Nonlinear acoustic theory on flowing liquid containing multiple microbubbles coated by a compressible visco-elastic shell: Low and high frequency cases

Kanagawa, Tetsuya; Honda, Mitsuhiro; Kikuchi, Yusei

doi:10.1063/5.0101219

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.

I. INTRODUCTION

Medical ultrasound has several important applications such as diagnosis via contrast images, tumor ablation therapy using high intensity focused ultrasound, and extracorporeal shock wave lithotripsy.^1–8 Since the combination of micro (or nano) bubbles and ultrasound enhances an efficiency in medical practices,^9,10 interaction of cavitating bubble oscillations and propagation of waves (i.e., shock, ultrasound, and pressure wave) has extensively been investigated (see, e.g., recent analytical and/or numerical progresses^11–33). Using microbubbles as ultrasound contrast agents (UCAs) has been shown to improve the contrast of the blood carrying tissue.³⁴ UCAs were originally uncoated microbubbles used for echocardiography.³⁵ However, in order to improve the stability and biocompatibility of the bubbles, protein coated bubbles such as Albumin was proposed, and then, lipid-covered microbubbles were developed as new UCAs and are now used to detect liver cancer.³⁶ Further applications of coated microbubbles are expected to include therapeutic applications such as transportation in drug delivery systems.^37–43

Therefore, a precise understanding of the interaction between ultrasound propagation and oscillations of coated microbubbles as UCAs is desirable. Experiments on attenuation (or dissipation) and scattering were performed using various types of coatings to understand their properties,^10,14–51 and molecular dynamics simulations were performed to clarify the properties of the shells.^52,53 In addition, shell parameters were modeled based on intermolecular interactions.^54,55 Particularly, Sojahrood et al. investigated the dynamics of coated (or uncoated) bubbles through bifurcation structure,^56–59 sub- and superharmonic behaviors,^60–63 and the mechanics of nonlinear power dissipation.^64–66

Several physico-mathematical models have been derived or proposed based on the equation of motion for coated bubbles. Equations of motion of single uncoated bubbles such as the Rayleigh–Plesset equation and Keller–Miksis equation⁶⁷ have been modified previously.^68–83 Furthermore, a pioneering work toward modeling single coated bubbles has been reported by de Jong⁴⁵ and Church⁷⁰ proposed a more rigorous theoretical model by treating the coating shell as an incompressible Kelvin–Voigt model. Hoff et al.⁷⁵ made the assumption that the shell thickness is small enough compared with the bubble radius and derived a new model from the Church model.⁷⁰ Previously, Sarkar et al. studied the viscous effect of the shell and developed a visco-elastic interface model for the thin coated bubble.⁸⁰ Moreover, Marmottant et al.⁷⁷ developed a model considering the physical properties of a lipid monolayer coating on a gas microbubble. Following this, other nonlinear models have been proposed with more complex rheological behaviors, such as strain softening and hardening^78,83 or shear thinning.⁷² In addition, Liu et al. formulated the radial oscillations, translational motions, and deformations for double coated bubbles.⁸⁴ Gubaidullin and Fedorov⁷⁴ investigated heat transfer with coated bubbles. However, these studies^{10,45,58,69–88,112} were restricted to single coated bubbles or some (i.e., not a high number of) coated bubbles. Since multiple (i.e., a high number of) coated bubbles are used as contrast agents in practical applications, an understanding of multiple coated bubbles is desired. Although the basic theory for multiple coated bubbles in relation to ultrasound propagation should be established, our recent papers^89,90 are among few that report on multiple coated bubbles, from the viewpoint of multiphase fluid dynamics (i.e., the formulation based on continuum model). Note that although Ma et al.⁹¹ obtained the parameters of acoustic nonlinearity for ultrasound in liquids containing multiple coated bubbles, they did not discuss the dissipation and various effects of propagation of nonlinear ultrasound.

The goal of the present study is (a) the consideration of the compressibility of the shell and (b) unified treatment of both low-frequency long wave and high-frequency short wave bands. With regard to (a), we extend the previously established microscopic modeling into macroscopic dynamics of multiphase flows as a continuum approach (i.e., the extension of the dynamics of single coated bubbles to that of multiple coated bubbles by combining the equation of motion for single coated bubbles and the set of volumetric averaged equations for bubbly liquids). The most recent model⁶⁸ incorporated the compressibility of coated shells and showed that neglecting the compressibility leads to an underestimation of the shear modulus. Based on the results of this paper, we found the non-negligible contribution of shell compressibility to the ultrasound attenuation. We completed the theory by generalizing our previous work⁸⁹ to incorporate other previously neglected factors, i.e., shell compressibility, liquid compressibility, influence of drag force on bubbles, translation of bubbles, and thermal conductivity. Equation for thermal conduction at the bubble–liquid interface⁹² is used. To the best of our knowledge, this is the first report considering both multiple coated bubbles and thermal conduction. With regard to (b), we focused on both low-frequency long waves and high-frequency short waves, and derived the Korteweg–de Vries–Burgers (KdVB) and nonlinear Schrödinger (NLS) equations, respectively. Using both the low and high frequency modes is also important in a medical application. For the case of multiple un-coated bubbles, the KdVB and NLS equations have been derived independently (e.g., pioneering classical works^93–96). By changing the magnitude of the set of nondimensional ratios appropriately to low- and high-frequency bands,⁹⁷ we can systematically derive the KdVB and NLS equations (i.e., two types of nonlinear wave equations) from one set of basic equations in a unified manner. The resulting KdVB and NLS equations describe the weakly nonlinear propagation/evolution process of ultrasound in liquids containing multiple coated bubbles. The effect of the physical properties of the shell (e.g., compressiblity, shear modulus, viscosity) and thermal conductivity inside the bubbles on the propagation properties of ultrasound, i.e., dissipation, nonlinear, and advection effects, are clarified based on the KdVB and NLS equations. The coefficients in the dissipation, nonlinear, and advection terms of the KdVB and NLS equations are then compared for typical UCAs. The connection between the present study and our previous two studies on ultrasound in liquids containing multiple coated bubbles is as follows: (i) Although Ref. 89 originally derived only the KdVB equation without shell compressibility, the present study considers the unified derivation of the KdVB and NLS equations, making this the first derivation of an NLS equation for multiple coated bubbles. (ii) Reference 90 derived the KdVB equation incorporating the effect of shell compressibility but did not discuss the dependence of the initial void fraction as one of most important points in bubble dynamics. Here, we use two-fluid model averaged equations to incorporate the dependence of the initial void fraction on each coefficient in both the KdVB and NLS equations.

The remainder of this paper is organized as follows. Section II formulates the problem by introducing the updated equation of motion for coated bubbles with shell compressibility⁶⁸ and the volumetric averaged equations for bubbly liquids.⁹⁸ Section III derives the KdVB equation for the low-frequency long wave. Section IV derives the NLS equation for the high-frequency short wave. Section V discusses the dissipation, nonlinear, and advection effects in the resultant KdVB and NLS equations, and Sec. VI concludes the paper.

II. FORMULATION OF THE PROBLEM

A. Problem and assumptions

As shown in Fig. 1, uniform weakly nonlinear propagation of ultrasonic (or acoustic or pressure) waves in compressible liquids containing multiple spherical coated bubbles is studied theoretically. The number of coated bubbles (see Fig. 2) is significantly high, which can be formulated using the continuum model shown in (4) and (5) below. For simplicity, the following assumptions are used:

Two propagation modes, that is, low-frequency long wave (KdVB equation in Sec. III) and high-frequency short wave (NLS equation in Sec. IV), are investigated.
The shell coating the bubble is a visco-elastic body that obeys the Kelvin–Voigt model.
Although the shell is compressible,⁹⁹ we regard the thickness of shell as a constant, for simplicity.
The propagation of one-dimensional (plane) waves is treated.
The liquid is compressible.
The bubbles do not coalesce, break up,¹⁰⁰ appear, or disappear, and bubble–bubble interaction^101–105 is neglected.
The bubble oscillations are spherically symmetric.
Initially, the gas and liquid phases flow with a constant uniform velocity, and initial nonuniform velocity distributions of each phase¹⁰⁶ are not considered.
The effect of liquid viscosity is considered only at the bubble–liquid interface.
The phase change^99,107,108 is neglected.
The mass transport across the bubble–liquid interface¹⁰⁹ is neglected.

B. Basic equations

To describe the dynamics as the expansion and contraction of coated bubbles, we used the updated model, i.e., Chabouh et al.'s equation,⁶⁸ for spherically symmetric volumetric oscillations in a compressible liquid with coated bubbles. Furthermore, we incorporated the translation of bubbles [i.e., the last term in the right-hand side in (1)] into Chabouh model.⁶⁸ We then extended Chabouh equation⁶⁸ for single coated bubbles without translation to the case of multiple coated bubbles with translation as follows:

\begin{matrix} ρ_{L 0}^{*} (1 - \frac{1}{c_{L 0}^{*}} \frac{D_{G} R^{*}}{D t^{*}}) R^{*} \frac{D_{G}^{2} R^{*}}{D t^{* 2}} + \frac{3}{2} ρ_{L 0}^{*} (1 - \frac{1}{3 c_{L 0}^{*}} \frac{D_{G} R^{*}}{D t^{*}}) {(\frac{D_{G} R^{*}}{D t^{*}})}^{2} \\ = (1 + \frac{1}{c_{L 0}^{*}} \frac{D_{G} R^{*}}{D t^{*}}) P^{*} + \frac{R^{*}}{c_{L 0}^{*}} \frac{D_{G}}{D t^{*}} (p_{L}^{*} + P^{*}) + \frac{{(u_{G}^{*} - u_{L}^{*})}^{2}}{4} . \end{matrix}

(1)

Here, the pressures of surrounding liquid are described by the balance equation of normal stresses as follows:

P^{*} = (p_{G}^{*} - 2 \frac{σ_{1}^{*}}{R^{*} - d_{0}^{*}}) (1 - \frac{4 G^{*}}{3 K^{*} + 4 G^{*}} \frac{3 d_{0}^{*}}{R^{*}}) - p_{L}^{*} - \frac{4 μ_{L}^{*}}{R^{*}} \frac{D R^{*}}{D t^{*}} - 12 G^{*} \frac{3 K^{*}}{3 K^{*} + 4 G^{*}} \frac{d_{0}^{*}}{R^{*}} (1 - \frac{R_{0}^{*}}{R^{*}}) - 4 μ_{S}^{*} (1 - \frac{4 G^{*}}{3 K^{*} + 4 G^{*}}) \times \frac{d_{0}^{*}}{R^{*} (R^{*} - d_{0}^{*})} \frac{D R^{*}}{D t^{*}} - 8 μ_{K}^{*} \frac{3 G^{*}}{3 K^{*} + 4 G^{*}} \frac{d_{0}^{*}}{R^{* 2}} \frac{D R^{*}}{D t^{*}} - \frac{2 σ_{2}^{*}}{R^{*}} .

(2)

Here, as an independent variable, $t^{*}$ is the time; as dependent variables, $R^{*}$ is the bubble radius, $u_{G}^{*}$ and $u_{L}^{*}$ are the gas and liquid velocities, respectively, $p_{G}^{*}$ and $p_{L}^{*}$ are the volumetric-averaged gas and liquid pressures, respectively, and $P^{*}$ is the surface averaged liquid pressure; as constants, $G^{*}$ is the shear modulus of shell, $K^{*}$ is the bulk modulus of shell, $μ_{S}^{*}$ is the shear viscosity of the shell, $μ_{K}^{*}$ is the viscosity that describes the friction losses due to volume change of the shell, $ρ_{L 0}^{*}$ is the initial liquid density, $p_{G 0}^{*}$ and $p_{L 0}^{*}$ are the initial gas and liquid pressures, respectively [see also (12) below], $μ_{L}^{*}$ is the liquid viscosity, $c_{L 0}^{*}$ is the sound speed in the unperturbed liquid, $d_{0}^{*}$ is the initial shell thickness, $R_{0}^{*}$ is the initial bubble radius, and $σ_{1}^{*}$ and $σ_{2}^{*}$ are the surface tensions at the internal and external boundaries of the shell, respectively; the superscript * represents dimensional quantities, the subscript 0 denotes the initial unperturbed state, and the subscripts $G$ and $L$ represent the gas and liquid phases, respectively; $D_{i} / D t^{*} (i = G, L)$ are material derivatives for the gas and liquid phases, respectively,

\frac{D_{G}}{D t^{*}} = \frac{\partial}{\partial t^{*}} + u_{G}^{*} \frac{\partial}{\partial x^{*}}, \frac{D_{L}}{D t^{*}} = \frac{\partial}{\partial t^{*}} + u_{L}^{*} \frac{\partial}{\partial x^{*}},

(3)

where the independent variable $x^{*}$ is the spatial coordinate.

