Nonlinear radial oscillations of a gas bubble including thermal effects

Abstract

A system of integral differential equations determining the nonlinear oscillations of a spherical gas bubble including thermal and viscous damping effects is derived. This system is obtained systematically from the Navier–Stokes equations for the liquid and the gas phases and from the interface conditions by the method of matched asymptotics. The ratio of the thermal length (De/ω)1/2, in the gas and the mean bubble radius is the small expansion parameter. Here De is the reference thermal diffusivity of the gas and ω the frequency of oscillation. By using the well-tested approximate viscosity function in compressible boundary layer theory, the pressure inside the bubble can be related to its radius by an integral equation. Coupling this integral equation with the Rayleigh–Plesset equation gives us a closed system determining the radial oscillations of the bubble. For small oscillations, the system is solved analytically and the results are in agreement with the linear theory. This system is then solved numerically for finite amplitude oscillations.