Ellipsoidal Oscillations of a Gas Bubble with Periodic Variation of the Surrounding Fluid Pressure

Ellipsoidal linear and nonlinear oscillations of a gas bubble under harmonic variation of the surrounding fluid pressure are studied. The system is considered under conditions in which periodic sonoluminescence of the individual bubble in a standing acoustic wave is observable. A mathematical model of the bubble dynamics is suggested; in this model, the variation of the gas/fluid interface shape is described correct to the square of the amplitude of the deformation of the spherical shape of the bubble. The character of the air bubble oscillations in water is investigated in relation to the initial bubble radius and the fluid pressure variation amplitude. It is shown that nonspherical oscillations of limited amplitude can occur outside the range of linearly stable spherical oscillations. In this case, both oscillations with a period equal to one or two periods of the fluid pressure variation and aperiodic oscillations can be observed.     

Forced oscillations of a gas bubble in a spherical volume of a compressible liquid

A spherically symmetric problem of oscillations of a single gas bubble at the center of a spherical flask filled with a compressible liquid under the action of pressure oscillations on the flask wall is considered. A system of differential-difference equations is obtained that extends the Rayleigh-Plesset equation to the case of a compressible liquid and takes into account the pressure-wave reflection from the bubble and the flask wall. A linear analysis of solutions of this system of equations is performed for the case of harmonic oscillations of the bubble. Nonlinear resonance oscillations and nearly resonance nonharmonic oscillations of the bubble caused by harmonic pressure oscillations on the flask wall are analyzed. Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 111–118, March–April, 1999.     

Evolution of deviations from the spherical shape of a vapor bubble in supercompression

This paper considers the evolution of small deviations of a cavitation bubble from a spherical shape during its single compression under conditions of experiments on acoustic cavitation of deuterated acetone. Vapor motion in the bubble and the surrounding liquid is defined as a superposition of the spherical component and its non-spherical perturbation. The spherical component is described taking into account the nonstationary heat conductivity of the liquid and vapor and the nonequilibrium nature of the vaporization and condensation on the interface. At the beginning of the compression process, the vapor in the bubble is considered an ideal gas with a nearly uniform pressure. In the simulation of the high-rate compression stage, realistic equations of state are used. The non-spherical component of motion is described taking into account the effect of liquid viscosity, surface tension, vapor density in the bubble, and nonuniformity of its pressure. Estimates are obtained for the amplitude of small perturbations (in the form of harmonics of degree n = 2, 3, ... with the wavelength λ = 2πR/n, where R is the bubble radius) of the spherical shape of the bubble during its compression until reaching extreme values of pressure, density, and temperature. These results are of interest in the study of bubble fusion since the non-sphericity of the bubble prevents its strong compression.     

Forced oscillations of two gas bubbles in a fluid in the vicinity of bubble contact

For a theoretical derivation of bubble coalescence conditions, nonlinear forced oscillations of two closely spaced spherical bubbles subjected to the action of a periodic external pressure field are considered. The equations, asymptotic with respect to a small distance between the bubble surfaces, are derived to describe the approach of the bubbles under the action of (i) the Bjerknes attraction force averaged over the oscillation period and (ii) the viscous drag. It is shown that due to nonlinear interaction of the viscous drag with the radial and translational oscillations of the bubbles a unidirectional repulsive force is generated, which prevents the approach of the bubbles. The coalescence of the bubbles is possible when the nondimensional parameter combined from the amplitude and frequency of the external pressure field, the bubble radius, and the fluid viscosity is greater than a certain critical value. The obtained coalescence condition is qualitatively confirmed by experiments.     

Evolution of distortions of the spherical shape of a cavitation bubble in acoustic supercompression

The evolution of small perturbations of the spherical shape of a vapor bubble in the process of its single strong expansion and compression in deuterated acetone is studied. In the mathematical model used the motion of vapor and liquid is broken down into the spherical component and its small nonspherical perturbation. The spherical component is described by the fluid dynamics equations with account for time-dependent heat conduction and evaporation and condensation on the liquid-vapor interface using equations of state constructed from experimental data. In describing the nonspherical component the liquid viscosity and the surface tension are taken into account, while the effect of the bubble content is disregarded. Certain simple analytical formulas are presented which describe the bubble radius at the moment of maximum expansion, its variation in the compression stage, and the evolution of the bubble sphericity distortion in compression.     

