Modified Rayleigh-Plesset Equation Research Articles

Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations

Shock wave structure in a bubbly mixture composed of a cluster of gas bubbles in a quiescent liquid with initial void fractions around 10% inside a 3D rectangular domain excited by a sudden increase in the pressure at one boundary is investigated using the front tracking/finite volume method. The effects of bubble/bubble interactions and bubble deformations are, therefore, investigated for further modeling. The liquid is taken to be incompressible while the bubbles are assumed to be compressible. The gas pressure inside the bubbles is taken uniform and is assumed to vary isothermally. Results obtained for the pressure distribution at different locations along the direction of propagation show the characteristics of one-dimensional unsteady shock propagation evolving towards steady-state. The steady-state shock structures obtained by the present direct numerical simulations, which show a transition from A-type to C-type steady-state shock structures, are compared with those obtained by the classical Rayleigh–Plesset equation and by a modified Rayleigh–Plesset equation accounting for bubble/bubble interactions in the mean-field theory.

Collapse of a bubble in an electric field

A low-Mach-number analysis is presented of the collapse of a bubble in an electric field, which is assumed to be homogeneous, but may be unsteady. Ellipsoidal shape deformations are accounted for in the analysis, but are assumed to be small. It is shown that the presence of an electric field leads to additional terms in a modified Rayleigh-Plesset equation. This differential equation for the bubble radius and a corresponding equation for ellipsoidal shape deformations have been integrated numerically. The results indicate that a bubble can be made to collapse by instantaneously switching on an electric field. Also, nonharmonic volumetric oscillations are observed for time-dependent electric fields of sufficiently large amplitude. It is shown that the rate of a collapse driven by external pressure variations due, for instance, to acoustic forcing can be accelerated.

RANS Simulation of Ducted Marine Propulsor Flow Including Subvisual Cavitation and Acoustic Modeling

High-fidelity Reynolds-averaged Navier Stokes (RANS) simulations are presented for the ducted marine propulsor P5206, including verification and validation (V&V) using available experimental fluid dynamics data, and subvisual cavitation, and acoustics analysis using the modified Rayleigh-Plesset equation along the bubble trajectories with a far-field form of the acoustic pressure for a collapsing spherical bubble. CFDSHIP-IOWA is used with the blended k−ω∕k−ε turbulence model and extensions for a relative rotating coordinate system and overset grids. The intervals of V&V analysis for thrust, torque, and profile averaged radial velocity just downstream of rotor tip are reasonable in comparison with previous results. The flow pattern displays the interaction and merging of the tip-leakage and trailing edge vortices. In the interaction region, multiple peaks and vorticity are smaller, whereas in the merging region, there is better agreement with the experiment. The tip-leakage vortex core position, size, circulation, and cavitation patterns for σi=5 also show good agreement with the experiment, although the vortex core size is larger and the circulation in the interaction region is smaller. The simulations indicate globally minimum Cp=−σi=−8.8 on the suction side of the rotor tip at 84% chord from the leading edge and locally minimum Cp=−6.4 in the tip-leakage vortex at 8% chord downstream of the trailing edge, whereas EFD indicates σi=11 and the location in the tip-leakage vortex core 50% chord downstream of the trailing edge. Subvisual cavitation and acoustics analysis show that bubble dynamics may partly explain these discrepancies.

High-frequency dynamics of ultrasound contrast agents.

Ultrasound contrast agents enhance echoes from the microvasculature and enable the visualization of flow in smaller vessels. Here, we optically and acoustically investigate microbubble oscillation and echoes following insonation with a 10 MHz center frequency pulse. A high-speed camera system with a temporal resolution of 10 ns, which provides two-dimensional (2-D) frame images and streak images, is used in optical experiments. Two confocally aligned transducers, transmitting at 10 MHz and receiving at 5 MHz, are used in acoustical experiments in order to detect subharmonic components. Results of a numerical evaluation of the modified Rayleigh-Plesset equation are used to predict the dynamics of a microbubble and are compared to results of in vitro experiments. From the optical observations of a single microbubble, nonlinear oscillation, destruction, and radiation force are observed. The maximum bubble expansion, resulting from insonation with a 20-cycle, 10-MHz linear chirp with a peak negative pressure of 3.5 MPa, has been evaluated. For an initial diameter ranging from 1.5 to 5 microm, a maximum diameter less than 8 microm is produced during insonation. Optical and acoustical experiments provide insight into the mechanisms of destruction, including fragmentation and active diffusion. High-frequency pulse transmission may provide the opportunity to detect contrast echoes resulting from a single pulse, may be robust in the presence of tissue motion, and may provide the opportunity to incorporate high-frequency ultrasound into destruction-replenishment techniques.

