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This study investigated the weakly nonlinear propagation of pressure waves in compressible, flowing water with spherical microbubbles, considering various forces. Previous theoretical studies on nonlinear pressure waves in bubbly flows did not consider the forces acting on the bubbles, although the validity of ignoring these forces has not been demonstrated. We focused on every possible force such as drag, gravity, buoyancy, and Bjerknes (acoustic radiation) forces acting on bubbles and studied their effects on pressure waves in a one-dimensional setting. Using a singular perturbation method, the Korteweg–de Vries–Burgers equation describing wave propagation was derived. The following results were obtained: (i) Bjerknes force on the bubbles enhanced the nonlinearity, dissipation, and dispersion of the waves; (ii) Drag, gravity, and buoyancy forces acting on the bubbles increased wave dissipation; (iii) Thermal conduction had the most substantial dissipation effect, followed by acoustic radiation, drag, buoyancy, and gravity. We confirmed that the dissipation due to forces on gas bubbles was quantitatively minor. [ABSTRACT FROM AUTHOR]

Using microbubbles coated by a thin shell as ultrasound contrast agents for ultrasound diagnosis improves image resolution. Since numerous microbubbles are used in clinical practice, understanding the acoustic properties of liquids containing multiple microbubbles is important. However, interactions between ultrasound and numerous coated microbubbles have not been fully investigated theoretically. Additionally, ultrasound contrast agents with shells made of various materials have been developed. Recently, an equation of motion that considers the anisotropy of the shell was proposed [Chabouh et al., "Spherical oscillations of encapsulated microbubbles: Effect of shell compressibility and anisotropy," J. Acoust. Soc. Am. 149, 1240 (2021)], and the effect of shell anisotropy on the resonance of the oscillating bubble was reported. In this study, we derived a nonlinear wave equation describing ultrasound propagation in liquids containing numerous coated microbubbles based on the method of multiple scales by expanding Chabouh's equation of motion for the single bubble. This was achieved by considering shell anisotropy in the volumetric average equation for the liquid and gas phases. Shell anisotropy was observed to affect the advection, nonlinearity, attenuation, and dispersion of ultrasound. In particular, the attenuation effects increased or decreased depending on the anisotropic shell elasticity. [ABSTRACT FROM AUTHOR]

The importance of viscoelasticity of biological media that are used in medical ultrasounds has been discussed in the literature. Furthermore, the use of microbubbles in biological media drastically improves the efficiency of both diagnostic and therapeutic ultrasounds. Weakly nonlinear wave equations for ultrasound propagation in liquids containing microbubbles have long been studied, although the viscoelasticity of the liquid phase has been ignored for simplicity. In this study, we derived a nonlinear wave equation for ultrasound propagation in a viscoelastic liquid containing microbubbles by considering the effect of the elasticity of the liquid. Additionally, we evaluated how the elasticity of the liquid modifies the nonlinear, dissipation, and dispersion effects of the ultrasound in a few tissue models (i.e., liver, muscle, breast cancer, fat, and skin models and that without shear elasticity). The results revealed that liquid shear elasticity decreases the nonlinear and dissipation effects and increases the dispersion effect, and this tendency is more significantly observed in the breast cancer tissue compared with other tissues. Furthermore, we numerically solved the nonlinear wave equation and investigated the changes in ultrasonic wave evolution with and without shear elasticity. [ABSTRACT FROM AUTHOR]

Microbubbles coated by visco-elastic shells are important for ultrasound diagnosis using contrast agents, and the dynamics of single coated bubbles has been investigated in the literature. However, although a high number of contrast agents are used in practical situations, there has long been an absence of a nonlinear acoustic theory for multiple coated bubbles, except for our recent work by Kikuchi and Kanagawa ["Weakly nonlinear theory on ultrasound propagation in liquids containing many microbubbles encapsulated by visco-elastic shell," Jpn. J. Appl. Phys. 60, SDDD14 (2021)], under several assumptions to be excluded. Aiming for generalization, in this study, we theoretically investigate weakly nonlinear propagation of ultrasound in liquid containing multiple bubbles coated by a visco-elastic shell with compressibility. Leveraging the method of multiple scales, both the Korteweg–de Vries–Burgers (KdVB) equation for a low-frequency long wave and nonlinear Schrödinger (NLS) equation for a high-frequency short wave are derived from the volumetric averaged equations for bubbly liquids based on a two-fluid model and the up-to-date model for single coated bubbles with shell compressibility. Neglected factors in our previous paper, i.e., compressibility of the shell and liquid, drag force acting on bubbles, bubble translation, and thermal conduction, are incorporated in the present KdVB and NLS equations; the proposed model will be regarded as a generic physico-mathematical model. The results show that shell compressibility attenuated ultrasound strongly and decreased nonlinearity of ultrasound. Finally, we compared the magnitudes of six dissipation factors (shell compressibility, shell viscosity, liquid compressibility, liquid viscosity, thermal effect, and drag force) for five typical ultrasound contrast agents, and a similar tendency between KdVB and NLS equations was revealed. [ABSTRACT FROM AUTHOR]

