Nonlinear Bubble Oscillations Research Articles

Nonlinear oscillations of gas bubbles in elastic solids

This work was part of a project whose long-range goal is the development of a device for estimating the size of bubbles in human tissue. Calibration standards are expected to consist of gas bubbles in polymer gels. Accordingly, equations in Prosperetti [J. Acoust. Soc. Am. 56, 878–885 (1974)] that describe forced oscillations of a gas bubble in a liquid have been modified to apply to the case of a gas bubble in a linearly elastic solid. The changes in the analytic expressions for the resonance solutions are modest, and the major features are preserved (e.g., hysteresis for main, subharmonic, and harmonic frequency regions). More precise equations for the liquid case in Kamath and Prosperetti [J. Acoust. Soc. Am. 85, 1538–1548 (1989)] have also been modified to treat the elastic solid case. Following these authors, a Galerkin spectral method was used to generate numerical solutions to the governing partial differential equations. Situations in which two or more stable solutions exist for a given set of parameters may be discussed.

Rectified diffusion during nonlinear gas bubble oscillations

Mass transfer of the dissolved gas across a bubble wall during nonlinear spherically symmetric oscillations is studied. Use of an earlier comprehensive model [Prosperetti et al., J. Acoust. Soc. Am. 83, 502–514 (1988)] to determine the bubble dynamics avoids the need to use ad hoc polytropic assumptions. A numerical technique based on a pseudospectral method is used to determine the dissolved gas concentration in the liquid. This formulation does not require the assumptions made in the classic theory of Eller and Flynn [J. Acoust. Soc. Am. 37, 493–503 (1965)]. In some conditions results indicate a significant difference in growth rate predictions when compared to the Eller-Flynn model. [Work supported by NSF.]

Resonance in nonlinear bubble oscillations

In two recent papers (Longuet-Higgins 1989a,b) the author showed that the shape oscillations of bubbles can emit sound like a monopole source, at second order in the distortion parameter ε. In the second paper (LH2) it was predicted that the emission would be amplified when the second harmonic frequency 2σn of the shape oscillation approaches the frequency ω of the breathing mode. This ‘resonance’ would however be drastically limited by damping due to acoustic radiation and thermal diffusion. The predictions were confirmed by further numerical calculations in Longuet-Higgins (1990a).Ffowcs Williams & Guo (1991) have questioned the conclusions of LH2 on the grounds that near resonance there is a slow (secular) transfer of energy between the shape oscillation and the volumetric mode which tends to diminish the amplitude of the shape oscillation and hence falsify the perturbation analysis. They have also argued that the volumetric mode never grows sufficiently to produce sound of the stated order of magnitude. In the present paper we show that these assertions are unfounded. Ffowcs Williams & Guo considered only undamped oscillations. Here we show that when the appropriate damping is included in their analysis the secular transfer of energy becomes completely insignificant. The resulting pressure pulse (figure 5 below) is found to be essentially identical to that calculated in LH2, figure 3. Moreover, in the initial-value problem considered in LH2, the excitation of the volumetric mode takes place not by a secular energy transfer but by a resonance during the first few cycles of the shape oscillation. This accounts for the amplification near resonance found in Longuet-Higgins (1990a). Finally, it is pointed out that the initial energy of the shape oscillations is far greater than is required to produce the O(ε2) volume pulsations that were studied in LH2, and which were used for a comparison with field data. This acoustic radiation was not calculated by Ffowcs Williams & Guo.

Studies of non-linear bubble oscillations in a simulated acoustic field

Gas bubbles in liquids are capable of forced oscillations in a sound field. Photographic study of these non-linear oscillations is difficult. It is technically most feasible at low acoustic frequencies, where lower framing rates are required, and where the resonance bubble size is larger. However, the acoustic pressures required to elicit a non-linear response from the bubble might, if generated at audible frequencies, be damaging to the hearing. This paper outlines an experimental solution to the problem, whereby a volume of liquid containing bubbles is vibrated vertically. The oscillating pressure in the water mimics an acoustic field at the vibration frequency and the bubble responds to it as such. The apparatus has minimal audible emissions, yet the oscillating pressure is of such an amplitude as to generate highly non-linear oscillations in the bubble. The experiment provides a good example of forced oscillation, as well as illustrating a subtle solution to a difficult experimental problem. Radius-time data from a given bubble were obtained using high-speed and stroboscopic photography, and were compared with the numerical predictions obtained from the Rayleigh-Plesset equation.

When bubbles go bad…. Chaotic shape oscillations of air bubbles in water

Using optical scattering techniques, the nonlinear oscillations of single gas bubbles are observed. A single bubble is acoustically levitated and driven by a standing-wave pressure field at kilohertz frequencies. The scattered light intensity from a single plane wave (TEM00) is observed as a function of time for both radial and nonradial oscillations. The resulting time series are analyzed using some techniques from dynamical systems theory. These measurements are accompanied by high-speed photographs in an attempt to determine the presence and character of shape oscillation modes. [Work supported by ONR, NCPA, and DFG.]

