Suppressing chaotic oscillations of a spherical cavitation bubble through applying a periodic perturbation

1/2 order subharmonic waves of two cavitation bubbles

In the work, the 1/2 order subharmonic wave of two coupling cavitation bubbles is investigated numerically via Fourier spectrum analysis. By analyzing the dynamics of bubble, we find that the mutual interaction between bubbles can affect the appearance of 1/2 order subharmonic. The results of parameter dependence show that the intensity of 1/2 order subharmonic would be promoted or inhibited with the increase of mutual interaction. The higher the driving amplitude or the smaller the distance between bubbles, the stronger the mutual interaction is, and also the greater the promotion or suppression of the 1/2 order subharmonic is. Moreover, while the 1/2 order subharmonic occurs, the energy of bubble would alternate between two different peaks, and the temperature inside bubble has a similar fluctuation while the bubble collapses. This qualitative analysis suggests that the bubble's dynamics for multi-bubble case is complex. Understanding the generation of subharmonic of bubble's dynamics is of great significance for helpful applying of cavitation bubble.

Numerical investigation of the effect of bubble properties on the linear resonance frequency shift due to inter-bubble interactions in ultrasonically excited lipid coated microbubbles

Ultrasonically excited microbubbles (MBs) have numerous applications in various fields, such as drug delivery, and imaging. Ultrasonically excited MBs are known to be nonlinear oscillators that generate secondary acoustic emissions in the media when excited by a primary ultrasound wave. The propagation of acoustic waves in the liquid is limited to the speed of sound, resulting in each MB receiving the primary and secondary waves at different times depending on their distance from the ultrasound source and the distance between MBs. These delays are referred to as primary and secondary delays, respectively. A previous study demonstrated that the inclusion of secondary delays in a model describing the interactions between MBs exposed to ultrasound results in an increase in the linear resonance frequency of MBs as they approach each other. This work investigates the impact of various MB properties on the change in linear resonance frequency resulting from changes in inter-bubble distances. The effects of shell properties, including the initial surface tension, surface dilatational viscosity of the shell monolayer, elastic compression modulus of the shell, and the initial radius of the MBs, are examined. MB size is a significant factor influencing the rate of linear resonance frequency increase with increasing concentration. Moreover, it is found that the shell properties of MBs play a negligible role in the rate of change in linear resonance frequency of MBs as the inter-bubble distances change.The findings of this study have implications for various applications of MBs in the biomedical field. By understanding the impact of inter-bubble distances and shell properties on the linear resonance frequency of MBs, the utilization of MBs in applications reliant on their resonant behavior can be optimized.

Mass transfer of microbubble in liquid under multifrequency acoustic excitation - A theoretical study

Microbubble's mass transfer under external acoustic excitation holds immense potential across various technological fields. However, the current state of acoustic technology faces limitations due to inadequate control over bubble size in liquids under external excitation. Here, we conducted numerical investigations of the mass transfer behavior of microbubbles in liquids under multifrequency acoustic excitations with different frequencies (in the MHz range), pressure amplitudes (in the range of several atmospheric pressures), and amplitude ratios. We identified various pressure threshold regions for the growth of gas bubbles (radii range from a few microns to tens of microns) and observed common intersections between single and multifrequency excitations that enable effective control of the growth intervals and final size of bubbles by adjusting the ratio of pressure amplitude and frequency value. Allocating power to the lower frequency component of multifrequency acoustic excitation is recommended to facilitate mass transfer or diffusion, as small-frequency acoustic excitation has a more significant effect than the higher frequency in the growth region. Our study provides a better understanding of the dynamics of bubbles under complex excitations and has practical implications for developing methods to control and promote bubble-related processes.

Effect of a modulated acoustic field on the dynamics of a vibrating charged bubble

We investigate the effect of amplitude-modulated acoustic irradiation on the dynamics of a charged bubble vibrating in a liquid. We show that the potential V(x) of the bubble, and the number and stability of its equilibria, depend on the magnitude of the charge it carries. Under high-frequency amplitude-modulation, a modulation threshold, Gth, was found for the onset of increased bubble amplitude oscillations. For some pressure field values, charge can facilitate the control of chaotic dynamics via reversed period-doubling bifurcation sequences. There is evidence for peak-shouldering and shock waves. The Mach number increases rapidly with the drive amplitude G. In the supersonic regime, for G>1.90Pa, the high-frequency modulation raises both Blake's and the transient cavitation thresholds. We found a decrease in the bubble's maximum charge threshold, and threshold modulation amplitude for the occurrence Vibrational resonance (VR). VR occurs due to the modulated oscillatory pressure field, and the influence on VR of the electrostatic charge, and other parameters of the system are investigated. In contrast to the cases of VR reported earlier, where the amplitude G of the high-frequency driving is typically much higher than the amplitude of the low-frequency driving (Ps), the VR resonance peaks occur here at relatively low G values (0 Collapse