Chirped‐excitation contrast imaging

Ultrasound contrast agent oscillation and echoes are investigated optically and acoustically following insonation with a high time‐frequency bandwidth pulse. Results of a numerical evaluation of the modified Rayleigh–Plesset equation are used to predict the dynamics of a microbubble and compared with experimental observations. Two confocally aligned transducers transmitting at 10 MHz and receiving at 5 MHz are used, and a chirp waveform is transmitted with −6‐dB bandwidth from 7.5 to 12.5 MHz. With a simple bandpass filter, subharmonic echo components can be detected using the return from a single pulse, as the power ratio of subharmonic components to fundamental components ranges up to −7 dB. A high‐speed camera system, which provides 2D frame images and streak images, is used in optical experiments. From the optical observations of a single microbubble, nonlinear oscillation, destruction, and radiation force are observed. For an initial diameter ranging from 1.5 to 5 μm, a maximum diameter below 8 μm is...

Advantages in using multifrequency excitation of contrast microbubbles for enhancing echo particle image velocimetry techniques: Initial numerical studies using rectangular and triangular waves

Accurate measurement of velocity profiles, multiple velocity vectors and local shear stress in arteries is very important for a variety of cardiovascular diseases. We have recently developed an ultrasound based velocimetry technique, termed echo particle image velocimetry (echo PIV). This method takes advantage of the nonlinear backscatter characteristics of ultrasound contrast microbubbles when exposed to certain ultrasonic forcing conditions. Preliminary in vitro, animal and clinical studies have shown significant promise of this method for measuring multiple velocity components with good temporal (up to 2 ms) and spatial (<1 mm) resolution. However, there is still difficulty in maximizing the nonlinearity of bubble backscatter using conventional Gaussian-pulse excitation techniques because: (1) significant harmonic components may not be produced at modest pressure amplitudes; and (2) the higher incident pressure amplitudes required to induce nonlinear behavior may cause bubble destruction. We present here a potential solution to this problem through the use of multifrequency excitation, where rectangular and triangular pulses with four harmonics are used to drive the bubble. The nonlinear behavior of the microbubble, as well as fragility and backscatter, were studied through numerical modeling via a modified Rayleigh-Plesset equation. Results show that the rectangular wave is effective in improving the visibility of microbubbles, with effective scattering cross-section area significantly higher (up to 35 times) than the widely-used Gaussian waveform. However, velocity and acceleration analysis of the bubble wall shows that the rectangular wave may threaten bubble stability. Due to lower wall velocity and acceleration, the triangular wave should decrease the potential for bubble destruction yet maintain relatively high second harmonic backscatter components. The impact of higher harmonics was studied by examining backscatter differences from incident rectangular and triangular pulses with four and two harmonics. Results indicate that a two-frequency excitation (which may be easier to implement practically) may be sufficient to induce nonlinear behavior of the microbubbles at modest incident pressures. These predictions provide support for the use of multifrequency driving to enhance echo PIV applications. (E-mail: Robin.Shandas@colorado.edu)

A bubble fission model for collapsing cavitation bubbles

A bubble fission model that takes into account energy dissipation during violent collapses of cavitation bubbles is introduced by assuming volume conservation and continuity of the gas pressure during instantaneous breakup, invoking either Rayleigh–Taylor instability or high speed microjet formation. In particular, a modified Rayleigh–Plesset equation is derived for the volumetric motion of the fission fragments in rebound. Results obtained under a pressure signal typical for cavitation bubbles confirm the scaling law of Brennen [C. E. Brennen, “Fission of collapsing cavitation bubbles,” J. Fluid Mech. 472, 153 (2002)] and the observations in experiments.