In this study, the weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid was theoretically investigated. This liquid contains several small uniformly distributed spherical polydisperse gas bubbles. The polydispersity considered here represents various types of initial bubble radii, and the liquid contains multiple bubbles, each with an initial radius. Using the method of multiple scales, we first derived the Korteweg–de Vries–Burgers (KdVB) equation with a correction term as a nonlinear wave equation. This equation describes the long-range wave propagation with weak nonlinearity, low frequency, and long wavelength in the polydisperse bubbly liquid using the basic equations in a two-fluid model. The utilization of the two-fluid model incorporates the dependence of an initial void fraction on each coefficient in the nonlinear, dissipation, and dispersion terms in the KdVB equation. Furthermore, unlike previous studies on waves in polydisperse bubbly liquids, we achieved the formulation without assuming an explicit form of the polydispersity function. Consequently, we discovered the contribution of polydispersity to the various effects of wave propagation, that is, the nonlinear, dissipation, and dispersion effects. In particular, the dispersion effect of the waves was found to be strongly influenced by polydispersity. [ABSTRACT FROM AUTHOR]

In this study, weakly nonlinear pressure waves in quiescent compressible liquids comprising several uniformly-distributed spherical microbubbles, at moderately high-frequency and short-wavelength, are theoretically investigated. The energy equation at the bubble–liquid interface and the effective polytropic exponent are utilized to clarify thermal effects inside bubbles on wave dissipation. In addition, thermal conduction is investigated in detail using four temperature-gradient models. The following results are drawn: (i) Nonlinear Schrödinger equation is derived as an effective equation, wherein three types of dissipation factors, i.e., liquid viscosity, liquid compressibility, and thermal conduction, are unified into a linear combination as the dissipation coefficient. This is different from our previous result treating the low-frequency and long-wavelength case [Kamei et al., Phys. Fluids 33, 053302 (2021)], i.e., two types of dissipation terms appeared and did not unify into a linear combination. (ii) Dissipation due to thermal conduction is more than four times larger than that due to other dissipation factors. (iii) Dissipation due to thermal conduction at the bubble–liquid interface is considerably larger than that due to thermal conduction through the bubbly liquid. (iv) It is found that the dissipation effect in the short-wave case is smaller than that in the long-wave case. [ABSTRACT FROM AUTHOR]

To clarify the effect of the drag force acting on bubbles and translation of bubbles on pressure waves, the weakly nonlinear (i.e., finite but small-amplitude) propagation of plane pressure waves with a thermal conduction in compressible water flows containing many spherical bubbles is theoretically investigated for moderately high-frequency and short-wavelength case. This work is an extension of our previous report [Yatabe et al., Phys. Fluids, 33, 033315 (2021)], wherein we elucidated the same for low-frequency and long-wavelength case. Based on our assumptions, the main results of this study are as follows: (i) using the method of multiple scales, the nonlinear Schrödinger type equation was derived; (ii) as in the previous long wave case, the translation of bubbles increased the nonlinear effect of waves, and the drag force acting on the bubbles resulted in the dissipation effect of waves; (iii) the increase in the nonlinear effect of the waves owing to the translation in the present short wavelength case is larger than that in the previous long wavelength case; (iv) the dissipation effect caused by the drag force was smaller than that caused by the liquid viscosity, acoustic radiation (i.e., liquid compressibility), and thermal conduction; (v) we then succeeded the comparison of the four dissipation factors (i.e., liquid viscous damping, thermal conduction, acoustic radiation, and drag force) on pressure waves. [ABSTRACT FROM AUTHOR]