Key Words

• Acoustic waves
• Amplitude modulation
• Bubble oscillator
• Electrostatic charges
• Vibrational resonance

Collapse

MESH Headings Collapse

Grants Collapse

Affiliation(s)

O T Kolebaje Department of Physics, Adeyemi Federal University of Education, Ondo, Ondo State, Nigeria; Department of Physical Sciences, Redeemer's University, P.M.B. 230, Ede, Nigeria
U E Vincent Department of Physical Sciences, Redeemer's University, P.M.B. 230, Ede, Nigeria; Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom.
B E Benyeogor Department of Physical Sciences, Redeemer's University, P.M.B. 230, Ede, Nigeria
P V E McClintock Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

Collapse

Cavitation bubble structures below a soft boundary in an ultrasonic field

We studied the layer structure of bubbles just below water/air and water/EPE (Expand aple poly ephylene) interfaces using high-speed photography. The layer structure was generated by floating spherical clusters, the source bubbles of which were identified to come from the attachment of bubble nuclei at the interface, the floating of bubbles in the bulk liquid, or bubbles generated on the surface of the ultrasonic transducer. The boundary shape affected the layer structure, which assumed a similar profile below the water/EPE interface. We developed a simplified model composed of a bubble column and bubble chain to describe interface impacts and the interaction of bubbles in a typical branching structure. We found that the resonant frequency of the bubbles is smaller than that of an isolated single bubble. Moreover, the primary acoustic field plays an important role in the generation of the structure. A higher acoustic frequency and pressure were found to shorten the distance between the structure and the interface. A hat-like layer structure of bubbles was more likely to exist in the low-frequency (28 and 40 kHz) intense inertial cavitation field, in which bubbles oscillate violently. By contrast, structures composed of discrete spherical clusters were more likely to form in the relatively weak cavitation field at 80 kHz, in which stable and inertial cavitation coexisted. The theoretical predictions were in good agreement with the experimental observations.

GPU-accelerated study of the inertial cavitation threshold in viscoelastic soft tissue using a dual-frequency driving signal

Inertial cavitation thresholds under two forms of ultrasonic excitation (the single- and dual-frequency ultrasound modes) are studied numerically. The Gilmore-Akulichev model coupled with the Zener viscoelastic model is used to model the bubble dynamics. The threshold pressures are determined with two criteria, one based on the bubble radius and the other on the bubble collapse speed. The threshold behavior is investigated for different initial bubble sizes, acoustic signal modes, frequencies, tissue viscosities, tissue elasticities, and all their combinations. Due to the large number of parameters and their many combinations (around 1.5 billion for each threshold criterion), all simulations were executed on graphics processing units to speed up the calculations. We used our own code written in the C++ and CUDA C languages. The results obtained demonstrate that using the dual-frequency signal mode can help to reduce the inertial cavitation threshold (in comparison to the single-frequency mode). The criterion based on the bubble size gives a lower threshold than the criterion using the bubble collapse speed. With an increase of the elasticity, the threshold pressure also increases, whereas changing the viscosity has a very small impact on the optimal threshold, unlike the elasticity. A detailed analysis of the optimal ultrasound frequencies for a dual-frequency driving signal found that for viscosities less than 0.02 Pa·s, the first optimal frequency, in general, is much smaller than the second optimal frequency, which can reach 1 MHz. However, for high viscosities, both optimal frequencies are similar and varied in the range 0.01-0.05 MHz. Overall, this study presents a detailed analysis of inertial cavitation in soft tissue under dual-frequency signal excitation. It may be helpful for the further development of different applications of biomedical ultrasound.