Numerical modeling of microbubble backscatter to optimize ultrasound particle image velocimetry imaging: initial studies

We have developed a promising non-invasive ultrasound-based method for performing particle image velocimetry (PIV) in vivo. This method, termed echo PIV, provides multi-component blood velocity data with good (∼2 ms) temporal resolution. The method takes advantage of the non-linear ultrasound backscatter characteristics of small gas-filled microbubbles (ultrasound contrast) that are seeded into the blood stream. In this study, we use a numerical model to explore potential areas to focus future work in echo PIV. Ultrasound backscatter from encapsulated microbubbles was modeled using a modified Rayleigh–Plesset equation (Church model, 1995), taking into account the protein/lipid shell layer as a thick, mass-conserving incompressible fluid surrounded by incompressible blood-like fluid. The equation of motion was solved numerically to characterize the fundamental and second harmonic components of the backscattered pressure. Results show a significant advantage in using the second harmonic component for echo PIV, especially for small bubble sizes less than 3 μm in diameter at 2.2 MHz frequency. The effect of the shell thickness ranging from 10 to 500 nm on the vibration amplitude of the bubble was examined and it is shown that the presence of the shell requires mechanical index (MI)>0.2 of incident pressure amplitude to improve bubble detectability. Analysis of the effect of pulse length shows a tradeoff between axial resolution (short pulse length) and bubble detectability (longer pulse length) will most likely be required. The effect of varying MI between 0.1 and 0.6 was also studied at a center frequency of 2.2 MHz and the results indicate that the resonance of the second harmonic is maximized for bubbles with diameter of approximately 2.75 μm. Bubble non-linearities at MI>0.2 induced a resonant frequency shift away from the integer multiple of the incident frequency in the second harmonic backscatter. For a given bubble size, there is a combination of optimal incident frequency and mechanical index range that maximizes the ratio of the second harmonic compared to the fundamental. This resonant frequency decreases with increasing bubble radius. Further, a narrow bandwidth pulse is shown to increase signal strength. Both of these effects may cause conflict with factors governing spatial resolution. Optimization of the incident frequency, microbubble size and mechanical index to enhance bubble detectability will depend on the particular clinical application. These theoretical predictions provide further understanding of the physics behind our echo PIV technique, and should be useful for guiding the design of echo PIV systems.

A Newtonian rheological model for the interface of microbubble contrast agents

A quantitative model of the dynamics of an encapsulated microbubble contrast agent will be a valuable tool in contrast ultrasound (US). Such a model must have predictive ability for widely varying frequencies and pressure amplitudes. We have developed a new model for contrast agents, and successfully investigated its applicability for a wide range of operating parameters. The encapsulation is modeled as a complex interface of an infinitesimal thickness. A Newtonian rheology with surface viscosities and interfacial tension is assumed for the interface, and a modified Rayleigh–Plesset equation is derived. The rheological parameters (surface tension and surface dilatational viscosity) for a number of contrast agents (Albunex®, Optison® and Quantison®) are determined by matching the linearized model dynamics with experimentally obtained attenuation data. The model behavior for Optison® (surface tension 0.9 N/m and surface dilatational viscosity 0.08 msP) was investigated in detail. Specifically, we have carried out a detailed interrogation of the model, fitted in the linear regime, for its nonlinear prediction. In contrast to existing models, the new model is found to capture the characteristic subharmonic emission of Optison® observed by Shi et al. (1999). A detailed parametric study of the bubble behavior was executed using the ratio of scattering to attenuation (STAR). It shows that the encapsulation drastically reduces the influence of resonance frequency on scattering cross-section, suggesting possible means of improvement in imaging at off-resonant frequencies. The predictive capability of the present model indicates that it can be used for characterizing different agents and designing new ones. (E-mail: sarkar@me.udel.edu)

An interfacial rheological model of ultrasound contrast agents: Theory and experiments