Weakly nonlinear propagation of pressure waves in initially quiescent compressible liquids uniformly containing many spherical microbubbles is theoretically studied based on the derivation of the Korteweg–de Vries–Burgers (KdVB) equation. In particular, the energy equation at the bubble–liquid interface [Prosperetti, J. Fluid Mech. 222, 587 (1991)] and the effective polytropic exponent are introduced into our model [Kanagawa et al., J. Fluid Sci. Technol. 6, 838 (2011)] to clarify the influence of thermal effect inside the bubbles on wave dissipation. Thermal conduction is investigated in detail using some temperature-gradient models. The main results are summarized as follows: (i) Two types of dissipation terms appeared; one was a well-known second-order derivative comprising the effect of viscosity and liquid compressibility (acoustic radiation) and the other was a newly discovered term without differentiation comprising the effect of thermal conduction. (ii) The coefficients of the KdVB equation depended more on the initial bubble radius rather than on the initial void fraction. (iii) The thermal effect contributed to not only the dissipation effect but also to the nonlinear effect, and nonlinearity increased compared with that observed by Kanagawa et al. (2011). (iv) There were no significant differences among the four temperature-gradient models for milliscale bubbles. However, thermal dissipation increased in the four models for microscale bubbles. (v) The thermal dissipation effect observed in this study was comparable with that in a KdVB equation derived by Prosperetti (1991), although the forms of dissipation terms describing the effect of thermal conduction differed. (vi) The thermal dissipation effect was significantly larger than the dissipation effect due to viscosity and compressibility. [ABSTRACT FROM AUTHOR]

Theoretical investigation of the effects of a translation of bubbles and a drag force acting on bubbles on the wave propagation in bubbly flows has long been lacking. In this study, we theoretically and numerically investigate the weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flows that contain uniformly distributed spherical gas bubbles with translation and drag forces. First, we assume that the gas and liquid phases flow at independent velocities. Then, the drag force and virtual mass force are introduced in an interfacial transport across the bubble–liquid interface in the momentum conservation equations. Furthermore, we consider the translation and spherically symmetric oscillations as bubble dynamics and deploy a two-fluid model to introduce the translation and drag forces. Bubbles do not coalesce, break up, extinct, or appear. For simplicity, the gas viscosity, thermal conductivities of the gas and liquid, and phase change and mass transport across the bubble–liquid interface are ignored. The following results are then obtained: (i) Using the method of multiple scales, two types of Korteweg–de Vries–Burgers equations with a correction term due to the drag force are derived. (ii) The translation of bubbles enhances the nonlinear effect of waves, and the drag force acting on bubbles contributes the nonlinear and dissipation effects of waves. (iii) The results of long-period numerical analysis verify that the temporal evolution of the wave (not flow) dissipation due to the drag force differs from that caused by the acoustic radiation. [ABSTRACT FROM AUTHOR]

The two-fluid model with bubble oscillations, proposed by Egashira et al. (2004), can explain the properties of cavitating bubbly flow and pressure wave propagation in the bubbly liquid. However, the viscous effect as well as energy conservation leading to temperature changes inside the bubble with bubble oscillations have not yet been considered. Hence, this study aimed to incorporate the viscous (bulk viscosity and drag) and thermal effects to the previously proposed two-fluid model with bubble oscillations. Bulk viscosity was considered by averaging the shear stress term in the single-phase momentum conservation for a Newtonian fluid, and the drag was introduced by transforming the interfacial shear stress. We derived the averaged energy conservation for a general two-phase flow with a thermal conduction inside bubbles and heat transfer between the two phases, and limited this equation to that for a bubbly flow by closing the interfacial temperature gradient term via constitutive equations for a single bubble. Furthermore, we investigated the stability of our proposed one-dimensional model equations using the dispersion analysis. This analysis provided the following insight: (i) The difference in the temperature gradient models had a slight effect on the stability of the proposed model equations; (ii) the thermal conduction inside the bubbles was dominant in the thermal damping in bubbly flows rather than the heat transfer between the two phases; (iii) incorporating both the bulk viscosity and drag stabilized the proposed model equations. Our results provide insights into the development of mathematical models to investigate the thermal effects in bubbly flow with bubble oscillations, such as cavitating bubbly flow and wave propagation in bubbly liquids. • Two-fluid model with bubble oscillations was analytically derived by volume average. • Interfacial term of energy conservation was closed by temperature gradient model. • Stability analysis revealed that thermal conduction inside bubble was dominant. • Viscosity and drag play an important role in stability of our derived two-fluid model. [ABSTRACT FROM AUTHOR]

