Shape oscillations drops

In order to allow for a variation of interfacial area from that of an equivalent sphere, the eccentricity of the ellipsoidal drop must be taken into account. The area ratio of Eq. (44) does not exceed unity by a serious amount until an eccentricity of 1.5 is attained. An experimental plot of eccentricity as ordinate vs. equivalent spherical drop diameter as abscissa may result in a straight line (G7, Kl, K3, S12). A parameter is yet to be developed by which the lines can be predicted without recourse to experiment. Eccentricity is not an accurate shape description of violently oscillating drops and should therefore be used only for drop size below the peak diameter (region B of Fig. 5). 

The conditions under which fluid particles adopt an ellipsoidal shape are outlined in Chapter 2 (see Fig. 2.5). In most systems, bubbles and drops in the intermediate size range d typically between 1 and 15 mm) lie in this regime. However, bubbles and drops in systems of high Morton number are never ellipsoidal. Ellipsoidal fluid particles can often be approximated as oblate spheroids with vertical axes of symmetry, but this approximation is not always reliable. Bubbles and drops in this regime often lack fore-and-aft symmetry, and show shape oscillations. 

General criteria for determining the shape regimes of bubbles and drops are presented in Chapter 2, where it is noted that the boundaries between the different regions are not sharp and that the term ellipsoidal covers a variety of shapes, many of which are far from true ellipsoids. Many bubbles and drops in this regime undergo marked shape oscillations, considered in Section F. Where oscillations do occur, we consider a shape averaged over a small number of cycles. 

Fig. 7.12 Simple model to show nature of shape oscillations for bubbles and drops in free 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). 

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

These flow transitions lead to a complex dependence of transfer rate on Re and system purity. Deliberate addition of surface-active material to a system with low to moderate k causes several different transitions. If Re < 200, addition of surfactant slows internal circulation and reduces transfer rates to those for rigid particles, generally a reduction by a factor of 2-4 (S6). If Re > 200 and the drop is not oscillating, addition of surfactant to a pure system decreases internal circulation and reduces transfer rates. Further additions reduce circulation to such an extent that shape oscillations occur and transfer rates are increased. Addition of yet more surfactant may reduce the amplitude of the oscillation and reduce the transfer rates again. Although these transitions have been observed (G7, S6, T5), additional data on the effect of surface active materials are needed. 

The assumption of transfer by a purely turbulent mechanism in the Handlos-Baron model leads to the prediction that the internal resistance is independent of molecular diffusivity. However, such independence has not been found experimentally, even for transfer in well-stirred cells or submerged turbulent jets (D4). In view of this fact and the neglect of shape and area oscillations, models based upon the surface stretch or fresh surface mechanism appear more realistic. For rapid oscillations in systems with Sc 1, mass transfer rates are described by identical equations on either side of the drop surface, so that the mass transfer results embodied in Eqs. (7-54) and (7-55) are valid for the internal resistance if is replaced by p. Measurements of the internal resistance of oscillating drops show that the surface stretch model predicts the internal resistance with an average error of about 20% (B16, Yl). Agreement of the data for drops in liquids with Eq. (7-56) considerably improves if the constant is increased to 1.4, i.e.. 

There is some evidence that isolated drops may shake themselves apart if shape oscillations become sufficiently violent (L7). It has been suggested (El, Gil, H22) that breakup occurs when the exciting frequency of eddy shedding matches the natural frequency of the drop. However, other workers (S7) have found that oscillations give way to random wobbling before breakup occurs. While it is possible that resonance may produce breakup in isolated cases, this mechanism appears to be less important than the Taylor instability mechanism described above. 

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. 

