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Viscous irrotational theories and the force on an expanding bubble - A cell-model analysis

By J. C. Padrino and D. D. Joseph

Department of Aerospace Engineering and Mechanics, University of Minnesota, 110 Union St. SE, Minneapolis, MN 55455, USA

(Last posted 26 July 2008)


The dynamics of a bounded viscous incompressible fluid surrounding a spherical bubble in rectilinear motion simultaneously experiencing volume changes is examined by means of two viscous irrotational theories, namely, viscous potential flow and the dissipation method. The forces that the liquid produces on the bubble and on the outer spherical boundary of the liquid are determined from these two approaches at the instant when the bubble is concentric with the outer surface. Viscous potential flow involves surface integration of the irrotational normal stress; the dissipation method stems from the mechanical energy balance, including the dissipation integral, evaluated in potential flow. In the inner boundary, zero tangential stress is enforced. Two choices for the tangential stress condition on the outer boundary are considered: Zero tangential stress or irrotational tangential stress. In a sense, this is an extension to include viscous effects of the inviscid analysis by Sherwood [Int. J. Multiphase Flow, 25, 705, 1999]. The potential flow that follows from Sherwood's work is used in the derived formulae to compute the drag. To the added-mass forces associated with the bubble acceleration and rate of change of the bubble radius determined by Sherwood, a viscous contribution is added here that depends upon the instantaneous bubble velocity and the inner and outer instantaneous radii of the bubble-liquid cell. When the outer radius is taken to infinity, the expressions for the drag yield results given in the literature. If the inner and outer radii are held fixed, results from the cell model may be used to approximate the drag on a bubble moving in a bubbly flow with the same volume fraction as the cell. The analysis yields two results for the viscous drag on the bubble contingent on the boundary condition applied on the outer sphere. These formulas have been presented in the literature, although regarded as contradictories. By emphasizing the role of the tangential stress on the outer boundary, it is shown that both results are valid as they depend on the choice of the outer dynamic boundary condition. These results agree to first order in the volume fraction. The terminal rise velocity of a bubble swarm is derived using the drag from the viscous irrotational theories. Results for the drag coefficient and bubble rise velocity are compared with other theoretical results as well as data from numerical simulations and experiments.