Influence of rigid wall on the nonlinear pulsation of nearby bubble

This paper mainly focuses on the nonlinear pulsation of a bubble near the rigid wall. Dynamics of near-wall bubble and free bubble are discussed and compared in details. Investigation reveals as the driving acoustic pressure amplitude increases, nonlinear pulsation of bubble becomes intense gradually. Besides, decreasing the viscosity of host liquid is advantageous for the nonlinear pulsation of bubble. Bifurcation diagrams of bubble radius show acoustic reflection of the rigid wall makes the initial bifurcation appear at low driving acoustic amplitude and on bubble with small ambient radius, and makes the bifurcation still exist for bubble in high-viscosity liquids. That indicates the rigid wall will produce enhancement on the nonlinearity of nearby bubble. As the bubble approaches the wall, the enhancement becomes strong. Moreover, research on the influence of driving frequency shows the rigid wall makes the frequency band corresponding to chaos around the resonant frequency of free bubble shift downward.

A single oscillating bubble in liquids with high Mach number

The oscillation characteristics of a single bubble and its induced radiation pressure and the dissipated power are essential for a wide range of applications. For bubble oscillations with high Mach number, the influence of the liquid compressibility is significantly strong and should be fully considered. In the present paper, the bubble wall motion equation with the second-order Mach number is employed for investigating a free oscillating bubble in the liquid with numerical and experimental verifications. For the purpose of comparisons, the revised Keller-Miksis equation up to the first-order Mach number is solved with the same conditions (e.g. the initial conditions and the ambient pressure). Through our simulations, comparing with the predictions by the first-order equation, we find that: (1) The bubble radius, the bubble wall radial velocity and the bubble wall radial acceleration predicted by the second-order equation with high Mach number are significantly different respectively, and the dimensionless differences increase with the increase of the Mach number. (2) The valid prediction range of the second-order equation is much larger. (3) The dissipated power predicted by the second-order equation with high Mach number is smaller.

On the threshold of 1/2 order subharmonic emissions in the oscillations of ultrasonically excited bubbles

The pressure threshold for 1/2 order subharmonic (SH) emissions and period doubling during the oscillations of ultrasonically excited bubbles is thought to be minimum when the bubble is sonicated with twice its resonance frequency (f_r). This estimate is based on studies that simplified or neglected the effects of thermal damping. In this work, the nonlinear dynamics of ultrasonically excited bubbles is investigated accounting for the thermal dissipation. Results are visualized using bifurcation diagrams as a function of pressure. Here we show that, and depending on the gas, the pressure threshold for 1/2 order SHs can be minimum at a frequency between 0.5f_r≤f≤0.6f_r. In this frequency range, the generation of 1/2 order SHs are due to the occurrence of 5/2 order ultra-harmonic resonance. The stability of such oscillations is size dependent. For an air bubble immersed in water, only bubbles bigger than 1 μm in diameter are able to emit non-destructive SHs in these frequency ranges.

Nonlinear dynamics and bifurcation structure of ultrasonically excited lipid coated microbubbles

In many applications, microbubbles (MBs) are encapsulated by a lipid coating to increase their stability. However, the complex behavior of the lipid coating including buckling and rupture sophisticates the dynamics of the MBs and as a result the dynamics of the lipid coated MBs (LCMBs) are not well understood. Here, we investigate the nonlinear behavior of the LCMBs by analyzing their bifurcation structure as a function of acoustic pressure. We show that, the LC can enhance the generation of period 2 (P2), P3, higher order subharmonics (SH), superharmonics and chaos at very low excitation pressures (e.g. 1 kPa). For LCMBs sonicated by their SH resonance frequency and in line with experimental observations with increasing pressure, P2 oscillations exhibit three stages: generation at low acoustic pressures, disappearance and re-generation. Within non-destructive oscillation regimes and by pressure amplitude increase, LCMBs can also exhibit two saddle node (SN) bifurcations resulting in possible abrupt enhancement of the scattered pressure. The first SN resembles the pressure dependent resonance phenomenon in uncoated MBs and the second SN resembles the pressure dependent SH resonance. Depending on the initial surface tension of the LCMBs, the nonlinear behavior may also be suppressed for a wide range of excitation pressures.

Acoustic vibrational resonance in a Rayleigh-Plesset bubble oscillator

The phenomenon of vibrational resonance (VR) has been investigated in a Rayleigh-Plesset oscillator for a gas bubble oscillating in an incompressible liquid while driven by a dual-frequency force consisting of high-frequency, amplitude-modulated, weak, acoustic waves. The complex equation of the Rayleigh-Plesset bubble oscillator model was expressed as the dynamics of a classical particle in a potential well of the Liénard type, thus allowing us to use both numerical and analytic approaches to investigate the occurrence of VR. We provide clear evidence that an acoustically-driven bubble oscillates in a time-dependent single or double-well potential whose properties are determined by the density of the liquid and its surface tension. We show both theoretically and numerically that, besides the VR effect facilitated by the variation of the parameters on which the high-frequency depends, amplitude modulation, the properties of the liquid in which the gas bubble oscillates contribute significantly to the occurrence of VR. In addition, we discuss the observation of multiple resonances and their origin for the double-well case, as well as their connection to the low frequency, weak, acoustic force field.