A new interface model with zero thickness and intrinsic surface rheological parameters has been developed for the encapsulation of a microbubble contrast agent. A modified Rayleigh–Plesset equation with these parameters (surface tension γ and surface dilatational viscosity κs) was obtained. The rationale for the zero-thickness interface lies in the anisotropy and the inhomogeneity in the structure of the encapsulation. The parameters of the model were determined using in vitro attenuation data for various contrast agents (Albunex, Optison, Sonazoid, and Quantison). The linearized dynamics was used to match data from a controlled set of experiments performed at smaller pressure amplitudes (&amp;lt;0.1 MPa). The model was investigated for its ability to predict the nonlinear response at higher amplitudes (&amp;gt;1.0 MPa) for two test cases: Optison (Mallinckrodt, St. Louis, MO) with γ=0.9 N/m, κs=0.08 msP and Sonazoid (Amersham Health, Oslo, Norway) with γ=0.6 N/m, κs=0.01 msP. In contrast to existing models, the new model captured the distinct characteristics of subharmonic emissions—initiation and rapid growth—for these agents. The present model seems to be a reliable descriptor of contrast agent behaviors, and therefore, may be used to characterize different agents and to design next generation agents by tuning the model parameters. [Work supported by DOD.]

On the destruction of microbubble ultrasound contrast agents

In recent years, the use of microbubble ultrasound (US) contrast agents as carriers in drug and gene delivery applications has intensified the need for a clear understanding of the processes involved in their destruction. In this study, an analysis of the conditions in the shell of a contrast agent particle has been made, based on the full numerical solution of a modified Rayleigh–Plesset equation. The results indicate that extremely high shell stresses may be expected under typical clinical conditions. Examination of previous experimental evidence in the light of these findings suggests that the shells are almost invariably disrupted, even if they are not visibly destroyed. This has some serious implications, both for targeted delivery processes and reliable assessment of the potential for harmful bioeffects. At present, neither the model nor the experimental data provide an adequate description of contrast agent behaviour. This is due primarily to the lack of information regarding the mechanical response of the shell material and the restriction of the model to the case of small, spherically symmetrical oscillations. Methods for addressing these deficiencies in future work are proposed. (E-mail: e_stride@meng.ucl.ac.uk)

Chaotic oscillation of a bubble in a weakly viscous dielectric fluid under electric fields

The dynamics of a bubble in a weakly viscous dielectric fluid under electric fields is studied. The dynamical equations for the volume and shape mode oscillations are derived using the domain perturbation method with first-order accuracy in deformation. For the volume mode oscillation, we obtain the modified Rayleigh–Plesset equation which includes a forcing term due to the effect of electric field. For the shape mode oscillations, the Prosperetti–Seminara equation [Phys. Fluids 21, 1465 (1978)] is also extended. The dynamical equations are analyzed with two types of electric fields: the uniform field and the axisymmetric straining field. Equilibrium analysis is performed to find the equilibrium points in the phase planes and their stabilities in static electric fields. Then, the effects of time-periodic electric fields on the bubble dynamics are considered at two levels of viscosity effect [the inviscid limit and the case of Re=O(10)]. The nonlinear dynamics theory is used for analysis of the complicated volume and shape mode oscillations.

Optical and Acoustical Dynamics of Microbubble Contrast Agents inside Neutrophils

Acoustically active microbubbles are used for contrast-enhanced ultrasound assessment of organ perfusion. In regions of inflammation, contrast agents are captured and phagocytosed by activated neutrophils adherent to the venular wall. Using direct optical observation with a high-speed camera and acoustical interrogation of individual bubbles and cells, we assessed the physical and acoustical responses of both phagocytosed and free microbubbles. Optical analysis of bubble radial oscillations during insonation demonstrated that phagocytosed microbubbles experience viscous damping within the cytoplasm and yet remain acoustically active and capable of large volumetric oscillations during an acoustic pulse. Fitting a modified version of the Rayleigh-Plesset equation that describes mechanical properties of thin shells to optical radius-time data of oscillating bubbles provided estimates of the apparent viscosity of the intracellular medium. Phagocytosed microbubbles experienced a viscous damping approximately sevenfold greater than free microbubbles. Acoustical comparison between free and phagocytosed microbubbles indicated that phagocytosed microbubbles produce an echo with a higher mean frequency than free microbubbles in response to a rarefaction-first single-cycle pulse. Moreover, this frequency increase is predicted using the modified Rayleigh-Plesset equation. We conclude that contrast-enhanced ultrasound can detect distinct acoustic signals from microbubbles inside of neutrophils and may provide a unique tool to identify activated neutrophils at sites of inflammation.

