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Numerical Investigation of Non Linear Oscillations of Gas Bubbles in Liquids
Abstract
Forced oscillations of a spherical gas bubble in an incompressible, viscous liquid (water) are calculated numerically. The information gathered is mainly displayed in the form of frequency response curves of the steady-state solutions showing the harmonics, subharmonics, and ultraharmonics. Bubbles oscillating ultraharmonically at frequencies below the main resonance may emit half the driving frequency. This fact gives rise to a new explanation for the occurrence of the first subharmonic in the spectrum of the cavitation noise in ultrasonic cavitation.
Subject Classification: [43]30.70, [43]30.75.
... For the sake of simplicity, the pressure of the gas in the interface with the liquid is frequently modeled as inversely proportional to a power of the radius of the bubble itself, introducing an exponent proportional to the so-called polytropic index (this index is set equal to 1 in the case of an isothermal phenomenon, while it coincides with the specific heat ratio in the adiabatic case). 5,7,9,21,22 Through this simplification of the problem, the radius equation is self-consistent and can be integrated on its own. Another approach is to provide a more accurate description of the phenomena that take place inside the bubble by coupling the Rayleigh-Plesset or the Keller-Miksis equation to a set of partial differential equations (PDEs) or ordinary differential equations (ODEs) that take into account the details. ...
... These are ordinary differential equations that show the same mathematical structure as a damped and forced non-linear oscillator, and therefore, they also present the same phenomena: small amplitude oscillations, transients, asymptotic solutions, resonances, highly non-linear behaviors, and deterministic chaos. 3,9,21,23 In particular, given the non-linearity, it will not be possible to determine analytical solutions for these ODEs, unless the bubble is a gas-free cavity. In this regard, as already announced in the introduction, one faces the problem of closing equation (1) or (2), since knowledge of the gas pressure is required in order to proceed with the integration. ...
... The approximated Eqs. (30)-(32) were obtained, requiring that the gas velocity at the boundary (r ¼ R) coincides with _ R. Thus, in order to determine C 1 , one has to consider the remaining conditions regarding temperature and heat flux at the interface gas-liquid (21). To this aim, we refer to the model for T L and q L presented in Eq. (18) and the integration constants C 1 and B 3 turn out to be ...
The paper contains a preliminary study on the role that dynamic pressure might play in the dynamics of a gas bubble oscillating in a liquid. To this aim, we introduce a mathematical model, proposed under the homobaricity hypothesis and deduced from the 14-moment theory of rational extended thermodynamics through significant simplifications, that makes the equations easily integrable over long time intervals. In the presence of a gas with high bulk viscosity, relevant effects can be observed in different physical conditions: isothermal or adiabatic regimes, small amplitude oscillations, non-linear oscillations, resonances, and sonoluminescence. To make the study more realistic, we always refer to carbon dioxide gas, which on the one hand could present high values of bulk viscosity and on the other hand is known for its peculiar behaviors in the framework of cavitation and gas bubbles.
... Presently, many researchers focused on the bubble dynamics in the infinite fluid [7][8][9]. The interaction between the bubble and the free surface in the confined-area fluids is enhanced, resulting in changes in the dynamics and the time of bubble collapse of the free surface. ...
... where P d is the liquid side pressure at the surface; P out is the gas side pressure at the surface between the droplet and the external gas; R d is the instantaneous radius of the droplet containing a bubble. Combined the above equations, the kinetic equation for the bubble within a droplet can be expressed as Equation (8). ...
... During the first stage of the bubble collapse, the bubble wall is nearly spherical with the center of mass moving vertically upward, and the droplet surface being no splash. Among them, frames (1)- (8) show the first stage of the bubble collapse, in which the droplet shape hardly changes with the growth of the bubble. Frames (9)- (13) show the second stage of the bubble collapse, in which the centroid of the bubble moves obviously. ...
In the present paper, the cavitation bubble dynamics model for a single bubble oscillating within a droplet is improved based on the classical Rayleigh–Plesset bubble dynamics equation and the effects of liquid surface tension and viscosity are both considered. In the aspect of the experiment, the collapsing dynamic process of a bubble within a droplet is carried out by building a high-speed photography experimental platform. In addition, the numerical solution of the dynamic equation for the collapse time of the bubble within the droplet is also carried out. The findings are given as follows: (1) The bubble dynamic equation considering liquid surface tension and viscosity of bubble within droplet is proposed. (2) The surface of liquid droplets induced by the bubble motion could be divided into three modes: no splashing, scattered splashing, composite splash consisting of scattered and flaky splash. (3) The bubble interface during the first collapsing stage could be divided into three types: spherical, conical, and fungiform. (4) The numerical solution shows an accurate prediction of the bubble collapse time within the droplet especially under the condition of medium radius ratio.
... Fig. 2 provides the frequency response curves of an individual bubble when the pressure amplitudes are as follows: P a = 0.005, 0.1, 0.3, 0.6 and 0.9 bar. According to Lauterborn [55], when the frequency of the sound field f is close to a rational number "m/n" (m, n = 1, 2, 3,…) times the linear resonance frequency f res , all the resonances in Fig. 2 can be classified into four types according to the "order" (the inverse of "m/n"): Fig. 2 demonstrated that with P a increasing, all resonances lean towards the lower frequencies [10,23,47,56,57]. Furthermore, the detailed work on the comprehensive oscillation characteristics of the 2nd and 3rd order harmonically resonant bubbles could be referred to Ref. [58]. ...
... By a novel method [59], Sojahrood et al. comprehensively characterized the evolution of 2nd, 3rd harmonic and subharmonics as a function of pressure [60]. The onset of the first subharmonic (1/2 order) resonance was proposed as a method to determine the cavitation threshold [55]. One of the main applications of the 1/2 order subharmonic emissions is in the medical ultrasound. ...
... In addition, the position of the resonance radii corresponding to the main, 2nd (3rd), and 1/2 order resonance frequencies of the coupled bubbles gradually moves towards the left with P a increasing, which is also reflected in the R 01 -R 02 planes in the following sections. This owing to the fact that all the resonance frequencies lean towards lower frequencies due to the "bending phenomenon" in nonlinear dynamics as the P a increases [37,55]. Fig. 4 provides the schematic diagram of six types of translational motions of bubbles under the action of F B : (a) pair with invariant interbubble distance, (b) stable pair after attraction, (c) coalescence pair after mutual attraction, (d) stable pair after mutual repulsion, (e) constantly repulsive pair, and (f) chaotic pair. ...
The translation behaviors of oscillating bubbles are closely related to the polymerizations and dispersions between them, which are crucial for the ultrasonic cavitation effect. In this study, six types of translational motion of bubbles with a wide range of sizes (2-100 μm) in the R01-R02 plane are investigated. Our results demonstrate that in addition (to the 2nd order harmonic), the 1/2 order subharmonic can change the bubble pairs from the three states of the attraction, stable after attraction, and repulsion to that of the repulsion, coalescence, and attraction, respectively. Furthermore, within the range of the main resonance radius and the 1/2 order subharmonic resonance radius, the chaotic bubble pairs with alternating attractive and repulsive forces appear in the region between the coalescence pairs and stable pairs after attraction. Finally, the corresponding physical mechanisms of the chaotic translational motions are also revealed.
... From the data presented in Fig. 3A, the insonication frequency at which the subharmonic response is largest shifts to lower frequencies for bubbles in their 'elastic' state (σ 0 = 0.01); for example, from f = 11 MHz at 10 % excursion to f = 10 MHz at 21 % excursion for the bubble in Fig. 3A. In addition to the expected increase in the subharmonic peak amplitude with increasing radial excursion, the behaviour represented for the bubbles that start off in the elastic state mirrors the response typical of the overall radial amplitude [20,24,33,40]. ...
... excursion) has been well examined. From both experimental [24,33,41] and modelling work [20,37,40,42,43], the pressuredependent frequency of maximum response f MR from an individual microbubble shift towards decreasing frequencies as the transmit pressure is increased. The subharmonic response of an individual micronsized bubble, in particular phospholipid-coated bubbles, has also been well investigated [26,36,39,[44][45][46][47]. ...
Phospholipid encapsulated ultrasound contrast agents have proven to be a powerful addition in diagnostic imaging and show emerging applications in targeted therapy due to their resonant and nonlinear scattering. Microbubble response is affected by their intrinsic (e.g. bubble size, encapsulation physics) and extrinsic (e.g. boundaries) factors. One of the major intrinsic factors at play affecting microbubble vibration dynamics is the initial phospholipid packing of the lipid encapsulation. Here, we examine how the initial phospholipid packing affects the subharmonic response of either individual or a system of two closely-placed microbubbles. We employ a finite element model to investigate the change in subharmonic resonance under ‘small’ and ‘large’ radial excursions. For microbubbles ranging between 1.5 and 2.5 µm in diameter and in its elastic state (σ0 = 0.01 N/m), we demonstrate up to a 10 % shift towards lower frequencies in the peak subharmonic response as the radial excursion increases. However, for a bubble initially in its buckled state (σ0 = 0 N/m), we observe a maximum shift of 8 % towards higher frequencies as the radial excursion increases over the same range of bubble sizes – the opposite trend. We studied the same scenario for a system of two individual microbubbles for which we saw similar results. For microbubbles that are initially in their elastic state, in both cases of a) two identically sized bubbles and b) a bubble in proximity to a smaller bubble, we observed a 6 % and 9 % shift towards lower frequencies respectively; while in the case of a neighboring larger bubble no change in subharmonic resonance frequency was observed. Microbubbles that are initially in a buckled state exert no change, 5 % and 19 % shift towards higher frequencies, in two-bubble systems consisting of a) same-size, b) smaller, and c) larger neighboring bubble respectively. Furthermore, we examined the effect of two adjacent bubbles with non-equal initial phospholipid states. The results presented here have important implications in ultrasound contrast agent applications.
