Shape oscillations

D. Zardi and G. Seminara. Chaotic mode competition in the shape oscillations of pulsating bubbles. J. Fluid Mech., 286 257-276, 1995. 

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. 

For circulating fluid particles without shape oscillations the internal resistance varies with time in a way similar to that discussed in Chapter 5 for fluid spheres. The occurrence of oscillation, with associated internal circulation, always has a strong effect on the internal resistance. If the oscillations are sufficiently strong to promote vigorous internal mixing, the resistance within the particle becomes constant. 

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. 

Theoretical predictions relating to the orientation and deformation of fluid particles in shear and hyperbolic flow fields are restricted to low Reynolds numbers and small deformations (B7, C8, T3, TIO). The fluid particle may be considered initially spherical with radius ciq. If the surrounding fluid is initially at rest, but at time t = 0, the fluid is impulsively given a constant velocity gradient G, the particle undergoes damped shape oscillations, finally deforming into an ellipsoid (C8, TIO) with axes in the ratio where... 

E aspect ratio averaged over shape oscillations... 

From another standpoint, the superposition of all of these shape oscillations can be viewed as a natural basis for the diffuseness of the nuclear surface. 

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. 

This result is well known. More interesting is the coupling between the radial and shape oscillations when R 0. 

Hence we can see from this example of a bubble with weak time-periodic oscillations of volume that the qualitative condition for stability Gk < 0 does not apply unless Gk is constant or only very slowly varying with time (and in the latter case, strictly only for an interval of time that is short compared with the characteristic time scale for variation of Gk ). If the coefficient Gk varies at a rate that is comparable with the natural frequency of shape oscillation for a bubble of constant volume, then the bubble may undergo a resonant oscillation of shape to very large amplitudes even though Gk < 0 for all times. 

Wurster, B. R. Mohn. 1987. Spike-shaped oscillations in the absence of measurable changes in cyclic AMP concentration in a mutant of Dictyostelium discoideum. J. Cell Science 87 723-30. 

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]. 

Fig. 9.17. The stable flip-flop producing a square-shape oscillation output...

Ehrengruber, M.U., Goates, T.D. and Deranleau, D.A. (1995). Shape oscillations a fundamental response of human neutrophils stimulated by chemotactic peptides FEBS Lett. 359, 229-232. 

Ehrengruber, M.U., Deranleau, D.A. and Coates, T.D. (1996). Shape oscillations of human neutrophil leukocytes Characterization and relationship to cell motility. J Exp. Biol. 199, 741-747. 

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. 

Abstract A bquid droplet may go through shape oscillation if it is forced out of its equilibrium spherical shape, while gas bubbles undergo both shape and volume oscillations because they are compressible. This can happen when droplets and bubbles are exposed to an external flow or an external force. Liquid droplet oscillation is observed during the atomization process when a liquid ligament is first separated from a larger mass or when two droplets are collided. Droplet oscillations may change the rate of heat and mass transport. Bubble oscillations are important in cavitation problems, effervescent atomizers and flash atomization where large number of bubbles oscillate and interact with each other. This chapter provides the basic theory for the oscillation of liquid droplet and gas bubbles. 

A liquid droplet free from any other forces except its surface tension forces tends to remain in equilibrium, spherical shape. Oscillations occur when a liquid droplet is forced out of its equilibrium shape. An initially spherical inviscid droplet wifli radius R that is perturbed by C will oscillate according to [1]... 

Fig. 5.6 Shape oscillation of a 1 cm diameter water droplet on a vibrating plate. The lower shapes are those after half period. From (a) to (f) values of n (harmonic mode) increases from 2 to 7 [31] (Courtesy of JPST)...

Gas bubbles are relevant to various aspects of the atomization and sprays. In flashing process or flash atomization, bubbles are formed inside the liquid which significantly alter the atomization process (see Chap. 10). Also in effervescent atomizers, high-pressure air is injected inside a liquid and disperses as small bubbles. In addition, bubbles are formed in cavitating nozzles, which significantly alter the atomization process. Gas bubbles go through volume oscillations in addition to shape oscillation discussed in the previous section. In this section, dynamic evolution and stability of a spherical bubble undergoing volume oscillation is discussed. 

As the amplitude of the volume oscillation increases, the radial motion of the bubble becomes unstable, and small disturbances cause the bubble to undergo various shape oscillations [42]. As the bubble radius increases, the threshold for the instability decreases. Excitation of different shapes is referenced to as dancing bubble motion or erratic motion . 

Similar to energy transfer between different shapes of a liquid droplet, coupling between the volume oscillation and different shape oscillations occur for bubbles in acoustic fields [43]. Interaction between modes can lead to chaotic response of the bubble to the external forcing. For a large enough bubble, the spectrum of distortion modes is dense, and several distortion modes attribute to the shape. Development of chaos depends on the number of excited shape modes. 

Onset of shape oscillation of a bubble also excites the translational motion of it. Since the forces on the bubble, which govern its translational motion, depend on the shape of the bubble, shape oscillation induces an imbalance on the position of the bubble, and its center of mass starts to move. When equations governing volume oscillations, shape oscillations, and translational motion are solved simultaneously, it reveals that any perturbation in any of these motions, if large enough, can excite other motiOTis. These motions are coupled leading to nonlinear behavior of bubbles under forced oscillation [44]. Excitation of translational motion was also interpreted as Self-propulsion of asymmetrically vibrating bubbles by Benjamin and Ellis [45] who used a nonlinear analysis to explain the so-called erratic motion of bubble in acoustic fields. 

Shape oscillation and translational motion of gas bubbles inside a liquid medium is also achieved in the context of forced vibration of hquid containers with dispersed gas bubbles. Vibration induces an acceleration and effectively a buoyancy force which oscillates in time. Depending on the direction of the applied force, bubbles move in the liquid medium, and since the pressure variation around the bubble is... 

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