Quasi-one-dimensional steady-state cavitating nozzle flows

Quasi-one-dimensional cavitating nozzle flows are considered by employing a homogeneous bubbly liquid flow model. The nonlinear dynamics of cavitating bubbles is described by a modified Rayleigh–Plesset equation that takes into account bubble/bubble interactions by a local homogeneous mean-field theory and the various damping mechanisms by a damping coefficient, lumping them together in the form of viscous dissipation. The resulting system of quasi-one-dimensional cavitating nozzle flow equations is then uncoupled leading to a nonlinear third-order ordinary differential equation for the flow speed. This equation is then cast into a nonlinear dynamical system of scaled variables which describe deviations of the flow field from its corresponding incompressible single-phase value. The solution of the initial-value problem of this dynamical system can be carried out very accurately, leading to an exact description of the hydrodynamic field for the model considered.A bubbly liquid composed of water vapour–air bubbles in water at 20 °C for two different area variations is considered, and the initial cavitation number is chosen in such a way that cavitation can occur in the nozzle. Results obtained, when bubble/bubble interactions are neglected, show solutions with flow instabilities, similar to the flashing flow solutions found recently by Wang and Brennen. Stable steady-state cavitating nozzle flow solutions, either with continuous growth of bubbles or with growth followed by collapse of bubbles, were obtained when bubble/bubble interactions were considered together with various damping mechanisms.

Nonlinear scatter from the ultrasound contrast agent Sonazoid: Theory and numerical simulations

A model for the contrast agent Sonazoid is presented. The model is based on a modified Rayleigh–Plesset equation, with the shell added as an incompressible viscoelastic solid. Different models for the shell’s nonlinear stress–strain relationship are discussed. Shell elasticity and viscosity were estimated from experiments, and the model was used to simulate bubble radius and scattered sound. Pulse-inversion imaging was simulated. The amplitude was increased from 20 to 500 kPa, giving scattered pulses that changed from inverted to time-shifted versions of each other. This effect was best seen in the cross correlation of the responses. The nonlinear response was strongest for frequencies below bubble resonance. Compared to unshelled bubbles, Sonazoid required higher drive amplitudes to respond nonlinearly. The Sonazoid bubbles responded similarly to unshelled bubbles of smaller diameter. A subharmonic response was found in a limited frequency band and only for diameters above 3 μm. An amplitude threshold was found around 500 kPa, depending on diameter and frequency. This threshold is higher than for unshelled bubbles. A chaotic oscillation pattern was found for some diameter–frequency combinations, in agreement with published results. For some situations where the cw results predicted a subharmonic mode, this mode was not obtained from finite-length pulses.

Could optical radiation pressure be used to move a sonoluminescing bubble to the pressure antinode of an acoustic standing wave?

It is generally recognized that in SBSL, the bubble is displaced slightly above the antinode of the acoustic standing wave which drives the bubble oscillations. This is because the acoustic radiation force must balance the average buoyancy of the bubble. For scientific purposes and to simulate SBSL in microgravity, it would be desirable to reposition the bubble at the antinode. One strategy to achieve that repositioning where the buoyancy force would be balanced entirely by the average optical radiation pressure of a downward propagating laser beam has been numerically investigated. Optical levitation of small bubbles has been previously demonstrated [B. T. Unger and P. L. Marston, J. Acoust. Soc. Am. 83, 970–975 (1988)]; however, it is not trivially applied to SBSL; with cw illumination of a quiescent bubble, the laser power required increases as the cube of the bubble radius. To simulate the large-radius phase of SBSL bubble oscillations, a modified Rayleigh–Plesset equation was numerically integrated. The required power for cw and pulsed laser beams was estimated. The average power is significantly reduced if the beam is pulsed to coincide with the large-radius phase of the oscillations so as to maximize the relevant optical cross section.