... 22 Despite their assumption, the models consider the effects of pressure and temperature enabling them to predict hydrodynamic phenomena witnessed in experiments, such as non-linear bubble oscillations, stability of bubble oscillations, and the bifurcation of bubble oscillation modes. [28][29][30] Furthermore, the Rayleigh-Plesset model appears to be relatively the most favorable model, as it considers the influence of liquid surface tension, viscosity, and nucleation inertia on bubble evolution. In fact, most commercially available models, namely, the Kunz et al., 31 32 and Zwart-Gerber-Belamri (ZGB) 33 models, are derived and implemented based on a simplified Rayleigh-Plesset model. ...
... Using Eq. (28) and the assumption of force equilibrium at the observed node, an iterative equation is deduced, as shown in Eq. (30), that is, then solved using the Jacobi iterative method until convergence to obtain the final position using Eq. (31) ...
One of the many ways of cavitation utilized for process intensification is through acoustically inducing it. As acoustic cavitation gained traction in recent industrial works, numerical modeling became an important study tool to scrutinize and optimize acoustic cavitation applications. However, available hydrodynamic cavitation models are found incapable of accurately predicting acoustic cavitation structures and flow features. This could source from the oversimplification of the Rayleigh–Plesset equation or from obscure effects of empirical model constants. To address this issue, new mass transfer source terms for Zwart–Gerber–Belamri model were derived based on the consideration of Rayleigh–Plesset's second-order derivatives. In addition, a design of experiments statistical approach, coupled with Monte Carlo simulations, was implemented to assess the influence of empirical model constants on the model's performance by examining variations in amplitude and frequency responses. Moreover, a set of optimized model constants was obtained: evaporation constant = 17.359 88, condensation constant = 0.1, Bubble Radius = 25 × 10 ⁻⁶ m, and Nucleation Site Volume Fraction = 5 × 10 ⁻⁴ , to obtain a maximum pressure and frequency of 3.62 bar and 4128.73 Hz, respectively. The new model, with the new constants, was configured into ANSYS Fluent 22.1 and validated against experimental values. The new model resulted with maximum pressure and frequency of 3.48 bar and 4894.56 Hz, respectively, validating the statistical model and showing drastic improvement in qualitatively and quantitatively capturing acoustic cavitation.
... Radial oscillation amplitude increases and the bubble does not anymore exhibit only one resonance. Depending on the pressure amplitude, the bubble resonance curve may exhibit multiple harmonics and SH resonance peaks (Werner Lauterborn, 1976;Werner Lauterborn & Kurz, 2010;A.J. Sojahrood, Haghi, Li, et al., 2020). In this section different resonance frequencies of a bubble oscillator is studied. ...
... Increasing the pressure amplitude to 40 kPa results in further increase in the maximum bubble radius; however, the frequency at which maximum bubble radius occurs shifts to 0.87f r . This called the pressure dependent resonance frequency (PDf r ) and is studied in detail in (Werner Lauterborn, 1976;MacDonald et al., 2004;. (A.J. showed that the maximum nondestructive P sc [when R max /R 0 ≤2 (Flynn & Church, 1988)] can increase up to 9 folds when the bubble is sonicated with its PDf r . ...
The oscillations of ultrasonically excited bubbles are complex and chaotic. The bubble oscillator exhibits fundamental, super-harmonic, and subharmonic resonance frequencies that depend on the acoustic pressure amplitude and frequency. Acoustic energy is dissipated upon passing through a bubbly medium by thermal, viscous, and radiation damping. Nonlinear bubble oscillations and their relation to the ultrasound energy dissipation mechanisms need to be understood for optimizing and controlling bubble-related applications, yet due to its complexity this relationship has not been studied comprehensively. In this chapter, the linear and nonlinear resonance frequencies of the bubble oscillator are discussed for a range of exposure regimes. The pressure-dependent nonlinear and chaotic bubble oscillations as a function of applied acoustic pressure are introduced and then classified. The energy dissipation terms are derived in both the linear and nonlinear regimes for free and encapsulated bubbles. The differences between linear and nonlinear regimes are highlighted by analyzing the dissipated energies at bubble resonances. Furthermore, the evolution of the damping mechanisms as a function of pressure is presented and classified. It is shown that at optimized exposure parameters that depend on the bubble resonances, a specific category of energy dissipation can be maximized or minimized. The impact of selecting the appropriate exposure parameters for ultrasound applications is discussed.
... horns (local peaks in the bubble expansion) appear as light-blue wedges. The main resonance frequency of an 8 μm bubble is approximately 430 kHz represented by the biggest wedge. The smaller wedges accumulated towards the lower frequency ranges are the harmonics of the main resonance caused by the non-linearity of the system (Klapcsik et al., 2018;W. Lauterborn, 1976;W. Lauterborn & Holzfuss, 1991). For the details of the resonance properties of a single bubble, the reader is referred to publications (Klapcsik & Hegedűs, 2017;W. Lauterborn, 1976;A.J. Sojahrood et al., 2019;A. J. Sojahrood et al., 2015). Observe that the left-hand sides of the resonance horns in Fig. 4.11 match with the left-hand side ...
... smaller wedges accumulated towards the lower frequency ranges are the harmonics of the main resonance caused by the non-linearity of the system (Klapcsik et al., 2018;W. Lauterborn, 1976;W. Lauterborn & Holzfuss, 1991). For the details of the resonance properties of a single bubble, the reader is referred to publications (Klapcsik & Hegedűs, 2017;W. Lauterborn, 1976;A.J. Sojahrood et al., 2019;A. J. Sojahrood et al., 2015). Observe that the left-hand sides of the resonance horns in Fig. 4.11 match with the left-hand sides of the stripes in Fig. 4.10. This sudden change in the amplitude of the oscillation and the chemical yield is due to the hysteric behavior of the resonances. Although the appearanc ...
This chapter introduces state-of-the-art modelling techniques of chemical kinetics inside a single spherical oscillating bubble placed in an infinite domain of liquid water. The initial content of the bubble is pure oxygen and water vapor. The reaction mechanism that governs chemical kinetics inside the bubble takes into account many aspects that are usually neglected in previous sonochemical investigations. First, at the collapse state of a bubble, the pressure inside can reach several hundreds of atmospheres; thus, the incorporation of the pressure dependence of reactions in which a third body plays a role is mandatory. Second, third body efficiencies are also taken into account. Third, the backward reactions are computed via thermodynamic equilibrium conditions. Fourth, reactions that have non-Arrhenius temperature dependence can be described by two sets of Arrhenius constants. These reactions are identified and modelled properly. As more experimental data have been accumulated over the decades, the Arrhenius constants of certain reactions have been changed even by orders of magnitude. Therefore, it is also important to employ up-to-date values of the Arrhenius constants. The behavior of the proposed model is demonstrated with reaction condition sets (pressure amplitude, frequency and bubble size) typically used during the experiments. The production of important chemical species (e.g., hydrogen or free radicals) are investigated from energy efficiency points of view (yield in mole per unit dissipated power of the bubble).
... The dynamic behavior of spherical cavitation bubbles described by Keller equation is compatible with the theoretical results of large-scale bubbles, such as underwater explosions, as presented in the research paper of Keller and Kolodner (1956). For small and medium-sized bubbles, such as cavitation bubbles, the results of Keller equation are consistent with those of Lauterborn (1976) based on R-P equation modified by Noltingk and Neppiras (1950). Consequently, Herring and Gilmore's theoretical model is extended to Keller's theoretical model, and Keller equation has become widely used as the primary equation for predicting bubble dynamics in a compressible fluid. ...
Bubbles play crucial roles in various fields, including naval and ocean engineering, chemical engineering, and biochemical engineering. Numerous theoretical analyses, numerical simulations, and experimental studies have been conducted to reveal the mysteries of bubble motion and its mechanisms. These efforts have significantly advanced research in bubble dynamics, where theoretical study is an efficient method for bubble motion prediction. Since Lord Rayleigh introduced the theoretical model of single-bubble motion in incompressible fluid in 1917, theoretical studies have been pivotal in understanding bubble dynamics. This study provides a comprehensive review of the development and applicability of theoretical studies in bubble dynamics using typical theoretical bubble models across different periods as a focal point and an overview of bubble theory applications in underwater explosion, marine cavitation, and seismic exploration. This study aims to serve as a reference and catalyst for further advancements in theoretical analysis and practical applications of bubble theory across marine fields.
