Waves In Bubbly Liquids Research Articles

Weakly nonlinear propagation of pressure waves in bubbly liquids with a polydispersity based on two-fluid model equations

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.

Numerical modelling and theoretical analysis of the acoustic attenuation in bubbly liquids

The propagations of acoustic waves in bubbly liquids have been extensively investigated through experimental and theoretical methods, a computational fluid dynamics (CFD) method was introduced to investigate the acoustic attenuation through bubbles in this study. Numerical ‘measurements’ of attenuation coefficient (α) and phase velocity (V) were conducted using a homogeneous cavitation model, which were modeled from experimental schemes and compared with the theoretical results. At extremely small bubble volume fraction (β around 10−11), new formulas of the theoretical α (αtheo) were proposed respectively for linear and transient oscillations, and a new formula of the numerical α (αnum) was proposed for transient oscillations. Results showed that αnum matched precisely with αtheo for linear, nonlinear and transient oscillations. At medium β (around 10−4), the relative difference of αnum between the VOF and present methods was less than 1.6%, while it reached 15.4% after replacing the bounded Keller-Miksis equation (KME) in the present method with the KME. However, the traditional theoretical α and V matched precisely with the predictions by the present method with the KME. Thus new theoretical α and V were proposed based on the bounded KME, and the relative differences between αtheo and αnum were less than 1%. It can be concluded that the bounded KME should be used in both numerical and theoretical predictions.

Application of Argument Principle for Study a Structure of Roots of Transcendental Equations

In study of the structure of shock waves in bubbly liquids we need to find integral curve connecting two singular points of the system of differential equations. We linearized the system of basic equations around equilibrium state. The condition of existence of nontrivial solution of the linear equations leads to the characteristic transcendental equation with respect to exponential index. The structure of roots of such equation is studied by using argument principle [1]. It is shown that for compression waves such equation has one real root.

Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

Thermal effect inside bubbles for weakly nonlinear pressure waves in bubbly liquids: Theory on short waves

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.

An exhaustive theoretical analysis of thermal effect inside bubbles for weakly nonlinear pressure waves in bubbly liquids

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.

Weakly nonlinear theory on pressure waves in bubbly liquids with a weak polydispersity

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.

Theoretical Elucidation on an Effect of Thermal Conduction on High-Speed and High-Frequency Nonlinear Pressure Waves in Bubbly Liquids

Theoretical Elucidation on an Effect of Thermal Conduction on High-Speed and High-Frequency Nonlinear Pressure Waves in Bubbly Liquids

Theoretical Study on an Effect of Thermal Conductivity on Nonlinear Propagation of Pressure Waves in Bubbly Liquids

Theoretical Study on an Effect of Thermal Conductivity on Nonlinear Propagation of Pressure Waves in Bubbly Liquids

Theoretical Study on an Effect of Liquid Viscosity and Thermal Conductivity on Weakly Nonlinear Propagation of Short Pressure Waves in Bubbly Liquids

This study theoretically clarifies an effect of the liquid viscosity and the thermal conductivity on weakly nonlinear propagation of pressure waves in a liquid containing many spherical microbubbles. As in our preceding paper (Kamei et al., J. JSCE, Ser. A2, 75 (2019), 499) focusing on a long wave, by the use of the method of multiple scales, a nonlinear Schrödinger equation describing the long-range propagation of an envelope wave of short carrier wave is derived from the basic equations incorporating the liquid viscosity and the thermal conductivity. As a result, as in our preceding long wave case, the liquid viscosity and the thermal conductivity affect the dissipation effect, and a nonlinear effect in the adiabatic process decreases in comparison with the previous study (Kanagawa et al., J. Fluid Sci. Technol., 6 (2011), 838). On the other hand, unlike in our preceding long wave case, a dispersion effect in the adiabatic process decreases in comparison with the previous study.

