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.     

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.     

On the application of LDA to bubbly flow in the wobbling regime

 Considerations for applying LDA to bubbly flows with bubbles about 3 to 4 mm in diameter were investigated by means of detailed experiments in the model geometry of a train of bubbles. Both forward scatter and backscatter LDA were studied. The validity of phase discrimination via burst amplitude was tested and special attention was paid to the impact of bubble interface response to the laser beams. Forward and backscatter measurements can be compared well. In both configurations, predominantly the liquid phase is “seen” by LDA. A bubble itself only leads to a velocity realization in special conditions. In those cases the Doppler shift is determined by the motion of the bubble interface which consists of the motion of the center of gravity of the bubble as well as shape oscillations. In backscatter bubbles only give velocity realizations when their “cheeks” pass through the measuring volume virtually perpendicularly. It is shown that the bubble-caused velocity realization frequency is very low for bubbles of the size used. Phase discrimination on burst amplitude does not hold. In ambient cases such as bubble columns one can assume that only the liquid phase is being studied. Received: 4 May 1998/Accepted: 30 September 1998     

Radial oscillations of soluble vapor-gas bubbles in a liquid

Soluble vapor-gas bubbles performing small radial oscillations in a liquid are considered. The heat- and mass-transfer processes and temperature and concentration inhomogeneities in the vaporgas mixture are taken into account. Expressions for the damping rate of radial oscillations of soluble vapor-gas bubbles are obtained. In [1–3] the dynamics of vapor-gas bubble oscillations were considered for a gas insoluble in liquid.     

Effect of weak compressibility of a fluid on bubble-bubble interaction in strong acoustic fields

The effect of weak compressibility of a fluid on the interaction between spherical bubbles in a strong acoustic field is considered. A small parameter ɛ which represents the ratio of the characteristic velocity of radial oscillations of the bubbles to the speed of sound in the fluid is used as a parameter characterizing the fluid compressibility. The equations governing the interaction between two bubbles are derived with an accuracy O(ɛ) in the case in which the ratio of the characteristic velocities of their translational and radial motions is of the order of ɛ. It is shown that neglecting the fluid compressibility effect due to the bubble interaction can lead to either enhancement or attenuation of their radial oscillations following the main compression stage, variation in the oscillation frequency, the bubble approach velocity, and the velocity of the spatial motion of the coupled pair, and the bubble approach and collision rather than their moving away from one another with the formation of a coupled pair.     

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.     

Measurements with a flow direction boundary-layer probe in a two-dimensional laminar separation bubble

 Measurements with a directional sensitive hot-wire probe have been carried out in a two-dimensional laminar separation bubble caused by an adverse pressure gradient. The probe has three parallel, in plane wires and can be traversed in the boundary layer in all spatial directions. The central wire, operated as a conventional hot-wire in CTA mode, and two surrounding resistance wires measure the instantaneous magnitude and direction of the flow, respectively. The probe is calibrated and operated in a similar way as a single hot-wire probe for boundary layer measurements. The frequency response is high enough for measurements of naturally occurring instability waves in the bubble. The flow direction intermittency was measured inside the bubble and regions with reversed flow were mapped out. Prior to reattachment periodical oscillations of the flow direction are found associated with shedding of vortical structures from the bubble. Received: 13 March 1998/Accepted: 22 April 1998     

Numerical Simulation of Three-Dimensional Bubble Oscillations by a Generalized Vortex Method

