Nonlinear Bubble Dynamics Research Articles

Bridging scales in multiscale bubble growth dynamics with correlated fluctuations using neural operator learning

The intricate process of bubble growth dynamics involves a broad spectrum of physical phenomena from microscale mechanics of bubble formation to macroscale interplay between bubbles and surrounding thermo-hydrodynamics. Traditional bubble dynamics models including atomistic approaches and continuum-based methods segment the bubble dynamics into distinct scale-specific models. To bridge the gap between microscale stochastic fluid models and continuum-based fluid models for bubble dynamics, we develop a composite neural operator model to unify the analysis of nonlinear bubble dynamics across microscale and macroscale regimes by integrating a many-body dissipative particle dynamics (mDPD) model with a continuum-based Rayleigh–Plesset (RP) model through a novel neural network architecture, which consists of a deep operator network for learning the mean behavior of bubble growth subject to pressure variations and a long short-term memory network for learning the statistical features of correlated fluctuations in microscale bubble dynamics. Training and testing data are generated by conducting mDPD and RP simulations for nonlinear bubble dynamics with initial bubble radii ranging from 0.1 to 1.5 micrometers. The results show that the trained composite neural operator model can accurately predict bubble dynamics across scales, with a 99% predictive accuracy for the time evolution of the bubble radius under varying external pressure while containing correct size-dependent stochastic fluctuations in microscale bubble growth dynamics. The composite neural operator is the first deep learning surrogate for multiscale bubble growth dynamics that can capture correct stochastic fluctuations in microscopic fluid phenomena, which sets a new direction for future research in multiscale fluid dynamics modeling.

ULTRA-SR Challenge: Assessment of Ultrasound Localization and TRacking Algorithms for Super-Resolution Imaging.

With the widespread interest and uptake of super-resolution ultrasound (SRUS) through localization and tracking of microbubbles, also known as ultrasound localization microscopy (ULM), many localization and tracking algorithms have been developed. ULM can image many centimeters into tissue in-vivo and track microvascular flow non-invasively with sub-diffraction resolution. In a significant community effort, we organized a challenge, Ultrasound Localization and TRacking Algorithms for Super-Resolution (ULTRA-SR). The aims of this paper are threefold: to describe the challenge organization, data generation, and winning algorithms; to present the metrics and methods for evaluating challenge entrants; and to report results and findings of the evaluation. Realistic ultrasound datasets containing microvascular flow for different clinical ultrasound frequencies were simulated, using vascular flow physics, acoustic field simulation and nonlinear bubble dynamics simulation. Based on these datasets, 38 submissions from 24 research groups were evaluated against ground truth using an evaluation framework with six metrics, three for localization and three for tracking. In-vivo mouse brain and human lymph node data were also provided, and performance assessed by an expert panel. Winning algorithms are described and discussed. The publicly available data with ground truth and the defined metrics for both localization and tracking present a valuable resource for researchers to benchmark algorithms and software, identify optimized methods/software for their data, and provide insight into the current limits of the field. In conclusion, Ultra-SR challenge has provided benchmarking data and tools as well as direct comparison and insights for a number of the state-of-the art localization and tracking algorithms.

Charged Cavitation Multibubbles Dynamics Model: Growth Process

The nonlinear dynamics of charged cavitation bubbles are investigated theoretically and analytically in this study through the Rayleigh–Plesset model in dielectric liquids. The physical and mathematical situations consist of two models: the first one is noninteracting charged cavitation bubbles (like single cavitation bubble) and the second one is interacting charged cavitation bubbles. The proposed models are formulated and solved analytically based on the Plesset–Zwick technique. The study examines the behaviour of charged cavitation bubble growth processes under the influence of the polytropic exponent, the number of bubbles N, and the distance between the bubbles. From our analysis, it is observed that the radius of charged cavitation bubbles increases with increases in the distance between the bubbles, dimensionless phase transition criteria, and thermal diffusivity, and is inversely proportional to the polytropic exponent and the number of bubbles N. Additionally, it is evident that the growth process of charged cavitation bubbles is enhanced significantly when the number of bubbles is reduced. The electric charges and polytropic exponent weakens the growth process of charged bubbles in dielectric liquids. The obtained results are compared with experimental and theoretical previous works to validate the given solutions of the presented models of noninteraction and interparticle interaction of charged cavitation bubbles.