Cavitation damage and generated noise in high water base fluids.

Cavitation damage tests in high-water-base-fluids (HWBF) are carried out with an ultrasonic vibratory cavitation test apparatus, and cavitation noise spectra are measured. The results are compared with those for other fire resistant fluids (water-glycol fluid and phosphate ester), petroleum oil, and water. The mass loss in HWBF is about one-half of that in water, which is triple or quadruple of that in the water-glycol fluid, and several decuple in phosphate ester. For the tested liquids, the mass loss in petroleum oil was the lowest. Sound pressure levels in the frequency domain 40 kHz in HWBF are lower, compared with that in water. The values of SPL and mass loss in 0.3% HWBF are smaller than those in 5% and 20% HWBF and water. Harmonic resonance and subharmonic components in noise spectra detected in the water-glycol fluid, phosphate ester and petroleum oil may be caused by nonlinear oscillation of bubbles induced by pressure variation with resonant frequency.

A multiscale analysis of nonlinear oscillations of gas bubbles in compressible liquids II. First and second subharmonics and harmonics

Approximate analytical solutions for the steady-state oscillations in regions of first and second subharmonics, and first and second harmonics, are presented to second order in expansion. We give frequency response curves, from which the influence is obvious of the compressibility of the liquid upon the behaviour of the bubble at different frequencies of the outer acoustic field.

Occurrence of transient cavitation in pulsed sawtooth ultrasonic fields.

Thus far, studies conducted to assess the safety of diagnostic ultrasound have employed sinusoidal sound fields. To evaluate the influence of nonlinearly distorted acoustic fields, this article compares the responses of microbubbles of variable size, exposed to (1) a sinusoidal pulse and (2) a sawtooth pulse. The nonlinear oscillations of a spherical bubble in a viscous compressible liquid stimulated into motion by an ultrasonic pulse are predicted, using a theoretical model for bubble dynamics. The maximum gas pressures inside the bubble when it collapses under the influence of a sinusoid or a sawtooth are deduced. Experimental work on Drosophila larvae exposed to sinusoidal and to sawtooth fields is consistent with the theoretical analysis.

Effect of polymer additives on the generation of subharmonic and harmonic bubble oscillations in an ultrasonically irradiated liquid

A study of non-linear oscillation of bubbles in polymer solutions subjected to a pulsating pressure has been performed. The equation of the bubble oscillation, the equation of the natural frequency of the bubble, and the pressure equation are derived. The frequency response curves and the maximum pressure in a Polyox solution, a PAM solution, a CMC solution, and water are found. The effects of polymer concentration on onset curves for the subharmonic and harmonic resonances are clarified.

Nonlinear bubble dynamics

The standard approach to the analysis of the pulsations of a driven gas bubble is to assume that the pressure within the bubble follows a polytropic relation of the form p=p0(R0/R)3κ, where p is the pressure within the bubble, R is the radius, κ is the polytropic exponent, and the subscript zero indicates equilibrium values. For nonlinear oscillations of the gas bubble, however, this approximation has several limitations and needs to be reconsidered. A new formulation of the dynamics of bubble oscillations is presented in which the internal pressure is obtained numerically and the polytropic approximation is no longer required. Several comparisons are given of the two formulations, which describe in some detail the limitations of the polytropic approximation.

A numerical method for long time solutions of integro-differential systems in multiphase flow

Abstract A numerical method to solve a class of intego-differential equations asociated with multiphase flow problems is described. The system has at least two time scales, a normal time and a long time. As an initial value problem, the solution is aperiodic in the normal time and approaches a periodic oscillation over a long time. The numerical method is designed to give accurate calculations over the long time. The method is efficient, saving storage space and computational time. Sample numerical calculations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the method. The system considered describes the forced nonlinear oscillations of a gas bubble in a liquid. We present a systematic numerical study of the phenomenon of the slow growth of the mean bubble radius to a steady value over a long time. The correlation between the phenomena of a sudden burst of large amplitude and resonance in the forced oscillations (at subcritical frequencies) is also studied.

Nonlinear oscillation of gas bubble with internal phenomena.

The pressure response of a small air bubble in water is calculated numerically in an oscillatory pressure field. The following phenomena are taken into account in the calculation ; mist formation inside the bubble due to homogeneous condensation, diffusion between vapor and noncondensable gas inside the bubble and so on. The internal condition adapts itself easily to the oscillatory pressure field by the internal phenomena, which includes mist formation, evaporation and deposition onto the bubbler wall, so that the bubble tends to oscillate synchronously. The frequency response curves for bubbles are calculated. The curve is shifted towards low frequency, and the maximum bubble radius becomes large. When the pressure amplitude is increased, the periodic bubble oscillation bifurcates to non-periodic oscillation, which is a deterministic chaos.

