Motion of a pair of bubbles in a liquid of low viscosity

Lagrange's equations are used to examine the long-range interaction of bubbles. The Lagrange function equals the kinetic energy of an ideal liquid flowing around a bubble. The generalized external forces include the upthrust and the viscous resistance to flow around each buble. The azimuthal angle is increased by the long-range interaction. The locus for the relative motion is calculated for: 1) the case in ~hieh the relative speed is fairly high, which allows one to neglect the effects of viscosity on the collision time. 2) low relative speed, where the viscous forces determine the motion. Estimates are given for the differential effective cross-section for elastic scattering and the coalescence cross-section.We are indebted to V. G. LevichandV. V. Tolmachev for discussions.