Modified Rayleigh-Plesset Equation Research Articles

Radius evolution for the synchronous collapse of a circular vapor bubble cluster

The radius evolution for the synchronous collapse motion of a circular vapor bubble cluster in an ideal and infinite fluid is studied. With the assumption that all bubbles are evenly distributed around the circumference and have the same initial conditions, we establish the associated governing equation for this case from a modified Rayleigh–Plesset equation. The inverted analytical solution in the form of t=t(r) and analytical approximations in the form of r=r(t) for the radius evolution to the initial value problem of the dimensionless governing equation are derived. These novel analytical approximations extend the previous ones for a single bubble by adding the effects of surface tension and multiple bubbles. In addition, we point out that the analytical results and dynamic behaviors for multiple bubbles will degenerate into the corresponding ones for single bubbles when the radius of the bubble cluster approaches to infinity.

Analytical solutions and dynamic behaviors to synchronous oscillation of same bubbles at vertices of cuboid and rectangle

The present work focuses on the nonlinear dynamics of the synchronous oscillating multiple bubbles in two typical spatial locations, namely, cuboid and rectangle arrangements. The governing equation for such synchronous oscillating multiple bubbles is derived from a modified Rayleigh–Plesset equation. Theoretical results including the collapse time and analytical solution (in three forms) for multiple vapor bubbles, as well as the maximum/minimum radii, oscillation period, and analytical solution in the form of Weierstrass elliptic function for multiple gas-filled ones, are provided. On the basis of these results, we not only study the dynamic characteristics of multi-bubbles straightforwardly but also carefully observe a series of evolution behaviors of bubbles when the number of bubbles decreases gradually on the order of 8→4→2→1. It should be pointed out that we also compare the multi-bubble behaviors between the general cuboid/rectangle arrangements and the corresponding cube/square arrangements under two reasonable restrictions, respectively. Furthermore, the limiting behaviors of the synchronous oscillating multiple gas-filled bubbles are discussed as the initial pressure of the gas in bubble approaches to zero.

Numerical simulation method of nonlinear contrast-enhanced ultrasound imaging

<sec>Contrast-enhanced ultrasound imaging (CEUS) based on the acoustic nonlinearity of ultrasonic microbubble has received great attention in recent years. Compared with conventional linear ultrasound imaging, nonlinear CEUS can further improve the imaging resolution while overcoming the challenge of clutter filtering. Simulation, acting as an effective tool for research on new mechanisms and technologies of ultrasound imaging, has been a long-term focus of computational acoustics. In the community of biomedical ultrasound, common sound field simulation tools are mainly based on finite element method (FEM), analytical method, <i>k</i>-space pseudospectral method and finite-difference time-domain method (FDTD), which are relatively mature solutions for simulating the nonlinear characteristics of tissue. However, it is still not trivial to simulate nonlinear CEUS by using the prevailing methods, as the nonlinearity of microbubble is often not considered.</sec><sec>In this paper, we propose a simulation method of nonlinear CEUS imaging that successfully combines the microbubble nonlinearity and classic <i>k</i>-space pseudospectral method. Specifically, forced oscillation response of the microbubble is computed based on the modified Rayleigh-Plesset equation and such a nonlinear response is further dealt as an additional source for analyzing the nonlinear component propagation and CEUS imaging. To investigate the performance of the proposed method, B-mode images of single microbubble and clustered microbubbles are simulated based on plane wave imaging. The plane wave based CEUS imaging can thus be carried out with different compounding angles and different contrast pulse sequencing (CPS) strategies (pulse inversion, amplitude modulation, pulse inversion & amplitude modulation, and probe element alternation). Different soft-tissue and mechanical parameters of the microbubble can be adjusted by using the proposed nonlinear simulation strategy, thus providing efficient solution for CEUS simulation. Such a method can evaluate the performances of different CPS strategies, and further contribute to the CEUS development.</sec>

Pressure oscillations due to a sudden, finite-volume, underwater air release

The underwater release of air from the sudden rupture of a finite canister was studied by time-resolved pressure and interfacial geometry measurements. Two distinct dynamical regimes for the air were identified: an inertial regime, in which the air oscillated with increasing frequency, and a capillary pinch-off regime, in which emerging air bubbles pinched off from the air remaining in the canister. The temporal scales for these regimes were identified, and the inertial regime was shown to follow Rayleigh–Plesset dynamics, and was characterized by an empirically defined length scale associated with the air volume. The behavior of the two regimes in frequency-space was modeled by a modified Rayleigh–Plesset equation accounting for volume losses, turbulent damping, and a time-dependent capillary forcing.