... Subsequently, the bubble sank from the wall to bottom and eventually penetrated, while generating a high-speed micro-jet penetrated the bubble towards the wall. The Rayleigh-Plesset equation is obtained as [4,45]: ...
In orthopedic surgery, cortical bone cutting usually involves washing and cooling with physiological saline. However, how the saline changes the cutting behaviors of bone ultrasonic vibration cutting remains challenging. Hence, this paper simulates the clinical ultrasonic cutting condition in orthopedics to reveal the cutting behaviors of bone ultrasonic vibration orthogonal cutting immersed in physiological saline. The dynamic equation and motion process curves of ultrasonic cavitation bubbles were established. The results showed that the bone cutting immersed in physiological saline significantly improved the surface quality, reduced surface roughness and mechanical damage, and avoided large brittle cracks propagation. In saline immersed cutting, the physiological saline changes the mechanical behaviors of bone materials, resulting in plastic behaviors for the material removal and crack deflection. This study establishes the influence of physiological saline on the ultrasonic vibration cutting performance, providing guidance for orthopedic bone cutting surgery methods.
... Francescutto and Nabergoj 29 derived explicit formulations for the frequency response of the primary resonance, the first and second SuHs, and the first and second SHs of a spherical bubble in an incompressible viscous liquid under the action of a sound field. They adjusted the order of the excitation term using the multi-scale method and analyzed it with the steady-state results obtained by scholars such as Lauterborn, 11 Prosperetti, 30 and Samek. 31 Samek 32,33 applied the multi-scale method to investigate the primary resonance, SuH resonance, and SH resonance of a spherical bubble based on the Keller equation, obtaining a second-order solution. ...
In this paper, the dimensionless oscillation equation of a cylindrical bubble is analyzed using the multi-scale method, Lyapunov stability theory, and the Routh–Hurwitz stability criterion. The corresponding second-order analytical solution and stability criterion are obtained. By examining the cases of second-order super-harmonic resonance and 1/2-order sub-harmonic resonance, the harmonic resonance characteristics of cylindrical bubbles and the influencing factors are revealed. The conclusions are summarized as follows: (1) Super-harmonic resonance can exhibit up to three solutions, along with unstable phenomena such as jump and hysteresis. Sub-harmonic resonance, however, shows at most two solutions simultaneously, without jump or hysteresis phenomena. (2) As the acoustic excitation amplitude increases, both the response amplitude and the unstable zone significantly enlarge. An increase in nonlinear coefficients can reduce the response amplitude and increase instability. (3) When the acoustic excitation amplitude reaches a certain threshold, the oscillation mode of the bubble shifts from periodic to chaotic. Under the same initial conditions, the chaos threshold for sub-harmonic resonance is higher than that for super-harmonic resonance.
... 5 Later on, the said equation was solved by Lauterborn. 6 Further, modi¯cation to the equation was done by Keller and Miksis, 7 who included acoustic radiation in the equation, and presented that this solution is free from the defects that occurred in the solution of Lauterborn which failed to be convergent to a periodic solution. ...
This research is focused on studying the behavior of bubbles in one-dimensional nozzle flows. The study delves into the complex flow of cavitating bubbles within geometries of varying cross-sectional areas, using a second-grade fluid. The cavitation phenomenon poses a potential threat, as it can harm surfaces upon bubble collapse or interaction with adjacent boundaries. Thus, the proposed model unveils the intricate behavior of cavitating bubbles in response to nozzle shape and fluid properties. The governing equations are first modeled using the Rayleigh–Plesset equation along with continuity and momentum equations for bubbly liquids. They are then simplified and nondimensionalized to solve by means of the RK method. The influence of several nondimensional parameters, including the Reynolds number, fluid parameter, cavitation number and the number of bubbles, on the flow behavior of second-grade fluid through various channels is illustrated using graphs. To validate the results, the Newtonian formulation was applied in the current setting, and it was found that the obtained results matched the available literature. Also, critical values of void fractions for the radius of the bubble are presented. The non-Newtonian properties of the fluid are observed a significant outcome for the reduction of cavitation damage and noise.
... Lauterborn [31] computed the velocity and pressure exerted by a freely oscillating spherical microbubble in a viscous, incompressible fluid as ...
Sonoporation is a non-invasive method that uses ultrasound for drug and gene delivery for therapeutic purposes. Here, both Finite Element Method (FEM) and Lattice Boltzmann Method (LBM) are applied to study the interaction physics of microbubble oscillation and collapse near flexible tissue. After validating the Finite Element Method with the nonlinear excited lipid-coated microbubble as well as the Lattice Boltzmann Method with experimental results, we have studied the behavior of a three-dimensional compressible microbubble in the vicinity of tissue. In the FEM phase, the oscillation microbubble with a lipid shell interacts with the boundary. The range of pressure and ultrasound frequency have been considered in the field of therapeutic applications of sonoporation. The viscoelastic and interfacial tension as the coating properties of the microbubble shell have been investigated. The presence of an elastic boundary increases the resonance frequency of the microbubble compared to that of a free microbubble. The increase in pressure leads to an expansion in the range of the microbubble’s motion, the velocity induced in the fluid, and the shear stress on the boundary walls of tissue. An enhancement in the surface tension of the microbubble can influence fluid flow and reduce the shear stress on the boundary. The multi-pseudo-potential interaction LBM is used to reduce thermodynamic inconsistency and high-density ratio in a two-phase system for modeling the cavitation process. The three-dimensional shape of the microbubble during the collapse stages and the counter of pressure are displayed. There is a time difference between the occurrence of maximum velocity and pressure. All results in detail are presented in the article bodies.
... The spherically symmetric oscillations of a gas bubble in 19 a liquid being forced by a periodic pressure field are suscep- 20 tible to parametric instabilities which, for a sufficiently large 21 driving pressure, will cause this symmetry to break, giving 22 rise to distinct and possibly complex surface shape oscillation 23 patterns. To model such patterns the shape distortion is often 24 expressed as an infinite sum of spherical harmonics, reducing 25 to Legendre polynomials in axisymmetry, the amplitudes of 26 which are referred to as shape modes and which have their 27 own distinctive oscillation frequencies [1]. In experimental 28 studies of shape deformation, an often encountered difficulty 29 is due to translational instability. ...
The self-propulsion (translational instability) of a gas bubble in a liquid undergoing parametrically induced axisymmetric shape distortion due to being forced by a temporally sinusoidal, spatially constant acoustic field is investigated. Employing a model which accounts for the nonlinear coupling between the spherical oscillations, the axial translation and shape deformation of the bubble, the parametric excitement of two neighboring shape modes by the fundamental resonance, at the same driving frequency is studied. It is shown that provided pertinent driving pressure threshold values are exceeded, the respective shape modes are excited on different timescales. The growth of the shape mode on the faster timescale saturates giving rise to sustained constant amplitude oscillations, while the growth of the shape mode on the slower timescale is both modulated and unbounded. During the growth of the second shape mode, growing, oscillatory bubble translation is also observed.
... The effects of phase inversion, amplitude modulation, and amplitude-modulated pulse inversion methods are studied by simulating the scattered echoes of microbubbles. Models are used to formulate microbubble behavior [39][40][41][42][43][44][45][46][47]. When the pressure around the microbubble changes, the bubble repeatedly contracts and expands in the radial direction, forming a spring-mass system that oscillates with nonlinearity. ...
Sonazoid, an ultrasound contrast agent, has been covered by insurance in Japan since January 2007 for the diagnosis of hepatic mass lesions and is widely used for diagnosing not only primary liver cancer but also liver metastases such as those from breast cancer and colorectal cancer. Contrast-enhanced ultrasound for breast mass lesions has been covered by insurance since August 2012 after phase II and phase III clinical trials showed that the diagnostic performance was significantly superior to that of B-mode and contrast-enhanced magnetic resonance imaging. This paper describes the principles of imaging techniques in contrast-enhanced ultrasonography including the filter, pulse inversion, amplitude modulation, and amplitude-modulated pulse inversion methods. The pulse inversion method, which visualizes the second-harmonic component using the nonlinear scattering characteristics of the contrast agent, is widely used regardless of the contrast agent and target organ because of its high resolution. Sonazoid has a stiffer shell and requires a higher acoustic amplitude than Sonovue to generate nonlinear vibrations. The higher transmitted sound pressure generates more tissue harmonic components. Since pulse inversion allows visualization of the tissue harmonic components, amplitude modulation and amplitude-modulated pulse inversion, which include few tissue harmonic components, are primarily used. Amplitude modulation methods detect nonlinear signals from the contrast agent in the fundamental band. The mechanism of the amplitude modulation is considered to be changes in the echo signal’s phase depending on the sound pressure. Since the tissue-derived component is minor in amplitude modulation methods, good contrast sensitivity can be obtained.
