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

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