Analytical study on the dynamic characteristics of multiple gas-filled spherical bubbles in typical spatial locations

The dynamic characteristics of multiple gas-filled spherical bubbles in three types of typical spatial locations are investigated analytically through a modified Rayleigh–Plesset equation. In the first type, two bubble centers form a one-dimensional straight line; the second type consists of any number of bubbles whose centers form a regular polygon in a two-dimensional plane; and in the third type, the bubble centers form a regular polyhedron in three-dimensional space. We show that physically these cases correspond qualitatively to periodic oscillations. Analytical expressions are derived for the maximum and minimum radii, based on which the oscillation amplitude and period are studied analytically. Parametric analytical solutions are also obtained. The influences of physical parameters on the multibubble motion are determined with the aid of these analytical results. We also study the limiting behavior of the analytical results for multiple bubbles, with the corresponding results for single bubbles being obtained as the distance between bubble centers approaches infinity.

Numerical simulation of a confined cavitating gas bubble driven by ultrasound

This work investigates the flow disturbance generated by an ultrasonically driven gas bubble confined in a narrow gap over one acoustic cycle. Here, we provide a more accurate representation of ultrasonic cleaning by implementing a volume-of-fluid model in OpenFOAM that simulates the ultrasound as a sinusoidally time-varying pressure boundary condition. A modified Rayleigh–Plesset equation is solved to select an acoustic forcing that instigates bubble collapse. Simulations reveal the interaction between the inflow from the acoustic forcing and the flow deflected by the confining walls intensifies the strength of the self-piercing micro-jet(s), and consequently of the unsteady boundary layer flow, compared to the traditional collapse near a single rigid wall. Depending on the gap height and the position of bubble inception inside the gap, three distinct collapse regimes involving dual-jets or directed-jets are identified, each resulting in a different shear-stress footprint on the confining boundaries. Plots of the spatiotemporal evolution of the shear flow (that is difficult to measure experimentally) reveal peak shear-stress magnitudes at collapse that are double those reported for an undriven laser-induced bubble in similar geometric confinement. This twofold increase is attributed to the ultrasonic signal driving the collapse. Surprisingly, in our simulations we have not encountered a transferred-jet regime previously observed for an unforced bubble collapsing in a similar configuration. This unexpected finding highlights the different physics involved in modeling acoustically driven bubbles compared with the conventional laser-induced bubbles used in experiments.

Three-dimensional oscillation of an acoustic microbubble between two rigid curved plates

Understanding the near boundary acoustic oscillation of microbubbles is critical for the effective design of ultrasonic biomedical devices and surface cleaning technologies. Accordingly, this study investigates the three-dimensional microbubble oscillation between two curved rigid plates experiencing a planar acoustic field using boundary integral method (BIM). The numerical model is validated via comparison with the nonlinear oscillation of the bubble governed by the modified Rayleigh-Plesset equation and with the axisymmetric model for an acoustic microbubble in infinite fluid domain. Then, the influence of the wave direction and horizontal standoff distance (h) on the bubble dynamics (including jet velocity, jet direction, centroid movement, total energy, and Kelvin impulse) were evaluated. It was concluded that the jet velocity, the maximum radius and the total energy of the bubble are not significantly influenced by the wave direction, while the jet direction and the high-pressure region depend strongly on it. More importantly, it was found that the jet velocity and the high-pressure region around the jet in acoustic bubble are drastically larger than their counterparts in the gas bubble.

