Expansion and Contraction of Planar, Cylindrical, and Spherical Underwater Gas Bubbles

The one-dimensional motion of a planar gas bubble, the axially symmetric motion of a cylindrical bubble, and the spherically symmetric motion of a spherical bubble in a liquid are considered. Following J. B. Keller and I. I. Kolodner [J. Appl. Phys. 27, 1152–1161 (1956)], the velocity potential in the liquid is assumed to satisfy the wave equation, while the pressure is assumed to be uniform throughout the bubble. In each case there is an equilibrium bubble radius ā. First the small-amplitude motions about equilibrium are found. In the planar case the bubble radius a (t) tends monotonically to ā; in the cylindrical case a (t) performs a finite number of damped oscillations and then tends monotonically to ā; and in the spherical case a (t) tends in an oscillatory way to ā. Next, large-amplitude motions are treated. In the planar case they are determined analytically and it is found that a (t) still tends monotonically to ā. In the spherical case an ordinary differential equation is derived for a (t) which has been solved numerically by Keller and Kolodner (op. cit.), and which yields damped oscillations. In the cylindrical case a pair of nonlinear integrodifferential equations is derived for a (t) and for the velocity potential. These equations are solved numerically and yield a finite number of oscillations followed by a monotonic approach to equilibrium, in agreement with the small-amplitude theory.