Effects of time delay on the translational motion of two interacting bubbles

A model coupling the radial pulsation and translational motion of two interacting bubbles is established to investigate their translation, accounting for time-delay effects. The results demonstrate that due to time-delay effects, bubbles that originally approach each other do so more slowly, those that initially move apart separate more rapidly, and bubbles undergoing dynamic equilibrium exhibit an increased equilibrium spacing. The translational motion states of two bubbles can be altered by considering time-delay effects. As the driving acoustic pressure increases, the approaching region expands, while the separating and dynamic equilibrium regions shrink, with all regions shifting toward smaller bubble radii. Analysis of the secondary Bjerknes force reveals its dominant role in bubble translational motion. When the time-averaged secondary Bjerknes force remains constant, the inter-bubble distance maintains a dynamic equilibrium spacing. The results provide an explanation for understanding the formation of bubble clouds.