Modeling of Cavitation Bubble Cloud with Discrete Lagrangian Tracking

In this paper, a Lagrangian-Eulerian (LE) two-way coupling model is developed to numerically study the cavitation bubble cloud. In this model, the gas-liquid mixture is treated directly as a continuous and compressible fluid and the governing equations are solved by methods in Eulerian descriptions. An isobaric closure exhibiting better consistency properties is applied to evaluate the pressure of gas-liquid mixture. The dispersed gas/vapor bubbles are tracked in a Lagrangian fashion, and their compression and expansion are described by a modified Rayleigh-Plesset equation, which considers the close-by flow properties other than these of the infinity for each bubble. The performance of the present method is validated by a number of benchmark tests. Then, this model is applied to study how the bubble cloud affects the shape and propagation of a pressure wave when the pressure pulse travels through. In the end, a three-dimensional simulation of a vapor cloud’s Rayleigh collapse is carried out, and the induced extreme pressure is discussed in detail. The total bubble number’s influence on the extreme collapse pressure and the size distribution of bubbles during the collapse are also analyzed.

Data assimilation for modeling cavitation bubble dynamics

The original or modified Rayleigh Plesset equation (RPE) is often used to analyze cavitation bubble dynamics. The prediction accuracy of these equations is governed by the initial values of the physical parameters. However, even for higher fidelity models, deviations from experimental measurements are observed due to the models' underlying assumptions. Here, we present a novel state-observer data assimilation technique designed to fuse time-resolved cavitation bubble diameter measurements with a governing model to yield enhanced spatiotemporal prediction of the cavitation bubble dynamics. This technique places an observer variable in the original or modified RPE and uses a proportional–integral–derivative (PID) control law on the difference between the predicted and measured cavitation bubble diameter. The data-assimilated modeling most accurately estimates the bubble diameter and far-field pressure as the deviation of bubble diameter and far-field pressure predictions from measurements decrease by up to 90% and 60%, respectively. Although the assimilated model is not a substitude for high fidelity models, this technique overcomes the inherent model assumptions, and make the model’s outputs more robust with respect to the physical parameters' initial values.

A new cavitation model considering inter-bubble action

The process of cavitation involves generation, growth, coalescence, and collapse of small bubbles and is tremendously influenced by bubble–bubble interactions. To understand these interactions, a new cavitation model based on the transport equation is proposed herein. The modified Rayleigh–Plesset equation is analyzed to determine the bubble growth rate by assuming equal-sized spherical bubble clouds. The source term in the transport equation is then derived according to the bubble growth rate with the bubble-bubble interaction. The proposed model is validated by various test simulations, including microscopic bubble cloud evolution as well as macroscopical two- and three-dimensional cavitating flows. Compared with previous models, namely the Kunz and Zwart cavitation models, the newly proposed model does not require adjustable parameters and generally results in better predictions both microscopic and macroscopical cases. This model is more physical.

Analytical investigation of the nonlinear dynamics of empty spherical multi-bubbles in hydrodynamic cavitation

The nonlinear dynamics of empty multi-bubbles with the same distance and initial conditions are studied analytically through a modified Rayleigh–Plesset equation. The collapse time and analytical solution are derived under various initial conditions. In particular, when considering a positive initial vibration velocity, the exact analytical expression for the maximal radius is obtained by solving a cubic algebraic equation. To the best of our knowledge, this is the first time that a parabolic function has been used to construct the parametric analytical solution for this case. This type of function is able to simulate the collapse motion whereby the bubble radius first grows to the maximal radius and then decays to zero. The limiting behavior of the resulting analytical results for multi-bubbles (including the collapse time, analytical solution, and maximal radius) is also investigated, enabling the corresponding analytical results for single bubbles to be deduced in the limit as the distance between the multi-bubbles approaches infinity. In addition, the dynamical characteristics and qualitative analysis of these bubbles and the effects of the relevant physical parameters are studied.

Acoustic response for nonlinear, coupled multiscale model containing subwavelength designed microstructure instabilities.

