Viscosity capillary waves

The subject of surface viscosity is a somewhat complicated one it has been reviewed by several groups [95,96], and here we restrict our discussion to its measurement via surface shear and scattering from capillary waves. 

Another approach to measurement of surface tension, density, and viscosity is the analysis of capillary waves or ripples whose properties are governed by surface tension rather than gravity. Space limitations prevent more than a summary presentation here readers are referred to several articles [123,124]. 

Fig. 4 General solution for the dispersion equation on water at 25 °C. The damping coefficient a vs. the real capillary wave frequency o> , for isopleths of constant dynamic dilation elasticity ed (solid radial curves), and dilational viscosity k (dashed circular curves). The plot was generated for a reference subphase at k = 32431 m 1, ad = 71.97 mN m-1, /i = 0mNsm 1, p = 997.0kgm 3, jj = 0.894mPas and g = 9.80ms 2. The limits correspond to I = Pure Liquid Limit, II = Maximum Velocity Limit for a Purely Elastic Surface Film, III = Maximum Damping Coefficient for the same, IV = Minimum Velocity Limit, V = Surface Film with an Infinite Lateral Modulus and VI = Maximum Damping Coefficient for a Perfectly Viscous Surface Film...

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...

The experimental methods for the determination of liquid viscosity are similar to those used for gases ( 8.VII F) (i) transpiration, through capillaries, (ii) torque on rotating cylinders, or the damping of oscillating solid discs or spheres, in the liquid, (iii) fall of solid spheres through the liquid, (iv) flow of liquid through an aperture in a plate, (v) capillary waves. Methods (i) and (ii) are mostly used for absolute, the others for comparative, measurements. 

Insoluble monolayers on an aqueous substrate have been investigated by means of the capillary wave method for many years. Lucassen and Hansen (1966) in their pioneering work neglected the surface viscosity and considered only pure elastic films. Subsequent studies showed that the surface elasticity of real surface films is a complex quantity, and both the equilibrium surface properties and the kinetic coefficients of relaxation processes in the films influence the characteristics of surface waves. However, it has been discovered recently that the real situation is even more complicated and the macroscopic structure of surface films influences the dependency of the damping coefficient of capillary waves on the area per molecule (Miyano and Tamada 1992, 1993, Noskov and Zubkova 1995, Noskov et al. 1997, Chou and Nelson 1994, Chou et al. 1995, Noskov 1991, 1998, Huhnerfuss et al. this issue). Some peculiarities of the experimental data can be explained, if one takes into account the capillary wave scattering by two-dimensional particles (Noskov et al. 1997). 

In the absence of dissipation and with uniform surface tension, plane small-amplitude capillary waves will propagate undamped and unamplified. Viscosity and surface tension gradients lead to the damping of capillary waves. In the following section we shall discuss the damping due to the presence of surface-active substances, which because of the wave shape are not uniformly distributed, giving rise to a spatially nonuniform surface tension. Of interest in... 

Let us examine yet another example of a surfactant influencing surface tension the suppression of waves on the water surface by a layer of oil spread over the water. In Section 17.4, we have considered capillary waves on a dean water surface without regard for the viscosity of the liquid. When a layer of oil is present, it is obviously necessary to take viscosity into account. In addition, the presence of the surfactant changes the surface tension S of the interface, resulting in the appearance of a surface gradient. 

First, we consider the influence of the liquid s viscosity on the damping of plane capillary waves on deep water. Suppose the liquid has low viscosity so that viscous effects only manifest themselves inside a thin boundary layer near the interface. Hence, outside the boundary layer, the liquid flow is potential, and the potential is described by the Laplace equation, while the liquid flow near the surface is described by the boundary layer equations with the accompanying condition that the tangent viscous stress at the free interface must be zero. The solution of this problem can be found in [2]. The main difference from the case of a non-viscous liquid is the appearance of a coefficient of the form exp(—jSjt) in the... 

Derive the dispersion relation for capillary waves on the surface of a fluid with nonzero viscosity. Since this is a dissipative system, the energy approach used in the text is not applicable and one must start from the hydrodynamic equations for an incompressible fluid. Assume that that amplitude of the wave and its velocity are small, linearize the hydrodynamic equations and calculate both the real and imaginary parts of the dispersion relation for these (possibly overdamped) waves. 

Time-resolved quasi-elastic laser scattering (QELS) experiments [306, 307] allow the measurement of the dispersion law (the relation between the frequency of capillary waves, co, and their wavelength, In/k). To derive the dispersion law theoretically, one must consider the dynamics of the two degrees of freedom - the position of the surface, z = (x, y, t), and the fluid velocity. In the limit of small viscosity, such a theory [291] gives... 

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