A numerical method is implemented for simulating the simultaneous three-dimensional volume and shape oscillations of a compressible vapor or gas bubble suspended in an inviscid ambient fluid in the presence of interfacial tension. The flow generated by the bubble expansion, contraction, and deformation is represented by an interfacial distribution of potential dipoles supplemented by a point source situated inside the bubble, accounting for changes in the bubble volume. The mathematical formulation is completed by setting the strength of the point source proportional to the integral of the density of the double-layer potential over the interface. The motion of marker points distributed over the interface is computed using a boundary-element implementation of Baker's generalized vortex method in which the normal component of the interfacial velocity is computed in terms of tangential derivatives of the vector potential associated with the dipoles, whereas the tangential component of the interfacial velocity is computed in terms of the surface gradient of the scalar harmonic potential. The density of the double-layer distribution is computed by solving an integral equation of the second kind using an iterative method, while the evolution of the interfacial distribution of the harmonic potential is computed using Bernoulli's equation for irrotational flow. The onset of interfacial irregularities due to numerical instabilities is prevented by truncating the Fourier–Legendre spectrum of the interfacial distribution of the harmonic potential. With smoothing implemented, the numerical method is capable of describing simultaneous volume and shape oscillations for an indefinite period of time. Received 7 September 2001 and accepted 30 April 2002 Published online 30 October 2002 RID="*" ID="*" This research was supported by a grant provided by NASA. Communicated by J.R. Blake     

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.     

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.     

Bubble velocity jump discontinuity in polyacrylamide solutions: a photographic study

 It has been established that in the presence of elastic as well as surface tension forces, a jump can occur in the bubble velocity-bubble volume plot. It was proposed that this phenomenon was the result of an elastic instability at the bubble interface, which drastically changes the local boundary conditions. The origin of this change was mainly attributed to the magnitude of the elastic forces, which can extract surface active impurities at the gas-liquid interface. The purpose of this paper is to present photographic evidence of this hypothesis and to confirm past results. Received: 4 December 1998 Accepted: 17 February 1999     

Buoyancy-driven motion of a two-dimensional bubble or drop through a viscous liquid in the presence of a vertical electric field

The effect of an electric field on the buoyancy-driven motion of a two-dimensional gas bubble rising through a quiescent liquid is studied computationally. The dynamics of the bubble is simulated numerically by tracking the gas–liquid interface when an electrostatic field is generated in the vertical gap of the rectangular enclosure. The two phases of the system are assumed to be perfect dielectrics with constant but different permittivities, and in the absence of impressed charges, there is no free charge in the fluid bulk regions or at the interface. Electric stresses are supported at the bubble interface but absent in the bulk and one of the objectives of our computations is to quantify the effect of these Maxwell stresses on the overall bubble dynamics. The numerical algorithm to solve the free-boundary problem relies on the level-set technique coupled with a finite-volume discretization of the Navier–Stokes equations. The sharp interface is numerically approximated by a finite-thickness transition zone over which the material properties vary smoothly, and surface tension and electric field effects are accounted for by employing a continuous surface force approach. A multi-grid solver is applied to the Poisson equation describing the pressure field and the Laplace equation governing the electric field potential. Computational results are presented that address the combined effects of viscosity, surface tension, and electric fields on the dynamics of the bubble motion as a function of the Reynolds number, gravitational Bond number, electric Bond number, density ratio, and viscosity ratio. It is established through extensive computations that the presence of the electric field can have an important effect on the dynamics. We present results that show a substantial increase in the bubble’s rise velocity in the electrified system as compared with the corresponding non-electrified one. In addition, for the electrified system, the bubble shape deformations and oscillations are smaller, and there is a reduction in the propensity of the bubble to break up through increasingly larger oscillations.     

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     

Particle tracking velocimetry of the flow field around a collapsing cavitation bubble

The velocity field in the vicinity of a laser-generated cavitation bubble in water is investigated by means of particle tracking velocimetry (PTV). Two situations are explored: a bubble collapsing spherically and a bubble collapsing aspherically near a rigid wall. In the first case, the accuracy of the PTV method is assessed by comparing the experimental data with the flow field around the bubble as obtained from numerical simulations of the radial bubble dynamics. The numerical results are matched to the experimental radius–time curve extracted from high-speed photographs by tuning the model parameters. Trajectories of tracer particles are calculated and used to model the experimental process of the PTV measurement. For the second case of a bubble collapsing near a rigid wall, both the bubble shape and the velocity distribution in the fluid around the bubble are measured for different standoff parameters γ at several instants in time. The results for γ > 1 are compared with the corresponding results of a boundary-integral simulation. For both cases, good agreement between simulation and experiment is found.     