... Microbubbles driven at sufficiently high pressure amplitudes can undergo highly non-linear oscillations, generating higher order harmonic responses of the bubble (Lauterborn, 1976;Prosperetti, 1974), as well as subharmonic behavior (Neppiras, 1969). In the context of viscoelastic fluids, Allen and Roy showed that the presence of elasticity can enhance the second harmonic response of an Upper Convected Maxwell fluid (Allen and Roy, 2000b). ...
Understanding the ultrasound pressure-driven dynamics of microbubbles confined in viscoelastic materials is relevant for multiple biomedical applications, ranging from contrast-enhanced ultrasound imaging to ultrasound-assisted drug delivery. The volumetric oscillations of spherical bubbles are analyzed using the Rayleigh-Plesset equation, which describes the conservation of mass and momentum in the surrounding medium. Several studies have considered an extension of the Rayleigh-Plesset equation for bubbles embedded into viscoelastic media, but these are restricted to a particular choice of constitutive model and/or to small deformations. Here, we derive a unifying equation applicable to bubbles in viscoelastic media with arbitrary complex moduli and that can account for large bubble deformations. To derive this equation, we borrow concepts from finite-strain theory. We validate our approach by comparing the result of our model to previously published results and extend it to show how microbubbles behave in arbitrary viscoelastic materials. In particular, we use our viscoelastic Rayleigh-Plesset model to compute the bubble dynamics in benchmarked viscoelastic liquids and solids.
... For each frequency within a certain frequency range, the amplitude of the bubble oscillation has two stable values and one unstable value, as introduced in Refs. [36,37]. In any specific case, which steady state is ultimately reached depends on the initial conditions and the direction of frequency variation. ...
We study theoretically and numerically sound attenuation in bubble-containing media when the bubbles are freely oscillating at high Mach numbers. This paper expands one of the main forms of bubble-related acoustic damping factors by extending the previous theories to higher Mach numbers, further improves the theories of nonlinear sound propagation in bubble-containing media. A nonlinear sound propagation model incorporating second-order liquid compression terms is developed, expressing the sound velocity and density in the medium as a function of the driving pressure, and taking into account the higher-order liquid compression effects on sound propagation. The correctness of the proposed model is verified by comparing with a linear model and a nonlinear model containing only low-order Mach number terms. When the bubble oscillates at a high Mach number, radiation damping, which is directly related to Mach number, becomes the main damping component affecting sound attenuation. The higher the driving amplitude, the stronger the nonlinear effect, and the greater the impact of high-order liquid compression effects on the sound attenuation, the more necessary it is to use the proposed model to calculate the sound attenuation. For high Mach numbers, varying the bubble radius and bubble number density, respectively, the difference between the proposed model and the model containing only low-order Mach number terms in capturing the pressure-dependent attenuation is calculated. Due to stronger radiation damping in smaller bubbles, the effect of compressibility becomes more important. The smaller the bubble radius, the greater the half-quality factor of the curve related to the difference in attenuation calculated by the two models, the more necessary it is to calculate the pressure-dependent attenuation using the proposed model. Here, the half-quality factor is defined as the corresponding frequency bandwidth when the curve falls from the maximum value to 22 times. Without considering the coupling effect between bubbles, the half-quality factor of the curve is not affected by the bubble number density.
... In comparison to Boyle's Law, which does not account for inertia or damping, it is able to predict bubble overshoot, rebound and oscillation, which are to be expected in any lightly damped system when rate of pressure change is anything but very slow compared to the resonant frequency of the bubble. The Rayleigh Plesset equation also produces realistic outputs when the absolute pressure is near or below zero (and indeed for those intervals when the liquid is in tension) 20,21 . In comparison, when the external pressure becomes zero, Boyle's Law predicts that the gas volume increases without bounds. ...
The expansion and potential rupture of the swim bladder due to rapid decompression, a major cause of barotrauma injury in fish that pass through turbines and pumps, is generally assumed to be governed by Boyle’s Law. In this study, two swim bladder expansion models are presented and tested in silico. One based on the quasi-static Boyle’s Law, and a Modified Rayleigh Plesset Model (MRPM), which includes both inertial and pressure functions and was parametrised to be representative of a fish swim bladder. The two models were tested using a range of: (1) simulated and (2) empirically derived pressure profiles. Our results highlight a range of conditions where the Boyle’s Law model (BLM) is inappropriate for predicting swim bladder size in response to pressure change and that these conditions occur in situ, indicating that this is an applied and not just theoretical issue. Specifically, these conditions include any one, or any combination, of the following factors: (1) when rate of pressure change is anything but very slow compared to the resonant frequency of the swim bladder; (2) when the nadir pressure is near or at absolute zero; and (3) when a fish experiences liquid tensions (i.e. negative absolute pressures). Under each of these conditions, the MRPM is more appropriate tool for predicting swim bladder size in response to pressure change and hence it is a better model for quantifying barotrauma in fish.
... Early work done by Ashwell et al. [26] and Lauterborn [27] showed that nonlinear oscillations of gas bubbles forced into volume pulsations by an acoustic field in liquids might occur at frequencies equal to half of the driving frequency, the subharmonic. Furthermore, Leighton et al. [23] showed that it is possible to accurately size and resolve the bubbles in liquids by interrogating the subharmonic emitted by the bubble, which can be directly related to the resonance of these bubbles. ...
The influence of porosity on the mechanical behaviour of composite laminates represents a complex problem that involves many variables. Therefore, the evaluation of the type and volume content of porosity in a composite specimen is important for quality control and for predicting material behaviour during service. A suitable way to evaluate the porosity content in composites is by using nonlinear ultrasonics because of their sensitivity to small cracks. The main objective of this research work is to present an imaging method for the porosity field in composites. Two nonlinear ultrasound techniques are proposed using backscattered signals acquired by a phased array system. The first method was based on the amplitude of the half-harmonic frequency components generated by microbubble reflections, while the second one involved the frequency derivative of the attenuation coefficient, which is proportional to the porosity content in the specimen. Two composite samples with induced porosity were considered in the experimental tests, and the results showed the high accuracy of both methods with respect to a classic C-scan baseline. The attenuation coefficient results showed high accuracy in defining bubble shapes in comparison with the half-harmonic technique when surface effects were neglected.
... Figs. 16 and 17 show the dependence of the acoustically driven bubble's response amplitude on the ambient or equilibrium bubble radius, 0 . This is of interest because the compressibility function = − 0 ∕ is related to 0 [2,58]. Fig. 16(a) shows the response amplitude as a function of the equilibrium bubble radius 0 at four different modulation amplitudes . ...
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<G<10Pa) compared to the acoustic driving pressure Ps∼105 Pa. The optimal parameter values for enhanced response could be useful in acoustic cavitation applications.
... Here R denotes the bubble radius, n is the polytropic constant, and p˚and Rd enote equilibrium values. Such models have been studied extensively in [18,21,22,14,16,17,28]. For reviews of the subject, see e.g. ...
We study the radial relaxation dynamics toward equilibrium and time-periodic pulsating spherically symmetric gas bubbles in an incompressible liquid due to thermal effects. The asymptotic model ([A. Prosperetti, J. Fluid Mech., 1991] and [Z. Biro and J. J. L. Velazquez, SIAM J. Math. Anal., 2000]) is one where the pressure within the gas bubble is spatially uniform and satisfies an ideal gas law, relating the pressure, density and temperature of the gas. The temperature of the surrounding liquid is taken to be constant and the behavior of the liquid pressure at infinity is prescribed to be constant or periodic in time. In arXiv:2207.04079, for the case where the liquid pressure at infinity is a positive constant, we proved the existence of a one-parameter manifold of spherical equilibria, parameterized by the bubble mass, and further proved that it is a nonlinearly and exponentially asymptotically stable center manifold. In the present article, we first refine the exponential time-decay estimates, via a study of the linearized dynamics subject to the constraint of fixed mass. We obtain, in particular, estimates for the exponential decay rate constant, which highlight the interplay between the effects of thermal diffusivity and the liquid viscosity. We then study the nonlinear radial dynamics of the bubble-fluid system subject to a pressure field at infinity which is a small-amplitude and time-periodic perturbation about a positive constant. We prove that nonlinearly and exponentially asymptotically stable time-periodically pulsating solutions of the nonlinear (asymptotic) model exist for all sufficiently small forcing amplitudes. The existence of such states is formulated as a fixed point problem for the Poincar\'e return map, and the existence of a fixed point makes use of our (constant mass constrained) exponential time-decay estimates of the linearized problem.
... [4,21,22,29,[38][39][40] The dynamics of an acoustically driven microbubble is well described by the celebrated Rayleigh-Plesset equation. [41] Linearization of the Rayleigh-Plesset equation shows that, in the limit of small amplitudes of oscillation, the acoustically driven bubble can be modeled as a simple mass-spring system where the inertia of the surrounding liquid (mass) oscillates against the compressible gas core (spring). [42,43] The corresponding bubble eigenfrequency is inversely proportional to its size, as was shown nearly a century ago by Minnaert. ...