Analytical solution for the Rayleigh–Plesset equation by Weierstrass elliptic equation

The nonlinear dynamics of cavitation bubbles are studied theoretically through the Rayleigh–Plesset equation. A direct approach is first proposed to construct an analytical solution for cavitation bubbles. In particular, when considering the gas bubbles, we show the connection between the bubble motion and Weierstrass elliptic equation with the help of Sandman transformation. To the best of our knowledge, this is the first time that an elliptic function has been used to construct the parametric analytical solution for this case. The present findings would be significant in understanding the complex behavior of cavitation bubbles in practical applications, especially when considering the surface tension.

Conditional moment methods for polydisperse cavitating flows

The dynamics of cavitation bubbles are important in many flows, but their small sizes and high number densities often preclude direct numerical simulation. We present a computational model that averages their effect on the flow over larger spatiotemporal scales. The model is based on solving a generalized population balance equation (PBE) for nonlinear bubble dynamics and explicitly represents the evolving probability density of bubble radii and radial velocities. Conditional quadrature-based moment methods (QBMMs) are adapted to solve this PBE. A one-way-coupled bubble dynamics problem demonstrates the efficacy of different QBMMs for the evolving bubble statistics. Results show that enforcing hyperbolicity during moment inversion (CHyQMOM) provides comparable model-form accuracy to the traditional conditional method of moments and decreases computational costs by about ten times for a broad range of test cases. The CHyQMOM-based computational model is implemented in MFC, an open-source multi-phase and high-order-accurate flow solver. We assess the effect of the model and its parameters on a two-way coupled bubble screen flow problem.

Application of Koopman operator theory to the control of nonlinear bubble dynamics

Microbubbles are widely used in biomedicine for ultrasound contrast imaging and are a promising vehicle for targeted drug and gene delivery. The ability to control the oscillations of bubbles through the applied ultrasound can improve the effectiveness of these treatments, for example, by enhancing the acoustic echo or inciting bubble rupture at precise locations. Koopman operator theory has gained interest in the past decade as a framework for rigorously transforming nonlinear dynamics on the state space into linear dynamics on abstract function spaces, which preserves the underlying nonlinear dynamics of the system. These spaces can be approximated purely through machine learning and data-driven methodologies, which then enables the application of classical linear control strategies to strongly nonlinear systems. Here, we use a Koopman linear quadratic regulator (LQR) to control the nonlinear dynamics of spherical bubbles, as described by the well-known Rayleigh-Plesset equation, with two novel objectives: (1) stabilization of the bubble at a nonequilibrium radius and (2) simple harmonic oscillation at amplitudes large enough to incite nonlinearities. Control is implemented through both broadband and single-frequency transducers that are modulated by the Koopman LQR controller. We repeat these results using Koopman model predictive control (MPC), which allows for the implementation of constraints. This work is a step towards controlling nonspherical shape modes of encapsulated microbubbles.

Nonlinear dynamics and acoustic emissions of interacting cavitation bubbles in viscoelastic tissues

The cavitation-mediated bioeffects are primarily associated with the dynamic behaviors of bubbles in viscoelastic tissues, which involves complex interactions of cavitation bubbles with surrounding bubbles and tissues. The radial and translational motions, as well as the resultant acoustic emissions of two interacting cavitation bubbles in viscoelastic tissues were numerically investigated. Due to the bubble–bubble interactions, a remarkable suppression effect on the small bubble, whereas a slight enhancement effect on the large one were observed within the acoustic exposure parameters and the initial radii of the bubbles examined in this paper. Moreover, as the initial distance between bubbles increases, the strong suppression effect is reduced gradually and it could effectively enhance the nonlinear dynamics of bubbles, exactly as the bifurcation diagrams exhibit a similar mode of successive period doubling to chaos. Correspondingly, the resultant acoustic emissions present a progressive evolution of harmonics, subharmonics, ultraharmonics and broadband components in the frequency spectra. In addition, with the elasticity and/or viscosity of the surrounding medium increasing, both the nonlinear dynamics and translational motions of bubbles were reduced prominently. This study provides a comprehensive insight into the nonlinear behaviors and acoustic emissions of two interacting cavitation bubbles in viscoelastic media, it may contribute to optimizing and monitoring the cavitation-mediated biomedical applications.

