Bjerknes forces between two bubbles in a viscous fluid

A refined expression for the secondary Bjerknes force between two spherical gas bubbles in a viscous incompressible fluid is derived, allowing for the dipole oscillations of the bubbles and acoustic streaming around them. The investigation is motivated by the fact that the existing theory [see, for example, E. A. Zabolotskaya, Sov. Phys. Acoust. 30, 365–368 (1984)] neglects both of these processes, taking into account only linear viscous effects due to the radial oscillations of the bubbles. More correct calculations reveal that, provided the viscous penetration depth in the surrounding fluid is comparable to the bubble size, the forces on the bubbles differ noticeably from values given by the former theory, and, in addition, are no longer equal and opposite to each other. Such conditions are shown by numerical examples to be quite realistic ones, even for low-viscosity fluids like water.