Microbubbles entrained in a piezo-driven drop-on-demand (DOD) printhead disturb the acoustics of the microfluidic ink channel and thereby the jetting behavior. Here, the resonance behavior of an ink channel as a function of the microbubble size and the number of bubbles is studied through theoretical modeling and experiments. The system is modeled as a set of two coupled harmonic oscillators: one corresponding to the compliant ink channel and one to the microbubble. The predicted and measured eigenfrequencies are in excellent agreement. It was found that the resonance frequency is independent of the bubble size as long as the compliance of the bubble dominates over that of the piezo actuator. An accurate description of the eigenfrequency of the coupled system requires the inclusion of the increased inertance of the entrained microbubble due to confinement. We show that the inertance of a confined bubble can be accurately obtained by using a simple potential flow approach. The model is further validated by the excellent agreement between the modeled and measured microbubble resonance curves. The present work therefore provides physical insight in the coupled dynamics of a compliant ink channel with an entrained microbubble.
... Due to the influence of gas compressibility, the gas jet pressure decreases with its volume increases along the axial and radial direction of the gas jet. When the ventilated cavity reaches the maximum shape, the internal pressure cannot be maintained, and the ventilated cavity begins to collapse (Crum and Prosperetti, 1983;Lauterborn, 1976;Qin et al., 2016;and Weiland and Vlachos, 2013). ...
To reduce the energy dissipation of the submerged water jet, a series of experiments of the submerged water jet wrapped in an annular gas jet are performed under different gas ventilation rates, annular sizes, water jet nozzle diameters, and water jet velocities in a transparent water tank. In the experiments, a ventilated cavity is created by the annular gas jet that encloses the submerged water jet. The submerged water jet is separated from the surrounding water within a certain distance after leaving the nozzle exit by the ventilated cavity, which contributes to the effective working length of the submerged water jet significantly increasing, referring to the energy dissipation decrease. Furthermore, the geometry of the ventilated cavity changes periodically, i.e., the cavity length and diameter decrease after increasing to the peak values in each cycle. Moreover, the ventilated cavity development process can be mainly divided into formation, collapse, and intermission stages. The maximum cavity length of the ventilated cavity decrease with the per unit time momentum ratio between the water jet and the gas jet. Namely, the per unit time momentum ratio between the water jet and the gas jet is the dominating parameter of the cavity geometry.
... The surface tension of the clean gasliquid interface is 0:072 N=m, the viscosity of the water is l ¼ 0:001 Pa s, and the ambient pressure is p 0 ¼ 10 5 Pa. Figure 9 shows the bubble behavior for f a ¼ 1MHz and Dp a ¼ f300; 600g kPa. With Dp a ¼ 300 kPa, although above the Blake pressure, 71 the bubble collapse occurs at low Mach numbers, whereas with Dp a ¼ 600 kPa, the bubble undergoes an inertial collapse with Mach numbers at which the compressibility of the liquid cannot be neglected. Figure 10 shows the pressure amplitude, Dpðr; tÞ ¼ pðr; tÞ Àp 1 ðtÞ, and the velocity, u(r, t), of the acoustic emissions associated with the collapse of the lipid-coated microbubble, indicated with "P" in Fig. 9. ...
Despite significant progress in understanding and foretelling pressure-driven bubble dynamics, models that faithfully predict the emitted acoustic waves and the associated shock formation of oscillating or collapsing bubbles have received comparably little attention. We propose a numerical framework using a Lagrangian wave tracking approach to model the acoustic emissions of pressure-driven bubbles based on the Kirkwood–Bethe hypothesis and under the assumption of spherical symmetry. This modeling approach is agnostic to the equation of the state of the liquid and enables the accurate prediction of pressure and velocity in the vicinity of pressure-driven bubbles, including the formation and attenuation of shock fronts. We validate and test this new numerical framework by comparison with solutions of the full Navier–Stokes equations and by considering a laser-induced cavitation bubble as well as pressure-driven microbubbles in excitation regimes relevant to sonoluminescence and medical ultrasound, including different equations of state for the liquid. A detailed analysis of the bubble-induced flow field as a function of the radial coordinate r demonstrates that the flow velocity u is dominated by acoustic contributions during a strong bubble collapse and, hence, decays predominantly with [Formula: see text], contrary to the frequently postulated decay with [Formula: see text] in an incompressible fluid.
... Since the pioneering research on the collapse of an empty cavity in a liquid by Lord Rayleigh [1] and the extension of bubble dynamics by Plesset [2], researchers have extended the knowledge of single bubble oscillation under driving pressure [3][4][5][6][7][8][9][10][11]. Another research direction based on bubble dynamics is the study of the propagation of pressure waves in bubbly liquids, which has been performed by various researchers such as van Wijngaarden [12][13][14][15] and Caflisch group [16][17][18]. ...
With promising applications in medical diagnosis and therapy, the behavior of shell-encapsulated ultrasound contrast agents (UCAs) has attracted considerable attention. Currently, second-generation contrast agents stabilized by a phospholipid membrane are widely used and studies have focused on the dynamics of single phospholipid shell-encapsulated microbubbles. To improve the safety and the efficiency of the methods using the propagation or targeted ultrasound, a better understanding of the propagation of ultrasound in liquids containing multiple encapsulated microbubbles is required. By incorporating the Marmottant-Gompertz model into the multiple scale analysis of two-phase model, this study derived a Korteweg-de Vries-Burgers equation as a weakly nonlinear wave equation for one-dimensional ultrasound in bubbly liquids. It was found that the wave propagation characteristics changed with the initial surface tension, highlighting two notable features of the phospholipid shell: buckling and rupture. These results may provide insights into the suitable state of microbubbles, and better control of ultrasound for medical applications, particularly those that require high precision.
... And considerable progress has been made in the research, such as the oscillation curve and frequency response. To be specific, Lauterborn [15] investigated the nonlinear oscillation response curve of vapor bubbles in the incompressible liquid and summarized the characteristics of main resonance, harmonic resonance, subharmonic resonance and ultra-harmonic resonance during the oscillation. Zhang and Li [16] studied the nonlinear oscillation of bubbles under the dual-frequency acoustic excitation and found that bubbles have two special resonance forms (i.e. of combination resonance and simultaneous resonance) under the dual-frequency excitation. ...
In the present paper, resonance characteristics of the vapor bubble oscillating in an acoustic field are investigated analytically. The analytical solution of the non-dimensional perturbation of the instantaneous bubble radius during the transient process in the initial oscillation stage is explicitly obtained and physically analyzed at the resonance situation based on the Laplace transform method. And the typical oscillation behaviors obtained from the analytical solution are thoroughly exhibited and analyzed in the time and frequency domains. In addition, the corresponding oscillation behaviors at the non-resonance situation are also investigated for the purpose of comparisons. Through our investigation, several essential conclusions can be drawn as follows: (1) The analytical solution of the non-dimensional perturbation of the instantaneous bubble radius can be divided into four terms according to the physical meaning. Among them, it is the term related to the acoustic field that causes the progressively violent bubble oscillation. (2) The vapor bubble with a smaller equilibrium radius could respond faster and more significantly to the acoustic field during the oscillation. (3) The bubble oscillation characteristics always exhibit significant differences at the resonance and non-resonance situations in both the time and frequency domains, even if the difference between the natural frequency of the oscillating vapor bubble and the angular frequency of the acoustic field is greatly small.
... Notably, the equilibrium radius of the bubble takes on a wide range of values from below a micrometre up to several millimetres [46,[51][52][53]96] depending on the generating mechanism [52]. The magnitudes of the dimensionless parameters were obtained using the following constant values: κ = 1.3, σ = 0.0725 N/m, µ th = 0.001 Pa.s, ρ = 998 kgm −3 , µ l = 0.001 P a.s, P 0 = 1 atm and R 0 = 10 mm. ...
We examine the impacts of time-delay and phase shift between two acoustic driving forces on vibrational resonance (VR) phenomena in the oscillations of a spherical gas bubble. Using the approximate method of direct separation of the motions, we obtain the equation of slow motion and the response amplitude, and we validate the theoretical predictions with numerical simulations. We find that the response amplitude of the system at the lower frequency varies periodically with respect to the phase shift. When the phase shift consists of an even number of periods, it can be optimized to enhance the system’s response in the relevant parameter space of the high-frequency driving force. In addition to the enhancement of the VR peak by variation of the phase shift, our results show that the time-delay also plays a significant role in the bubble’s response to dual-frequency acoustic driving fields. It and can be exploited either to suppress drastically, or to modulate, the resonance peaks, thereby controlling the resonances. Our analysis shows further that cooperation between the time-delay and the amplitude of the high-frequency component of the acoustic waves can induce multiple resonances. These results could potentially be exploited to control and enhance ultrasonic cleaning processes by varying the time-delay parameter in the presence of phase shifted dual-frequency acoustic waves. Moreover, it could be employed to achieve improved accuracy in ultrasonic biomedical diagnosis and tumour therapy, as well as for targeted delivery of reagents transported within bubbles.
