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 \frac{U}{\sqrt{gD}}=-\frac{8}{3}\frac{\nu(1+8s)}{\sqrt{gD^3}}+ \frac{\sqrt{2}}{3}\left[ 1-2s-\frac{16s\sigma}{\rho gD^2}+ \frac{32v^2}{gD^3}(1+8s)^2\right]^{1/2}, \nonumber where $R = D/2$ is the radius of the cap, $\rho$ and $\nu$ are the density and kinematic viscosity of the liquid, $\sigma$ is surface tension, $r(\theta) = R(1 + s\theta^2)$ and $s = r''(0)/D$ is the deviation of the free surface from perfect sphericity $r(\theta)=R$ near the stagnation point $\theta = 0$. 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 $\nu$ vanish; if $s$ alone is zero, \[\frac{U}{\sqrt{gD}}= -\frac{8}{3}\frac{\nu}{\sqrt{gD^3}}+\frac{\sqrt{2}}{3} \left[ 1+\frac{32\nu^2}{gD^3}\right]^{1/2},\] 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).