Free oscillations of drops and bubbles: the initial-value problem
TL;DR: In this paper, the authors study the initial value problem posed by the small amplitude free oscillations of free drops, gas bubbles, and drops in a host liquid when viscous effects cannot be neglected.
Abstract: We study the initial-value problem posed by the small-amplitude (linearized) free oscillations of free drops, gas bubbles, and drops in a host liquid when viscous effects cannot be neglected. It is found that the motion consists of modulated damped oscillations, with the damping parameter and frequency approaching only asymptotically the results of the normal-mode analysis. The connexion with the normal-mode method is demonstrated explicitly and the experimental relevance of our results is discussed.
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TL;DR: In this paper, the basic equations for nonlinear bubble oscillation in sound fields are given, together with a survey of typical solutions, and three stability conditions for stable trapping of bubbles in standing sound fields: positional, spherical and diffusional stability.
Abstract: Bubbles in liquids, soft and squeezy objects made of gas and vapour, yet so strong as to destroy any material and so mysterious as at times turning into tiny light bulbs, are the topic of the present report. Bubbles respond to pressure forces and reveal their full potential when periodically driven by sound waves. The basic equations for nonlinear bubble oscillation in sound fields are given, together with a survey of typical solutions. A bubble in a liquid can be considered as a representative example from nonlinear dynamical systems theory with its resonances, multiple attractors with their basins, bifurcations to chaos and not yet fully describable behaviour due to infinite complexity. Three stability conditions are treated for stable trapping of bubbles in standing sound fields: positional, spherical and diffusional stability. Chemical reactions may become important in that respect, when reacting gases fill the bubble, but the chemistry of bubbles is just touched upon and is beyond the scope of the present report. Bubble collapse, the runaway shrinking of a bubble, is presented in its current state of knowledge. Pressures and temperatures that are reached at this occasion are discussed, as well as the light emission in the form of short flashes. Aspherical bubble collapse, as for instance enforced by boundaries nearby, mitigates most of the phenomena encountered in spherical collapse, but introduces a new effect: jet formation, the self-piercing of a bubble with a high velocity liquid jet. Examples of this phenomenon are given from light induced bubbles. Two oscillating bubbles attract or repel each other, depending on their oscillations and their distance. Upon approaching, attraction may change to repulsion and vice versa. When being close, they also shoot self-piercing jets at each other. Systems of bubbles are treated as they appear after shock wave passage through a liquid and with their branched filaments that they attain in standing sound fields. The N-bubble problem is formulated in the spirit of the n-body problem of astrophysics, but with more complicated interaction forces. Simulations are compared with three-dimensional bubble dynamics obtained by stereoscopic high speed digital videography.
586 citations
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TL;DR: Eulerian surface tension models play an increasingly important role in our capacity to understand and predict a wide range of multiphase flow problems The accuracy and robustness of these models have improved markedly in the past 20 years, so that they are now applicable to complex, threedimensional configurations of great theoretical and practical interest as discussed by the authors.
Abstract: Numerical models of surface tension play an increasingly important role in our capacity to understand and predict a wide range of multiphase flow problems The accuracy and robustness of these models have improved markedly in the past 20 years, so that they are now applicable to complex, three-dimensional configurations of great theoretical and practical interest In this review, I attempt to summarize the most significant recent developments in Eulerian surface tension models, with an emphasis on well-balanced estimation, curvature estimation, stability, and implicit time stepping, as well as test cases and applications The advantages and limitations of various models are discussed, with a focus on common features rather than differences Several avenues for further progress are suggested
309 citations
Cites background from "Free oscillations of drops and bubb..."
...Prosperetti derived closed-form solutions for the Laplace transform of shape evolution both for planar and spherical interfaces (Prosperetti 1980, 1981)....
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TL;DR: In this paper, a simplified model of the oscillations of a gas bubble in a slightly compressible liquid is discussed by means of simplified model based on the assumption of a spatially uniform internal pressure.
Abstract: Several aspects of the oscillations of a gas bubble in a slightly compressible liquid are discussed by means of a simplified model based on the assumption of a spatially uniform internal pressure. The first topic considered is the linear initial-value problem for which memory effects and the approach to steady state are analysed. Large-amplitude oscillations are studied next in the limit of large and small thermal diffusion lengths obtaining, in the first case, an explicit expression for the internal pressure, and, in the second one, an integral equation of the Volterra type. The validity of the assumption of uniform pressure is then studied analytically and numerically. Finally, the single-bubble model is combined with a simple averaged-equation model of a bubbly liquid and the propagation of linear and weakly nonlinear pressure waves in such a medium is considered.
272 citations
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TL;DR: In this article, the authors investigated the oscillations of moderate to large raindrops using a seven-story fall column with shape data obtained from multiple-strobe photographs, with additional measurements at intermediate distances to assess the role of aerodynamic feedback as the source of drop oscillations.