We defined uncoated bubbles by setting $d_{0}^{*} = 0$ ⁠. For the case of uncoated bubbles without translation, (1) coincides with the Keller–Miksis equation.⁶⁷ We treated R^* not only as a time variable but also as a field variable [i.e., $R^{*} = R^{*} (x^{*}, t^{*})$ ], to extend (1) to the case of multiple bubbles. Since $R^{*}$ ⁠, $p_{G}^{*}$ ⁠, $p_{L}^{*}$ ⁠, $P^{*}$ ⁠, $u_{G}^{*}$ ⁠, and $u_{L}^{*}$ in (1) are unknown variables, we should use conservation equations as a continuum model to close the set of equations. As volumetric-averaged equations based on a two-fluid model,⁹⁸ the conservation laws of mass and momentum for the gas and liquid phases are introduced as

\frac{\partial}{\partial t^{*}} (α ρ_{G}^{*}) + \frac{\partial}{\partial x^{*}} (α ρ_{G}^{*} u_{G}^{*}) = 0,

(4)

\frac{\partial}{\partial t^{*}} [(1 - α) ρ_{L}^{*}] + \frac{\partial}{\partial x^{*}} [(1 - α) ρ_{L}^{*} u_{L}^{*}] = 0,

(5)

\frac{\partial}{\partial t^{*}} (α ρ_{G}^{*} u_{G}^{*}) + \frac{\partial}{\partial x^{*}} (α ρ_{G}^{*} u_{G}^{* 2}) + α \frac{\partial p_{G}^{*}}{\partial x^{*}} = F^{*} + D^{*},

(6)

\frac{\partial}{\partial t^{*}} [(1 - α) ρ_{L}^{*} u_{L}^{*}] + \frac{\partial}{\partial x^{*}} [(1 - α) ρ_{L}^{*} u_{L}^{* 2}] + (1 - α) \frac{\partial p_{L}^{*}}{\partial x^{*}} + P^{*} \frac{\partial α}{\partial x^{*}} = - F^{*} - D^{*},

(7)

where α is the void fraction (i.e., volumetric fraction of gas phase; $0 < α < 1$ ⁠), $F^{*}$ is the interfacial momentum transport, and $D^{*}$ is the drag force. The following model of virtual mass force for $F^{*}$ is used:^110,111

F^{*} = - β_{1} α ρ_{L}^{*} (\frac{D_{G} u_{G}^{*}}{D t^{*}} - \frac{D_{L} u_{L}^{*}}{D t^{*}}) - β_{2} ρ_{L}^{*} (u_{G}^{*} - u_{L}^{*}) \frac{D_{G} α}{D t^{*}} - β_{3} α (u_{G}^{*} - u_{L}^{*}) \frac{D_{G} ρ_{L}^{*}}{D t^{*}},

(8)

where for a spherical bubble, the values of the coefficients β₁, β₂, and β₃ are 1/2. For the case of spherical bubbles, D^* is given as follows:^112–114

D^{*} = - \frac{3}{8 R^{*}} α C_{D} ρ_{L}^{*} (u_{G}^{*} - u_{L}^{*}) | u_{G}^{*} - u_{L}^{*} |,

(9)

where $C_{D}$ is the drag coefficient for a single spherical bubble.

To describe thermal conduction at the bubble–liquid interface, the following equation for thermal effect inside bubbles⁹² is introduced:

\frac{D p_{G}^{*}}{D t^{*}} = \frac{3}{R^{*}} [{(κ - 1) λ_{G}^{*} \frac{\partial T_{G}^{*}}{\partial r^{*}} |}_{r^{*} = R^{*}} - κ p_{G}^{*} \frac{D R^{*}}{D t^{*}}],

(10)

where $T_{G}^{*}$ is the temperature of the gas phase, κ is the ratio of specific heats, $r^{*}$ is the radial distance from the center of the bubble, and $λ_{G}^{*}$ is the thermal conductivity of the gas inside the bubble. For the temperature-gradient as the first term on the right-hand side of (10), we use the model proposed by Sugiyama et al.,¹¹⁵

{\frac{\partial T_{G}^{*}}{\partial r^{*}} |}_{r^{*} = R^{*}} = \frac{Re ({\tilde{L}}_{P}^{*}) (T_{0}^{*} - T_{G}^{*})}{{| {\tilde{L}}_{P}^{*} |}^{2}} + \frac{Im ({\tilde{L}}_{P}^{*})}{ω_{B}^{*} {| {\tilde{L}}_{P}^{*} |}^{2}} \frac{D_{G} T_{G}^{*}}{D t^{*}},

(11)

where $T_{0}^{*}$ is the initial temperature. Furthermore, the resonant frequency of the linear spherical symmetric oscillation of a single bubble, $ω_{B}^{*}$ ⁠, is obtained from (1),

\begin{matrix} ω_{B}^{* 2} = \frac{3 K^{*} + 4 G^{*}}{ρ_{L 0}^{*} R_{0}^{* 2}} {4 G^{*} \frac{3 K^{*}}{3 K^{*} + 4 G^{*}} \frac{R_{0}^{* 3} - {(R_{0}^{*} - d_{0}^{*})}^{3}}{R_{0}^{* 3}} \\ + (3 γ_{e} p_{G 0}^{*} - \frac{2 σ_{1}^{*}}{R_{0}^{*} - d_{0}^{*}}) [1 - \frac{4 G^{*}}{3 K^{*} + 4 G^{*}} \frac{R_{0}^{* 3} - {(R_{0}^{*} - d_{0}^{*})}^{3}}{R_{0}^{* 3}}]} \\ \times {[(3 γ_{e} p_{G 0}^{*} - \frac{2 σ_{1}^{*}}{R_{0}^{*} - d_{0}^{*}}) \frac{R_{0}^{* 3} - {(R_{0}^{*} - d_{0}^{*})}^{* 3}}{R_{0}^{* 3}} + 4 G^{*} + 3 K^{*} \frac{{(R_{0}^{*} - d_{0}^{*})}^{3}}{R_{0}^{* 3}}]}^{- 1} - \frac{2 σ_{2}^{*}}{ρ_{L 0}^{*} R_{0}^{* 3}} - {(\frac{2 μ_{e 0}^{*}}{ρ_{L 0}^{*} R_{0}^{* 2}})}^{2}, \end{matrix}

(12)

with the effective polytropic exponent $γ_{e}$ ⁠, the initial effective viscosity $μ_{e 0}^{*}$ ⁠, and the complex numbers $Γ_{N}$ and $α_{N}$ are given by

γ_{e} = Re (\frac{Γ_{N}}{3}),

(13)

μ_{e 0}^{*} = μ_{L}^{*} + Im (\frac{p_{G 0}^{*} Γ_{N}}{4 ω_{B}^{*}}),

(14)

Γ_{N} = \frac{3 α_{N}^{2} κ}{α_{N}^{2} + 3 (κ - 1) (α_{N} coth α_{N} - 1)},

(15)

α_{N} = \sqrt{\frac{κ ω_{B}^{*} p_{G 0}^{*} R_{0}^{* 2}}{2 (κ - 1) T_{0}^{*} λ_{G}^{*}}} (1 + i),

(16)

where $i$ denotes the imaginary unit, and the complex number with the length dimension, ${\tilde{L}}_{P}^{*}$ in (11), is given as

{\tilde{L}}_{P}^{*} = \frac{R_{0}^{*} (α_{N}^{2} - 3 α_{N} coth α_{N} + 3)}{α_{N}^{2} (α_{N} coth α_{N} - 1)} .

(17)

Finite amplitude propagation of pressure waves in liquids containing a large number of oscillating coated bubbles with a finite amplitude is the present target. For simplicity, we utilize the linear resonant frequency for the single bubble $ω_{B}^{*}$ to distinguish low- and high-frequency bands as shown in (33), we provide some comments for the linear resonant frequency for interacting bubbles. Guédra¹⁰² proposed an analytical form for the resonance frequency of the interacting bubbles by using $ω_{B}^{*}$ and coupling strength. Yasui et al.⁸⁸ proposed a simple parameter called the coupling strength describing an approximation of the pressure of other bubbles at the location of each bubble, which is calculated from the distance between the bubbles and the number of bubbles. In a forthcoming paper, we will incorporate these effects in $ω_{B}^{*}$ ⁠.

Equations (1), (4)–(7), and (10) contain ten unknown variables, i.e., $R^{*}$ ⁠, $p_{G}^{*}$ ⁠, $p_{L}^{*}$ ⁠, $P^{*}$ ⁠, $u_{G}^{*}$ ⁠, $u_{L}^{*}$ ⁠, α, $ρ_{G}^{*}, ρ_{L}^{*}$ ⁠, and $T_{G}^{*}$ ⁠, and should be closed by the Tait equation of state for the liquid phase, the equation of state for an ideal gas, and the conservation equation of mass inside the bubble, as follows:

p_{L}^{*} = p_{L 0}^{*} + \frac{ρ_{L 0}^{*} c_{L 0}^{* 2}}{n} [{(\frac{ρ_{L}^{*}}{ρ_{L 0}^{*}})}^{n} - 1],

(18)

\frac{p_{G}^{*}}{p_{G 0}^{*}} = \frac{ρ_{G}^{*}}{ρ_{G 0}^{*}} \frac{T_{G}^{*}}{T_{0}^{*}},

(19)

\frac{ρ_{G}^{*}}{ρ_{G 0}^{*}} = {(\frac{R_{0}^{*}}{R^{*}})}^{3},

(20)

where n is the material constant (e.g., n = 7.15 for water).

Note that our previous paper⁸⁹ omitted the compressibility of the shell and liquid, drag force acting on bubbles, translation of bubbles, and thermal conductivity.

C. Multiple scales analysis

Independent variables are first nondimensionalized as

t = \frac{t^{*}}{T^{*}}, x = \frac{x^{*}}{L^{*}},

(21)

where $T^{*}$ and $L^{*}$ are the typical period and wavelength, respectively. Using t and x, the near-field, far-field I, and far-field II are described by extended independent variables in the method of multiple scales as follows:¹¹⁶

t_{0} = ϵ^{0} t, x_{0} = ϵ^{0} x, (near - field),

(22)

t_{1} = ϵ^{1} t, x_{1} = ϵ^{1} x, (far - field I),

(23)

t_{2} = ϵ^{2} t, x_{2} = ϵ^{2} x, (far - field II),

(24)

where ϵ is a nondimensional wave amplitude $(0 < ϵ ≪ 1$ ⁠, i.e., a weakly nonlinear problem¹¹⁶). The differential operators are immediately expanded as

\frac{\partial}{\partial t} = \sum_{i = 0}^{2} ϵ^{i} \frac{\partial}{\partial t_{i}}, \frac{\partial}{\partial x} = \sum_{i = 0}^{2} ϵ^{i} \frac{\partial}{\partial x_{i}} .

(25)

It is worth noting that far-field II is not used in Sec. III.

The dependent variables are now regarded as functions of (22)–(24) and expanded in a power series in ϵ as follows:

\frac{α}{α_{0}} = 1 + ϵ α_{1} + ϵ^{2} α_{2} + ϵ^{3} α_{3} + O (ϵ^{4}),

(26)

\frac{u_{G}^{*}}{U^{*}} = u_{G 0} + ϵ u_{G 1} + ϵ^{2} u_{G 2} + ϵ^{3} u_{G 3} + O (ϵ^{4}),

(27)

\frac{u_{L}^{*}}{U^{*}} = u_{L 0} + ϵ u_{L 1} + ϵ^{2} u_{L 2} + ϵ^{3} u_{L 3} + O (ϵ^{4}),

(28)

\frac{R^{*}}{R_{0}^{*}} = 1 + ϵ R_{1} + ϵ^{2} R_{2} + ϵ^{3} R_{3} + O (ϵ^{4}),

(29)

\frac{ρ_{L}^{*}}{ρ_{L 0}^{*}} = {\begin{matrix} 1 + ϵ^{2} ρ_{L 1} + ϵ^{3} ρ_{L 2} + O (ϵ^{4}), (KdVB), \\ 1 + ϵ^{5} ρ_{L 1} + ϵ^{6} ρ_{L 2} + O (ϵ^{7}), (NLS), \end{matrix}

(30)

\frac{p_{L}^{*}}{ρ_{L 0}^{*} U^{* 2}} = p_{L 0} + ϵ p_{L 1} + ϵ^{2} p_{L 2} + ϵ^{3} p_{L 3} + O (ϵ^{4}),

(31)

\frac{T_{G}^{*}}{T_{0}^{*}} = 1 + ϵ T_{G 1} + ϵ^{2} T_{G 2} + ϵ^{3} T_{G 3} + O (ϵ^{4}),

(32)

where $U^{*} (\equiv L^{*} / T^{*})$ is the typical propagation speed of a wave, and the nondimensional pressures for the gas and liquid phases of O(1) are given as $p_{G 0} = p_{G 0}^{*} / (ρ_{L 0}^{*} U^{* 2})$ and $p_{L 0} = p_{L 0}^{*} / (ρ_{L 0}^{*} U^{* 2})$ ⁠. The ratio of the initial densities of the gas and liquid phases is $ρ_{G 0}^{*} / ρ_{L 0}^{*} = O (ϵ^{3})$ ⁠. The effect of initial polydispersity^117–119 is neglected in this paper for simplicity. The expansion of liquid density is categorized into two cases.⁹⁷