... In comparison, the 6.75 V torus bubble shows oscillations with large amplitude at the first harmonic of 5.3 MHz, I a1 = 11.87 µA, along with a subharmonic detected at 2.6 MHz and 4.82 µA. The emergence of subharmonics and the shifting of the first-harmonic (main resonance) peak to a lower frequency are indications of nonlinear oscillations [47,48,[64][65][66]. This gradual loss of torus stability with voltage is more prominent for the 460-nm pore and is discussed in Sec. ...
The transition from nucleate to film boiling on micro/nanotextured surfaces is of crucial importance in a number of practical applications, where it needs to be avoided to enable safe and efficient heat transfer. Previous studies have focused on the transition process at the macroscale, where heat transfer and bubble generation are activated on an array of micro/nanostructures. In the present study, we narrow down our investigation scale to a single nanopore, where, through localized Joule heating within the pore volume, single-bubble nucleation and transition are examined at nanosecond resolution using resistive pulse sensing and acoustic sensing. Akin to macroscale boiling, where heterogeneous bubbles can nucleate and coalesce into a film, in the case of nanopores also, patches of heterogeneous bubbles nucleating on the cylindrical pore surface can form a torus-shaped vapor film blanketing the entire pore surface. In contrast to conventional pool boiling, nanopore boiling involves a reverse transition mechanism, where, with increased heat generation, film boiling reverts to nucleate boiling. With increasing bias voltage across the nanopore, the Joule heat production increases within the pore, leading to destabilization and collapse of the torus-shaped vapor film.
... Derived from mass and momentum conservation principles, the model uses the bubble radius as a generalized coordinate with the at-rest or unperturbed equilibrium bubble radius as a boundary condition. The resulting equation governing the bubble radial dynamics is expressed as [35][36][37]: ...
An acoustic technique designed to increase the gas-dissolution rate of a bubble in a liquid medium is here presented. The increased gas-dissolution rate is achieved by increasing the bubble's surface-to-volume ratio via bubble fragmentation. This was accomplished by attaching an electroacoustic transducer to the system or load in which bubbles travel and exciting the transducer at the frequency of resonance. We have demonstrated that the electric resonance of the transducer attached to the system corresponds in frequency to the mechanical resonance which allows for achieving such a state without the use of an internally placed hydrophone to certify the resonance state. The acoustic bubble fragmentation technique proved to be able to increase the dissolution rate 4 to 5 times of bubbles with initial diameters between 150 and 550 lm in distilled water and in medical grade saline solution. Possible mechanisms responsible for the results obtained are discussed as well as additional advantages of the technique. The technique has potential medical applications such as an on-site emergency treatment for scuba divers suspected to be at risk of developing decompression sickness and in any other clinical scenario where the risk of an air embolism might be present.
... 3-4 MHz for d = 2 µm in Fig. 3A) also exhibit the same trend as the primary resonance peaks; albeit at a lower amplitude. Indeed, the presence of these harmonic peaks is a well-known and established feature of resonant bubble systems [56,59,60]. ...
Ultrasound-driven microbubbles, typically between 1-8 μ m in diameter, are resonant scatterers that are employed as diagnostic contrast agents and emerging as potentiators of targeted therapies. Microbubbles are administered in populations whereby their radial dynamics – key to their effectiveness - are greatly affected by intrinsic (e.g. bubble size) and extrinsic (e.g. boundaries) factors. In this work, we aim to understand how two neighbouring microbubbles influence each other. We developed a finite element model of a system of two individual phospholipid-encapsulated microbubbles vibrating in proximity to each other to study the effect of inter-bubble distance on microbubble radial resonance response. For the case of two equal-sized and identical bubbles, each bubble exhibits a decrease between 7-10% in the frequency of maximum response (fMR) and an increase in amplitude of maximum response (AMR) by 9-11% as compared to its isolated response in free-space, depending on the bubble size examined. For a system of two unequal-sized microbubbles, the large bubble shows no significant change, however the smaller microbubble shows an increase in fMR by 7-11% and a significant decrease in AMR by 38-52%. Furthermore, in very close proximity the small bubble shows a secondary off-resonance peak at the corresponding fMR of its larger companion microbubble. Our work suggests that frequency-dependent microbubble response is greatly affected by the presence of another bubble, which has implications in both imaging and therapy applications. Furthermore, our work suggests a mechanism by which nanobubbles show significant off-resonance vibrations in the clinical frequency range, a behaviour that has been observed experimentally but heretofore unexplained.
... However, more recent studies have shown that at a low frequency of 250 kHz, which is an order of magnitude below the MB resonance frequency, enhanced MB oscillations are observed and can increase sonoporation efficacy (Ilovitsh et al., 2018;Ilovitsh et al., 2020). The physical mechanism accounting for this phenomenon is known as the Blake threshold effect, which states that when exciting a MB well below its resonance frequency, beyond a defined threshold pressure (Blake threshold), the MB will expand extensively (Harkin et al., 1999;Blake, 1949;Lauterborn, 1976). These high amplitude oscillations can transform the TMB into mechanical therapeutic warheads that poke large holes in cell membranes ( Fig. 1. ...
Ultrasound insonation of microbubbles can be used to form pores in cell membranes and facilitate the local trans-membrane transport of drugs and genes. An important factor in efficient delivery is the size of the delivered target compared to the generated membrane pores. Large molecule delivery remains a challenge, and can affect the resulting therapeutic outcomes. To facilitate large molecule delivery, large pores need to be formed. While ultrasound typically uses megahertz frequencies, it was recently shown that when microbubbles are excited at a frequency of 250 kHz (an order of magnitude below the resonance frequency of these agents), their oscillations are significantly enhanced as compared to the megahertz range. Here, to promote the delivery of large molecules, we suggest using this low frequency and inducing large pore formation through the high-amplitude oscillations of microbubbles. We assessed the impact of low frequency microbubble-mediated sonoporation on breast cancer cell uptake by optimizing the delivery of 4 fluorescent molecules ranging from 1.2 to 70 kDa in size. The optimal ultrasound peak negative pressure was found to be 500 kPa. Increasing the pressure did not enhance the fraction of fluorescent cells, and in fact reduced cell viability. For the smaller molecule sizes, 1.2 kDa and 4 kDa, the groups treated with an ultrasound pressure of 500 kPa and MB resulted in a fraction of 58% and 29% of fluorescent cells respectively, whereas delivery of 20 kDa and 70 kDa molecules yielded 10% and 5%, respectively. These findings suggest that low-frequency (e.g., 250 kHz) insonation of microbubbles results in high amplitude oscillation in vitro that increase the uptake of large molecules. Successful ultrasound-mediated molecule delivery requires the careful selection of insonation parameters to maximize the therapeutic effect by increasing cell uptake.
... This model was extended by Poritsky [11] to account for liquid viscosity. The resulting equation (1.2) is a generalized form of the Rayleigh equation, also known as the Rayleigh-Plesset-Noltingk-Neppiras-Poritsky equation [12], which describes the motion of an oscillating bubble in a viscous, incompressible fluid of infinite extent: ...
Encapsulated microbubbles (EMBs) were originally developed as contrast agents for ultrasound imaging but are more recently emerging as vehicles for intravenous drug and
gene delivery. Ultrasound can excite nonspherical oscillations or shape modes, that can
enhance the acoustic signature of an EMB and also incite rupture, which promotes drug
and gene delivery at targeted sites (e.g., tumors). Therefore, the ability to control shape
modes can improve the efficacy of both the diagnosis and treatment mediated by EMBs.
This work uses optimal control theory to determine the ultrasound input that maximizes
a desired nonspherical EMB response (e.g., to enhance scattering or rupture) while minimizing the total acoustic input in order to enhance patient safety and reduce unwanted
side effects. For enhancing rupture, the optimal control problem is applied to models of
both a free gas bubble and an EMB that account for small amplitude shape deformations.
These models are solved subject to a cost function that maximizes the incidence of rupture and minimizes the acoustic energy input. The optimal control problem is solved numerically through pseudospectral collocation methods using commercial optimization software, TOMLAB. Single frequency and broadband acoustic forcing schemes are explored and compared. The results show that encapsulation greatly increases the acoustic effort required to incite rupture. Furthermore, the acoustic effort required to incite rupture depends both on the form of the acoustic forcing (single frequency vs. broadband) and on the shape mode that is forced to become unstable. Optimal control has also been applied to maximize sound emission or bubble echo, which can improve image contrast.
Following prior work, we achieve this by maximizing the kinetic energy at bubble collapse while imposing a limit on the intensity of the incident ultrasound, as required for patient safety. The results show that, for a sufficiently large limit on acoustic driving,
the nonspherical shape modes limit the maximum kinetic energy that can be achieved at
collapse and also the radiated sound from the EMB. At low values of the driving limit,
shape modes have little to no effect on the radiated sound. However, as the driving limit
is increased, the degree to which shape modes reduce the radiated sound increases and
can be significant.