Effect of viscoelastic properties of a liquid on the dynamics of small oscillations of a gas bubble

A series of papers has been devoted to questions of gas bubble dynamics in viscoeiastic liquids. Of these papers we mention [1–4]. The radial oscillations of a gas bubble in an incompressible viscoeiastic liquid have been studied numerically in [1, 2] using Oldroyd's model [5]. Anexact solution was found in [3], and independently in [4], for the equation of small density oscillations of a cavity in an Oldroyd medium when there is a periodic pressure change at infinity. The analysis of bubble oscillations in a viscoeiastic liquid is complicated by properties of limiting transitions in the rheological equation of the medium. These properties are of particular interest for the problem under investigation. These properties are discussed below, and characteristics of the small oscillations of a bubble in an Oldroyd medium are investigated on the basis of a numerical analysis of the exact solution obtained in [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 82–87, May–June, 1976.The authors are grateful to V. N. Nikolaevskii for useful advice and for discussing the results.     

Forced nonlinear oscillations of a gas bubble in a large spherical flask (resonator) filled with a fluid

Radial oscillations of a gas bubble in a large spherical flask filled with a fluid are considered. We derive an equation of the change of the bubble radius by the known law of pressure variation at the boundary of the liquid volume (the law of motion of the piston) for a period of time during which, repeatedly reflected from the piston, the leading front of the reflected-from-the bubble perturbations reaches the bubble. For further calculations of the change of the bubble radius, recurrent relations which include the wave reflected from the bubble in the previous cycle and its subsequent reflection from the piston are obtained. Under harmonic action of the piston on the fluid-bubble system, a certain periodic regime with a package of bubble oscillations is established. Institute of Mechanics, Ural Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 77–87, September–October, 1998.     

Oscillations of a gas-vapor bubble in an acoustic field

It is shown that at large vapor contents, as a result of the combined action of phase transitions and capillary effects, the small radially symmetric oscillations of gas-vapor bubbles in an acoustic field are unstable in amplitude. The critical vapor concentration in the bubble separating regions of qualitatively different bubble behavior in the acoustic field is determined. Expressions are obtained for the decay rate of the radial oscillations of the gas-vapor bubble and the growth rate characterizing the rate of increase of oscillation amplitude in the region of instability. It is shown that adding only a slight amount of gas to the vapor bubble leads to a marked decrease in the growth rate. It is found that in the particular case of a vapor bubble the tine growth rate characterizing the development of the instability is of the same order as the second resonance frequency of the vapor bubble. This may serve to explain why in the case of vapor bubble oscillations the second resonance effect, which has been established in a number of theoretical studies and is widely discussed in the literature, has not yet been experimentally confirmed. The problem of spherically symmetrical processes around gasvapor bubbles was posed in [1], and their small oscillations are investigated in detail in [2–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 79–33, May–June, 1986.The authors are grateful to R. I. Nigmatulin for useful discussions.     

Optimization of shock-wave action on a spherical bubble in an ideal incompressible liquid

The possibility of controlling the oscillations of a spherical gas bubble in an ideal incompressible liquid is subjected to theoretical analysis. Liquid surface tension forces are not taken into account. The optimization process realizing a maximum of the radius amplitude and a maximum of the gas pressure in the bubble for a given impulsive change of pressure at infinity is considered. A shock-resonance bubble oscillation procedure giving stepwise pressure changes at the extrema of the radius is constructed. This problem is of interest in connection with the investigation of cavitation erosion [1] and processes in biological tissues [2–4]. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–178, September–October, 1988.     