Slip velocity measurements in an upward bubbly flow by combined LDA and electrodiffusional techniques

 An experimental technique for the measurement of the local slip velocity of spherical bubbles is reported. It is based on the measurement of the local liquid velocity by an electrodiffusional method, and the bubble velocity by a specially adapted LDA (Laser Doppler anemometer) with a short measuring volume. The bubble velocity is measured taking into account the shift between the bubble centre and the centre of the LDA measuring volume. The slip velocity is obtained by subtracting the liquid velocity from the bubble velocity at the point corresponding to the bubble centre. The technique is applicable for flows with high velocity gradients. Results of the slip velocity measurements in an upward bubbly flow at laminar pipe Reynolds numbers are presented. Received: 25 July 1996/Accepted: 13 April 1998     

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.     

Cavity and flow measurements of reproducible bubble entrainment following drop impacts

High-speed motion pictures of air–water interface dynamics of drop impacts that reproducibly make bubbles are presented. The pictures show previously unobserved details of the phenomenon. Measurements are compared with available computational methods. Experimental and numerical results agree with each other on the overall shape of the interface and the occurrence of bubble detachment. Measurements, however, show that the cavity depth stagnates before bubble entrapment. This behavior is not predicted by simulation. Also discussed are the presence of a jet that strikes the new bubble after formation and the possible effect of droplet surface oscillations on bubble entrainment. Received: 25 April 2000 / Accepted: 26 April 2001     

Nucleation and growth of a gas bubble in magma

The dynamics of a “collective” gas bubble in the magma melt during its decompression was numerically studied on the basis of a complete mathematical models of an explosive volcanic eruption. It is shown that the bubble size distribution obtained for the nucleation process has one peak, which allows considering a “collective” bubble. The main stages of bubble growth due to gas diffusion and changes in the viscosity of the medium are determined. It is shown that the high viscosity of the melt makes possible the transition from the Rayleigh equation to a simpler relation for the radial velocity of the bubble.     

Behavior of a small single bubble rising in a rotating flow field

A small single bubble was generated with a single-hole nozzle facing upward in a water bath contained in a rotating cylindrical vessel. The bubble size falls in the surface tension force dominant regime. The vertical, radial, and tangential migration velocities of the bubble were measured with two CCD cameras and a high-speed video camera. The tangential velocity component of water flow was measured with particle image velocimetry. A helical motion of the bubble was observed under every experimental condition. The direction of the helical motion was the same as that of the tangential velocity component. This helical motion is associated with the large initial shape deformation of the bubble near the nozzle exit and the subsequent regular shedding of vortices behind it. The period and amplitude of the helical motion were obtained by analyzing the trajectory of the bubble. These quantities were non-dimensionalized by the volume equivalent bubble diameter and the terminal bubble velocity in the vertical direction and correlated as functions of the Eotvos number. Empirical equations were proposed for the period and amplitude. Originally published in the Journal of JSEM, Vol. 4, No. 2, pp. 38–45 (2004).     

Evolution of a Small Distortion of the Spherical Shape of a Gas Bubble under Strong Expansion-Compression

The evolution of a small distortion of the spherical shape of a gas bubble which undergoes strong radial expansion-compression upon a single oscillation of the ambient liquid pressure under a harmonic law are analyzed by numerical experiments. It is assumed that the distortions of the spherical bubble shape are axisymmetric and have the form of individual spherical surface harmonics with numbers of 2–5. Bubble-shape oscillations prior to the beginning of expansion are taken into account. Generally, the distortion value during bubble expansion-compression depends on the phase of bubble-shape oscillation at the beginning of the expansion (initial phase). Emphasis is placed on the dependence of the maximum distortions in the initial phase at certain characteristic times of bubble expansion-compression on the amplitude of the external excitation, liquid viscosity, and distortion mode (harmonic number). The parameters of the problem are typical of the stable periodic sonolumiescence of an individual air bubble in water at room temperature. An exception is the liquid pressure oscillation amplitude, which is varied up to values that are five times the static pressure. That large excitation amplitudes are beyond the stability threshold of periodic oscillations of spherical bubbles. Their consideration is of interest from the point of view of increasing the compression ratio of the bubble gas, i. e., increasing the maximum temperature and density achievable in the final compression stage.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 17–28, July–August, 2005.