic disturbance of such a film-covered surface results in a surface tension that varies from point to point on the surface because of the fluctuations in surface concentration. Consequently, in addition to a transverse stress being developed, a finite tangential surface stress is also present. The solution to this dispersion equation has two roots, one of which corresponds to the capillary waves (transverse motion) and one of which corresponds to longitudinal or dilational waves derived from the transverse stress. The dispersion relation (D( o)) obtained for a film at the interface between two media is... 

Evolution equation for transverse (capillary-gravity) waves... 

The capillary-wave method corresponds to small frequency fluctuations. For pure transverse wave motion, Eq. (259) leads to... 

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. 

Where l is the length of the capillary, a is the radius of the capillary, Jo, is a Bessel function of the first kind and k = sj / r /poS) = 8, is the viscous skin depth, which is the distance at which the amplitude of the vorti-city (transverse) wave has attenuated by a factor of the natural logarithm e . Inserting (3) into (1) and using Poisson s equation for the charges distribution we can solve for the FDSP Helmholtz-Smoluchowski equation [Reppert et al., 2001],... 

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