The scaling relations of the nondimensional ratios among the physical parameters appropriate for the low frequency and long wave and the high frequency and short wave are defined as follows:⁹⁷

(\frac{R_{0}^{*}}{L^{*}}, \frac{ω^{*}}{ω_{B}^{*}}, \frac{U^{*}}{c_{L 0}^{*}}) \equiv {\begin{matrix} (O (\sqrt{ϵ}), O (\sqrt{ϵ}), O (\sqrt{ϵ})) \equiv (Δ \sqrt{ϵ}, Ω \sqrt{ϵ}, V \sqrt{ϵ}), (KdVB), \\ (O (1), O (1), O (ϵ^{2})) \equiv (Δ, Ω, V ϵ^{2}), (NLS), \end{matrix}

(33)

where $ω^{*}$ is a typical angular frequency of an incident wave, and Δ, Ω, and V are constants of O(1). Furthermore, the shell shear viscosity, viscosity describing the friction losses due to volume change, shell shear modulus, shell bulk modulus, and shell thickness are nondimensionalized as follows (see also Ref. 89):

\begin{matrix} \frac{μ_{S}^{*}}{ρ_{L 0}^{*} U^{* 2} T^{*}} = {\begin{matrix} O (1) = μ_{S}, (KdVB), \\ O (ϵ) = ϵ μ_{S}, (NLS), \end{matrix} \\ \frac{μ_{K}^{*}}{ρ_{L 0}^{*} U^{* 2} T^{*}} = {\begin{matrix} O (1) = μ_{K}, (KdVB), \\ O (ϵ) = ϵ μ_{K}, (NLS), \end{matrix} \\ \frac{G^{*}}{ρ_{L 0}^{*} U^{* 2}} = G, \frac{K^{*}}{ρ_{L 0}^{*} U^{* 2}} = K, \frac{d_{0}^{*}}{R_{0}^{*}} = ϵ d_{0}, \end{matrix}

(34)

where $μ_{S}$ ⁠, $μ_{K}$ ⁠, G, K, and $d_{0}$ are constants of O(1).

The nondimensional liquid viscosity $μ_{L}^{}$ is defined as follows:⁹⁷

\frac{μ_{L}^{*}}{ρ_{L 0}^{*} U^{* 2} T^{*}} = {\begin{matrix} O (ϵ) = ϵ μ_{L}, (KdVB), \\ O (ϵ^{2}) = ϵ^{2} μ_{L}, (NLS), \end{matrix}

(35)

where $μ_{L}$ is a constant of O(1). The nondimensionalization of initial effective viscosity and some parameters appearing in the energy equation are as follows:^113,120,121

\frac{μ_{e 0}^{*}}{ρ_{L 0}^{*} U^{* 2} T^{*}} = {\begin{matrix} O (ϵ) = ϵ μ_{e 0}, (KdVB), \\ O (ϵ^{2}) = ϵ^{2} μ_{e 0}, (NLS), \end{matrix}

(36)

\frac{3 (κ - 1) λ_{G}^{*}}{p_{G 0}^{*} ω^{*} R_{0}^{*}} \frac{Re ({\tilde{L}}_{P}^{*}) T_{0}^{*}}{{| {\tilde{L}}_{P}^{*} |}^{2}} = {\begin{matrix} O (ϵ) = ϵ ζ_{STM 1}, (KdVB), \\ O (ϵ^{2}) = ϵ^{2} ζ_{STM 1}, (NLS), \end{matrix}

(37)

\frac{3 (κ - 1) λ_{G}^{*}}{p_{G 0}^{*} ω^{*} R_{0}^{*}} \frac{ω^{*} Im ({\tilde{L}}_{P}^{*}) T_{0}^{*}}{ω_{B}^{*} {| {\tilde{L}}_{P}^{*} |}^{2}} = {\begin{matrix} O (ϵ) = ϵ ζ_{STM 2}, (KdVB), \\ O (ϵ^{3}) = ϵ^{3} ζ_{STM 2}, (NLS), \end{matrix}

(38)

where $μ_{L}$ ⁠, $ζ_{STM 1}$ ⁠, and $ζ_{STM 2}$ are constants of O(1). The definition of the drag coefficient is as follows:¹¹⁴

C_{D} = {\begin{matrix} O (\sqrt{ϵ}) \equiv Λ \sqrt{ϵ}, (KdVB), \\ O (1) = 8 μ_{L}^{*} / | u_{G}^{*} - u_{L}^{*} | ρ_{L}^{*} R^{*}, (NLS), \end{matrix}

(39)

where Λ is a constant of O(1).

III. LOW FREQUENCY AND LONG WAVE

First, we focus on the low frequency and long wave that can be regarded as a weakly dispersive band in comparison with the analysis in Sec. IV and derive the KdVB equation. Our previous KdVB equation⁹⁰ is successfully generalized by considering the additional effects, i.e., of drag force acting bubbles, thermal effect inside bubbles, and initial flow velocitiy; furthermore, the present KdVB equation is derived from two-fluid model (not mixture) equations to incorporate α₀ dependence.¹²³ The derivation is performed by using the same mathematical procedure in our previous papers.^89,90

A. $O (ϵ)$ ⁠: Linear propagation at near-field

Substituting (25)–(39) into (1) and (4)–(7) and equating coefficients of like powers of ϵ in the set of resultant equations, we have the following set of linearized first-order equations:

\frac{D α_{1}}{D t_{0}} - 3 \frac{D R_{1}}{D t_{0}} + \frac{\partial u_{G 1}}{\partial x_{0}} = 0,

(40)

α_{0} \frac{D α_{1}}{D t_{0}} - (1 - α_{0}) \frac{\partial u_{L 1}}{\partial x_{0}} = 0,

(41)

β_{1} (\frac{D u_{G 1}}{D t_{0}} - \frac{D u_{L 1}}{D t_{0}}) - 3 p_{G 0} \frac{\partial R_{1}}{\partial x_{0}} + p_{G 0} \frac{\partial T_{G 1}}{\partial x_{0}} = 0,

(42)

(1 - α_{0}) \frac{D u_{L 1}}{D t_{0}} - α_{0} β_{1} (\frac{D u_{G 1}}{D t_{0}} - \frac{D u_{L 1}}{D t_{0}}) - α_{0} u_{0} \frac{D α_{1}}{D t_{0}} + u_{0} (1 - α_{0}) \frac{\partial u_{L 1}}{\partial x_{0}} + (1 - α_{0}) \frac{\partial p_{L 1}}{\partial x_{0}} = 0,

(43)

[3 (γ_{e} - 1) p_{G 0} - \frac{Δ^{2}}{Ω^{2}}] R_{1} + p_{G 0} T_{G 1} - p_{L 1} + \frac{4 μ_{e 0}^{2}}{Δ^{2}} = C_{1},

(44)

\frac{D T_{G 1}}{D t_{0}} + 3 (κ - 1) \frac{D R_{1}}{D t_{0}} = 0.

(45)

The following symbol is introduced as

C_{1} = \frac{- 3 d_{0}}{3 K + 4 G} [4 G p_{G 0} (3 γ_{e} - 1) - 3 K (3 γ_{e} p_{G 0} - σ_{1}) + {(3 γ_{e} p_{G 0} - σ_{1})}^{2} - 12 G K],

(46)

and this also appears in Sec. IV, where

σ_{1} = \frac{2 σ_{1}^{*}}{ρ_{L 0}^{*} U^{* 2} R_{0}^{*}} .

(47)

Eliminating α₁, $u_{G 1}$ ⁠, $u_{L 1}$ ⁠, and $p_{L 1}$ from (40)–(45), the single linear wave equation for first-order perturbations of bubble radius, R₁, is obtained

\frac{D^{2} R_{1}}{D t_{0}^{2}} - v_{p}^{2} \frac{\partial^{2} R_{1}}{\partial x_{0}^{2}} = 0,

(48)

with the phase velocity $v_{p}$ ⁠,

v_{p} = \sqrt{\frac{3 α_{0} (1 - α_{0} + β_{1}) κ p_{G 0} - (1 - α_{0}) β_{1} s_{4}}{3 (1 - α_{0}) β_{1} α_{0}}},

(49)

with the constant coefficient,

s_{4} = 3 (γ_{e} - κ) p_{G 0} - \frac{Δ^{2}}{Ω^{2}} .

(50)

The definition of the linear Lagrange derivative $D / D t_{0}$ is

\frac{D}{D t_{0}} = \frac{\partial}{\partial t_{0}} + u_{0} \frac{\partial}{\partial x_{0}} .

(51)

For simplicity, the initial velocities of both phases are assumed to be equal (i.e., $u_{G 0} = u_{L 0} = u_{0}$ ⁠). However, the perturbations of each velocity are not equal [i.e., $u_{G i} \neq u_{L i} (i = 1, 2)$ ].

Setting $v_{p} = 1$ gives the explicit form of $U^{*}$ as

U^{*} = \sqrt{\frac{3 α_{0} (1 - α_{0} + β_{1}) κ p_{G 0}^{*} / ρ_{L 0}^{*} - (1 - α_{0}) β_{1} \hat{s_{4}}}{3 β_{1} α_{0} (1 - α_{0})}},

(52)

with

\hat{s_{4}} = 3 (γ_{e} - κ) \frac{p_{G 0}^{*}}{ρ_{L 0}^{*}} - R_{0}^{* 2} ω_{B}^{* 2} .

(53)

Focusing on the right-running wave alone, a phase function $φ_{0}$ is defined as

φ_{0} \equiv x_{0} - (v_{p} + u_{0}) t_{0} .

(54)

All first-order perturbations (i.e., α₁, $u_{G 1}, u_{L 1}, p_{L 1}$ ⁠, and $T_{G 1}$ ⁠) are expressed with regard to R₁ as follows:

α_{1} = s_{1} R_{1} = 3 (1 - α_{0}) - \frac{κ {(1 - α_{0})}^{2} (3 α_{0} - Δ^{2} / Ω^{2})}{α_{0} (1 - α_{0} + β_{1}) κ - (γ_{e} - κ) (1 - α_{0}) β_{1}} R_{1},

(55)

u_{G 1} = s_{2} R_{1} = v_{p} (s_{1} - 3) R_{1},

(56)

u_{L 1} = s_{3} R_{1} = - v_{p} \frac{α_{0}}{1 - α_{0}} s_{1} R_{1},

(57)

p_{L 1} = s_{4} R_{1} - C_{1} + \frac{4 μ_{e 0}^{2}}{Δ^{2}},

(58)

T_{G 1} = s_{5} R_{1} = - 3 (κ - 1) R_{1} .

(59)

As a result, the dissipation and dispersion effects of waves and the effect of bubble translation and drag force do not appear in the leading order of approximation. The resultant physics of wave propagation are governed by the linear wave equation in (48) (including linear advection in $D / D t_{0}$ ⁠).

B. $O (ϵ^{2})$ ⁠: Nonlinear propagation at far-field

As in the case of Sec. III A, from the second-order approximation, the following single inhomogeneous equation for R₂ is derived:

\begin{matrix} \frac{D^{2} R_{2}}{D t_{0}^{2}} - v_{p}^{2} \frac{\partial^{2} R_{2}}{\partial x_{0}^{2}} = - \frac{1}{3} \frac{D K_{1}}{D t_{0}} + \frac{1}{3 α_{0}} \frac{D K_{2}}{D t_{0}} + \frac{u_{0}}{3 α_{0} (1 - α_{0})} \frac{\partial K_{2}}{\partial x_{0}} + \frac{1 - α_{0} + β_{1}}{3 (1 - α_{0}) β_{1}} \frac{\partial K_{3}}{\partial x_{0}} + \frac{1}{3 α_{0} (1 - α_{0})} \frac{\partial K_{4}}{\partial x_{0}} \\ + \frac{1}{3 α_{0}} \frac{\partial^{2} K_{5}}{\partial x_{0}^{2}} - \frac{p_{G 0} [α_{0} (1 - α_{0}) + β_{1}]}{3 β_{1} α_{0} (1 - α_{0})} \int \frac{\partial^{2} K_{6}}{\partial x_{0}^{2}} d t_{0} \\ = K (R_{1}; φ_{0}, t_{1}, x_{1}), \end{matrix}

(60)

where the explicit forms of the inhomogeneous terms $K_{i}$ (⁠ $i = 3, 4, 5, 6$ ⁠) are presented in Appendix A, and $K_{1}$ and K₂ are the same as the counterparts in Ref. 118. The solvability condition to the inhomogeneous equation in (60),⁹⁷ K = 0 is imposed. By using (25), the independent variables $φ_{0}$ ⁠, t₁, and x₁ in (60) are restored into t and x

\frac{\partial R_{1}}{\partial t} + (v_{p} + u_{0}) \frac{\partial R_{1}}{\partial x} + ϵ (Π_{0} \frac{\partial R_{1}}{\partial x} + Π_{1} R_{1} \frac{\partial R_{1}}{\partial x} - Π_{2} \frac{\partial^{2} R_{1}}{\partial x^{2}} + Π_{3} \frac{\partial^{3} R_{1}}{\partial x^{3}} + Π_{4} R_{1} + Π_{5} | R_{1} | R_{1}) = 0.