Theoretical Elucidation of Weakly Nonlinear Interaction Between Long and Short Ultrasonic Waves in Bubbly Liquids

Theoretical Elucidation of Weakly Nonlinear Interaction Between Long and Short Ultrasonic Waves in Bubbly Liquids

Derivation of Nonlinear Schrödinger Equation for Long-Range Propagation of Acoustic Waves in Bubbly Liquids with a Polydispersity of Bubble Size

Derivation of Nonlinear Schrödinger Equation for Long-Range Propagation of Acoustic Waves in Bubbly Liquids with a Polydispersity of Bubble Size

1Pa4-1 Theoretical Study on Nonlinear and Thermal Effects of High-Speed Pressure Waves in Bubbly Liquids

1Pa4-1 Theoretical Study on Nonlinear and Thermal Effects of High-Speed Pressure Waves in Bubbly Liquids

Theoretical Study on an Effect of Thermal Conductivity at Bubble-Liquid Interface on Weakly Nonlinear Propagation of Pressure Waves in Bubbly Liquids

Theoretical Study on an Effect of Thermal Conductivity at Bubble-Liquid Interface on Weakly Nonlinear Propagation of Pressure Waves in Bubbly Liquids

Theoretical Study on Nonlinear Interaction between Long and Short Waves in Bubbly Liquids

Theoretical Study on Nonlinear Interaction between Long and Short Waves in Bubbly Liquids

Theoretical Elucidation of an Relationship of Thermal Conduction at Bubble-Liquid Interface on Pressure Waves in Bubbly Liquids and an Interaction between Nonlinear and Thermal Effects

Theoretical Elucidation of an Relationship of Thermal Conduction at Bubble-Liquid Interface on Pressure Waves in Bubbly Liquids and an Interaction between Nonlinear and Thermal Effects

Weakly Nonlinear Analysis on Pressure Waves in Bubbly Liquids with a Polydispersity of Bubble Size

Weakly Nonlinear Analysis on Pressure Waves in Bubbly Liquids with a Polydispersity of Bubble Size

Effect of a mass flux at bubble-liquid interface on propagation properties of weakly nonlinear waves

Toward the establishment and realization of medical applications accompanied with a phase change such as a drug delivery system, a formulation of the effect of mass transfer (i.e., mass flux) at the bubble-liquid interface on a ultrasound propagation process has been desired. Since a large amplitude ultrasound is utilized in such an application, the consideration of both the mass flux and the wave nonlinearity is essential. Although Fuster and Montel [J. Fluid Mech.(2015)] investigated linear wave propagation, nonlinear analysis has not been performed. In this paper, we theoretically clarify the effect of mass flux at the bubble-liquid interface on weakly nonlinear (i.e., finite but small amplitude) propagation of pressure waves in bubbly liquids. The effect of total evaporation or condensation as a mass flux across the bubble interface is taken into account as a simple classical model. Although various types of dissipation are installed, the effect of viscosity of gas inside the bubble is neglected. The set of basic equations for bubbly flows is composed of the conservation laws of mass, momentum, and energy, equation of motion for radial oscillations of bubble wall, equations of state, and so on. From the method of multiple scales, some nonlinear wave equations (e.g., KdV-Burgers equation) are derived. We then discuss the effects of mass flux on the propagation processes of nonlinear waves.

Derivation of an amplitude equation for pressure wave propagation in polydisperse bubbly liquids

A weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent water uniformly containing small gas bubbles is theoretically investigated. In the present study, the bubbles do not coalesce, break up, appear, and disappear. The bubbles are spherical, and these oscillations are spherically symmetric. In addition, the viscosity of gas inside the bubbles and the thermal conductivities of the both phases are neglected. Although abovementioned assumptions were used in our previous studies [e.g., Kanagawa et al., J. Fluid Sci. Technol.(2010); Kanagawa, J. Acoust. Soc. Am. (2015)] and classical studies [e.g., van Wijngaarden, J. Fluid Mech. (1968)], we here incorporate polydispersity of bubbly liquids. From the method of multiple scales, an amplitude (or a nonlinear wave) equation describing a long-range propagation of waves in bubbly liquids is derived from the basic equations in a two-fluid model. By comparing the present result with the previous results assuming monodispersity, we qualitatively and quantitatively discuss an effect of polydispersity on the wave propagation process in bubbly liquids.

Influence of thermodynamics inside bubble and flow nonuniformity on weakly nonlinear waves in bubbly liquids

An effect of the liquid viscosity and the thermal conductivity and nonuniform distribution of initial flow velocity on weakly nonlinear propagation of pressure waves in an initially quiescent liquid containing many small spherical gas bubbles is theoretically elucidated. Based on the derivation of the KdV–Burgers equation for a long wave and the nonlinear Schrodinger equation for a short carrier wave, the following results are obtained: (i) the installation of the energy equation affects nonlinear, dispersion, and dissipation effects; (ii) the liquid viscosity and the thermal conductivity lead to change considerably the explicit form of the coefficient in the dissipation term; and (iii) the weak nonuniform effect appears in a far field.