A numerical study of the complex dynamic behavior of a reactive gas bubble in water

It is recognized that, in nuclear reactors, oxygen-hydrogen gas bubbles may form and this formation can lead to an explosion hazard. In this study, numerical investigations are conducted into the collapse and explosion of an isolated oxygen-hydrogen bubble immersed in water. The mathematical model of the bubble's radial motion is based on the compressible form of the modified Rayleigh-Plesset equation of the bubble dynamics. The model also takes into account, in differential form, the heat transfer between the gas and the surrounding inert water. Parametric study on the present model is primarily aimed at delineating the effects of exothermicity of the reactive gas mixture and the initial bubble diameter on the bubble's volume and thermal oscillations under various sustained liquid pressure fields. The exothermicity of the bubble's gas content is varied by changing the mole fraction of the mixture of stoichiometric oxygen-hydrogen with the inert gas, argon, as a diluent. The results show that if the incident pressure pulse is not of sufficient strength, the mixture within the bubble does not ignite and the bubble radius remains below the initial equilibrium value throughout the pulsation period. Under the above conditions, due to the low heat capacity of argon compared to that of oxygen and hydrogen, the maximum gas temperature attained by the bubble increases with the increase in argon content or, equivalently, with the decrease in oxygen-hydrogen content. If the incident pressure pulse is strong enough, the bubble is seen to explode and the maximum gas temperature that the bubble attains depends directly upon the initial oxygen-hydrogen content. When the bubble undergoes ignition, with the increase of exothermicity, the maximum bubble radius increases and the period of the volume oscillation becomes longer. Also, with decreasing initial bubble radius, the liquid threshold pressure for bubble explosion increases, while the period of bubble oscillation decreases.

Bubble hydrodynamics and the confined electron model for sonoluminescence

The origin of sonoluminescence, the conversion of acoustic energy into ultraviolet/visible radiation in liquids, has remained elusive. It is proposed that the emission arises from electrons confined to voids in the hot, dense fluid formed during the final stages of bubble collapse. Such electrons are produced by high-temperature ionization of the bubble constituents. A hard sphere-based model was developed for the fluid structure, thermodynamics, and confined electron emission. The model is consistent with the observed spectral distributions, power output, and time scale associated with emission from single cavitating rare gas bubbles. Effective temperatures during emission in the 200- to 700-nm spectral window are predicted to range from 20 000 to 60 000 K. The consistency of the confined electron model for single bubble sonoluminescence with the observed dynamics of bubble collapse is also explored. The key aspect of this approach is the treatment of the collapsed bubble as a true fluid. The hydrodynamics of bubble collapse are modeled with a modified Rayleigh–Plesset equation using improved formulations for the equation of state and specific heat ratio. Results are presented for the rare gases and related to experimental observations.

Compressibility in bubble dynamics and scattering

The Rayleigh–Plesset equation describes the oscillations of a spherical bubble wall under the assumption that the fluid surrounding the bubble is incompressible. Many modifications of this equation have been proposed to incorporate slight fluid compressibility, and though different in form, they are asymptotically equivalent. The different forms of the equations, however, reveal remarkably different properties. In some forms of the modified Rayleigh–Plesset equation, for example, spurious unstable solutions are present while in other cases they are not. Here, physically motivated restrictions on the form of the modified Rayleigh–Plesset equation are discussed. These restrictions are discussed in the context of causality requirements and higher-order corrections to the modified Rayleigh–Plesset equation.

Dynamic growth of a spherical bubble in a time-periodic electric field

The dynamics of a spherical bubble in a time-dependent electric field is investigated via the modified Rayleigh-Plesset equation where the effect of an electric field is added. The effect of an imposed electric field is found to be equivalent to the increase of the ambient pressure by the amount of 3/8e0e1E02(2S-1), where e0e1 is the electric permittivity of the gas inside the bubble, E0 the strength of the imposed electric field, S the permittivity ratio of the outside fluid to the inside gas. The effects of a time-periodic electric field have been studied by using two methods of analysis; the two-timing perturbation analysis for the regular dynamics near the stable steady solution and the Poincare map analysis for the global dynamics. It is revealed that an O(e1/3) response in the oscillation of bubble radius can be obtained from an O(e) resonant time-periodic forcing in the neighborhood of a stable steady solution. By the Poincare map analysis, it is also shown that the bubble can either undergo bounded oscillation, or else respond chaotically and grow very rapidly. The probability of escape to rapid growth is found to be a strong function of the forcing frequency, of which the optimal value is slightly lower than the intrinsic resonant frequency of oscillation under the steady electric field.