(61)

Finally, we have the KdVB equation with correction (i.e., fifth and sixth) terms

\frac{\partial R_{1}}{\partial τ} + Π_{1} R_{1} \frac{\partial R_{1}}{\partial ξ} - Π_{2} \frac{\partial^{2} R_{1}}{\partial ξ^{2}} + Π_{3} \frac{\partial^{3} R_{1}}{\partial ξ^{3}} + Π_{4} R_{1} + Π_{5} | R_{1} | R_{1} = 0,

(62)

via the variable transformation

τ \equiv ϵ t, ξ \equiv x - (v_{p} + u_{0} + ϵ Π_{0}) t .

(63)

Equation (62) describes the competition of nonlinearity, dissipation due to five factors, dispersion, thermal dissipation, and dissipation due to drag force. Here, $Π_{0}$ is the advection coefficient (i.e., the propagation speed in a moving coordinate), $Π_{1}$ is the nonlinear coefficient for the second order nonlinear effect, $Π_{2}$ is the dissipation coefficient, $Π_{3}$ is the dispersion coefficient, $Π_{4}$ is the dissipation coefficient due to thermal conduction,¹²⁰ and $Π_{5}$ is the nonlinear dissipation coefficient due to the drag force.¹¹⁴ The explicit form of the nonlinear coefficient $Π_{1}$ is presented in Appendix B, and the other coefficients $Π_{i} (i = 0, 2, 3, 4, 5)$ are given by

Π_{0} = - \frac{v_{p} (1 - α_{0}) V^{2} s_{4}}{6 α_{0}} - \frac{1}{3 α_{0}} \frac{2 μ_{e 0}^{2}}{Δ^{2}} - \frac{C_{2}}{6 α_{0}} - \frac{3 (κ - 1)}{2} \frac{[α_{0} (1 - α_{0}) + β_{1}] p_{G 0}}{3 β_{1} α_{0} (1 - α_{0})} ζ_{STM 2},

(64)

Π_{2} = \underset{liquid viscosity}{\underset{︸}{Π_{21}}} + \underset{liquid compressibility}{\underset{︸}{Π_{22}}} + \underset{shell viscosity}{\underset{︸}{Π_{23}}} + \underset{shell compressibility}{\underset{︸}{Π_{24}}} + \underset{thermal}{\underset{︸}{Π_{25}}},

(65)

Π_{3} = \frac{Δ^{2}}{6 α_{0}} > 0,

(66)

Π_{4} = \frac{p_{G 0} [α_{0} (1 - α_{0}) + β_{1}]}{2 β_{1} α_{0} (1 - α_{0}) v_{p}^{2}} (κ - 1) ζ_{STM 1} > 0,

(67)

Π_{5} = \frac{Λ}{16 v_{p} β_{1} Δ} (s_{3} - s_{2}) | s_{3} - s_{2} | > 0.

(68)

Here, $Π_{2}$ and $Π_{3}$ are positive, whereas $Π_{0}$ and $Π_{1}$ are negative. It should be noted that obtaining the analytical expression of the signs of $Π_{0}$ and $Π_{1}$ is challenging. In (64), the symbol C₂ is introduced as

C_{2} = \frac{3 d_{0}}{3 K + 4 G} [{(3 γ_{e} p_{G0} - σ_{1})}^{2} - 3 K (3 γ_{e} p_{G0} - σ_{1})] + [3 (γ_{e} - κ) - 1] \frac{12 G p_{G0} d_{0}}{3 K + 4 G} - (1 - \frac{12 G}{3 K + 4 G}) σ_{1} d_{0},

(69)

and this also appears in Sec. IV.

The three coefficients, the thermal conduction $Π_{4}$ ⁠, the drag force $Π_{5}$ ⁠, and other effects $Π_{2}$ ⁠, representing the dissipation are decomposed into

Π_{21} = \frac{2 μ_{L}}{3 α_{0}},

(70)

Π_{22} = \frac{V Δ^{3}}{6 α_{0} Ω^{2}},

(71)

Π_{23} = \frac{2 μ_{S}}{3 α_{0}} (1 - \frac{4 G}{3 K + 4 G}) d_{0},

(72)

Π_{24} = \frac{4 μ_{K}}{3 α_{0}} \frac{3 G}{3 K + 4 G} d_{0},

(73)

Π_{25} = \frac{(κ - γ_{e}) V Δ p_{G 0}}{2 α_{0}} .

(74)

The first, second, third, fourth, and fifth terms in (65), $Π_{2 i}$ (⁠ $i = 1, 2, 3, 4, 5$ ⁠), describe the dissipation due to the liquid viscosity at the interface, liquid compressibility, shell viscosity, shell compressibility, and thermal effect (not thermal conduction), respectively. The detailed discussion is presented in Sec. V A.

Note that our previous KdVB equation^89,90 neglected the thermal effect, drag force acting bubbles, bubble translation, and initial flow velocity.

IV. HIGH FREQUENCY AND SHORT WAVE

The most important difference between our previous⁹⁰ and present papers is the focus on the high frequency and short wave that can be regarded as a strongly dispersive band in comparison with the analysis in Sec. III A. Weakly nonlinear modulation of the carrier wave is described by an envelope wave. Although some previous studies derived the NLS equation for the case of liquids containing uncoated bubbles (e.g., Refs. 97 and 113), here, for the first time, we successfully derive the NLS equation for liquids containing multiple coated bubbles. The focus on the wave frequency band near the resonance frequency of bubble oscillations will be a spark in medical applications.

A. $O (ϵ)$ ⁠: Linear propagation of carrier wave at near-field

As in the leading order of approximation in the low frequency case (Sec. III A), we first have the following set of linear equations:

\frac{D α_{1}}{D t_{0}} - 3 \frac{D R_{1}}{D t_{0}} + \frac{\partial u_{G 1}}{\partial x_{0}} = 0,

(75)

α_{0} \frac{D α_{1}}{D t_{0}} - (1 - α_{0}) \frac{\partial u_{L 1}}{\partial x_{0}} = 0,

(76)

β_{1} (\frac{D u_{G 1}}{D t_{0}} - \frac{D u_{L 1}}{D t_{0}}) + p_{G 0} (\frac{\partial T_{G 1}}{\partial x_{0}} - 3 \frac{\partial R_{1}}{\partial x_{0}}) = 0,

(77)

\begin{matrix} (1 - α_{0}) \frac{D u_{L 1}}{D t_{0}} - α_{0} β_{1} (\frac{D u_{G 1}}{D t_{0}} - \frac{D u_{L 1}}{D t_{0}}) - α_{0} u_{0} \frac{D α_{1}}{D t_{0}} \\ + u_{0} (1 - α_{0}) \frac{\partial u_{L 1}}{\partial x_{0}} + (1 - α_{0}) \frac{\partial p_{L 1}}{\partial x_{0}} = 0, \end{matrix}

(78)

\frac{D^{2} R_{1}}{D t_{0}^{2}} - [\frac{3 (γ_{e} - 1) p_{G 0}}{Δ^{2}} - 1] R_{1} - \frac{p_{G 0}}{Δ^{2}} T_{G 1} + \frac{p_{L 1}}{Δ^{2}} = - \frac{C_{1}}{Δ^{2}},

(79)

3 (κ - 1) \frac{D R_{1}}{D t_{0}} + \frac{D T_{G 1}}{D t_{0}} = 0.

(80)

Here, C₁ is the same as that in (46). The difference between the present case (i.e., high-frequency short wave) and the previous case (low-frequency long wave; Sec. III A) is the presence of the second-order derivative of R₁ on the left-hand side of (79), because the present case has a strong dispersive effect compared with the previous case [see (44)]. As in the previous case, although the initial velocities of both phases are assumed to be the same, the perturbations of each velocity are not equal [i.e., $u_{G i} \neq u_{L i}$ (⁠ $i = 1, 2, 3$ ⁠)].

Equations (75)–(80) are reduced to the following linear equation with a dispersion term:

L_{1} [R_{1}] = 0,

(81)

\begin{matrix} L_{1} \equiv \frac{D^{2}}{D t_{0}^{2}} - \frac{3 p_{G 0} {[κ (1 - α_{0}) + β_{1}] α_{0} + β_{1} (κ - γ_{e})} + β_{1} Δ^{2} (1 - α_{0})}{3 β_{1} α_{0} (1 - α_{0})} \frac{\partial^{2}}{\partial x_{0}^{2}} - \frac{Δ^{2}}{3 α_{0}} \frac{\partial^{2}}{\partial x_{0}^{2}} (\frac{D^{2}}{D t_{0}^{2}}) . \end{matrix}

(82)

The dispersion term (i.e., last term) on the left-hand side of (82) due to the dispersion term in the linearized Keller equation (79) did not appear in the previous case [see (48) in Sec. III A].

We assumed the solution of (81) as a quasi-monochromatic wave train, which evolves into a slowly modulated wave packet^97,116

R_{1} = A (t_{1}, x_{1}; t_{2}, x_{2}) e^{i θ} + c . c .,

(83)

θ \equiv k x_{0} - Ω t_{0},

(84)

where A is the complex amplitude with a slow variation, which depends only on slow scales [i.e., t_i and x_i (⁠ $i = 1, 2$ ⁠)], and is obviously a constant in the near-field characterized by fast scales (i.e., t₀ and x₀); $k (\equiv k^{*} L^{*} = k^{*} U^{*} / ω_{B}^{*})$ is the nondimensionalized wavenumber; and $c . c .$ denotes the complex conjugate. Hereafter, two types of waves, $e^{i θ}$ corresponding to the carrier wave and A to the envelope, are considered.⁹⁷ The following linear dispersion relation connects Ω and k:

\begin{matrix} D (k, Ω) \equiv \frac{Δ^{2} k^{2}}{3 α_{0}} [1 - {(Ω - u_{0} k)}^{2}] - {(Ω - u_{0} k)}^{2} \\ + \frac{[β_{1} + κ (1 - α_{0})] α_{0} + β_{1} (κ - γ_{e})}{β_{1} α_{0} (1 - α_{0})} p_{G 0} k^{2} = 0, \end{matrix}

(85)

which is rewritten explicitly as

Ω = u_{0} k + k \sqrt{\frac{Δ^{2}}{3 α_{0} + Δ^{2} k^{2}} + \frac{3 p_{G 0} {[β_{1} + κ (1 - α_{0})] α_{0} + β_{1} (κ - γ_{e})}}{β_{1} α_{0} (1 - α_{0}) (3 α_{0} + Δ^{2} k^{2})}} .

(86)

The positive Ω in (86) corresponds to the right-running carrier wave. The nondimensional phase velocity $V_{p}$ is readily obtained as

V_{P} \equiv \frac{Ω}{k} = v_{p} + u_{0},

(87)

v_{p} \equiv \sqrt{\frac{Δ^{2}}{3 α_{0} + Δ^{2} k^{2}} + \frac{3 p_{G 0} {[β_{1} + κ (1 - α_{0})] α_{0} + β_{1} (κ - γ_{e})}}{β_{1} α_{0} (1 - α_{0}) (3 α_{0} + Δ^{2} k^{2})}} .

(88)

The nondimensional group velocity $V_{g}$ is also readily obtained as

V_{g} \equiv \frac{d Ω}{d k} = - \frac{\partial D / \partial k}{\partial D / \partial Ω} = v_{g} + u_{0}, v_{g} \equiv \frac{3 α_{0} v_{p}}{3 α_{0} + Δ^{2} k^{2}} .

(89)

We finally determine the typical propagation speed of a wave $U^{*}$ by setting $v_{p} = 1$ under Ω = 1,⁹⁷

U^{*} \equiv \sqrt{\frac{{[β_{1} + κ (1 - α_{0})] α_{0} + β_{1} (κ - γ_{e})} p_{G 0}^{*}}{β_{1} α_{0} (1 - α_{0}) ρ_{L 0}^{*}}} .

(90)

The typical length $L^{*} \equiv U^{*} T^{*} = U^{*} / ω_{B}^{*}$ is simultaneously determined.

Substituting (83) into (75)–(80) provides

α_{1} = b_{1} R_{1}, u_{G 1} = b_{2} R_{1}, u_{L 1} = b_{3} R_{1}, p_{L 1} = b_{4} R_{1} - C_{1}, T_{G 1} = b_{5} R_{1},

(91)

with

\begin{matrix} b_{1} = \frac{1 - α_{0}}{α_{0} (1 - α_{0} + β_{1})} [3 α_{0} β_{1} - \frac{(1 - α_{0}) b_{4}}{v_{p}^{2}}], b_{2} = v_{p} (b_{1} - 3), \\ b_{3} = - v_{p} \frac{α_{0} b_{1}}{1 - α_{0}}, b_{4} = Δ^{2} (v_{p}^{2} k^{2} - 1) - 3 (κ - γ_{e}) p_{G 0}, b_{5} = - 3 (κ - 1) . \end{matrix}

(92)

As a result, the dispersion effect appeared in the near-field, in contrast to Sec. III A.

