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.