Liquid vorticity computation in non‐spherical bubble dynamics

The purpose of this work is to compare efficiency of a number of numerical techniques of computation of liquid vorticity from non‐spherical bubble oscillations. The techniques based on the finite‐difference method (FDM), the collocation method (one with differentiating (CMd) the integral boundary condition and another without it (CM)) and the Galerkin method (GM) have been considered. The central‐difference approximations are used in FDM. Sinus functions are chosen as the basis in GM. Problems of decaying a small distortion of the spherical shape of a bubble and dynamics of a bubble under harmonic liquid pressure variation with various parameters are used for comparison. The FDM technique has been found to be most efficient in all the cases. Copyright © 2004 John Wiley & Sons, Ltd.     

Formation and Amplification of Shock Waves in a Bubble “Cord”

The structure and dynamics of the wave field generated by a bubble system in the form of an axial bubble cylinder (cord) excited by a plane shock wave propagating along the axis in an axisymmetric shock tube are numerically examined. It is shown that consecutive excitation of oscillations of the bubble zone results in formation of a quasi-steady shock wave in the cord and in the ambient liquid. Results of the numerical analysis of the maximum amplitude of the resulting wave as a function of problems parameters are described.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 46–52, September–October, 2005.     

Oscillations of a gas bubble in a non-Newtonian liquid under the action of an acoustic field

The evolution of the radius of a spherical cavitation bubble in an incompressible non-Newtonian liquid under the action of an external acoustic field is investigated. Non-Newtonian liquids having relaxation properties and also pseudoplastic and dilatant liquids with powerlaw equation of state are studied. The equations for the oscillation of the gas bubble are derived, the stability of its radial oscillation and its spherical form are investigated, and formulas are given for the characteristic frequency of oscillations of the cavitation hollow in a relaxing liquid. The equations are integrated numerically. It is shown that in a relaxing non-Newtonian liquid the viscosity may lead to the instability of the radial oscillations and the spherical form of the bubble. The results obtained here are compared with the behavior of a gas bubble in a Newtonian liquid.     

Parametric pulse excitation in distributed mechanical systems with nonstationary boundaries

In a distributed system whose parameters vary with time the natural oscillation modes are interconnected and so it is possible to get parametric excitation of several synchronized harmonic modes simultaneously. If the natural oscillation spectrum of such a system consists of almost equally spaced lines, then a periodic change of the parameters with time can lead to the excitation of pulse-type oscillations [1]. This phenomenon can occur both in systems whose size varies with time and in systems whose boundary properties are nonstationary. The present paper is devoted to a study of the instability in these systems.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 145–151, July–August, 1976.     

Free nonlinear oscillations of a thermally relaxing spherical gas bubble in an incompressible liquid

The quasi-adiabatic regime of free oscillation of a bubble in the presence of irreversible interphase heat transfer between the bubble and the ambient liquid is studied. On the basis of simplified model equations of a rarefield bubble mixture, a nonlinear-oscillation equation of the relaxation type is obtained. In constructing an exact particular solution of this equation, the heat transfer law associated with bubble compression is established. For studying the harmonic oscillations, the Krylov-Bogolyubov-Mitropol’skii asymptotic method is used. It is shown that, for a small bubble, the viscosity and heat transfer effects are of the same order. For a small bubble, the influence of these effects on the formation of the natural-oscillation frequency, which is small in the linear approximation, may be significant in the nonlinear formulation. For a large bubble, the influence of these effects is negligible in both approximations. For the approximate solution of the nonlinear equation, a uniformly valid second-order expansion is constructed.     

On Rectified Diffusion and Sonoluminescence

If acoustically driven, a gas-filled bubble may exist indefinitely even in an unsaturated liquid through a process known as “rectified diffusion.” When the oscillation period is small compared with the gaseous diffusion time, the radius of the steadily oscillating bubble can be determined by asymptotic methods, in the way pioneered by Eller and Flynn (1965). The next term in their expansion is evaluated here and is shown to be significant if the radius of the bubble is small or if the amplitude of its oscillations is large. For the identical level of saturation and the same conditions of excitation, multiple solutions are possible. As a result of resonance between overtones of the frequency of free bubble oscillation with the frequency of the acoustic drive, there generally exist, in addition to a stable large-radius, stable small-radius states. The relevance of the present results to sonoluminescence is briefly discussed. Received 3 January 1997 and accepted 14 April 1997     

