The rate at which a long bubble rises in a vertical tube

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Abstract
As a viscous fluid in a vertical tube drains under the effect of gravity, a finger of air rises in the tube. The shape of the interface and the rate at which the finger rises is determined numerically for different values of the dimensionless parameter G = ρgb2/T, where ρ is the density difference between the viscous fluid and the air, b is the radius of the tube, and T is the interfacial tension. A relationship between the Bond number G and the capillary number Ca = μU/T is found and compared with the perturbation result of Bretherton (1961), where μ is the viscosity of the fluid and U is the constant velocity at which the finger rises. The numerical results support Bretherton's conclusions for very small values of Ca and extend the relationship between G and Ca to a region where the perturbation expansion is no longer valid. The results are also valid for long bubbles rising in a tube.

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