Derivation of an effective equation for ultrasound propagation in a liquid containing many encapsulated bubbles

Abstract

Toward an establishment of basis for medical applications such as ultrasound diagnosis, the mathematical model for the propagation of ultrasound (or pressure waves) in a liquid containing microbubbles encapsulated by a viscoelastic shell is constructed from the theoretical viewpoint. For simplicity, the bubbles do not coalesce, break up, appear, and disappear; the bubbles are spherical, and these oscillations are spherically symmetric; the viscosity of gas inside the bubbles and the thermal conductivities of both phases are neglected: these assumptions are the same as those in our previous studies [e.g., Kanagawa et al., J. Fluid Sci. Technol. (2010); Kanagawa, J. Acoust. Soc. Am. (2015)]. In this paper, the Hoff model (equation of spherical shell bubble) is used for the equation of motion of the bubble in order to theoretically clarify the effect of viscosity and stiffness of the shell, instead of the Rayleigh-Plesset or Keller model. From the method of multiple scales, some linear and /or nonlinear wave equations (e.g., KdV-Burgers equation) as an effective equation are derived from the basic set of governing equations for liquids containing many encapsulated bubbles.