B. $O (ϵ^{2})$ ⁠: Linear propagation of envelope wave at far-field I

From the second-order of approximation, we have the following single inhomogeneous equation for R₂:

\begin{matrix} L_{1} [R_{2}] = M (R_{1}) \\ = - \frac{1}{3} \frac{D M_{1}}{D t_{0}} + \frac{1}{3 α_{0}} \frac{D M_{2}}{D t_{0}} + \frac{u_{0}}{3 α_{0} (1 - α_{0})} \frac{\partial M_{2}}{\partial x_{0}} + \frac{1 - α_{0} + β_{1}}{3 (1 - α_{0}) β_{1}} \frac{\partial M_{3}}{\partial x_{0}} + \frac{1}{3 α_{0} (1 - α_{0})} \frac{\partial \hat{M_{4}}}{\partial x_{0}} - \frac{Δ^{2}}{3 α_{0}} \frac{\partial^{2} \hat{M_{5}}}{\partial x_{0}^{2}} - \frac{[β_{1} + α_{0} (1 - α_{0})] p_{G 0}}{3 β_{1} α_{0} (1 - α_{0})} \int \frac{\partial^{2} M_{6}}{\partial x_{0}^{2}} d t_{0}, \end{matrix}

(93)

with

\hat{M_{4}} = M_{4} + α_{0} C_{1} \frac{\partial α_{1}}{\partial x_{0}},

(94)

\hat{M_{5}} = M_{5} - \frac{C_{2}}{Δ^{2}} R_{1} - \frac{C_{3}}{Δ^{2}},

(95)

where M_i (⁠ $i = 1, 2, 3, 4, 5, 6$ ⁠) are the same as the counterparts in our previous case for uncoated bubbles without drag force.¹¹³ Substituting (83) and (91) into (93), the single inhomogeneous term M can be obtained as

M = Γ A^{2} e^{2 i θ} + i (- \frac{\partial D}{\partial Ω}) [\frac{\partial A}{\partial t_{1}} + (v_{G} + u_{0}) \frac{\partial A}{\partial x_{1}} + i η A] e^{i θ} + c . c .,

(96)

with

η \equiv - \frac{1}{\partial D / \partial Ω} \frac{k^{2}}{3 α_{0}} C_{2},

(97)

where the real constant Γ is explicitly shown in Appendix B.

From the solvability condition for (96), the coefficient of $e^{i θ}$ on the right-hand side of (96) must be zero. Thus, we have a linear wave equation with a correction (i.e., third) term for the envelope wave A:

\frac{\partial A}{\partial t_{1}} + (v_{g} + u_{0}) \frac{\partial A}{\partial x_{1}} + i η A = 0.

(98)

Substituting (98) into (93) yields

L_{1} [R_{2}] = Γ A^{2} e^{2 i θ} + c . c .,

(99)

and its solution is obtained as follows:⁹⁷

R_{2} = c_{0} A^{2} e^{2 i θ} + c . c .,

(100)

with

c_{0} \equiv \frac{Γ}{D_{22}}, D_{22} \equiv D (2 k, 2 Ω) = - \frac{4 Δ^{2} v_{p}^{2} k^{4}}{α_{0}} .

(101)

Using (100) to set the second-order equations, we have the other second-order perturbations as follows:

(\begin{matrix} α_{2} \\ u_{G 2} \\ u_{L 2} \\ p_{L 2} \\ T_{G 2} \end{matrix}) = (\begin{matrix} c_{1} & d_{1} & e_{1} & 0 & 0 \\ c_{2} & d_{2} & e_{2} & 0 & 0 \\ c_{3} & d_{3} & e_{3} & 0 & 0 \\ c_{4} & d_{4} & e_{4} & {- C}_{3} & c_{s} \\ c_{5} & 0 & 0 & 0 & 0 \end{matrix}) (\begin{matrix} A^{2} e^{2 i θ} + c . c . \\ i (\partial A / \partial t_{1}) e^{i θ} + c . c . \\ A e^{i θ} + c . c . \\ 1 \\ | A |^{2} \end{matrix}),

(102)

with the new real constants e_i (⁠ $i = 1, 2, 3, 4$ ⁠) given by

e_{1} = - \frac{(1 - α_{0}) e_{3}}{α_{0} v_{p}} - \frac{b_{1} u_{0} η}{(u_{0} + v_{g}) v_{p} k} + \frac{(1 - α_{0}) b_{3} η}{(u_{0} + v_{g}) α_{0} v_{p} k},

(103)

e_{2} = e_{1} v_{p} + \frac{b_{1} - 3}{k} \frac{u_{0}}{u_{0} + v_{g}} η + \frac{b_{2}}{k} \frac{η}{u_{0} + v_{g}},

(104)

e_{3} = \frac{e_{4}}{v_{p}} + \frac{η}{(v_{p} + u_{0}) v_{p}} [α_{0} (3 - b_{5}) p_{G 0} - (1 - α_{0}) b_{4} - u_{0} (1 - α_{0}) b_{3}],

(105)

e_{4} = {- C}_{2} + 2 \frac{u_{0}}{u_{0} + v_{g}} v_{p} k η Δ^{2} .

(106)

Here, the explicit forms of the real constants are c_i (⁠ $i = 1, 2, 3, 4, 5$ ⁠), d_i (⁠ $i = 1, 2, 3, 4$ ⁠), and $c_{s}$ are the same as those in our previous uncoated case.¹¹³

Although the present result clarified the nonlinear propagation of carrier waves, the envelope wave was restricted to linear propagation. Furthermore, the wave propagation was not affected by the thermal effect, drag force, and bubble translation. We now proceed to the next order of approximation to elucidate the nonlinear propagation of envelope waves.

FIG. 1.

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Ultrasound propagation in liquid containing multiple spherical microbubbles coated by a visco-elastic shell.

FIG. 2.

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Conceptual illustration of an coated bubble, where R^* is the bubble radius, d₀^* is the initial shell thickness, and σ₁^* and σ₂^* are the surface tensions at the internal and external boundaries of the shell, respectively.

C. $O (ϵ^{3})$ ⁠: Nonlinear propagation of envelope wave and resultant NLS equation

The third-order of approximation clarifies the weakly nonlinear propagation process of the envelope wave. The single inhomogeneous equation for R₃ is described as

\begin{matrix} L_{1} [R_{3}] = N (R_{1}, R_{2}) \\ = - \frac{1}{3} \frac{D N_{1}}{D t_{0}} + \frac{1}{3 α_{0}} \frac{D N_{2}}{D t_{0}} + \frac{u_{0}}{3 α_{0} (1 - α_{0})} \frac{\partial N_{2}}{\partial x_{0}} + \frac{1 - α_{0} + β_{1}}{3 (1 - α_{0}) β_{1}} \frac{\partial N_{3}}{\partial x_{0}} \end{matrix}

+ \frac{1}{3 α_{0} (1 - α_{0})} \frac{\partial \hat{N_{4}}}{\partial x_{0}} - \frac{Δ^{2}}{3 α_{0}} \frac{\partial^{2} \hat{N_{5}}}{\partial x_{0}^{2}} - \frac{[β_{1} + α_{0} (1 - α_{0})] p_{G 0}}{3 β_{1} α_{0} (1 - α_{0})} \int \frac{\partial^{2} N_{6}}{\partial x_{0}^{2}} d t_{0},

(107)

where

\hat{N_{4}} = N_{4} - α_{0} (C_{2} R_{1} + C_{3}) \frac{\partial α_{1}}{\partial x_{0}} - α_{0} C_{1} (\frac{\partial α_{2}}{\partial x_{0}} + \frac{\partial α_{1}}{\partial x_{1}}),

(108)

\begin{matrix} \hat{N_{5}} = N_{5} + C_{4} \frac{R_{1}}{Δ^{2}} + C_{5} \frac{R_{1}^{2}}{Δ^{2}} - 3 \frac{4 G}{3 K + 4 G} \frac{p_{G 2} d_{0}}{Δ^{2}} \\ + 3 \frac{4 G}{3 K + 4 G} \frac{p_{G 1} R_{1} d_{0}}{Δ^{2}} + C_{6} \frac{R_{2}}{Δ^{2}} + \frac{C_{7}}{Δ^{2}} \\ - [4 μ_{S} (1 - \frac{4 G}{3 K + 4 G}) d_{0} + 8 μ_{K} \frac{3 G}{3 K + 4 G} d_{0}] \frac{D R_{1}}{D t_{0}} \frac{1}{Δ^{2}} \end{matrix},

(109)

with

C_{4} = \frac{3 d_{0}^{2}}{3 K + 4 G} [12 G γ_{e} p_{G0} - σ_{1}^{2} + {(3 γ_{e} p_{G0})}^{2} + 3 K (- 3 σ_{1} + 3 γ_{e} p_{G0} - 4 G)] + {(\frac{3 d_{0}}{3 K + 4 G})}^{2} \times (4 G + 3 γ_{e} p_{G0} - σ_{1}) {(3 γ_{e} p_{G0} - σ_{1} - 3 K)}^{2} .

(110)

Explicit forms of C₅, C₆, and C₇ are not shown, because they do not affect the results of the present study. The inhomogeneous term N can be rewritten as

N = Λ_{1} e^{3 i θ} + Λ_{2} e^{2 i θ} + Λ_{3} e^{i θ} + c . c .,

(111)

where $Λ_{i} (i = 1, 2, 3)$ are the complex variables comprising A; $Λ_{1}$ ⁠, and $Λ_{2}$ are not shown because they are not essential to deriving the NLS equation. Here, $Λ_{3}$ is explicitly given as

\begin{matrix} Λ_{3} = (- \frac{\partial D}{\partial Ω}) {i [\frac{\partial A}{\partial t_{2}} + (v_{g} + u_{0}) \frac{\partial A}{\partial x_{2}}] + ν_{3} \frac{\partial^{2} A}{\partial x_{1}^{2}} + ν_{1} | A |^{2} A + i ν_{2} A + ν_{01} A + i ν_{02} \frac{\partial A}{\partial x_{1}}}, \end{matrix}

(112)

and this should be zero from the solvability condition.

Combining (98) and (112) using (25), we have

\begin{matrix} i [\frac{\partial A}{\partial t} + (v_{g} + u_{0}) \frac{\partial A}{\partial x}] + ν_{3} \frac{\partial^{2} A}{\partial x_{1}^{2}} - ϵ η A \\ + ϵ^{2} (ν_{1} | A |^{2} A + i ν_{2} A + ν_{01} A + i \frac{ν_{02}}{ϵ} \frac{\partial A}{\partial x}) = 0. \end{matrix}

(113)

Finally, we can obtain the NLS equation with dissipation (i.e., fourth) and advection (i.e., second) terms

i \frac{\partial A}{\partial τ} + ν_{01} A + ν_{1} | A |^{2} A + i ν_{2} A + ν_{3} \frac{\partial^{2} A}{\partial ξ^{2}} = 0,

(114)

via the variable transformation

τ = ϵ^{2} t, ξ = ϵ [x - (v_{g} + u_{0} + \frac{η}{\hat{k}} + ϵ ν_{02}) t],

(115)

noting that complex amplitude A is assumed to be a sinusoidal solution [i.e., $A = exp (i \hat{k} ξ - i \hat{ω} τ)$ ⁠, where $\hat{k}$ and $\hat{ω}$ are the frequency and wavenumber of the envelope wave, respectively].

The second, third, fourth, and fifth terms on the left-hand side of (114) represent the advection (i.e., wave propagation in moving coordinate), third-order nonlinear, dissipation, and dispersion effects of the envelope wave, respectively; ν₀₂ represents the advection effect. The two advection coefficients, ν₀₁ and ν₀₂, are given by

ν_{01} = \frac{1}{3} \frac{1}{\partial D / \partial Ω} (C_{32} η - \frac{Δ^{2} k^{2} η^{2}}{α_{0}} - \frac{C_{4} k^{2}}{α_{0}}),

(116)

ν_{02} = \frac{1}{3} \frac{1}{\partial D / \partial Ω} (C_{31} - C_{32} v_{g} + \frac{2 Δ^{2} k^{2} v_{g} η^{2}}{α_{0}}),

(117)

with the real constants

C_{31} = v_{p} k (e_{2} + \frac{1 - α_{0}}{α_{0}} e_{3}) + \frac{k}{α_{0}} e_{4},

(118)

C_{32} = e_{2} k + \frac{1 - α_{0}}{α_{0}} e_{3} k .