Nonlinear dynamics and small damping signal control of chaos in a model of flow-induced oscillations of cylinders

Nonlinear dynamics of flow-induced oscillations of cylinders is investigated. The approach in our paper is made to introduce an harmonic forced vibration in the coupling term of the structural equation since this may be the consequence of approximating the potential force that could act as a periodic excitation. The method of multiple scales is used to determine the steady state responses. Amplitude and phase modulation equations as well as external force-response and frequency-response curves are obtained. We show that harmonic excitation can induce resonance phenomena in the oscillation of the structure for a range of frequencies of potential force, and also lock-in phenomena appear in the structure part. Also, we find that the structure can be damaged as the amplitude of the potential excitation increases. Numerical simulations confirm the existence of chaotic vibration in the system, a small damping signal control is used to suppress it since it may cause fatigue in the system. The model developed is expected to yield better results for structure in water.     

Instability of a spherical gas bubble in a variable-pressure field

Only a few studies, of which we mention [1–5], have been addressed to the problem of the stability of the accelerated motion of a spherical interface of two fluids. In the present paper we consider the problem of the stability of radial motion of the spherical boundary of a gas bubble in an incompressible inviscid liquid under the action of variable external pressure. Surface tension is not taken into account. We study the possibility of the existence of stable motions for broad classes of time dependence of the external pressure, namely for monotonic and periodic dependences. It is shown that stability is possible only for infinitely large bubble radii or for very specific assumptions concerning the initial conditions and the pressure-time dependence law.     

Curvilinear motion of an ellipsoidal bubble

The Lagrange equations are used to investigate the curvilinear motion of an ellipsoidal bubble. At a small inclination of the minor axis of the ellipsoid to the vertical, the ellipsoid begins to oscillate about the equilibrium position and its trajectory begins to swing. Analytic expressions are presented for the oscillation frequency and for the ratio of the swing amplitude to the amplitude of the oscillations of the ellipsoid. It is assumed that the bubble has the form of an axially symmetrical ellipsoid [1].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 90–93, May–June, 1971.     

Comparative Analysis of the Profile Evolutions of an Elastic Harmonic Wave Caused by the Second and Third Harmonics

The influence of the second and third harmonics on the evolution of a harmonic longitudinal wave propagating through a nonlinearly elastic material has been simulated for real composite materials (the most typical plots are presented for a granulated composite with copper granules and molybdenum matrix). The frequency and initial amplitude are varied beginning from conditionally small values (at which visible distortions appear after a great number of oscillations) to extremely large values (at which the profile becomes distorted already after the second or third oscillation). Four and three different stages of profile evolution due to the influence of the second and third harmonics, respectively, are observed. It is found out that the effect of the initial amplitude on the evolution process is weaker for the second harmonic, and the effect of the frequency on the evolution process is weaker for the third harmonics. It is also revealed that the ranges of frequencies and initial amplitudes within which the evolution caused by different harmonics is very intensive are different—the effect of the third harmonic is stronger at larger values of both parameters. The effects of both harmonics are tantamount within the boundary ranges where the second harmonic is already predominant and the third harmonic is at the early stage of development     

Dynamics of soluble gas bubbles

The problem of the mass, thermal and dynamic interaction between a bubble containing a soluble gas and a liquid is considered. It is shown that this problem can be reduced to the problem of the behavior of a vapor bubble with phase transitions investigated in detail in [1–3]. Expressions are obtained for the rate of decay of the radially symmetric oscillations of the bubbles due to the solubility of the gas in the liquid. The effective coefficients of mass transfer between the radially pulsating bubbles and the liquid are determined. A numerical solution is obtained for the problem of the radial motion of a bubble created by a sudden change of pressure in the liquid which, in particular, corresponds to the behavior of the bubbles behind the shock front when a shock wave enters a bubble screen.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 52–59, November–December, 1985.