Mathematical modelling for acoustic microstreaming produced by a gas bubble undergoing asymmetric oscillations

An exact solution is developed for the bubble-induced acoustic microstreaming in the case of a gas bubble undergoing asymmetric oscillations. The modeling is based on the decomposition of the solenoidal, first- and second-order, vorticity fields into poloidal and toroidal components. The result is valid for small amplitude bubble oscillations without restriction on the size of the viscous boundary layer $(2{\nu}/{\omega})^{(1/2)}$ in comparison to the bubble radius. The nonspherical distortions of the bubble interface are decomposed over the set of orthonormal spherical harmonics ${Y_n^m} ({\theta},{\phi})$ of degree $n$ and order $m$. The present theory describes the steady flow produced by the nonspherical oscillations $(n,{\pm}m)$ that occur at a frequency different from that of the spherical oscillation as in the case of a parametrically-excited surface oscillation. The three-dimensional aspect of the streaming pattern is revealed as well as the particular flow signatures associated to different asymmetric oscillations.