(119)

The explicit form of the nonlinear coefficient ν₁ is presented in Appendix B. The dispersion coefficient ν₃ is given by

ν_{3} = - \frac{9 α_{0} k v_{p} Δ^{2}}{2 {(3 α_{0} + Δ^{2} k^{2})}^{2}} (= \frac{1}{2} \frac{d v_{g}}{d k} = \frac{1}{2} \frac{d V_{g}}{d k}) < 0.

(120)

The dissipation coefficient ν₂ is represented by a linear combination of six factors as follows:

ν_{2} = \underset{liquid viscosity}{\underset{︸}{ν_{21}}} + \underset{liquid compressibility}{\underset{︸}{ν_{22}}} + \underset{shell viscosity}{\underset{︸}{ν_{23}}} + \underset{shell compressibility}{\underset{︸}{ν_{24}}} + \underset{drag force}{\underset{︸}{ν_{25}}} + \underset{thermal}{\underset{︸}{ν_{2 th}}},

(121)

where

ν_{21} = \frac{2 k^{2}}{3 α_{0} + Δ^{2} k^{2}} μ_{L},

(122)

ν_{22} = \frac{Δ^{2} - 3 (γ_{e} - κ) p_{G 0}}{2 (3 α_{0} + Δ^{2} k^{2})} Δ k^{2} V,

(123)

ν_{23} = \frac{2 k^{2}}{3 α_{0} + Δ^{2} k^{2}} μ_{S} (1 - \frac{4 G}{3 K + 4 G}) d_{0},

(124)

ν_{24} = \frac{4 k^{2}}{3 α_{0} + Δ^{2} k^{2}} μ_{K} \frac{3 G}{3 K + 4 G} d_{0},

(125)

ν_{25} = \frac{1}{\partial D / \partial Ω} \frac{k}{β_{1} Δ^{2}} (b_{2} - b_{3}) μ_{L},

(126)

ν_{2 th} = - \frac{1}{\partial D / \partial Ω} \frac{β_{1} + α_{0} (1 - α_{0})}{β_{1} α_{0} (1 - α_{0})} (κ - 1) p_{G 0} \frac{k^{2}}{Ω} ζ_{STM 1} .

(127)

The detailed discussion is presented in Sec. V A.

V. RESULTS AND DISCUSSION

For the case of the KdVB equation, Figs. 3–5 show the advection, nonlinear, and dissipation coefficients $Π_{0}, Π_{1}$ ⁠, and $Π_{2}$ ⁠, respectively, vs the initial void fraction α₀. For the case of the NLS equation, Fig. 7 shows the two advection coefficients ν₀₁ and ν₀₂, Fig. 8 shows the nonlinear coefficient ν₁, and Figs. 9 and 10 show the dissipation coefficient ν₂ (since $ν_{2 th}$ is significantly large, the dissipations except for thermal dissipation, $ν_{2} - ν_{2 th}$ are shown in Fig. 9 and $ν_{2 th}$ is shown in Fig. 10). A quantitative discussion is presented using the physical properties of five well-known UCAs, i.e., Albunex, Sonazoid, SonoVue, Levovist, and Optison, as summarized in Table I. In the following, uncoated bubbles (i.e., without shell) are defined by setting $d_{0}^{*} = 0$ ⁠.

FIG. 3.

$FIG. 3. Advection coefficient Π 0 vs the initial void fraction α0 for Sonazoid, Optison, Albunex, SonoVue, and Levovist: ϵ = 0.15, Ω = 1, p L 0 ∗ = 101 325 Pa, ρ L 0 ∗ = 1000 kg / m 3, c L 0 ∗ = 1500 m / s, μ L ∗ = 1 × 10 − 3 Pa · s, μ K ∗ = 0.7 μ S ∗, K ∗ = 0.7 G ∗, and R 0 ∗ = 1 μ m; the gas inside the bubble is air { κ = 1.40 and λ G ∗ = 0.025 [ W / ( m · K ) ]}; the other values are summarized in Table I. Although these values are also used in Figs. 4–13, different value of only ϵ is used to plot coefficients in the NLS equations (i.e., ν0, ν1, and ν2) in Figs. 7–13. The range is (a) 10 − 8 ≤ α 0 ≤ 4 × 10 − 8 and (b) 10 − 5 ≤ α 0 ≤ 4 × 10 − 5.$

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Advection coefficient $Π_{0}$ vs the initial void fraction α₀ for Sonazoid, Optison, Albunex, SonoVue, and Levovist: $\sqrt{ϵ} = 0.15$ ⁠, Ω = 1, $p_{L 0}^{*} = 101 325 Pa$ ⁠, $ρ_{L 0}^{*} = 1000 kg / m^{3}$ ⁠, $c_{L 0}^{*} = 1500 m / s$ ⁠, $μ_{L}^{*} = 1 \times 10^{- 3} Pa \cdot s$ ⁠, $μ_{K}^{*} = 0.7 μ_{S}^{*}$ ⁠, $K^{*} = 0.7 G^{*}$ ⁠, and $R_{0}^{*} = 1 μ m$ ⁠; the gas inside the bubble is air { $κ = 1.40$ and $λ_{G}^{*} = 0.025$ $[W / (m \cdot K)]$ }; the other values are summarized in Table I. Although these values are also used in Figs. 4–13, different value of only ϵ is used to plot coefficients in the NLS equations (i.e., ν₀, ν₁, and ν₂) in Figs. 7–13. The range is (a) $10^{- 8} \leq α_{0} \leq 4 \times 10^{- 8}$ and (b) $10^{- 5} \leq α_{0} \leq 4 \times 10^{- 5}$ ⁠.

FIG. 4.

$FIG. 4. Nonlinear coefficient Π 1 vs the initial void fraction α0 for five UCAs and without shell. Unlike the case of Π 0 (Fig. 3), although the dependence of α0 is quite minimal, a slight dependence on α0 exists.$

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Nonlinear coefficient $Π_{1}$ vs the initial void fraction α₀ for five UCAs and without shell. Unlike the case of $Π_{0}$ (Fig. 3), although the dependence of α₀ is quite minimal, a slight dependence on α₀ exists.

FIG. 5.

$FIG. 5. Dissipation coefficient Π 2 vs the initial void fraction α0 for five UCAs. Note that a slight dependence on α0 exists qualitatively.$

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Dissipation coefficient $Π_{2}$ vs the initial void fraction α₀ for five UCAs. Note that a slight dependence on α₀ exists qualitatively.

TABLE I.

Physical properties of the shell (i.e., the shear modulus $G_{S}^{*} [MPa]$ ⁠, shell viscosity $μ_{S}^{*} [Pa \cdot s]$ ⁠, and shell thickness $d_{S 0}^{*} [nm]$ ⁠) for five major contrast agents: Albunex,⁷⁰ Sonazoid,⁸⁰ SonoVue,⁵⁰ Levovist,⁴⁴ and Optison.⁶⁹ According to surface tensions in Church,⁷⁰ σ₁ = 0.04 [N/m] and σ₂ = 0.005 [N/m] are used. These values are used in Figs. 3–13.

UCA	$G_{S}^{*} [MPa]$	$μ_{S}^{*} [Pa \cdot s]$	$d_{S 0}^{*} [nm]$	Reference
Sonazoid	52	0.99	4	80
Albunex	15	0.05	15	70
SonoVue	20	0.6	4	50
Levovist	80	1.3	6	44
Optison	20.7	1.7	7.5	69

Note that the dependence of each coefficient on α₀ exists qualitatively, as is clear from the explicit forms of coefficients in (64), (65), (70)–(74), (116), (117), (120)–(126), (B1), and (B10). However, as shown in Figs. 4–11, the dependence is quantitatively quite minimum for the case of low α₀.

FIG. 6.

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Each component of dissipation coefficient $Π_{2}$ in (65): the liquid viscosity at the interface (i.e., the first term), liquid compressibility (i.e., the second term), shell viscosity (i.e., the third term), shell compressibility (i.e., the fourth term), and thermal effect (i.e., the fifth term). Five UCAs are exampled: (a) Albunex, (b) Levovist, (c) Sonazoid, (d) SonoVue, and (e) Optison. Note that a slight dependence on α₀ exists qualitatively.

FIG. 7.

$FIG. 7. Advection coefficients of the NLS equation vs the initial void fraction α0 for five UCAs: k = 1 and ϵ = 0.07; ϵ = 0.07 is used to plot coefficients in the NLS equation (i.e., ν0, ν1, and ν2) in Figs. 7–13. (a) ν01 and (b) ν02. Note that a slight dependence of ν02 on α0 exists qualitatively.$

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Advection coefficients of the NLS equation vs the initial void fraction α₀ for five UCAs: k = 1 and $ϵ = 0.07$ ⁠; $ϵ = 0.07$ is used to plot coefficients in the NLS equation (i.e., ν₀, ν₁, and ν₂) in Figs. 7–13. (a) ν₀₁ and (b) ν₀₂. Note that a slight dependence of ν₀₂ on α₀ exists qualitatively.

FIG. 8.

$FIG. 8. Nonlinear coefficient ν1 vs the initial void fraction α0 for five UCAs. Note that a slight dependence on α0 exists qualitatively.$

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Nonlinear coefficient ν₁ vs the initial void fraction α₀ for five UCAs. Note that a slight dependence on α₀ exists qualitatively.

FIG. 9.

$FIG. 9. Dissipation coefficient ν 2 − ν 2 th vs the initial void fraction α0 for five UCAs; remarking that, since the dissipation due to thermal conduction ν 2 th is quite large, ν 2 th is neglected in this figure and shown in Fig. 10 below. Note that a slight dependence on α0 exists qualitatively.$

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Dissipation coefficient $ν_{2} - ν_{2 th}$ vs the initial void fraction α₀ for five UCAs; remarking that, since the dissipation due to thermal conduction $ν_{2 th}$ is quite large, $ν_{2 th}$ is neglected in this figure and shown in Fig. 10 below. Note that a slight dependence on α₀ exists qualitatively.

FIG. 10.

$FIG. 10. Dissipation coefficient due to thermal conduction, ν 2 th in (126), which was omitted in Fig. 9, vs the initial void fraction α0 for five UCAs. The value is quite large compared with the other five dissipation factors (see Fig. 9). Note that a slight dependence on α0 exists qualitatively.$

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Dissipation coefficient due to thermal conduction, $ν_{2 th}$ in (126), which was omitted in Fig. 9, vs the initial void fraction α₀ for five UCAs. The value is quite large compared with the other five dissipation factors (see Fig. 9). Note that a slight dependence on α₀ exists qualitatively.

FIG. 11.

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Each component of dissipation coefficient ν₂ due to liquid viscosity at the interface (ν₂₁), liquid compressibility (ν₂₂), shell viscosity (ν₂₃), shell compressibility (ν₂₄), and drag force (ν₂₅); dissipation coefficient due to the thermal conduction was shown in Fig. 10 and (126). Five typical UCAs were compared: (a) Albunex, (b) Levovist, (c) Sonazoid, (d) SonoVue, and (e) Optison. Note that a slight dependence on α₀ exists qualitatively.

A. Dissipation

Based on the decomposition of dissipation (attenuation) factors, we obtain (65) and (121). We first focus on the compressibility and viscosity of the liquid and shell. Note that the KdVB equation in (62) contains the other dissipation terms, i.e., $Π_{4}$ and $Π_{5}$ in (67) and (68).

1. Four factors: Importance of shell compressibility

First, we discuss the four factors of dissipation, i.e., the liquid viscosity, liquid compressibility, shell viscosity, and shell compressibility. Figures 6 and 11 represent each component of $Π_{2}$ in the KdVB equation and ν₂ in the NLS equation, respectively: (i) orange curves represent the liquid viscosity at the interface $Π_{21}$ and ν₂₁, (ii) green curves represent the shell viscosity $Π_{23}$ and ν₂₃, (iii) red curves represent the shell compressibility $Π_{24}$ and ν₂₄ and (iv) purple curves represent the liquid compressibility $Π_{22}$ and ν₂₂.

The following tendency was observed for both the KdVB and NLS equations: The order of sizes of each dissipation factor differs depending on the UCAs. The dissipation due to the shell compressibility is always greater than that due to the shell viscosity. In contrast, the dissipation due to the liquid compressibility is always smaller than that due to the liquid viscosity. This reflects the importance of the consideration of shell compressibility.

The predominance of dissipation by the liquid phase (i.e., $Π_{21} + Π_{22}$ or $ν_{21} + ν_{22}$ ⁠) or that by the shell (i.e., $Π_{23} + Π_{24}$ or $ν_{23} + ν_{24}$ ⁠) is determined by the shell viscosity $μ_{S}^{*}$ ⁠, because $μ_{K}^{*} / μ_{S}^{*}$ is equal to $K^{*} / G^{*}$ ⁠.⁶⁸ Also, $μ_{K}^{*} = 0.7 μ_{S}^{*}$ ⁶⁸ is used.

2. Thermal effect and drag force

The discussion of dissipation due to thermal effects and drag force is complex compared with the dissipation due to compressibility and viscosity (Sec. V A 1). Dissipation due to thermal effect is represented by both $Π_{25}$ and $Π_{4}$ in the KdVB equation and only $ν_{2 th}$ in the NLS equation. Dissipation due to drag force is represented by $Π_{5}$ in the KdVB equation and ν₂₅ in the NLS equation (pink curve in Fig. 11).