Furthermore, we develop a new interfacial rheological model that characterizes the stabi�lizing shell of an EMB by applying Transient Network Theory (TNT). TNT is a statistically�based continuum theory developed by Vernerey et al. (2017) that can model the behavior of general viscoelastic materials, including purely elastic solids or purely viscous fluids.
The result is a novel analytical model of an axisymmetric microbubble encapsulated by
a viscoelastic membrane undergoing small nonspherical oscillations subjected to ultrasound. Legendre polynomials and associated Legendre polynomials are used to describe the amplitudes of the shape perturbations in the radial and tangential directions, respectively. Using this model, the shape stability of a microbubble coated by a viscoelastic shell is investigated over a range of ultrasound frequencies and amplitude relevant for medical applications. Stability diagrams are generated and compared to the EMB model by Liu et al. (2012) that was used for the above studies on optimal control. The new TNT model exhibits hysteresis effects that are characteristic of viscoelastic materials. This model is more general than previous EMB models and can be used to characterize a broader class of shell materials.
... Since the pioneering research on the collapse of an empty cavity in a liquid by Lord Rayleigh [1] and the extension of bubble dynamics by Plesset [2], many scientists have extended the knowledge of bubble behavior according to driving pressure [3][4][5][6][7][8][9][10][11]. Another research direction based on bubble dynamics is the study of the propagation of pressure waves in bubbly liquids, which has been performed various researchers such as van Wijngaarden [12][13][14][15] and Caflisch [16][17][18]. ...
With promising applications in medical diagnosis and therapy, the behavior of shell-encapsulated ultrasound contrast agents (UCAs) has attracted considerable attention. Currently, second-generation contrast agents stabilized by a phospholipid membrane are widely used and studies have focused on the dynamics of single phospholipid shell-encapsulated microbubbles. To improve the safety and the efficiency of the methods using the propagation or targeted ultrasound, a better understanding of the propagation of ultrasound in liquids containing multiple encapsulated microbubbles is required. By incorporating the Marmottant-Gompertz model into the multiple scale analysis of two-phase model, this study derived a Korteweg-de Vries-Burgers equation as a weakly nonlinear wave equation for one-dimensional ultrasound in bubbly liquids. It was found that the wave propagation characteristics changed with the initial surface tension, highlighting two notable features of the phospholipid shell: buckling and rupture. These results may provide insights into the suitable state of microbubbles, and better control of ultrasound for medical applications, particularly those that require high precision.
... The pressure at which this growth occurs is known as the Blake threshold. [57][58][59] At these low frequencies, and for the large oscillations, lipid-shelled MBs can be approximately modeled as clean gas MBs, and thus the MB shell properties do not play a significant role in the effect. Building on this discovery, in this paper we show that the Blake threshold effect can also trigger violent nanobubble oscillations, enabling their use as low energy cavitation nuclei for histotripsy. ...
Scaling down the size of microbubble contrast agents to the nanometer level holds the promise for noninvasive cancer therapy. However, the small size of nanobubbles limits the obtained bioeffects as a result of ultrasound cavitation, when operating near the nanobubble resonance frequency. Here we show that coupled with low energy insonation at a frequency of 80 kHz, well below the resonance frequency of these agents, nanobubbles serve as noninvasive therapeutic warheads that trigger potent mechanical effects in tumors following a systemic injection. We demonstrate these capabilities in tissue mimicking phantoms, where a comparison of the acoustic response of micro- and nano-bubbles after insonation at a frequency of 250 or 80 kHz revealed that higher pressures were needed to implode the nanobubbles compared to microbubbles. Complete nanobubble destruction was achieved at a mechanical index of 2.6 for the 250 kHz insonation vs. 1.2 for the 80 kHz frequency. Thus, the 80 kHz insonation complies with safety regulations that recommend operation below a mechanical index of 1.9. In vitro in breast cancer tumor cells, the cell viability was reduced to 17.3 ± 1.7% of live cells. In vivo, in a breast cancer tumor mouse model, nanobubble tumor distribution and accumulation were evaluated by high frequency ultrasound imaging. Finally, nanobubble-mediated low frequency insonation of breast cancer tumors resulted in effective mechanical tumor ablation and tumor tissue fractionation. This approach provides a unique theranostic platform for safe, noninvasive and low energy tumor mechanotherapy.
The present study concerns the numerical modeling of microbubble oscillation within an elastic microvessel, aiming to enhance the safety and efficacy of ultrasound-mediated drug delivery and diagnostic imaging. The success of such applications depends on a thorough understanding of microbubble–vessel interactions. Despite some progress, the critical impact of the stabilizing shell around gas core has remained underexplored. To address this, we developed a novel numerical approach that models the stabilizing shell. Additionally, there is novelty in modeling consequent vascular deformation in response to complicated spatiotemporal microbubble oscillations. The novel approach was implemented for shear stress evaluation as a critical factor in vascular permeability. Finally, our unique approach offered novel insights into microbubble–vessel interactions under diverse acoustic conditions. Results indicated substantial impact of shell properties and acoustic parameters on induced shear stress. With a fourfold increase in acoustic pressure amplitude, 15.6-fold and sixfold increases were observed in maximum shear stress at 1 and 3 MHz , respectively. Also, the peak shear stress could reach up to 15.6 kPa for a shell elasticity of 0.2 N/m at 2.5 MHz. Furthermore, decreasing microvessel/bubble size ratio from 3 to 1.5 increased maximum shear stress from 5.1 to 24.3 kPa. These findings are crucial for optimizing ultrasound parameters in clinical applications, potentially improving treatment outcomes while minimizing risk of vessel damage. However, while our model demonstrated high fidelity in reproducing experimental observations, it is limited by assumptions of vessel geometry and homogeneity of vessel properties. Future work can improve our findings through in vitro experimental measurements.
Koopman operator theory has gained interest as a framework for 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 through data-driven methodologies, which enables the application of classical linear control strategies to nonlinear systems. Here, a Koopman linear quadratic regulator (KLQR) was used to acoustically control the nonlinear dynamics of a single spherical bubble, as described by the well-known Rayleigh–Plesset equation, with several objectives: (1) simple harmonic oscillation at amplitudes large enough to incite nonlinearities, (2) stabilization of the bubble at a nonequilibrium radius, and (3) periodic and quasiperiodic oscillation with multiple frequency components of arbitrary amplitude. The results demonstrate that the KLQR controller can effectively drive a spherical bubble to radially oscillate according to prescribed trajectories using both broadband and single-frequency acoustic driving. This approach has several advantages over previous efforts to acoustically control bubbles, including the ability to track arbitrary trajectories, robustness, and the use of linear control methods, which do not depend on initial guesses.
A multiple‐frequency ultrasound (MFU) technique is proficient in enhancing the effect of acoustic cavitation compared to a single‐frequency ultrasound. This comprehensive review delves into the complex field of MFU and its profound impact on microbial inactivation in food processing. The exploration begins with an intricate examination of the mechanism of power ultrasound, elucidating the intricate interplay of acoustic cavitation and its diverse effects. Subsequently, the mechanism of MFU was provided, which is basically the enhanced cavitation obtained during its application. Delving into the core mechanisms of MFU, the review navigates through microbial inactivation, unraveling the ways in which MFU disrupts and eliminates microorganisms. The exploration extends to the synergistic potential of combined applications, where MFU is applied with other treatment techniques to enhance microbial inactivation. Beyond its microbial inactivation prowess, the review meticulously explores the far‐reaching effects of MFU on the nutritional and quality attributes of food products. Furthermore, the diverse applications of MFU were also reviewed. In addition, limitations and adverse effects, emphasizing the importance of optimizing parameters to balance microbial safety and food quality, were also discussed. As the review unfolds, it lays the groundwork for future research, identifying avenues for further exploration and innovation in this dynamic field. In essence, this review not only consolidates the current understanding of MFU but also guides future research endeavors in the quest for more efficient, sustainable, and quality‐preserving food processing technologies.
Ultrasound insonation of microbubbles can form transient pores in cell membranes that enable the delivery of non-permeable extracellular molecules to the cells. Reducing the size of microbubble contrast agents to...
The effects of sonication power on the ultrasonic cavitation and sonochemistry as well as the degradation of paracetamol were studied and compared for single- and dual-frequency sonoreactors. For the single-frequency sonication, a 500 kHz plate transducer was employed, with three different calorimetric powers of 8.4, 16.7 and 27.9±3.9 W. For the dual-frequency sonication, the plate transducer was perpendicularly coupled with a low-frequency 20 kHz ultrasonic horn, and three calorimetric powers of 27.9, 33.4, 44.6±3.9 W were studied. At all the studied powers, dual-frequency sonication led to a synergistic effect in the degradation of paracetamol, though varying the power of the horn did not affect the degradation rate. A comparison of the degradation data versus the yield of oxidants as well as the overall intensities of sonoluminescence and sonochemiluminescence suggested the degradation is by the action of oxidants near the surface of the bubbles as the major reaction mechanism. Despite the enhancement observed for the degradation, dual-frequency sonication had no significant effect on the yield of either of the oxidants, regardless of the applied power to the horn. In contrast, dual-frequency sonication decreased the overall sonoluminescence and sonochemiluminescence intensities at all powers studied, suggesting that the application of dual-frequency sonication reduces the size of cavitation bubbles. Normal distribution function analysis confirmed dual-frequency sonication resulted in smaller sonoluminescing bubbles, hence the reduction in the sonoluminescence intensity. The increase in degradation rate under DFUS is attributed to the increase in the transfer of paracetamol from the bulk towards the bubbles. As a result, the availability of the pollutant molecules in the vicinity of the bubbles to react with HO• would increase and consequently, the degradation rate would enhance under DFUS.
