Steady irrotational flow of inviscid liquid of density Ļ l around a spherical gas bubble which lies on the axis of a cylindrical pipe is investigated using the analysis of Smythe (Phys. Fluids 4 (1961) 756). The bubble radius b = qa is assumed small compared to the pipe radius a, and the interfacial tension between gas and liquid is Ī³ . Far from the bubble, in the frame in which the bubble is at rest, the liquid velocity along the pipe is v 0 , whereas the liquid velocity at points on the wall closest to the bubble is
The decrease in wall pressure as the bubble passes is therefore p = 1.776Ļ l v 2 0 q 3 . When the Weber number W = 2bv 2 0 Ļ l /Ī³ is small, the bubble deforms into an oblate spheroid with aspect ratio Ļ = 1 + 9W (1 + 1.59q 3 )/64. If the fluid viscosity Āµ is non-zero, and the Reynolds number Re = 2v 0 Ļ l b/Āµ is large, a viscous boundary layer develops on the walls of the pipe. This decays algebraically with distance downstream of the bubble, and an exponentially decaying similarity solution is found upstream. The drag D on the bubble is D = 12ĻĀµv 0 b(1 -2.21Re -1/2 )(1 + 1.59q 3 ) + 7.66Āµv 0 bRe 1/2 q 9/2 , larger than that given by Moore (J. Fluid Mech. 16 (1963) 161) for motion in unbounded fluid. At high Reynolds numbers the dissipation within the viscous boundary layers might dominate dissipation in the potential flow away from the pipe walls, but such high Reynolds numbers would not be achieved by a spherical air bubble rising in clean water under terrestrial gravity.