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

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.

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

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