To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller–Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity suppresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications. [ABSTRACT FROM AUTHOR]

Herein, the weak, nonlinear propagation of pressure waves in an initially quiescent liquid containing many small spherical gas bubbles is theoretically studied. We focus on the initial, polydispersity features of the bubble radius and bubble number density. Our analysis was not based on any assumptions about explicit polydispersity forms. Using equations based on a two-fluid model and the method of multiple scales with perturbation expansions, the Korteweg–de Vries–Burgers equation for a low-frequency long wave and nonlinear Schrödinger equation for a high-frequency short wave were derived. In both cases, the polydispersity contributes to the advection and dissipation effects of waves, and every coefficient in both equations includes the initial void fraction as one of the most important parameters, owing to the use of the two-fluid model. • Weakly nonlinear propagation of pressure waves was theoretically studied. • Waves in liquid containing many spherical microbubbles with a polydispersity. • Two-fluid model and perturbation expansions with multiple scales were used. • Korteweg–de Vries–Burgers/nonlinear Schrödinger equations were derived. • Polydispersity contributes to advection and dissipation effects of waves. [ABSTRACT FROM AUTHOR]

• Extension of acoustic theory for single encapsulated bubble to multiple bubbles. • Bubble encapsulated by a visco-elastic shell with compressibility. • Presence of shell affected advection, nonlinearity, and dissipation of ultrasound. • Ultrasound dissipation by compressibility and viscosities of shell and liquid. • Dissipation due to shell compressibility was the highest. Owing to its potential for application in ultrasound-based diagnosis and therapy, the dynamics of a microbubble encapsulated by shells has long been theoretically investigated. However, outside of our research group, previous theories have been restricted to the case of single encapsulated bubble, whereas in practical diagnostic scenarios, multiple encapsulated bubbles are used as ultrasound contrast agents. In this study, the most recent theory for a single encapsulated bubble incorporating shell compressibility was extended for multiple encapsulated bubbles. Using the method of multiple scales, weakly nonlinear wave equation for one-dimensional ultrasound in liquids containing multiple encapsulated microbubbles was derived from the set of volumetric averaged equations for bubbly flow. It was found that the shell compressibility significantly increased the advection and dissipation effects of ultrasound. Further, five types of dissipation effects were compared with each other, and showed that the dissipation effects corresponding to shell compressibility were the highest. [ABSTRACT FROM AUTHOR]

Changes are required on pages 9 and 10 of the original published paper.[1] The dashed curve (i.e., HT ht ) is incorrect in Figs. This trend is similar to the case of long wavelength.[2] Furthermore, HT ht is only dependent on I i SB 0 sb and only independent of I k i ." "The dissipation owing to the drag force, HT ht , is the smallest. [Extracted from the article]

• Three cases of nonlinear wave equation for pressure in polydisperse bubbly liquids. • Formulation of initial weak polydispersity. • Classification of two cases of nonlinear Schrödinger equation. • Contribution of polydispersity to advection effect. Weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid uniformly containing many spherical microbubbles is theoretically investigated, especially focusing on an initial small polydispersity of both the bubble radius and the number density of bubbles (i.e., void fraction), which appears in a field far from the sound source. Nonlinear waves in polydispersed bubbly liquids are classified into a form of three cases of nonlinear wave equation describing long-range propagation of waves. Using the method of multiple scales with perturbation expansions and the scaling relations of some nondimensional ratios, from the set of basic equations based on a two-fluid model, (i) for a low-frequency long wave, the Korteweg–de Vries–Burgers (KdVB) equation is derived and (ii) for a moderately high-frequency short wave, (ii-a) the NLS (nonlinear Schrödinger)-I (or LG (Landau–Ginzburg)-I) equation for a weak polydisperse medium and (ii-b) the NLS-II (or LG-II) equation in a strong polydisperse medium are derived in a unified manner. For all cases, polydispersity contributes to the advection effect of waves and induces variable coefficients into the KdVB, NLS-I, and NLS-II equations. Furthermore, the KdVB equation includes an inhomogeneous term owing to the polydispersity and the NLS-II equation includes second-order nonlinearity of polydispersity. The polydisperse effect is finally clarified quantitatively by focusing on the advection coefficients with a help of numerical examples. [ABSTRACT FROM AUTHOR]