GPU accelerated study of a dual-frequency driven single bubble in a 6-dimensional parameter space: The active cavitation threshold

The active cavitation threshold of a dual-frequency driven single spherical gas bubble is studied numerically. This threshold is defined as the minimum intensity required to generate a given relative expansion (R_max-R_E)/R_E, where R_E is the equilibrium size of the bubble and R_max is the maximum bubble radius during its oscillation. The model employed is the Keller-Miksis equation that is a second order ordinary differential equation. The parameter space investigated is composed by the pressure amplitudes, excitation frequencies, phase shift between the two harmonic components and by the equilibrium bubble radius (bubble size). Due to the large 6-dimensional parameter space, the number of the parameter combinations investigated is approximately two billion. Therefore, the high performance of graphics processing units is exploited; our in-house code is written in C++ and CUDA C software environments. The results show that for (R_max-R_E)/R_E=2, the best choice of the frequency pairs depends on the bubble size. For small bubbles, below 3μm, the best option is to use just a single frequency of a low value in the giant response region. For medium sized bubbles, between 3μm and 6μm, the optimal choice is the mixture of low frequency (giant response) and main resonance frequency. For large bubbles, above 6μm, the main resonance dominates the active cavitation threshold. Increasing the prescribed relative expansion value to (R_max-R_E)/R_E=3, the optimal choice is always single frequency driving with the lowest value (20kHz here). Thus, in this case, the giant response always dominates the active cavitation threshold. The phase shift between the harmonic components of the dual-frequency driving (different frequency values) has no effect on the threshold.

Simulation Study on the Influence of Multifrequency Ultrasound on Transient Cavitation Threshold in Different Media

Through the introduction of multifrequency ultrasound technology, remarkable results have been achieved in tissue ablation and other aspects. By using the nonlinear dynamic equation of spherical bubble, the effects of the combination mode of multifrequency ultrasound, the peak negative pressure and its duration, the phase angle difference, and the polytropic index on the transient cavitation threshold in four different media of water, blood, brain, and liver are simulated and analyzed. The simulation results show that under the same frequency difference and initial bubble radius, the transient cavitation threshold of the high-frequency, triple-frequency combination is higher than that of the low-frequency, triple-frequency combination. When the lowest frequency of triple frequencies is the same, the larger the frequency difference, the higher the transient cavitation threshold. When the initial bubble radius is small, the frequency difference has little effect on the transient cavitation threshold of the triple-frequency combination. With the increase of initial bubble radius, the influence of frequency difference on the transient cavitation threshold of the higher frequency combination of triple frequency is more obvious than that of the lower frequency combination of triple frequency. When the duration of peak negative pressure or peak negative pressure of the multifrequency combined ultrasound is longer than that of the single-frequency ultrasound, the transient cavitation threshold of the multifrequency combined ultrasound is lower than that of the single-frequency ultrasound; on the contrary, the transient cavitation threshold of the multifrequency combined ultrasound is higher than that of the single-frequency ultrasound. When the phase angle difference of multifrequency excitation is zero, the corresponding transient cavitation threshold is the lowest, while the change of the polytropic index has almost no effect on the transient cavitation threshold for the multifrequency combination. The research results can provide a reference for multifrequency ultrasound to reduce the transient cavitation threshold, which is of great significance for the practical application of cavitation.

Feedforward attractor targeting for non-linear oscillators using a dual-frequency driving technique

A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.

