Oscillations of Drops

Finally, it must be noted that tensides that are adsorbed at the interface cause a stiffening of the interface. They hinder or even stop the inner circulation and oscillation of drops, and reduce the mass transfer intensity. Moreover, they form a barrier against the mass transfer, so that a further resistance term should be considered in the overall mass transfer process [28] in Eq. (9.33). Since the nature and concentration of tensides in industrial processes cannot be predicted, such phenomena cannot be taken into consideration during equipment calculations. 

An original method involves quadrupole oscillations of drops K The drop (a) in a host liquid (P) is acoustically levitated. This can be achieved by creating a standing acoustic wave the time-averaged second order effect of this wave gives rise to an acoustic radiation force. This drives the drop up or down in p, depending on the compressibilities of the two fluids, till gravity and acoustic forces balance. From then onwards the free droplet is, also acoustically, driven into quadrupole shape oscillations that are opposed by the capillary pressure. From the resonance frequency the interfacial tension can be computed. The authors describe the instrumentation and present some results for a number of oil-water interfaces. 

A variant, with reminiscences to sec. 1.5, is based upon the capillary instability of jets, a topic that has drawn recent interest because of the increasing application of ink jet printers. Such printers are based on the deflection of a liquid jet in an electric field. The idea goes back to Sweet ), and has given rise to much printing technology. For the present purpose, oscillations in the jet are not produced by an elliptic orifice, but applied externally, say piezo-electrically. Dynamic surface or interfacial tensions can be obtained from, for Instance, the (quadrupole) oscillations of drops that have just broken away from the jet, or from the oscillations in the jet just before breaking. Measurements can be carried out down to lO" s" l... 

A. Prosperetti, Free oscillations of drops and bubbles the initial-value problem, J. Fluid Mech. 100, 333, 1980. 

T. S. Lundgren and N. N. Mansour, Oscillations of drops in zero gravity with weak viscous effects, J. Fluid Mech. 194, 479, 1991. 

E. H. Trinh, D. B. Thiessen, and R. G. Holt, Driven and freely decaying nonlinear shape oscillations of drops and bubbles immersed in a liquid, experimental results, J. Huid Mech. 364, 253, 1998. 

A stone dropped in a pond pushes the water downward, which is countered by elastic forces in the water that tend to restore the water to its initial condition. The movement of the water is up and down, but the crest of the wai c produced moves along the surface of the water. This type of wave is said to be transverse because the displacement of the water is perpendicular to the direction the wave moves. When the oscillations of the wave die out, there has been no net movement of water the pond is just as it was before the stone was dropped. Yet the wave has energy associated with it. A person has only to get in the path of a water wave crashing onto a beach to know that energy is involved. The stadium wave is a transverse wave, as is a wave in a guitar string. 

The conclusion to be drawn from the above examples and many others is that softness in a boiling system, preceding the boiling channel inlet, may cause flow oscillations of low frequency. It is probably the pressure perturbations arising from the explosive nature of nucleate boiling that initiates the oscillation, and the reduced burn-out flux which follows probably corresponds to the trough of the flow oscillation, as a reduction in flow rate always drops the burn-out flux in forced-convection boiling. 

Recently, the size and shape of a liquid droplet at the molten tip of an arc electrode have been studied,12151 and an iterative method for the shape of static drops has been proposed. 216 Shapes, stabilities and oscillations of pendant droplets in an electric field have also been addressed in some investigations. 217 218 The pendant drop process has found applications in determining surface tensions of molten substances. 152 However, the liquid dripping process is not an effective means for those practical applications that necessitate high liquid flow rates and fine droplets (typically 1-300 pm). For such fine droplets, gravitational forces become negligible in the droplet formation mechanism. 

Violent oscillations of the axially symmetric type can be induced in single drops formed at a nozzle. Drops of chlorobenzene (Dg = 0.985 cm) were so formed, and allowed to fall in water. At about five inches below the nozzle two types of rupture were observed. A small droplet was formed at the front and hurled ahead of the drop by the next oscillation. A second mode of formation caused a droplet to be formed by inertial pinch at the rear of the oscillating drop. This rear-formed droplet was always larger than the very small one formed in front. There were, on occasion, two successive pinch-formed droplets from the rear. In a few instances both front and rear formation occurred, as shown in Fig. 13 in selected... 

Not until the above effects can be mathematically related can we expect to progress beyond the experimental stage. To predict such items as size of drop formed at a nozzle, terminal velocity, drag curves, changes of oscillations, and speed of internal circulation, one must possess experimental data on the specific agent in the specific system under consideration. Davies (Dl, D2) proposes the use of the equation... 

Fig. 7.11 Wake configurations for drops in water (highly purified systems), reproduced from Winnikow and Chao (W8) with permission, (a) nonoscillating nitrobenzene drop = 0.280 cm, Re = 515 steady thread-like laminar wake (b) nonoscillating m-nitrotoluene drop 4 = 0.380 cm. Re = 688 steady thread accompanied by attached toroidal vortex wake (c) oscillating nitrobenzene drop 4 = 0.380 cm. Re = 686 central thread plus axisymmetric outer vortex sheet rolled inward to give inverted bottle shape of wake (d) oscillating nitrobenzene drop = 0.454 cm. Re = 775 vortex sheet in c has broken down to form vortex rings (e) oscillating nitrobenzene drop d = 0.490 cm. Re = 804 vortex rings in d now shed asymmetrically and the drop exhibits a rocking motion.

In practice, this model is oversimplified since the exciting wake shedding is by no means harmonic and is itself coupled with the shape oscillations and since Eq. (7-30) is strictly valid only for small oscillations and stationary fluid particles. However, this simple model provides a conceptual basis to explain certain features of the oscillatory motion. For example, the period of oscillation, after an initial transient (El), becomes quite regular while the amplitude is highly irregular (E3, S4, S5). Beats have also been observed in drop oscillations (D4). If /w and are of equal magnitude, one would expect resonance to occur, and this is one proposed mechanism for breakage of drops and bubbles (Chapter 12). 

In general, oscillations may be oblate-prolate (H8, S5), oblate-spherical, or oblate-less oblate (E2, FI, H8, R3, R4, S5). Correlations of the amplitude of fluctuation have been given (R3, S5), but these are at best approximate since the amplitude varies erratically as noted above. For low M systems, secondary motion may become marked, leading to what has been described as random wobbling (E2, S4, Wl). There appears to have been little systematic work on oscillations of liquid drops in gases. Such oscillations have been observed (FI, M4) and undoubtedly influence drag as noted earlier in this chapter. Measurements (Y3) for 3-6 mm water drops in air show that the amplitude of oscillation increases with while the frequency is initially close to the Lamb value (Eq. 7-30) but decays with distance of fall. 

Near the point of drop release, transfer coefficients can be much different from those predicted, due to large amplitude oscillation and internal circulation induced by departure from the nozzle or tip (Al, G4, Y3). 

As for steady motion, shape changes and oscillations may complicate the accelerated motion of bubbles and drops. Here we consider only acceleration of drops and bubbles which have already been formed formation processes are considered in Chapter 12. As for solid spheres, initial motion of fluid spheres is controlled by added mass, and the initial acceleration under gravity is g y - l)/ y + ) (El, H15, W2). Quantitative measurements beyond the initial stages are scant, and limited to falling drops with intermediate Re, and rising... 

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