Nonperiodic arrangements of inclusions with incremental linear negative stiffness embedded within a host material offer the ability to achieve unique and useful material properties on the macroscale. In an effort to study such types of inclusions, the present paper develops a time-domain model to capture the nonlinear dynamic response of a heterogeneous medium containing a dilute concentration of subwavelength nonlinear inclusions embedded in a lossy, nearly incompressible medium. Each length scale is modeled via a modified Rayleigh-Plesset equation, which differs from the standard form used in bubble dynamics by accounting for inertial and viscoelastic effects of the oscillating spherical element and includes constitutive equations formulated with incremental deformations. The two length scales are coupled through the constitutive relations and viscoelastic loss for the effective medium, both dependent on the inclusion and matrix properties. The model is then applied to an example nonlinear inclusion with incremental negative linear stiffness stemming from microscale elastic instabilities embedded in a lossy, nearly incompressible host medium. The macroscopic damping performance is shown to be tunable via an externally applied hydrostatic pressure with the example system displaying over two orders of magnitude change in energy dissipation due to changes in prestrain. The numerical results for radial oscillations versus time, frequency spectra, and energy dissipation obtained from the coupled dynamic model captures the expected response for quasistatic and dynamic regimes for an example buckling inclusion for both constrained and unconstrained negative stiffness inclusions.

Using the Doppler effect to characterize microbubble shell viscoelasticity

The experimental testing and quality control of microbubble formulations for ultrasound applications requires high-throughput measurement of physicochemical properties, such as shell elasticity and viscosity. Single-bubble optical experiments are slow and expensive. Ultrasound attenuation experiments on microbubble populations can be done quickly, but the measurement is averaged over the ensemble, and it is often difficult to separate effects of microbubble size from shell viscoelasticity. Here, we discuss a relatively simple alternative method of using the Doppler effect to measure ultrasound radiation force-induced displacements of many individual microbubbles within a population. Our method involves insonifying a diluted suspension of freely floating microbubbles with a linear-array ultrasound probe driven by an open scanner. The microbubble displacements along the axis of the insonified region were acquired from the frequency shifts in the measured echo signals using the multi-gate spectral Doppler approach. These measurements were compared with theoretical peak microbubble displacements computed by combining a modified Rayleigh-Plesset equation and a balance of translational forces. The shell elasticity and viscosity producing the measured peak microbubble displacement was determined by assuming a resonant bubble size at each driving frequency. Our initial results are encouraging, and demonstrate a step toward high-throughput microbubble characterization using the Doppler effect.

Analysis of sheet cavitation with bubble/bubble interaction models

A new cavitation model based on bubble/bubble interactions has been recently proposed by the authors Adama Maiga, Coutier-Delgosha, and Buisine [“A new Cavitation model based on bubble-bubble interactions,” Phys. Fluids 30, 123301 (2018)]. It includes all nonlinear interaction terms between two bubbles, and the only input is the local velocity divergence, while the classical Rayleigh-Plesset or Gilmore models only take into account the linear interaction terms and require the local pressure evolution. In this previous publication, the model has been validated against cases of single bubbles growths and collapses. In the present paper, the same model is applied to configurations of high-speed cavitating flow on a Venturi profile and a two-dimensional foil section, where experiments based on X-ray imaging have been recently conducted. Based on the flow velocity fields available in the two cases, the local velocity divergence is calculated, and a one-dimensional mean divergence evolution from the leading edge to the wake of the cavity is extracted and used as an input for the model validation. The evolutions of the biggest bubble and the volume of vapor are both found in good agreement with the experiments. A typical velocity divergence evolution is then defined, and our model is compared to the modified Rayleigh-Plesset equation that includes the interactions between multiple bubbles. It confirms that the radius evolution of the biggest bubble is systematically better estimated by the new model due to its capability to continuously take into account the interactions of big bubbles with smaller bubbles. Eventually, the model is used to investigate the scale effect, by applying 6 different time scales of cavitation development covering more than 5 orders of magnitude. It is shown that the bubble size does not vary proportionally to the scale, but according to a law that is empirically determined.