In this paper, the interaction of multiple bubbles in the cavitation field is investigated in combination with the phenomenon of small bubbles hovering around large bubbles observed experimentally. A model composed of three bubbles is developed and the dynamic behavior of cavitation bubble is analyzed. By considering the time delay effect of the interaction between bubbles and the nonspherical oscillation of large bubbles, the modified bubble dynamic equations are obtained. Numerical results show that the nonspherical effect of large bubbles has little effect on the oscillation of cavitation bubble. The suppressive effect of large bubble on cavitation bubble is closely related to the radius of the large bubble. The larger the size of the large bubble, the stronger the inhibition. When the size of large bubble is close to the resonant radius, the oscillation of cavitation bubble appears coupled resonance response, and the maximum expansion radius of bubble appears resonance peak. The distribution of the secondary Bjerknes force with bubble radius and the separation distances is strongly influenced by driving frequencies or sound pressures. When the large bubble is in the submillimeter order, the strength of the secondary Bjerknes force and the acoustic response mode are different due to the different strength of the nonlinear response of the cavitation bubble. As the distance decreases, when the acoustic pressure increases to a certain value, the secondary Bjerknes force on the cavitation bubble decreases due to abnormal acoustic absorption. The secondary Bjerknes force on cavitation bubble is likely to be repulsive at different separation distances. The theoretical results accord well with experimental phenomenon.
This paper presents a full numerical model accounting for the heat transfer and phase-change by combining the modified Keller–Miksis equation with the second order term of compressibility of liquid, partial differential equations (PDEs), and Hertz–Knudsen–Langmuir equation. Then, a simplified model for studying the dynamics of the cavitation bubble or bubble excited by the acoustic waves is proposed. The major contribution is to simplify the full model with PDEs to a set of coupled ordinary differential equations (ODEs). Specifically, two energy PDEs are converted to three ODEs by coupling the boundary conditions. The comparison among the full model and other simplified models is used to validate the accuracy and superiority of the simplified model, from which the application range of the proposed simplified model can be determined.
One of the main interests in developing microfluidic platforms is the lab-on-a-chip (LOC) applications, including the use of microdevices for the culture of prokaryotic and eukaryotic cells for different applications. In this context, the trapping and control of Micro/nano-scale bubbles have been considered as it becomes a field of growing concern addressing many challenges in LOC applications. In the present work, we present a novel microfluidic bioreactor based on the micropillar design that reduces stresses caused by bubbles on the cell growth in the microchip. The combination of one chamber with 3-pillars and 10-pillars design with the channel heights of 270 µm and 2 mm were evaluated. Computational fluid dynamics simulations (CFD) were conducted to investigate the flow dynamics in the new designs. Numerical values of velocity were compared with those obtained by experimental flow tests to evaluate the performance of the proposed microdevice. Finally, the functionality of the new milli-bioreactor design was validated by culturing human induced pluripotent stem cells (hiPSC) for 5 days. Cell growth, viability and sustain pluripotency were successfully proved using a new platform architecture. This multidisciplinary research provides new insights into the chemical engineering and operability of micro and milli-bioreactors with prospects for future industrial applications.
In addition to the issues of water quality and economic losses, leakages in pipe networks are a potential danger to public safety. Early detection of water leaks in pipe networks is crucial for implementing countermeasures to reduce structural damage and water losses. The method of data collection based on sensors installed in pipeline networks has gained a lot of attention due to its potential for application in real-time monitoring systems for leak detection. However, many models developed on data collected from simulations, engineered tests or field experiments, and few have been validated using real network data. To make further investigations in real water distribution systems (WDSs), machine learning (ML)-based leak detection models were developed and validated using signal data collected by piezoelectric accelerometers installed in WDSs over several cities of China. Nonlinear features describing the non-stationary characteristics contained in leak signals are extracted from the signals to build feature sets as inputs to the common classifiers, i.e., support vector machine (SVM), decision tree (DT), and K-nearest neighbor (KNN). In addition, the spectrograms obtained by the short-time Fourier transform processing are used for the convolutional neural network SqueezeNet. The effects of different features, different dataset compositions, and different models for leak detection are compared and analyzed. SqueezeNet has the best performance with 95.15% accuracy in identifying leaks, and KNN is the best of the three classifiers with superior sensitivity and 88.17% accuracy in identification. This paper demonstrates the effectiveness and practicability of using machine learning models to identify leaks in real pipe networks. The application of machine learning models can perform effective leak detection with minimal human intervention, thus providing a meaningful contribution to the study of leak detection in real WDSs. The proposed model aims to identify leakages under operational conditions in real-life pipe networks with higher efficiency and reduced manual involvement. This paper demonstrates the applicability of accelerometers based on microelectromechanical systems for leak detection and establishes real network-based machine learning models; thereby, providing meaningful contribution to the literature regarding leak detection in a real WDS.
This paper presents measurements of the acoustic emission from gas bubbles of controlled sizes “seeded” into water and glycerol‐water mixtures and subjected to sound fields over a wide range of frequencies and intensities up to, and beyond, the transient cavitation threshold. The emission from unprepared liquids was also recorded for comparison. A mechanical device has been developed for generating small bubbles of uniform sizes at controlled rates. The results suggest that there are at least two mechanisms for generating signals below the excitation frequency (f 0): (1) Subharmonics of f 0 may be produced by forced oscillations of bubbles whose radial resonance frequencies are submultiples of f 0; (2) bubbles of any size may be shock excited to emit signals at their radial resonance frequencies. Large bubbles of certain preferred sizes are normally present, and their resonances probably account for many low‐frequency signals that are not integral submultiples of f 0. It also seems certain that some other mechanism unconnected with resonant bubbles contributes to the relatively strong signals observed at the first subharmonic (f 0/2).
Observations of acoustic cavitation have been in helium I and II using an extensional‐mode vibrator capable of very high intensities at 10 kHz. Remote monitoring of the drive oscillatory velocity was achieved. In helium I clouds of small vapor bubbles were visible, as well as larger bubbles near the expected radial resonance size. For comparison, observations were also made using two other normal liquids—nitrogen and water—at temperatures near their boiling points. In helium II observation of acoustic streaming with emission of white noise suggested the presence of even smaller vapor bubbles. A strong subharmonic signal with low threshold excitation has not been fully explained.
The onset of acoustic cavitation has been studied using the first peak of subharmonic generation as an indicator of cavitation onset in the liquid. The findings are discussed in terms of the most satisfactory explanation of this subharmonic generation. The results of altering certain parameters of the liquid such as surface tension, viscosity, gas content and nucleation on subharmonic peak formation are given and commented on.
The nonlinear oscillations of a spherical gas bubble in an incompressible, viscous liquid subject to the action of a sound field are investigated by means of an asymptotic method. Approximate analytical solutions for the steady-state oscillations are presented for the fundamental mode, for the first and second subharmonics, and for the first and second harmonics to second order in the expansion. These results are compared with some numerical ones and a very good agreement is found.
Under proper conditions, bubbles driven by a sound field will pulsate periodically with a frequency equal to one‐half the frequency of the sound field. This frequency component is the subharmonic of order 1 2 and is generated when the acoustic pressure amplitude exceeds a threshold value. The threshold for subharmonic generation is calculated by means of a theory that relates the presence of the subharmonic to properties of Mathieu's equation. It is found that, for a given bubble, the threshold is a function of the driving frequency and is a minimum when the driving frequency is close to twice the resonance frequency of the bubble. In addition, solutions of a nonlinear equation of motion for the bubble wall, obtained on a computer, illustrate the growth of subharmonics and are used to determine the steady‐state amplitude of the subharmonic for a sequence of values of various parameters. [Work supported by Acoustics Programs, Office of Naval Research.]
Ultrasoniccavitation in liquids gives rise to a sound oscillating field at a frequency one‐half of the ultrasound‐driving frequency. The appearance of such a subharmonic may be used to detect the cavitation. The authors investigate experimentally such a possibility.
Measurements have been made of the low-frequency acoustic emission from bubbles in liquids irradiated by sound. A mechanical vibrator has been developed to generate the bubbles at controlled sizes and densities. The measurements suggest that the well-known signals at subharmonics of the fundamental are caused by forced radial oscillation of bubbles that are resonant at the subharmonic frequencies. As well as the subharmonics, other low-frequency resonances have been detected; these are due to shock-excitation of bubbles of certain preferred sizes that are normally present in the liquid. The damping of radial oscillations of large bubbles in viscous liquids has also been measured.