Rise velocity of a spherical cap bubble

The theory of viscous potential flow is applied to the problem of finding the rise velocity U of a spherical cap bubble (see Davies & Taylor 1950; Batchelor 1967). The rise velocity is given by {U}{√gD}=-8/3{ν(1+8s)}{√gD^3}+ {√2}{3}[ 1-2s-16sσ/ρ gD^2+ 32v^2/gD^3(1+8s)^2]1/2, where $R = D/2$ is the radius of the cap, ρ and ν are the density and kinematic viscosity of the liquid, σ is surface tension, r(θ) = R(1 + sθ²) and $s = r''(0)/D$ is the deviation of the free surface from perfect sphericity r(θ)=R near the stagnation point θ = 0 . The bubble nose is more pointed when $s < 0$ and blunted when $s > 0.$ A more pointed bubble increases the rise velocity; the blunter bubble rises slower. The Davies & Taylor (1950) result arises when s and ν vanish; if s alone is zero, \[{U}{√gD}= -8/3{ν}{√gD^3}+{√2}{3} [ 1+32ν^2/gD^3]1/2,\] showing that viscosity slows the rise velocity. This equation gives rise to a hyperbolic drag law \[C_D =6+32/R_e,\] which agrees with data on the rise velocity of spherical cap bubbles given by Bhaga & Weber (1981).