MULTIPLE-SCALE AND NUMERICAL ANALYSES FOR THE NONLINEAR OSCILLATIONS OF A GAS BUBBLE SURROUNDED BY A MAXWELL'S FLUID

In this work, we have revisited theoretically the linear and nonlinear oscillations of a single bubble immersed in a Maxwell's fluid under the action of an acoustic pressure field. We adopt the above rheological model to identify in a simple manner the viscoelastic influence and the impact of the external acoustic field on the motion of the bubble. The governing equations are reduced to a modified Rayleigh–Plesset equation, which is solved together with an ordinary differential equation needed to predict the rheological influence of the normal stresses on the interface of the bubble. The resulting dimensionless governing equations were solved numerically; however, for small deviations from the equilibrium radius of the bubble, we add a frequency analysis by using the multiple-scale method to find the influence of the viscoelastic parameters for those conditions near to resonance, together with the estimation of the bending curve, which characterizes the well-known bending phenomenon. For a single value of the dimensionless Weber number, we have identified that for low Reynolds and Deborah numbers, the oscillations are periodic, with some harmonic ones well defined. However, as we increase the above dimensionless parameters, strong nonlinearities appear, and they are more notable when the effect of the viscous damping is reduced. Furthermore, the effect of the nonlinear terms of the governing equations depends strongly on the amplitudes of the oscillations: when the multiple scale analysis is used and we consider small deviations of the dimensionless equilibrium radius, we obtain that the resonance conditions for the amplitude of the bubble are reduced if the Deborah number is increased. On the other hand, for moderate values of the deviations of the equilibrium radius and retaining the validity of the multiple scale analysis, the foregoing behavior is also conserved. In this last case, stronger amplitudes of the radius of the bubble are obtained also for increasing values of the Deborah number.

Enhanced microbubble contrast agent oscillation following 250\u2009kHz insonation

Microbubble contrast agents are widely used in ultrasound imaging and therapy, typically with transmission center frequencies in the MHz range. Currently, an ultrasound center frequency near 250 kHz is proposed for clinical trials in which ultrasound combined with microbubble contrast agents is applied to open the blood brain barrier, since at this low frequency focusing through the human skull to a predetermined location can be performed with reduced distortion and attenuation compared to higher frequencies. However, the microbubble vibrational response has not yet been carefully evaluated at this low frequency (an order of magnitude below the resonance frequency of these contrast agents). In the past, it was assumed that encapsulated microbubble expansion is maximized near the resonance frequency and monotonically decreases with decreasing frequency. Our results indicated that microbubble expansion was enhanced for 250 kHz transmission as compared with the 1 MHz center frequency. Following 250 kHz insonation, microbubble expansion increased nonlinearly with increasing ultrasonic pressure, and was accurately predicted by either the modified Rayleigh–Plesset equation for a clean bubble or the Marmottant model of a lipid-shelled microbubble. The expansion ratio reached 30-fold with 250 kHz at a peak negative pressure of 400 kPa, as compared to a measured expansion ratio of 1.6 fold for 1 MHz transmission at a similar peak negative pressure. Further, the range of peak negative pressure yielding stable cavitation in vitro was narrow (~100 kPa) for the 250 kHz transmission frequency. Blood brain barrier opening using in vivo transcranial ultrasound in mice followed the same trend as the in vitro experiments, and the pressure range for safe and effective treatment was 75–150 kPa. For pressures above 150 kPa, inertial cavitation and hemorrhage occurred. Therefore, we conclude that (1) at this low frequency, and for the large oscillations, lipid-shelled microbubbles can be approximately modeled as clean gas microbubbles and (2) the development of safe and successful protocols for therapeutic delivery to the brain utilizing 250 kHz or a similar center frequency requires consideration of the narrow pressure window between stable and inertial cavitation.

The acoustical signals produced by antibubble formation.

An antibubble is an unusual object: a submerged water drop encapsulated in a thin shell of air that is stable underwater for 10-100 s. They are often thought of as the inverse of a soap bubble because they are a spherical shell of air in water in contrast to a shell of water in air. Antibubbles may be formed when water droplets impact the surface of surfactant-covered water, within a limited range of drop radius and drop impact velocity. In this paper, the range of drop size and impact velocity over which large antibubbles (radius 1-3 mm) are generated by the impact of falling drops is characterised, and the relationship of these parameters to the size of the antibubble formed is shown. Measurements of the two acoustical signals that may be produced as an antibubble is formed by drop impact are reported, and their relationship to the antibubble radius and shell thickness is established. Acoustical measurements taken are interpreted in the context of a modified Rayleigh-Plesset equation that provides a good fit to the frequency data for air shells greater than 100 μm in thickness. However, these results highlight the need for future work on the damping mechanisms associated with these larger antibubbles.