For the case of the NLS equation, the dissipation due to thermal conduction $ν_{2 th}$ is the largest, and that due to the drag force $ν_{25}$ is the smallest in ν₂. In the case of the KdVB equation, the dissipation owing to thermal effect $Π_{25}$ is the smallest in $Π_{2}$ ⁠. Although ν₂ is represented by a linear combination of every factor of dissipation, there are three different dissipation terms (coefficients $Π_{2}, Π_{4}$ ⁠, and $Π_{5}$ ⁠) in the KdVB equation in (62). It is worth noting that (i) $Π_{2}$ is decomposed into five factors and (ii) although $Π_{2}$ is the coefficient of the second-order derivative, $Π_{4}$ and $Π_{5}$ are the coefficients without derivatives in (62).

The gas inside contrast agents can be regarded as isothermal behavior in clinical practice.⁴⁰ In fact, the dissipation coefficient owing to thermal conduction, i.e., $Π_{4}$ and $ν_{2 th}$ ⁠, vanishes in the isothermal case. Furthermore, the difference of other components of dissipation coefficients between the isothermal and non-isothermal cases was negligible.

3. Comparison among UCAs

In the dissipation coefficients $Π_{2}$ and ν₂, Optison with the largest shell viscosity $μ_{S}^{*}$ (see Table I) shows the largest dissipation and Albunex with the smallest $μ_{S}^{*}$ shows the smallest dissipation (Figs. 5 and 9, respectively). The order of size of each factor of $Π_{2}$ and ν₂ is the same in all the UCAs notably, although the effect of drag force is included in ν₂ and not in $Π_{2}$ ⁠, this order is not affected.

In comparison with the case of uncoated bubbles, it turns out to be increased in all the UCAs, as summarized in Table II for the case of the KdVB equation and Table III for the case of NLS equation, including the advection and nonlinear coefficients.

TABLE II.

Ratio of uncoated bubbles to coated bubbles: $α_{0} = 10^{- 5}$ and $R_{0}^{*} = 1 μ m$ for KdVB equation.

UCA	Albunex	Sonazoid	SonoVue	Levovist	Optison
Advection $Π_{0}$	0.41	0.69	0.71	0.49	0.63
Nonlinearity $Π_{1}$	0.70	0.88	0.90	0.73	0.42
Dissipation $Π_{2}$	1.4	5.6	3.9	6.9	14

UCA	Albunex	Sonazoid	SonoVue	Levovist	Optison
Advection $Π_{0}$	0.41	0.69	0.71	0.49	0.63
Nonlinearity $Π_{1}$	0.70	0.88	0.90	0.73	0.42
Dissipation $Π_{2}$	1.4	5.6	3.9	6.9	14

TABLE III.

The ratio of uncoated bubbles to coated bubbles: $α_{0} = 10^{- 5}$ and $R_{0}^{*} = 1 μ m$ for NLS equation.

UCA	Albunex	Sonazoid	SonoVue	Levovist	Optison
Advection $ν_{01}$	9.3	5.7	8.6	0.64	17
Advection $ν_{02}$	66	62	66	38	91
Nonlinearity $ν_{1}$	0.34	0.71	0.76	0.39	0.59
Dissipation $ν_{2} - ν_{2 th}$	1.2	4.7	3.4	5.9	12
Dissipation $ν_{2}$	0.086	0.69	0.79	0.16	0.45

UCA	Albunex	Sonazoid	SonoVue	Levovist	Optison
Advection $ν_{01}$	9.3	5.7	8.6	0.64	17
Advection $ν_{02}$	66	62	66	38	91
Nonlinearity $ν_{1}$	0.34	0.71	0.76	0.39	0.59
Dissipation $ν_{2} - ν_{2 th}$	1.2	4.7	3.4	5.9	12
Dissipation $ν_{2}$	0.086	0.69	0.79	0.16	0.45

B. Nonlinearity

To discuss the size of nonlinearity, we observed the absolute value of the nonlinear coefficients, i.e., $| Π_{1} |$ in Fig. 4 and $| ν_{1} |$ in Fig. 8, and compared the size of all the UCAs (except for the uncoated bubbles). Then, the same order was observed in both $| Π_{1} |$ and $| ν_{1} |$ ⁠. Furthermore, Albunex, having the highest shell thickness $d_{S 0}^{*}$ ⁠, has the smallest $| Π_{1} |$ and $| ν_{1} |$ ⁠.

C. Advection

We discuss the advection coefficients, $| Π_{0} |$ in the KdVB equation (Fig. 3) and ν₀₁ and ν₀₂ in the NLS equation (Fig. 7), among the five UCAs. In the range of $10^{- 8} \leq α_{0} \leq 10^{- 5}$ ⁠, Albunex has the smallest $| Π_{0} |$ ⁠; however, Levovist has the smallest ν₀₁ and ν₀₂. Conversely, SonoVue has the largest $| Π_{0} |$ ⁠, although Optison has the largest ν₀₁ and ν₀₂.

D. Case of high void fractions

Figure 12 provides an example of dependence of coefficients $Π_{0}$ and ν₁ on the initial void fraction α₀. Note that although Fig. 12 shows only a visible range of α₀ dependence (i.e., high void fraction case), α₀ dependence qualitatively exists regardless of the size of α₀.

FIG. 12.

$FIG. 12. Example of dependence of coefficients on the large initial void fraction: (a) Π 0 and (b) ν1. Note that although this figure shows only a visible range of α0 dependence (i.e., high void fraction case), α0 dependence qualitatively exists regardless of the size of α0.$

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Example of dependence of coefficients on the large initial void fraction: (a) $Π_{0}$ and (b) ν₁. Note that although this figure shows only a visible range of α₀ dependence (i.e., high void fraction case), α₀ dependence qualitatively exists regardless of the size of α₀.

Unlike the case of $Π_{0}$ ⁠, although the dependence of the other coefficients on α₀ is quite small, noting that slight dependence of α₀ exists. In fact, ν₁ for large α₀ slightly depends on α₀ [Fig. 12(b)].

E. Dependence of physical properties

Figure 13 shows the dependence of physical properties (i.e., shell rigidity $G_{S}^{*}$ ⁠, surface tension $σ_{1}^{*}$ ⁠, and shell thickness $d_{S 0}^{*}$ ⁠) on the nonlinear coefficients $Π_{1}$ and ν₁. Notably, the range of physical properties (i.e., horizontal axis) does not correspond to a specific contrast agent; however, it corresponds to five typical contrast agents shown in Table I.

FIG. 13.

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Dependence of the nonlinear coefficients $Π_{1}$ and ν₁ on the physical properties (i.e., the shell rigidity $G_{S}^{*}$ ⁠, surface tension $σ_{1}^{*}$ ⁠, and shell thickness $d_{S 0}^{*}$ ⁠). The range of physical properties corresponds to five typical contrast agents (Table I). (a) $Π_{1}$ vs $G_{S}^{*}$ ⁠, (b) ν₁ vs $G_{S}^{*}$ ⁠, (c) $Π_{1}$ vs $σ_{1}^{*}$ ⁠, (d) ν₁ vs $σ_{1}^{*}$ ⁠, (e) $Π_{1}$ vs $d_{S 0}^{*}$ ⁠, and (f) ν₁ vs $d_{S 0}^{*}$ ⁠.

Clearly, the nonlinear coefficient strongly depends on these physical properties. As values, the degree of change is larger for ν₁ than for $Π_{1}$ ⁠. Nevertheless, as is clear from explicit forms in (64)–(74) and (115)–(126), the remaining coefficients are also dependent on these physical properties.

VI. CONCLUSIONS

Apart from our previous research,^89,90 no theoretical studies have been reported on the dynamics of multiple coated bubbles. In this paper, we theoretically investigated the weakly nonlinear propagation process of one-dimensional ultrasound in a compressible flowing liquid uniformly containing multiple microbubbles coated by a visco-elastic body that obeys the Kelvin–Voigt model, especially focusing on (i) both low-frequency long wave (Sec. III) and high-frequency short wave (Sec. IV) bands and (ii) the incorporation of the compressibility of the shell.⁶⁸ As a result, the KdVB equation for the low frequency case and NLS equation for the high frequency case, which describe nonlinear evolutions of ultrasound, were successfully derived in a unified manner. Importantly, for the first time, we successfully derived the NLS equation for the case of multiple coated bubbles. Moreover, the compressibility of the liquid and shell, thermal conductivity, drag force acting on bubbles, and translation of bubbles, which were neglected in our previous study,⁸⁹ were incorporated into the present theory. Especially, in the incorporation of shell compressibility, the updated equation of motion for a single coated bubble with shell compressibility⁶⁸ was successfully extended to the case of multiple coated bubbles.

Focusing on the coefficients of advection, nonlinear, and dissipation terms in the resultant KdVB and NLS equations for five typical UCAs (i.e., Albunex, Sonazoid, SonoVue, Levovist, and Optison) and uncoated case, the following results were obtained: (i) The shell compressibility increased the dissipation effect but decreased the nonlinear effect; (ii) The dissipation due to thermal conduction was the largest of all the dissipation factors for the NLS equation; (iii) By comparing each dissipation factor for typical UCAs, both similarity and difference were observed; (iv) Non-negligible contribution of shell compressibility to ultrasound dissipation was identified although it depends on the types of contrast agents, e.g., the dissipation due to shell compressibility is approximately ten times larger than that of liquid compressibility for the case of Optison [Fig. 6(e)]; (v) Focusing on the dependence of nonlinear and advection effects on the initial void fraction, a similar tendency was observed for both the KdVB and NLS equations. From these results, we elucidated a basic understanding of the effect of the shell of coated bubbles on the dissipation, nonlinear, and advection effects of weakly nonlinear ultrasound, for both low and high frequencies. Particularly, we discovered that the effect of shell compressibility, which has long been neglected, is crucial.

Finally, as with any study, the limitations of the present study should be acknowledged and addressed in future work: (i) the buckling of the shell;⁷⁷ (ii) anisotropy of the shell;⁶⁸ (iii) asymmetric oscillations of nonspherical coated bubbles; (iv) the effect of visco-elasticity of liquids such as tissue;¹²⁵ (v) bubble–bubble interaction and the resultant change in eigenfrequency;^58,102 (vi) pressure dependence of the eigenfrequency of bubble oscillations;^81,82 and (vii) discussion of the nonlinear dissipation effect^63,64,66,124 for a high pressure amplitude case. Also, the KdVB and NLS equations derived here will be numerically solved under practical conditions conducive to diagnostic (and/or therapeutic) applications of ultrasound.

ACKNOWLEDGMENTS

This work was partially carried out with the aid of the JSPS KAKENHI (Nos. 18K03942 and 22K03898), based on the results obtained from a project subsidized by the New Energy and Industrial Technology Development Organization (NEDO) (No. JPNP20004), and Ono Charitable Trust for Acoustics. We would like to thank referees for their valuable comments and suggestions, and Editage (www.editage.com) for English language editing.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Tetsuya Kanagawa: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (lead); Project administration (lead); Supervision (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (lead). Mitsuhiro Honda: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – original draft (supporting); Writing – review and editing (supporting). Yusei Kikuchi: Data curation (supporting); Formal analysis (equal); Investigation (equal); Methodology (supporting); Validation (supporting).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX A: INHOMOGENEOUS TERMS

The inhomogeneous terms K_i (⁠ $i = 3, 4, 5, 6$ ⁠) in (60) are given by

\begin{matrix} K_{3} = p_{G 0} \frac{\partial}{\partial x_{1}} [3 R_{1} - T_{G 1}] + p_{G 0} α_{1} \frac{\partial}{\partial x_{0}} [3 R_{1} - T_{G 1}] + 3 \frac{\partial}{\partial x_{0}} [R_{1} (T_{G 1} - 2 R_{1})] - β_{1} \frac{D}{D t_{1}} (u_{G 1} - u_{L 1}) - β_{1} α_{1} \frac{D}{D t_{0}} (u_{G 1} - u_{L 1}) - β_{1} (u_{G 1} \frac{\partial u_{G 1}}{\partial x_{0}} - u_{L 1} \frac{\partial u_{L 1}}{\partial x_{0}}) \\ - β_{2} (u_{G 1} - u_{L 1}) \frac{D α_{1}}{D t_{0}} - \frac{3 Λ}{8 Δ} (u_{G 1} - u_{L 1}) | u_{G 1} - u_{L 1} |, \end{matrix}