Collective nonlinear behavior of interacting polydisperse microbubble clusters

Acoustically excited microbubbles (MBs) have shown to exhibit rich dynamics, enabling them to be employed in various applications ranging from chemistry to medicine. Exploiting the full potential of MBs for applications requires a good understanding of their complex dynamics. Improved understanding of MB oscillations can lead to further enhancement in optimizing their efficacy in many applications and also invent new ones. Oscillating MBs have been shown to generate secondary pressure waves that modify the dynamics of the MBs in their proximity. A modified Keller-Miksis equation is used to account for inter-bubble interactions. The oscillatory dynamics of each MB within clusters was computed by numerically solving the resulting system of coupled nonlinear second order differential equations in potential fluid flow. Frequency response analysis and bifurcation diagrams were employed to track the dynamics of interacting MBs. We start with investigating the effect of inter-bubble interactions for cases of three and four MBs over a wide range of acoustic and geometric parameters. Emergent collective behavior was observed which are dominated by the dynamics of the largest MB within the cluster. The emergent dynamics of smaller MBs within clusters can be characterized by constructive and destructive inter-bubble interactions. In constructive interactions, the radial oscillations of smaller MBs matched those of the largest MB and their oscillations are amplified. In destructive interactions, the oscillations of smaller bubbles are suppressed so that their oscillations match those of the largest MB. Furthermore, a special case of constructive interactions is presented where dominant MB (largest) can force smaller MBs into period doubling and subharmonic oscillations. The collective behavior is further investigated in large MB cluster and it is shown that largest MBs, even in small numbers can force smaller ones into period doubling and subharmonic oscillations.

A simple method to analyze the super-harmonic and ultra-harmonic behavior of the acoustically excited bubble oscillator

The bubble oscillator is a highly nonlinear system, which makes it difficult to generate a comprehensive understanding of its oscillatory behavior. One method used to investigate such complex dynamical systems is the bifurcation analysis. Numerous investigations have employed the method of bifurcation diagrams to study the effect of different control parameters on the bubble behavior. These studies, however, focused mainly on investigating the subharmonic (SH) and chaotic oscillations of the bubbles. Super-harmonic (SuH) and ultra-harmonic (UH) bubble oscillations remain under-investigated. One reason is that the conventional method used for generating bifurcation diagrams cannot reliably identify features that are responsible for the identification of SuH and UH oscillations. Additionally, the conventional method cannot distinguish between the UHs and SHs. We introduce a simple procedure to address this shortcoming. In this method, the maxima of the bubble oscillatory response were selected and plotted alongside the traditional bifurcation points for the corresponding control parameter. Results show that depending on the control parameters the conventional method or the method of maxima may miss intricate details of the oscillations. In order to have a comprehensive knowledge on the rich dynamics of the system, the two methods should be employed side by side. Through plotting the two bifurcation structures in tandem, the oscillatory behavior of the bubble was analyzed with more detail, and stable SuH and UH bubble oscillations were investigated. Based on this new analysis, the conditions for the generation and amplification of UH and SuH regimes are discussed.

Closed Loop Spatial and Temporal Control of Cavitation Activity with Passive Acoustic Mapping

Ultrasonically actuated microbubble oscillations hold great promise for minimally invasive therapeutic interventions. While several preclinical studies have demonstrated the potential of this technology, real-time methods to control the amplitude and type of microbubble oscillations (stable vs inertial acoustic cavitation) and ensure that cavitation occurs within the targeted region are needed for their successful translation to the clinic. In this paper, we propose a real-time nonlinear state controller that uses specific frequency bands of the microbubble acoustic emissions (harmonic, ultra-harmonic, etc.) to control cavitation activity (observer states). To attain both spatial and temporal control of cavitation activity with high signal to noise ratio, we implement a controller using fast frequency-selective passive acoustic mapping (PAM) based on the angular spectrum approach. The controller includes safety states based on the recorded broadband signal level and is able to reduce sensing inaccuracies with the inclusion of multiple frequency bands. In its simplest implementation the controller uses the peak intensity of the passive acoustic maps, reconstructed using the 3rd harmonic (4.896 × 0.019 MHz) of the excitation frequency. Our results show that the proposed real-time nonlinear state controller based on PAM is able to reach the targeted level of observer state (harmonic emissions) in less than 6 seconds and remain within 10 % of tolerance for the duration of the experiment (45 seconds). Similar response was observed using the acoustic emissions from single element passive cavitation detection, albeit with higher susceptibility to background noise and lack of spatial information. Importantly, the proposed PAM-based controller was able to control cavitation activity with spatial selectivity when cavitation existed simultaneously in multiple regions. The robustness of the controller is demonstrated using a range of controller parameters, multiple observer states concurrently (harmonic, ultra-harmonic, and broadband), noise levels (°6 to 12 dB SNR), and bubble concentrations (0.3 to 180 × 103 bubbles per microliter). More research in this direction under preclinical and clinical conditions is warranted.