In recent years, several theoretical and experimental attempts have been performed to develop methods based on oscillations of supported drops or bubbles. For example, Tian et al. used quadrupole shape oscillations in order to estimate the equilibrium surface tension, Gibbs elasticity, and surface dilational viscosity [203]. Pratt and Thoraval [204] used a pulsed drop rheometer for measurements of the interfacial tension relaxation process of some oil soluble surfactants. The pulsed drop rheometer is based on an instantaneous expansion of a pendant water drop formed at the tip of a capillary in oil. After perturbation an interfacial relaxation sets in. The interfacial pressure decay is followed as a function of time. The oscillating bubble system uses oscillations of a bubble formed at the tip of a capillary. The amplitudes of the bubble area and pressure oscillations are measured to determine the dilational elasticity while the frequency dependence of the phase shift yields the exchange of matter mechanism at the bubble surface [205,206]. 

Additional information on interfacial layers can be gained from rheological and ellipsometry experiments. There is quite a number of different experimental setups used to determine surface rheological parameters (27). New possibilities to determine surface dilational parameters arise from oscillating-drop experiments. Using axisymmetric drop shape analysis (ADSA) the change in interfacial ten-... 

The drop-shape oscillation technique as developed by Tian et al. (206, 207) is another technique suitable for closing the gap in the experimental methods for liquid/liquid interfaces. This method is based on the analysis of drop-shape oscillation modes and yields again the matter-ex-change mechanism and the dila-tional interfacial elasticity. The method is similar to the transient relaxation methods applicable only for comparatively low oscillation frequencies. 

E. Trinh and T. G. Wang, Large-amplitude free and driven drop shape oscillations experimental observations, J. Fluid Mech. 122, 315, 1982. 

P. L. Marston, Shape oscillation and static deformation of drops and bubbles driven by modulated radiation stresses theory, J. Acoust. Soc. Am. 67, 15, 1980. 

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. 

Feng, Z. C. Instability caused by the coupling between non-resonant shape oscillation modes of a charged conducting drop, J. Fluid Mech. 333, 1-21 (1997). 

Feng, Z. C., Y. H. Su Numerical simulations of the translational and shape oscillations of a liquid drop in an acoustic field, Phys. Huids 9, 519-529 (1997). 

The instrument shown schematically in Fig. 26 is suitable for slow oscillation experiments, as it was performed for the first time by Miller et al. in 1993. The frequency limit of the oscillations is given by the condition for the liquid meniscus shape, which has to be Laplacian. Under too fast deformations this condition is not fulfilled and hence the method does not provide reliable results. To reach higher frequencies of oscillation, the above mentioned oscillating drop or bubble experiments are suitable, because the shape of the menisci is spherical due to the small diameter. The instrument of Fig. 26 can be designed such that a pressure sensor and piezo translator are built in and the video system serves as optical control and determines the drop/bubble diameter accurately. 

In order to study the shape oscillations of a sessile drop, the experimental setup shown in Fig. 5 was designed and assembled. The setup consists of an electrowetting chip, a high-speed... 

As a drop becomes larger, it begins to distort in shape, oscillate, and finally break into smaller drops. Circulation occurs within it. Skelland provides a number of correlations for dispersed- and continuous-phase coefficients for various situations in the two phases, for example, a circulating drop with significant resistance in both phases, and an oscillating drop with resistance mainly within the drop. It is clear that... 

Drop and bubble methods have been developed significantly during the last years. These techniques provide access to dynamic properties of liquid interfaces. The drop and bubble shape technique as well as the fast oscillating drops and bubbles are described essentially as tools for dilational surface rheology. 

After the validation the influence of the non-Newtonian fluid properties on drop shape oscillations is investigated. For this purpose, droplets of Praestol 2500 0.8 % (P2500 0.8 %), Praestol 2500 0.3 % (P2500 0.3 %), and Praestol 2540 0.05 % (P2540) were simulated. For comparison, Newtonian droplets with the constant... 

Another oscillatory method makes use of a drop acoustically levitated in a liquid. The drop is made to oscillate in shape, and the interfacial tension can be calculated from the resonance frequency [113]. 

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. 

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