Cavitation-threshold Determination and Rheological-parameters Estimation of Albumin-stabilized Nanobubbles

Nanobubbles (NBs) are of high interest for ultrasound (US) imaging as contrast agents and therapy as cavitation nuclei. Because of their instability (Laplace pressure bubble catastrophe) and low sensitivity to US, reducing the size of commonly used microbubbles to submicron-size is not trivial. We introduce stabilized NBs in the 100–250-nm size range, manufactured by agitating human serum albumin and perfluoro-propane. These NBs were exposed to 3.34- and 5.39-MHz US, and their sensitivity to US was proven by detecting inertial cavitation. The cavitation-threshold information was used to run a numerical parametric study based on a modified Rayleigh-Plesset equation (with a Newtonian rheology model). The determined values of surface tension ranged from 0 N/m to 0.06 N/m. The corresponding values of dilatational viscosity ranged from 5.10−10 Ns/m to 1.10−9 Ns/m. These parameters were reported to be 0.6 N/m and 1.10−8 Ns/m for the reference microbubble contrast agent. This result suggests the possibility of using albumin as a stabilizer for the nanobubbles that could be maintained in circulation and presenting satisfying US sensitivity, even in the 3–5-MHz range.

Numerical analyses of nonlinear behavior of microbubble contrast agents in ultrasound field and effective parameters.

This research aims to investigate the effects of some parameters on the nonlinear dynamics of an encapsulated microbubble. In the present study, the radial behavior of an encapsulated microbubble with modified Rayleigh-Plesset equation presented by Marmottant, van der Meer, Emmer, Versluis, de Jong, Hilgenfeldt, and Lohse [J. Acoust. Soc. Am. 118(6), 3499-3505 (2005)] is solved with matlab programming software. Radial dynamics of an individual microbubble for a free and coated state and effects of the added shell are also investigated. Results show that dilatational interfacial viscosity ( ks) of the shell and changes of surface tension have significant effects on the amplitude of oscillations, bubble dynamics, and bubble surface velocity. Regarding the fact that the velocity of the bubble surface has a significant role in stability and collapse of the bubble, it can be noted that this velocity is reduced considerably by adding the shell. Because of extensive usage of nonlinear microbubble oscillations in medical imaging techniques, the nonlinear response of a lipid-coated contrast agent called "compression-only" behavior is studied and effective parameters on this behavior are investigated. The results of bubble radius excursion reveal the fact that excitation frequency, pressure, and dilatational interfacial viscosity of the bubble affect the compression-only behavior.

Cavitation in a hydraulic system: The influence of the distributor geometry on cavitation inception and study of the interactions between bubbles

Hydraulic systems are often subjected to pressure drops, which may lead to cavitation. In systems such as power steering, hoist loads, or ventricular assist devices, distributors are generally used. Significant pressure losses can happen in a distributor due to gap and overlap, which may lead to cavitation development. However, this issue is almost never included in the conception of the distributors. In this study, the multibubble model of the modified Rayleigh–Plesset equation is applied to the rotary distributor of an oil hydraulic system. The influence of the overlap length, the gap, the rotation speed, and distributor inlet pressure on the cavitation and particularly the interactions between bubbles at cavitation inception are studied. The study highlights a critical length of the overlap; over this value, the overlap length influences significantly the cavitation duration and the void fraction. More generally, some geometrical details have a strong influence on cavitation. Optimization of these details in engine parts, taking account the occurrence of cavitation, would be an appropriate solution to reduce its effects. The study also demonstrates that the growth of small bubbles may be delayed by the interactions with the nearby bigger ones, even if the ambient pressure is lower than their theoretical critical pressure. They eventually collapse at the first moments of the cavitation development. However, if the ambient pressure drops further, that is, beyond a critical pressure, a small bubble gains enough inertial energy to overcome these interaction phenomena and thus to grow. The growth of small bubbles increases the interactions between bubbles and slows down the growth of nearby bigger ones. The results show that the interactions between bubbles are of primary importance in the first moments of the cavitation development, which suggests that they should be taken into account in the definition of the critical pressure.