Numerical investigation of the inertial cavitation threshold by dual-frequency excitation in the fluid and tissue

Inertial cavitation thresholds, which are defined as bubble growth by 2-fold from the equilibrium radius, by two types of ultrasonic excitation (at the classical single-frequency mode and dual-frequency mode) were calculated. The effect of the dual-frequency excitation on the inertial cavitation threshold in the different surrounding media (fluid and tissue) was studied, and the paramount parameters (driving frequency, amplitude ratio, phase difference, and frequency ratio) were also optimized to maximize the inertial cavitation. The numerical prediction confirms the previous experimental results that the dual-frequency excitation is capable of reducing the inertial cavitation threshold in comparison to the single-frequency one at the same output power. The dual-frequency excitation at the high frequency (i.e., 3.1 + 3.5 MHz vs. 1.1 + 1.3 MHz) is preferred in this study. The simulation results suggest that the same amplitudes of individual components, zero phase difference, and large frequency difference are beneficial for enhancing the bubble cavitation. Overall, this work may provide a theoretical model for further investigation of dual-frequency excitation and guidance of its applications for a better outcome.

Chaotic oscillations of gas bubbles under dual-frequency acoustic excitation

Chaotic oscillation of bubbles in liquids reduces the efficiency of the sonochemical system and should be suppressed in the practical applications. In the present paper, a chaos control method based on the dual-frequency approach is numerically investigated and is proved to be an effective method even for cases with intensive energy input. It was found that the chaos could be successfully suppressed by the application of dual-frequency approach in a wide range of parameter zone (even with high acoustic pressure amplitude). Furthermore, influences of power allocation between two waves on the chaos control are quantitatively discussed with clear descriptions of the routes from stable oscillations to chaos.

Combination and simultaneous resonances of gas bubbles oscillating in liquids under dual-frequency acoustic excitation

The multi-frequency acoustic excitation has been employed to enhance the effects of oscillating bubbles in sonochemistry for many years. In the present paper, nonlinear dynamic oscillations of bubble under dual-frequency acoustic excitation are numerically investigated within a broad range of parameters. By investigating the power spectra and the response curves of oscillating bubbles, two unique features of bubble oscillations under dual-frequency excitation (termed as "combination resonance" and "simultaneous resonance") are revealed and discussed. Specifically, the amplitudes of the combination resonances are quantitatively compared with those of other traditional resonances (e.g. main resonances, harmonics). The influences of several paramount parameters (e.g., the bubble radius, the acoustic pressure amplitude, the energy allocation between two component waves) on nonlinear bubble oscillations are demonstrated.

The effect of high viscosity on the collapse-like chaotic and regular periodic oscillations of a harmonically excited gas bubble

In the last decade many industrial applications have emerged based on the rapidly developing ultrasonic technology such as ultrasonic pasteurization, alteration of the viscosity of food systems, and mixing immiscible liquids. The fundamental physical basis of these applications is the prevailing extreme conditions (high temperature, pressure and even shock waves) during the collapse of acoustically excited bubbles. By applying the sophisticated numerical techniques of modern bifurcation theory, the present study intends to reveal the regions in the excitation pressure amplitude-ambient temperature parameter plane where collapse-like motion of an acoustically driven gas bubble in highly viscous glycerine exists. We report evidence that below a threshold temperature the bubble model, the Keller-Miksis equation, becomes an overdamped oscillator suppressing collapse-like behaviour. In addition, we have found periodic windows interspersed with chaotic regions indicating the presence of transient chaos, which is important from application point of view if predictability is required.

Stable bubble oscillations beyond Blake's critical threshold

The equilibrium radius of a single spherical bubble containing both non-condensable gas and vapor is determined by the mechanical balance at the bubble interface. This expression highlights the fact that decreasing the ambient pressure below the so called Blake's critical threshold, the bubble has no equilibrium state at all. In the last decade many authors have tried to find evidence for the existence of stable bubble oscillation under harmonic forcing in this regime, that is, they have tried to stabilize the bubble motion applying ultrasonic radiation on the bubble. The available numerical results provide only partial proof for the existence as they are usually based on linearized or weakly nonlinear (higher order approximation) bubble models. Here, based on numerical techniques of the modern nonlinear and bifurcation theory, the existence of stable bubble motion has been proven without any restrictions in nonlinearities. Although the model, applied in this paper, is the rather simple Rayleigh-Plesset equation, the presented technique can be extended to more complex bubble models easily.