(A1)

\begin{matrix} K_{4} = \frac{D}{D t_{1}} [u_{0} α_{0} α_{1} - (1 - α_{0}) u_{L 1}] - (1 - α_{0}) \frac{\partial p_{L 1}}{\partial x_{1}} + α_{0} \frac{D}{D t_{0}} (α_{1} u_{L 1}) + α_{0} β_{1} \frac{D}{D t_{1}} (u_{G 1} - u_{L 1}) + α_{0} β_{1} α_{1} \frac{D}{D t_{0}} (u_{G 1} - u_{L 1}) + α_{0} β_{1} (u_{G 1} \frac{\partial u_{G 1}}{\partial x_{0}} - u_{L 1} \frac{\partial u_{L 1}}{\partial x_{0}}) + α_{0} β_{2} (u_{G 1} - u_{L 1}) \frac{D α_{1}}{D t_{0}} \\ - (1 - α_{0}) \frac{\partial u_{L 1}^{2}}{\partial x_{0}} + α_{0} α_{1} \frac{\partial p_{L 1}}{\partial x_{0}} - (1 - α_{0}) u_{0} \frac{D ρ_{L 1}}{D t_{0}} + \frac{3 Λ}{8 Δ} (u_{G 1} - u_{L 1}) | u_{G 1} - u_{L 1} | \\ + α_{0} {p_{L 1} + [{(\frac{ω_{B}^{*} R_{0}^{*}}{U^{*}})}^{2} - 3 (γ_{e} - 1) p_{G 0}] R_{1} - p_{G 1} T_{G 1} - C_{1} + \frac{4 μ_{e 0}^{2}}{Δ^{2}}} \frac{\partial α_{1}}{\partial x_{0}}, \end{matrix}

(A2)

\begin{matrix} K_{5} = - [\frac{Δ^{2}}{Ω^{2}} + 3 (2 - γ_{e}) p_{G 0}] R_{1}^{2} + Δ^{2} \frac{D^{2} R_{1}}{D t_{0}^{2}} - V Δ {[3 (γ_{e} - 1) p_{G 0} - \frac{Δ^{2}}{Ω^{2}}] \frac{\partial R_{1}}{\partial t_{0}} + p_{G 0} \frac{\partial T_{1}}{\partial t_{0}}} \\ + 3 p_{G 0} R_{1} T_{G 1} + 4 μ_{L} \frac{D R_{1}}{D t_{0}} - \frac{{(u_{G 1} - u_{L 1})}^{2}}{4} \\ + 4 [μ_{S} (1 - \frac{4 G}{3 K + 4 G}) + 2 μ_{K} \frac{3 G}{3 K + 4 G}] d_{S 0} \frac{D R_{1}}{D t_{0}} + C_{2} R_{1} + C_{3} + \frac{4 μ_{e 0}}{Δ^{2}} R_{1}, \end{matrix}

(A3)

\begin{matrix} K_{6} = - 3 (κ - 1) \frac{D R_{1}}{D t_{1}} - \frac{D T_{G 1}}{D t_{1}} - 3 \frac{D}{D t_{0}} [(κ - 1) T_{G 1} R_{1} + \frac{1}{2} (κ - 1) (3 κ - 4) R_{1}^{2}] \\ - 3 (κ - 1) u_{G 1} \frac{\partial R_{1}}{\partial x_{0}} - u_{G 1} \frac{\partial T_{G 1}}{\partial x_{0}} - ζ_{STM 1} T_{G 1} + ζ_{STM 2} \frac{\partial T_{G 1}}{\partial t_{0}}, \end{matrix}

(A4)

where C₂ was defined in (69).

APPENDIX B: EXPLICIT FORM OF SOME COEFFICIENTS

The nonlinear $Π_{1}$ coefficient of the KdVB equation in (62) is given as follows:

\begin{matrix} Π_{1} = \frac{1}{6} [k_{1} - \frac{k_{2}}{α_{0}} + \frac{(1 - α_{0} + β_{1}) k_{3}}{β_{1} (1 - α_{0})} + \frac{k_{4}}{α_{0} (1 - α_{0})} - \frac{2 Δ^{2} \hat{k_{5}}}{α_{0} Ω^{2}} + \frac{[α_{0} (1 - α_{0}) + β_{1}] p_{G 0}}{β_{1} α_{0} (1 - α_{0})} k_{6}] - \frac{{(s_{2} - s_{3})}^{2}}{12 α_{0} v_{p}}, \end{matrix}

(B1)

where explicit forms of $k_{i}$ (⁠ $i = 1, 2, 3, 4, 5, 6$ ⁠) are the same as counterparts in Ref. 46.

The nonlinear coefficient Γ in (96), (99), and (101) is given by

\begin{matrix} Γ = - \frac{2}{3} [v_{p} k m_{1} - \frac{v_{p} k m_{2}}{α_{0}} + \frac{1 - α_{0} + β_{1}}{(1 - α_{0}) β_{1}} k m_{3} + \frac{k m_{4}}{α_{0} (1 - α_{0})} \\ - \frac{2 Δ^{2} k^{2} m_{5}}{α_{0}} - \frac{β_{1} + α_{0} (1 - α_{0})}{β_{1} α_{0} (1 - α_{0}) Ω} p_{G 0} k^{2} m_{6}], \end{matrix}

(B2)

where

m_{1} = - 6 (b_{1} - 2) v_{p} k - 2 b_{2} (b_{1} - 3) k,

(B3)

m_{2} = - 2 α_{0} b_{1} b_{3} k,

(B4)

m_{3} = \hat{m} + p_{G 0} (3 b_{1} - b_{1} b_{5} + 6 b_{5} - 12) k,

(B5)

\begin{matrix} m_{4} = - α_{0} \hat{m} - 2 (1 - α_{0}) b_{3}^{2} k + α_{0} b_{1} b_{4} k - 2 α_{0} b_{1} b_{3} v_{p} k \\ + α_{0} b_{1} [- 3 (γ_{e} - κ) p_{G 0} + Δ^{2}] k, \end{matrix}

(B6)

\hat{m} = (β_{1} + β_{2}) (b_{2} - b_{3}) b_{1} v_{p} k - β_{1} (b_{2}^{2} - b_{3}^{2}) k,

(B7)

m_{5} = 1 - 3 b_{2} v_{p} k^{2} + \frac{3 (2 - γ_{e} - b_{5})}{2 Δ^{2}} p_{G 0} + \frac{5}{2} v_{p}^{2} k^{2} + \frac{{(b_{3} - b_{2})}^{2}}{4 Δ^{2}},

(B8)

m_{6} = 2 v_{p} b_{5}^{2} k + 6 [3 κ - 2 - \frac{κ (3 κ - 1)}{2}] .

(B9)

Finally, the nonlinear coefficient of the NLS equation ν₁ in (114) is given as follows:

\begin{matrix} ν_{1} = \frac{1}{3} \frac{1}{\partial D / \partial Ω} [k v_{p} n_{1} - \frac{k v_{p}}{α_{0}} n_{2} + \frac{(1 - α_{0} + β_{1})}{(1 - α_{0}) β_{1}} k n_{3} + \frac{k n_{4}}{α_{0} (1 - α_{0})} \\ - \frac{Δ^{2} k^{2} n_{5}}{α_{0}} - \frac{β_{1} + α_{0} (1 - α_{0})}{β_{1} α_{0} (1 - α_{0}) Ω} p_{G 0} k^{2} n_{6}] < 0, \end{matrix}

(B10)

where

\begin{matrix} n_{1} = - 3 v_{p} [c_{0} (4 - b_{1}) - c_{1} + 6 b_{1} - 10] k \\ + [c_{2} (3 - b_{1}) + b_{2} (3 c_{0} - c_{1} + 9 b_{1} - 18)] k, \end{matrix}

(B11)

n_{2} = - α_{0} (b_{1} c_{3} + b_{3} c_{1}) k,

(B12)

\begin{matrix} n_{3} = \hat{n} + p_{G 0} [(b_{5} - 3) c_{1} + 6 c_{0} b_{1} - 2 b_{1} b_{5} + 6 b_{1} (b_{5} - 2) + 3 c_{0} b_{5} + 3 c_{5} + 12 (5 - 3 b_{1})] k, \end{matrix}

(B13)

\begin{matrix} n_{4} = - α_{0} \hat{n} - α_{0} v_{p} (b_{3} c_{1} + b_{1} c_{3}) k - 2 (1 - α_{0}) b_{3} c_{3} k \\ + α_{0} [u_{0} b_{3} c_{1} + 6 b_{1} b_{3}^{2} - b_{4} c_{3} + b_{1} c_{4} - b_{1} (c_{4} - c_{s}) + u_{0} b_{1} c_{3} + 2 c_{1} b_{4}] k + α_{0} p_{G 0} [b_{1} c_{5} - 6 c_{1} (γ_{e} - κ) + 3 b_{1} c_{0} (γ_{e} - 1) - 6 b_{1} (2 - γ_{e} - b_{5})] k \\ + α_{0} Δ^{2} (2 c_{1} - b_{1} c_{0} - 2 b_{1}) k, \end{matrix}

(B14)

\begin{matrix} \hat{n} = (2 β_{1} - β_{2}) b_{1} (c_{2} - c_{3}) v_{p} k - (β_{1} - 2 β_{2}) (b_{2} - b_{3}) c_{1} v_{p} k \\ - b_{1} (b_{2} - b_{3}) [β_{1} (b_{2} + b_{3}) + β_{2} b_{2}] k - β_{1} (b_{2} c_{2} - b_{3} c_{3}) k, \end{matrix}

(B15)

\begin{matrix} n_{5} = - 3 + c_{0} (- v_{p}^{2} k^{2} + 2) + 2 b_{2} [b_{2} k - v_{p} (1 + 3 c_{0}) k] k \\ + 3 p_{G 0} \frac{c_{0} (2 - γ_{e} - b_{5}) + 3 γ_{e} + 6 b_{5} - c_{5} - 10}{Δ^{2}} + \frac{(b_{2} - b_{3}) (c_{2} - c_{3})}{2 Δ^{2}}, \end{matrix}

(B16)

\begin{matrix} n_{6} = - 3 v_{p} [(κ + 3) c_{0} + 9 κ^{2} (κ - 3) + 2 (13 κ - 4) - (κ - 1) c_{5}] k \\ - 2 b_{2} [3 (κ - 1) c_{0} + c_{5} - 3 (κ - 2) (3 κ + 1)] k . \end{matrix}

(B17)

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Author(s)

Nonlinear acoustic theory on flowing liquid containing multiple microbubbles coated by a compressible visco-elastic shell: Low and high frequency cases

I. INTRODUCTION

II. FORMULATION OF THE PROBLEM

A. Problem and assumptions

B. Basic equations

C. Multiple scales analysis

III. LOW FREQUENCY AND LONG WAVE

A. $O (ϵ)$ ⁠: Linear propagation at near-field

B. $O (ϵ^{2})$ ⁠: Nonlinear propagation at far-field

IV. HIGH FREQUENCY AND SHORT WAVE

A. $O (ϵ)$ ⁠: Linear propagation of carrier wave at near-field

B. $O (ϵ^{2})$ ⁠: Linear propagation of envelope wave at far-field I

C. $O (ϵ^{3})$ ⁠: Nonlinear propagation of envelope wave and resultant NLS equation

V. RESULTS AND DISCUSSION

A. Dissipation

1. Four factors: Importance of shell compressibility

2. Thermal effect and drag force

3. Comparison among UCAs

B. Nonlinearity

C. Advection

D. Case of high void fractions

E. Dependence of physical properties

VI. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: INHOMOGENEOUS TERMS

APPENDIX B: EXPLICIT FORM OF SOME COEFFICIENTS

References

Citing articles via

Resources

Explore

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Nonlinear acoustic theory on flowing liquid containing multiple microbubbles coated by a compressible visco-elastic shell: Low and high frequency cases

I. INTRODUCTION

II. FORMULATION OF THE PROBLEM

A. Problem and assumptions

B. Basic equations

C. Multiple scales analysis

III. LOW FREQUENCY AND LONG WAVE

A. O ( ϵ )⁠: Linear propagation at near-field

B. O ( ϵ 2 )⁠: Nonlinear propagation at far-field

IV. HIGH FREQUENCY AND SHORT WAVE

A. O ( ϵ )⁠: Linear propagation of carrier wave at near-field

B. O ( ϵ 2 )⁠: Linear propagation of envelope wave at far-field I

C. O ( ϵ 3 )⁠: Nonlinear propagation of envelope wave and resultant NLS equation

V. RESULTS AND DISCUSSION

A. Dissipation

1. Four factors: Importance of shell compressibility

2. Thermal effect and drag force

3. Comparison among UCAs

B. Nonlinearity

C. Advection

D. Case of high void fractions

E. Dependence of physical properties

VI. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: INHOMOGENEOUS TERMS

APPENDIX B: EXPLICIT FORM OF SOME COEFFICIENTS

References

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only

A. $O (ϵ)$ ⁠: Linear propagation at near-field

B. $O (ϵ^{2})$ ⁠: Nonlinear propagation at far-field

A. $O (ϵ)$ ⁠: Linear propagation of carrier wave at near-field

B. $O (ϵ^{2})$ ⁠: Linear propagation of envelope wave at far-field I

C. $O (ϵ^{3})$ ⁠: Nonlinear propagation of envelope wave and resultant NLS equation