Abstract
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.
Highlights
The pressure wave in a liquid containing many spherical microbubbles develops into a shock wave owing to the competition between a wave nonlinearity and a dissipation of the medium or into a solitary wave through competition between the wave nonlinearity and a dispersion due to bubble oscillations.1,2 As there are significant differences between shock and solitary waves, it is important to determine whether the pressure wave develops into the shock wave or the solitary wave
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 and the other was a newly discovered term without differentiation comprising the effect of thermal conduction. (ii) The coefficients of the Korteweg–de Vries–Burgers (KdVB) equation depended more on the initial bubble radius rather than on the initial void fraction. (iii) The thermal effect contributed to the dissipation effect and 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
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), 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
Summary
The pressure wave in a liquid containing many spherical microbubbles develops into a shock wave owing to the competition between a wave nonlinearity and a dissipation of the medium or into a solitary wave through competition between the wave nonlinearity and a dispersion due to bubble oscillations. As there are significant differences between shock and solitary waves, it is important to determine whether the pressure wave develops into the shock wave or the solitary wave. (KdVB) equation for a long wave with a low frequency is one of the most famous equations because it is derived in many theoretical studies; its solution agrees with the waveforms observed in several experiments. van Wijngaarden conducted pioneering work in deriving the Korteweg–de Vries (KdV) equation for pressure waves in a bubbly liquid but did not consider the dissipation effect. The purpose of this study is to derive the KdVB equation incorporating the four dissipation factors, i.e., the viscosity of the bubbly liquid and that at the bubble– liquid interface, thermal conduction at the bubble–liquid interface, and acoustic radiation due to liquid compressibility. Based on the derived KdVB equation, we focus on analyzing the thermal conduction at the interface and thermodynamics inside the bubble to theoretically clarify their effects on nonlinear wave propagation.
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