An exact solution is developed for bubble-induced acoustic microstreaming in the case of a gas bubble undergoing asymmetric oscillations. The modelling 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 non-spherical 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 non-spherical 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 with different asymmetric oscillations.

An analytical theory is developed that describes acoustic microstreaming produced by the interaction of an oscillating gas bubble with a viscoelastic particle. The bubble is assumed to undergo axisymmetric oscillation modes, which can include radial oscillation, translation and shape modes. The oscillations of the particle are excited by the oscillations of the bubble. No restrictions are imposed on the ratio of the bubble and the particle radii to the viscous penetration depth and the separation distance, as well as on the ratio of the viscous penetration depth to the separation distance. Capabilities of the developed theory are illustrated by computational examples. The shear stress produced by the acoustic microstreaming on the particle’s surface is calculated. It is shown that this stress is much higher than the stress predicted by Nyborg’s formula (1958 J. Acoust. Soc. Am. 30, 329–339), which is commonly used to evaluate the time-averaged shear stress produced by a bubble on a rigid wall.

An analytical theory is developed that describes acoustic microstreaming produced by two interacting bubbles. The bubbles are assumed to undergo axisymmetric oscillation modes, which can include radial oscillations, translation and shape modes. Analytical solutions are derived in terms of complex amplitudes of oscillation modes, which means that the modal amplitudes are assumed to be known and serve as input data when the velocity field of acoustic microstreaming is calculated. No restrictions are imposed on the ratio of the bubble radii to the viscous penetration depth and the distance between the bubbles. The interaction between the bubbles is considered both when the linear velocity field is calculated and when the second-order velocity field of acoustic microstreaming is calculated. Capabilities of the analytical theory are illustrated by computational examples.

Shape oscillations arising from the spherical instability of an oscillating bubble can be sustained in a stationary acoustic field. Describing such a steady state requires that nonlinear saturation effects are accounted for to counteract the natural exponential growth of the instability. In this paper, we analyse the establishment of finite-amplitude bubble shape oscillations as a consequence of nonlinear interactions between spherical and non-spherical modes. The set of coupled dynamical equations describing the volume pulsation and the shape oscillations is solved using a perturbation technique based on the Krylov–Bogoliubov method of averaging. A set of first-order differential equations governing the slowly varying amplitudes and phases of the different modes allows us to reproduce the exponential growth and subsequent nonlinear saturation of the most unstable, parametrically excited, shape mode. Solving these equations for steady-state conditions leads to analytical expressions of the modal amplitudes and derivations of the conditionally stable and absolutely stable thresholds for shape oscillations. The analysis of the solutions reveals the existence of a hysteretic behaviour, indicating that bubble shape oscillations could be sustained for acoustic pressures below the classical parametric threshold.