Nonlinear Dynamics and Acoustic Emissions of Interacting Cavitation Bubbles in Viscoelastic Tissues

The cavitation-mediated bioeffects are primarily associated with the dynamic behaviors of bubbles in viscoelastic tissues, which involves complex interactions of cavitation bubbles with surrounding bubbles and tissues. The radial and translational motions, as well as the resultant acoustic emissions of two interacting cavitation bubbles in viscoelastic tissues were numerically investigated. Due to the bubble–bubble interactions, a remarkable suppression effect on the small bubble, whereas a slight enhancement effect on the large one were observed at various acoustic pressures and initial radii of the bubbles. Moreover, as the initial distance between bubbles increases, the strong suppression effect is reduced gradually and it could effectively enhance the nonlinear dynamics of bubbles, exactly as the bifurcation diagrams exhibit a similar mode of successive period doubling to chaos. Correspondingly, the resultant acoustic emissions present a progressive evolution of harmonics, subharmonics, ultraharmonics and broadband components in the frequency spectra. In addition, with the elasticity and/or viscosity of the surrounding medium increasing, both the nonlinear dynamics and translational motions of bubbles were reduced prominently. This study provides a comprehensive insight into the nonlinear behaviors and acoustic emissions of two interacting cavitation bubbles in viscoelastic media, it may contribute to optimizing and monitoring the cavitation-mediated biomedical applications.

Nonlinear dynamics of acoustic bubbles excited by their pressure-dependent subharmonic resonance frequency: influence of the pressure amplitude, frequency, encapsulation and multiple bubble interactions on oversaturation and enhancement of the subharmonic signal

Nonlinear behavior of bubbles, and most importantly $$\frac{1}{2}$$ -order subharmonics (SH), are used to increase the contrast-to-tissue ratio (CTR) in diagnostic ultrasound (US) and to monitor bubble-mediated therapeutic US. It is shown experimentally and numerically that when bubbles are sonicated with a frequency that is twice their resonance frequency ( $$f_\mathrm{r}$$ ), SHs at half the excitation frequency are generated at the minimum excitation pressure. Thus, $$f=2f_\mathrm{r}$$ is defined as the SH resonance frequency. SHs increase rapidly with pressure and reach an upper limit of the achievable SH signal strength. Numerous studies have investigated the pressure threshold of SH oscillations; however, conditions to enhance the saturation level of SHs have not been investigated. In this paper, nonlinear dynamics of bubbles excited at frequencies in the range of $$f_\mathrm{r}<f<2f_\mathrm{r}$$ is studied for different sizes of bubbles (400 nm–8 $$\upmu \mathrm{m}$$ ). In agreement with previous studies, we show that the SH resonance frequency is pressure dependent and decreases as pressure increases. When a bubble is sonicated with its pressure-dependent SH resonance frequency, oscillations undergo a saddle node bifurcation from a P1 or P2 regime to a P2 oscillation regime with higher amplitude. The saddle node bifurcation is concomitant with over-saturation of the SH and UH amplitude and eventual enhancement of the upper limit of SH and UH strength (e.g., $$\approx $$ 7 dB in UH amplitude). This can increase the CTR and signal-to-noise ratio in applications. Here, we show that the highest non-destructive SH amplitude occurs when $$f\approxeq 1.6-1.8f_\mathrm{r}$$ . In the case of interacting bubbles, $$f_\mathrm{r}$$ and pressure threshold of SH emissions decrease with increasing concentration, and supersaturation amplitude decreases above an upper limit of concentration.

Classification of the major nonlinear regimes of oscillations, oscillation properties, and mechanisms of wave energy dissipation in the nonlinear oscillations of coated and uncoated bubbles

Acoustic waves are dissipated when they pass through bubbly media. Dissipation by bubbles takes place through thermal damping (Td), radiation damping (Rd), and damping due to the friction of the liquid (Ld) and friction of the coating (Cd). Knowledge of the contributions of Td, Rd, Ld, and Cd during nonlinear bubble oscillations will help in optimizing bubble and ultrasound exposure parameters for the relevant applications by maximizing a desirable outcome or oscillation pattern. In this work, we investigate the mechanisms of dissipation in bubble oscillations and their contribution to the total damping (Wtotal) in various nonlinear regimes. By using a bifurcation analysis, we have classified nonlinear dynamics of bubbles that are sonicated with their third superharmonic (SuH) and second SuH resonance frequency (fr), pressure dependent resonance frequency (PDfr), fr, subharmonic (SH) resonance (fsh = 2fr), pressure dependent SH resonance (PDfsh), and 1/3 order SH resonance, which are important exposure ranges for various applications. The corresponding Td, Rd, Ld, Cd, Wtotal, scattering to dissipation ratio, maximum wall velocity, and maximum backscattered pressure from non-destructive oscillations of bubbles were calculated and analyzed using the bifurcation diagrams. Universal ultrasound exposure parameter ranges are revealed in which a particular non-destructive bubble related phenomenon (e.g., wall velocity) is enhanced. The enhanced bubble activity is then linked to relevant ultrasound applications. This paper represents the first comprehensive analysis of the nonlinear oscillations regimes, the corresponding damping mechanisms, and the bubble related phenomena.