Abstract: The oscillations of moderate to large raindrops are investigated using a seven-story fall column with shape data obtained from multiple-strobe photographs. Measurements are made at a fall distance of 25 m for drops of D 5 2.5-, 2.9-, 3.6-, and 4.0-mm diameter, with additional measurements at intermediate distances to assess the role of aerodynamic feedback as the source of drop oscillations. Oscillations, initiated by the drop generator, are found to decay during the first few meters of fall and then increase to where the drops attained terminal speed near 10 m. Throughout the lower half of the fall column, the oscillation amplitudes are essentially constant. These apparently steady-state oscillations are attributed to resonance with vortex shedding. For D 5 2.5 and 3.6 mm, the mean axis ratio is near the theoretical equilibrium value, a result consistent with axisymmetric (oblate/prolate mode) oscillations at the fundamental frequency. For D 5 2.9 and 4.0 mm, however, the mean axis ratio is larger than the theoretical equilibrium value by 0.01 to 0.03, a characteristic of transverse mode oscillations. Comparison with previous axis ratio and vortex-shedding measurements suggests that the oscillation modes of raindrops are sensitive to initial conditions, but because of the prevalence of offcenter drop collisions, the predominant steady-state response in rain is expected to be transverse mode oscillations. A simple formula is obtained from laboratory and field measurements to account for the generally higher average axis ratio of raindrops having transverse mode oscillations. In the application to light to heavy rainfall, the ensemble mean axis ratios for raindrop sizes of D 5 1.5‐4.0 mm are shifted above equilibrium values by 0.01‐0.04, as a result of steady-state transverse mode oscillations maintained intrinsically by vortex shedding. Compared to the previous axis ratio formula based on wind tunnel measurements, the increased axis ratios for oscillating raindrops amount to a reduction of 0.1‐0.4 dB in radar differential reflectivity ZDR, and an increase of about 0.5 mm for a reflectivity-weighted mean drop size of less than about 3 mm.
256 citations
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TL;DR: In this article, an ultrasonic levitation device is described together with its various applications in the fields of fluid dynamics, material science, and light scattering, including surface waves on freely suspended liquids, the variations of the surface tension with temperature and contamination, the deep undercooling of materials with the temperature variations of their density and viscosity.
Abstract: An ultrasonic levitation device operable in both ordinary ground-based as well as in potential space-borne laboratories is described together with its various applications in the fields of fluid dynamics, material science, and light scattering. Some of the phenomena which can be studied by this instrument include surface waves on freely suspended liquids, the variations of the surface tension with temperature and contamination, the deep undercooling of materials with the temperature variations of their density and viscosity, and finally some of the optical diffraction properties of transparent substances.
241 citations
References
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TL;DR: An accurate method is presented for the numerical inversion of Laplace transform, which is a natural continuation to Dubner and Abate's method, and the error bound on the inverse f{t) becomes independent of t, instead of being exponential in t.
Abstract: An accurate method is presented for the numerical inversion of Laplace transform, which is a natural continuation to Dubner and Abate's method. (Dubner and Abate, 1968). The advantages of this modified procedure are twofold: first, the error bound on the inverse f{t) becomes independent of t, instead of being exponential in t; second, and consequently, the trigonometric series obtained for fit) in terms of F(s) is valid on the whole period 2T of the series. As it is proved, this error bound can be set arbitrarily small, and it is always possible to get good results, even in rather difficult cases. Particular implementations and numerical examples are presented.
953 citations
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TL;DR: In this paper, a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients.
Abstract: From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.
332 citations
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TL;DR: In this paper, the problem of describing free oscillations of a viscous liquid drop and of a bubble in a fluid is studied in detail, and it is shown that the oscillations are initially describable in terms of an irrotational approximation, and that the normal-mode results are recovered as t −* <».
Abstract: The problem of two viscous, incompressible fluids separated by a nearly spherical free surface is considered in general terms as an initial-value problem to first order in the perturbation of the spherical symmetry. As an example of the applications of the theory, the free oscillations of a viscous liquid drop and of a bubble in a viscous liquid are studied in some detail. It is shown that the oscillations are initially describable in terms of an irrotational approximation, and that the normal-mode results are recovered as t —* <». In between these asymptotic regimes, however, the motion is significantly different from either approximation.
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TL;DR: In this article, it was shown that for the diffraction of an arbitrary two-dimensional incident pulse by a wedge of angle n, the ratio of the resultant velocity potential to the corresponding value of the incident pulse at the corner of the wedge at any instant is equal to 2x/ (2x n) n; and that for a threedimensional pulse diffraction by a cone of solid angle u>, the ratio at the vertex of the cone is equal tO 4ir/ (47T ) co).
Abstract: By virtue of Green's Theorem, it is shown that for the diffraction of an arbitrary two-dimensional incident pulse by a wedge of angle n, the ratio of the resultant velocity potential to the corresponding value of the incident pulse at the corner of the wedge at any instant is equal to 2x/ (2x — n); and that for the diffraction of a threedimensional pulse by a cone of solid angle u>, the ratio at the vertex of the cone is equal tO 4ir/ (47T — co). Two-dimensional space. The statement concerning diffraction of a pulse by a wedge is evidently true in the special case of an incident plane Heaviside pulse which was solved by Keller and Blank [1]. It therefore also follows for all incident pulses which are superpositions of plane Heaviside pulses, or limits of such superpositions. Since this includes all incident pulses it yields the preceding statement. However, these considerations depend upon knowing the exact solution in a special case which the following proof does not require.** Let t — 0 be the instant at which the incident pulse — t„ fulfills the wave equation and the same initial conditions as that of
156 citations