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

A stable bubble trapped in a standing sound wave with frequency less than the resonance frequency of the bubble will be located at the velocity antinode. Steady streaming flow will develop around the bubble and is directly dependent on the bubble’s boundary. Four boundary conditions are possible: 1) nonpulsating; no slip, 2) nonpulsating; free shear, 3) pulsating; no slip, and 4) pulsating; free shear. To solve for these conditions, we expanded the equations of motion with the dimensionless lateral oscillation amplitude,$\epsilon$, using the singular perturbation method. The lateral oscillation amplitude is much smaller than the bubble radius $\epsilon<