The stability of a large gas bubble rising through liquid

Abstract

The upper surface of a large gas bubble rising steadily through liquid under gravity is a statically unstable interface, and if the liquid were stationary small sinusoidal disturbances to the interface with wavelength exceeding the critical value Λcdetermined by surface tension would grow exponentially. The existence of the deforming motion of the liquid adjoining the interface of a steadily rising bubble changes the nature of the problem of stability. It is shown that a small sinusoidal disturbance of the part of the interface that is approximately plane and horizontal remains sinusoidal, although with exponentially increasing wavelength. The amplitude of such a disturbance increases, from the instant at which Λ = Λcuntil Λ becomes comparable with the radius of curvature of the interface (R), and the largest amplification occurs for a disturbance whose initial wavelength is approximately equal to Λc. With a plausible guess at the disturbance amplitude and wavelength at which bubble break-up due to nonlinear effects is inevitable, it is possible to obtain an approximate numerical relation between the initial magnitude of the disturbance and the maximum value ofRfor which a bubble remains intact. This relation applies both to a spherical-cap bubble in a large tank and a bubble rising in a vertical tube in which the liquid far ahead of the bubble is stationary. The few published observations of the maximum size of spherical-cap bubbles are not incompatible with the theory, but lack of information about the magnitude of the ambient disturbances in the liquid precludes any close comparison.