Theoretical model for coupled radial and translational motion of two bubbles at arbitrary separation distances

A theoretical model is developed that describes nonlinear spherical pulsations and translational motions of two interacting bubbles at arbitrary separation distances between the bubbles. The derivation of the model is based on the multipole expansion of the bubble velocity potentials and the use of the Lagrangian formalism. The model consists of four coupled ordinary differential equations. Two of them are modified Rayleigh-Plesset equations for the radial oscillations of the bubbles and the other two describe the translational displacement of the bubble centers. The equations are not subject to the assumption that the distance between the bubbles is large compared to the bubble radii and hence make it possible to simulate the bubble dynamics starting from large separation distances up to contact between the bubbles providing that the deviation of the bubble shape from sphericity is negligible. Numerical simulations are carried out to demonstrate the capabilities of the developed model. It is shown that the correct modeling of the translational dynamics of the bubbles at small separation distances requires terms accurate up to ninth order in the inverse separation distance. Physical mechanisms are analyzed that lead to the change of the direction of the relative translational motion of the bubbles in finite-amplitude acoustic fields.