An iterative fullwave simulation approach to multiple scattering in media with randomly distributed microbubbles

Ultrasound contrast agents (UCA), such as microbubbles, enhance the scattering properties of blood, which is otherwise hypoechoic. The multiple scattering interactions of the acoustic field with UCA are poorly understood due to the complexity of the multiple scattering theories and the nonlinear microbubble response. The majority of bubble models describe the behavior of UCA as single, isolated microbubbles suspended in infinite medium. Multiple scattering models such as the independent scattering approximation can approximate phase velocity and attenuation for low scatterer volume fractions. However, all current models and simulation approaches only describe multiple scattering and nonlinear bubble dynamics separately. Here we present an approach that combines two existing models: (1) a full-wave model that describes nonlinear propagation and scattering interactions in a heterogeneous attenuating medium and (2) a Paul–Sarkar model that describes the nonlinear interactions between an acoustic field and microbubbles. These two models were solved numerically and combined with an iterative approach. The convergence of this combined model was explored in silico for 0.5 × 106 microbubbles ml−1, 1% and 2% bubble concentration by volume. The backscattering predicted by our modeling approach was verified experimentally with water tank measurements performed with a 128-element linear array transducer. An excellent agreement in terms of the fundamental and harmonic acoustic fields is shown. Additionally, our model correctly predicts the phase velocity and attenuation measured using through transmission and predicted by the independent scattering approximation.

Enhanced target detection and classification using two-pulse sonar methods

A number of two-pulse sonar techniques can be employed to separate linear and nonlinear scatterers. Twin Inverted Pulse Sonar (TWIPS) and Biased Pulse Summation Sonar (BiaPSS) are processes that exploit nonlinear bubble dynamics to perform such a classification, with clutter reduction. Consequently, these techniques can be used to enhance target detection in bubbly waters. TWIPS and BiaPSS rely upon bubbles being driven to large nonlinear pulsations, but this is dependent upon availability of a high-amplitude source. Previous studies could only access a narrow-band source and therefore used a low frequency so that, over the rarefaction cycle, most ocean bubbles would expand sufficiently (the size range of near-surface ocean bubbles under waves and wakes being large). However, recent access to a broadband high-power source has allowed exploitation of bubble resonances. This talk presents results of investigations using broadband linear sine sweeps, of which one is based upon the simulation of an Atlantic Bottlenose Dolphin (Tursiops truncatus) echolocation click from Capus et al. (2006). Insonifying a cloud of bubbles with a wide band of frequencies excites as many of those sizes at, or close to, their resonance. As such, the contrast between linear and nonlinear scatter is enhanced, improving the target detection capability.

Dynamics of a Bubble Cloud near a Rigid Wall

Simulations involving a large number of interacting bubbles in a bubble cloud are conducted to assess the effect of frequency and amplitude of a sinusoidal driving pressure on the bubble cloud behavior. As the pressure amplitude increases strong nonlinear bubble dynamics become more pronounced and higher pressures are generated at the wall. A resonance frequency, corresponding to the highest collective bubble behavior, results in a pressure peak orders of magnitudes higher than the excitation pressure. This frequency is significantly different from the natural frequency of a spherical cloud executing small amplitude oscillations.

Numerical simulations of gas microbubble dynamics in an acoustic field with the influence of rectified diffusion

A numerical method for simulation of the diffusion problem for a single gas bubble oscillating in unbounded weakly compressible liquid in an acoustic field is developed. The method enables computation of nonlinear dynamics of bubbles of variable mass. The total mass of the dissolved gas and the gas in the bubble is conserved by enforcing conservation in the discrete scheme. For this purpose a conservative scheme based on an integro-interpolative method is applied for computation of the diffusion flux. Generally, a study of the effect of rectified diffusion on bubble dynamics requires significant computational time. To reduce it, an approximation based on the assumption on quasistationary character of oscillations of the dissolved gas concentration is developed. This enables investigation of rectified diffusion during millions of periods of oscillations. The numerical results obtained by the proposed method are in good agreement with the available experimental data. Comparison of the bubble mass change using the presented scheme and the standard scheme, which does not conserve the total mass of the gas-liquid system, reveals that in the latter case the numerical error may accumulate and lead to physically incorrect results.

