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Rise velocity of a spherical cap bubble

Published online by Cambridge University Press:  02 July 2003

DANIEL D. JOSEPH
Affiliation:
University of Minnesota, Aerospace Engineering and Mechanics, 110 Union St. SE, Minneapolis, MN 55455, USA

Abstract

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).

Type
Papers
Copyright
© 2003 Cambridge University Press

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