A numerical study of thermal effects on nonlinear bubble oscillations

An integral-differential system was derived by Miksis and Ting describing the nonlinear oscillations of a gas bubble including thermal and viscous effects. Here, an efficient numerical scheme is presented to solve this system for long times relative to the forcing period. Examples are presented and compared with the Rayleigh–Plesset equation, which has no thermal damping. Numerical results show that the initial free oscillations are damped out much faster when thermal damping is included. Also shown is the fact that it is possible for a bubble to have its mean radius grow slowly with time. In this case for large time the bubble will approach a new constant mean radius, which is larger than the initial equilibrium radius. For the special case of small amplitude forcing, asymptotic periodic solutions for very large time are constructed and a method to systematically derive the higher-order terms is demonstrated.

A multiscale analysis of nonlinear oscillations of gas bubbles in liquids

The nonlinear oscillations of a spherical gas bubble in a compressible viscous liquid subject to the action of a sound field are investigated by the multiscale perturbation method. Approximate analytical solutions for the transient oscillations in the regions of the main resonance, first and second harmonic, and first and second subharmonic, are obtained to second order in the expansion.

Nonlinear oscillations of gas bubbles in viscoelastic fluids

Nonlinear oscillations of gas bubbles in viscoelastic fluids of a three-constant Oldroyd model are theoretically investigated. The equation of motion for a bubble in a viscoelastic fluid subject to a pulsating pressure, and the pressure equation at the bubble wall are obtained. By numerical calculations on these equations, the effects of relaxation time and retardation time on frequency response curves, and the relation between the maximum pressure at the bubble wall and the initial radius of the bubble are clarified.

Study of nonlinear oscillations of bubbles in Powell–Eyring fluids

To study cavitation phenomena in non-Newtonian fluids, problems about nonlinear oscillations of bubbles are theoretically examined by using a Powell–Eyring model as a rheological model. The equation of oscillation of bubbles in Powell–Eyring fluids subject to pulsating pressures and the equation of natural frequency of a bubble are derived. According to numerical calculations for the bubbles in polymer solutions, the effects of polymer concentration and pressure amplitude on frequency response curves are clarified. The relation between the impulse pressure occurring during the bubble oscillation and the ratio of the frequencies are also examined.

On Rayleigh's model of a freely oscillating bubble. I. Basic relations

Free nonlinear oscillations of a gas bubble in a noncompressible liquid are analysed. Excitation to the growth is considered in the form of an instantaneous delivery of heat into the bubble interior. The characteristic features of the pressure and velocity fields in the liquid are discussed.

Subharmonic emissions of a cavitating bubble

The nonlinear oscillations of a gas bubble pulsating in a liquid are critically dependent on the damping of these oscillations. We have used an accurate numerical formulation of the problem to investigate the subharmonic emissions from forced oscillations of the bubble. A systematic approach to chaotic behavior will be shown through successive bifurcation of the subharmonic emissions as the driving pressure is increased. [Work supported in part by the NSF and the ONR.]

An acoustic levitation technique for the study of nonlinear oscillations of gas bubbles in liquids

Abstract : A technique of acoustic levitation was developed for the study of individual gas bubbles in a liquid. Isopropyl alcohol and a mixture of glycerine and water (33-1/3% glycerine by volume) were the two liquids used in this research. Bubbles were levitated near the acoustic pressure antinode of an acoustic wave in the range of 20-22 kHz. Measurements were made of the levitation number as a function of the normalized radius of the bubbles. The levitation number is the ratio of the hydrostatic pressure gradient to the acoustic pressure gradient. These values were then compared to a nonlinear theory. Results were very much in agreement except for the region near the n=2 harmonic. An explanation for the discrepancy between theory and experiment appears to lie in the polytropic exponent associated with the gas in the interior of the bubble. (Author)

Nonlinear radial oscillations of a gas bubble including thermal effects

A system of integral differential equations determining the nonlinear oscillations of a spherical gas bubble including thermal and viscous damping effects is derived. This system is obtained systematically from the Navier–Stokes equations for the liquid and the gas phases and from the interface conditions by the method of matched asymptotics. The ratio of the thermal length (De/ω)1/2, in the gas and the mean bubble radius is the small expansion parameter. Here De is the reference thermal diffusivity of the gas and ω the frequency of oscillation. By using the well-tested approximate viscosity function in compressible boundary layer theory, the pressure inside the bubble can be related to its radius by an integral equation. Coupling this integral equation with the Rayleigh–Plesset equation gives us a closed system determining the radial oscillations of the bubble. For small oscillations, the system is solved analytically and the results are in agreement with the linear theory. This system is then solved numerically for finite amplitude oscillations.