Boundary effects on the streaming flow around a bubble located at the velocity antinode of a standing wave

This study uses the singular perturbation method to analyze the streaming flow around a pulsating bubble at the velocity antinode of a standing wave. The bubble radially and laterally oscillates with small nondimensional amplitudes of $ε`$ and $ε,$ respectively. The momentum equation is expanded using $ε$⁠. The frequency parameter $M$⁠, which is the ratio of the bubble radius to the viscous length, is included in the expanded equations as $OM−1$⁠. Four boundary conditions are solved: non-pulsating and pulsating assuming no-slip and shear-free boundaries. For the non-pulsating bubble, the streaming is on the order of $OM−1$ for the shear-free boundary. The flow has a quadrupole pattern, with direction from the equator to the poles. However, for the non-pulsating bubble with the no-slip boundary, the flow pattern is from the poles to the equator and the direction reverses after a critical value of $M=13.3$⁠. When bubble pulsation is introduced, the intensity of the streaming increases and is proportional to $M$⁠. The flow pattern is dipole with a direction from the south to the north pole for the shear-free boundary. For the non-slip boundary, the flow is quadrupole for small values of $M$ and varies with the phase shift $ϕ$⁠. As $M$ increases, the flow intensifies and becomes dipole. For both cases, the maximum velocity is at the phase shift angle $ϕ=135°$ and $M=10$⁠.