Wave number, capillary waves

The molecular collective behavior of surfactant molecules has been analyzed using the time courses of capillary wave frequency after injection of surfactant aqueous solution onto the liquid-liquid interface [5,8]. Typical power spectra for capillary waves excited at the water-nitrobenzene interface are shown in Fig. 3 (a) without CTAB (cetyltrimethy-lammonium bromide) molecules, and (b) 10 s after the injection of CTAB solution to the water phase [5]. The peak appearing around 10-13 kHz represents the beat frequency, i.e., the capillary wave frequency. The peak of the capillary wave frequency shifts from 12.5 to 10.0kHz on the injection of CTAB solution. This is due to the decrease in interfacial tension caused by the increased number density of surfactant molecules at the interface. Time courses of capillary wave frequency after the injection of different CTAB concentrations into the aqueous phase are reproduced in Fig. 4. An anomalous temporary decrease in capillary wave frequency is observed when the CTAB solution beyond the CMC (critical micelle concentration) was injected. The capillary wave frequency decreases rapidly on injection, and after attaining its minimum value, it increases... 

Let us now briefly mention an approach that is popular among physicists, and which comes down to correlating surface tensions to capillary waves. The underlying idea is that each fluid-fluid interface is subject to a superposition of a large number of thermal waves. The amplitudes of these waves are related to the inter-facial excess energy and the number and frequencies to the interfacial excess entropy, hence the total information obtainable yields F°, and hence y. The idea dates back to Mandelstam and has been taken up by others, including Frenkel and Buff et al. . ... 

The Schmidt number dependence, n, derived from gas exchange of He and CH4 in a wind tunnel experiment as a function of the friction velocity, (the wind speed extrapolated to the air-water interface via the drag coefficient). The break in the exponent from 0.7 to 0.5 occurs at the transition from a smooth surface to one dominated by capillary waves. Redrawn from Jahne et al. (1987). 

Fig. 22. (a) Snapshot of an interface between two coexisting phases in a binary polymer blend in the bond fluctuation model (invariant polymerization index // = 91, incompatibihty 17, linear box dimension L 7.5iJe, or number of effective segments N = 32, interaction e/ksT = 0.1, monomer number density po = 1/16.0). (b) Cartoon of the configuration illustrating loops of a chain into the domain of opposite type, fluctuations of the local interface position (capillary waves) and composition fluctuations in the bulk and the shrinking of the chains in the minority phase. Prom Miiller [109]... 

Under the conditions of scattering the phase velocity of capillary waves will deviate from the value for the medium without scatterers. Therefore, it is natural to introduce an effective wave number K for the former case and to choose the following trial functions for the coefficients C ... 

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. 

The so-called critical wave number kc, equal to the square root of the Bond number, thus separates the long-wavelength disturbances that are unstable from the shorter-wavelength disturbances that are stable because of the influence of capillary effects. We may note that the condition (6 102) shows that the thin-film analysis is valid provided Bo 0(1), as this is the condition for the wavelength of the disturbance to be large compared with the film thickness. 

Table 12-2. Growth rates versus wave number for capillary instability of a highly viscous thread...

Problem 12-3. Capillary Instability for a Thread in a Second Immiscible Fluid. In this problem, we consider the effect on capillary instability if, instead of being surrounded by air, the thread of liquid is surrounded by a second viscous immiscible fluid that is assumed to be unbounded in the radial direction. Derive a condition from which you could in principle, calculate the growth-rate parameter for an axisymmetric disturbance, a = a(k, Re, 7.) where k is the axial wave number and 7. is the ratio of the external fluid viscosity to the viscosity of the liquid thread. This condition can be simplified if either Re I or the thread is inviscid (though viscous effects still remain in the outer fluid). Evaluate a for several k values in each of these two cases. What is the qualitative effect of the second viscous fluid For example, is the range of unstable k values changed Is the fastest-growing linear mode changed relative to the case of a thread in air ... 

The mechanisms of formation of discrete segments of fluids in microfiuidic flow-focusing and T-junction devices, that we outlined above point to (i) strong effects of confinement by the walls of the microchannels, (ii) importance of the evolution of the pressure field during the process of formation of a droplet (bubble), (iii) quasistatic character of the collapse of the streams of the fluid-to-be-dispersed, and (iv) separation of time scales between the slow evolution of the interface during break-up and last equilibration of the shape of the interface via capillary waves and of the pressure field in the fluids via acoustic waves. These features form the basis of the observed - almost perfect -monodispersity of the droplets and bubbles formed in microfiuidic systems at low values of the capillary number. 

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