Bubble interactions near a free surface

A boundary integral method is used to calculate the highly non-linear motion of both one and a vertical column of two bubbles beneath a free surface; although the theory is developed for a finite number of bubbles. Cubic splines are used to represent the surface of the bubble and the infinite free surface; with a non-linear distribution of nodes being employed on the free surface in order to more accurately capture the motion of the free surface spike, which experiments show to be narrow and pronounced when bubbles are generated close to the boundary. Calculations show excellent agreement with experiments in both the one- and two-bubble cases.