The effects of surface tension and viscosity on the rise velocity of a large gas bubble in a closed, vertical liquid-filled tube

The rise of a large gas bubble or slug through a closed vertial tube of radius R and diameter D containing liquid has been calculated by potential flow theory. The effects of interfacial surface tension are explicitly accounted for by application of the Kelvin-Laplace equation and solving for the bubble shape. The solution is expressed in terms of the Stokes stream function which consists of an infinite series of Bessel functions. The resultant equations have been solved for the first six terms in the series. For negligible surface tension and negligible liquid viscocity, the bubble slip velocity is given by Us = 0.352 √gD and the radius of curvature at the nose Rc/R = 0.76. For air/water in a 2.54 cm dia tube, the inclusion of surface tension gives Us = 0.346 √gD, Rc/R = 0.71, which is consistent with experimental observation. The shape of the gas—liquid interface is hemispherical near the nose for large-diameter tubes, where surface tension is negligible. For small-diameter tubes, however, the surface deviates from hemispherical. It is also shown that for viscous liquids the potential flow solution may be applied with good results to a tube of effective radius Reff = R − vδ, where δ is the laminar wall film thickness and v is a function of the liquid properties and tube size.