Chaotic behavior of in-line bubbles rising with coalescences in non-Newtonian fluids: A multiscale analysis

The nonlinear dynamics of in-line bubbles rising with coalescence in non-Newtonian Carboxymethylcellulose sodium (CMC) fluids was studied through the techniques such as the multiresolution signal decomposition and the chaotic time series analysis. The temporary signals of bubble passages collected by an optical sensing device at different heights were investigated by a 12-level wavelet decomposition and the scalewise characteristics of bubble motion were extracted and analyzed. The chaotic time series analysis distinguished the periodicity or the deterministic chaos of bubble motion successsfully. The calculation of Kolmogorov entropy proves that in the ranges of experimental heights and gas flowrates, the bubble rising dynamics becomes more chaotic with the increase of height, and reaches the maximum chaotic extent in a certain height, while with the further increase of height, the chaotic extent decreases slowly. With the increase of gas flowrate, at the lower height, the bubble rising dynamics changes from periodicity to deterministic chaos, and at the higher heights it reaches the maximum chaotic extent in a certain gas flowrate; however, for both cases, it has little change in the higher gas flowrates. Moreover, with the increase of CMC concentration, the bubble rising dynamics becomes less chaotic when the height is beyond a certain value.

Numerical analysis of the dynamics of vapor bubbles

The nonlinear dynamics of vapor bubbles for the external pressure changing in steps is numerically analyzed based on a complete system of hydrodynamic equations without regard to the internal heat problem. Primary attention is focused on analyzing the applied numerical algorithms and on determining the conditions of complete vapor cavity collapse during first oscillation periods, depending on three parameters: Peclet number, Jacob number, and wave intensity.

Self focusing of acoustically excited Faraday ripples on a bubble wall

A theoretical explanation is presented to explain pattern formation during the generation of Faraday waves on a bubble wall. The theory derives the Hamiltonian formulation of the nonlinear bubble dynamics. The nonlinear Schrödinger equation for the envelope of surface modes on the bubble wall has been obtained. The solitary wave solution predicts that the shape distortions should be localized near the equator of the bubble.

Theory for acoustic propagation in marine sediment containing gas bubbles which may pulsate in a non‐stationary nonlinear manner

Whilst there is a considerable body of work in the literature on the theory of acoustic propagation in marine sediments, the incorporation of gas bubbles into such theories is done with the inclusion of assumptions which severely limit the applicability of those models to practical gas‐laden marine sediments. This paper provides a theory which does not assume that the bubble dynamics are linear, steady‐state and monochromatic. Such assumptions would be incompatible with many of the practical acoustic fields with which sediments are insonified today. These fields are necessarily high amplitude to provide adequate signal‐to‐noise ratios, given the high attenuation in gassy marine sediments; and often they utilise short pulses to obtain range resolution. This paper provides a theory appropriate for predicting the acoustically‐driven non‐stationary nonlinear dynamics of spherical gas bubbles embedded in a lossy elastic solid, and discusses how this could be incorporated into a nonlinear, time‐dependent propagation model.

Bubble nucleation and cavitation in soft tissue: The initial moments

In the absence of underlying pathology, biological tissue is expected to be free of gas bubbles. In order for cavitation to occur, the necessary bubbles must be nucleated by the acoustic field. Tiny vapor-filled bubbles continuously form and dissolve in regions of very low interfacial tension. In the presence of an acoustic wave, such microbubbles will act as cavitation nuclei. The probability of nucleation increases with both temperature and rarefactional pressure. Nonlinear propagation and diffraction of high-intensity focused ultrasound (HIFU) in tissue results in the formation of shocks and asymmetric waveforms with a lower peak but longer negative pressure phase than linear analysis predicts; the presence of shocks also greatly enhances energy deposition with a correspondingly more rapid temperature rise in the focal region. The effects of these processes on nucleation rate will be assessed. Post-nucleation bubble motion will then be explored using a modified version of the Keller-Miksis equation for nonlinear bubble dynamics combined with the linear Voigt model for viscoelastic media and a three-component version of the model to simulate a layer of degraded tissue between the gas bubble and the surrounding tissue. Various cavitational phenomena will be discussed. [Work supported by NSBRI SMS00402, RFBR, and DAMD17-02-2-0014.]

Nonlinear dynamics of gas bubbles in viscoelastic media

Understanding the behavior of cavitation bubbles driven by ultrasonic fields is an important problem in biomedical acoustics. The Keller–Miksis equation for nonlinear bubble dynamics is combined with the Voigt model for viscoelastic media. Using experimentally determined values, the effects of elasticity on bubble oscillations are studied. Inertial cavitation thresholds are determined using Rmax/R0=2, and subharmonic emissions are also estimated. The elasticity increases the threshold pressure for inertial cavitation, and subharmonic signals are significant only in a certain region of radii and driving pressures at a given frequency. These results should prove useful in cavitation detection and bubble-enhanced imaging work.