Rising bubbles

Abstract

The shape of a rising bubble, or of a falling drop, in an incompressible viscous fluid is computed numerically, omitting the condition on the tangential traction at the bubble or drop surface. When the bubble is sufficiently distorted, its top is found to be spherical and its bottom is found to be rather flat. Then the radius of its upper surface is in fair agreement with the formula of Davis & Taylor (1950). This distortion occurs when the effect of gravity is large while that of surface tension is small. When the effect of surface tension is large, the bubble is nearly a sphere.The shape is found, together with the flow of the surrounding fluid, by assuming that both are steady and axially symmetric, with the Reynolds number being large. The flow is taken to be a potential flow. The boundary condition on the normal component of normal stress, including the viscous stress, is satisfied, but not that on the tangential component. The problem is converted into an integro-differential set of equations, reduced to a set of algebraic equations by a difference method, and solved by Newton's method together with parameter variation.