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The motion of a spherical gas bubble in viscous potential flow

D.D. Joseph and J. Wang

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

(April, 2005)


A spherical gas bubble accelerates to steady motion in an irrotational flow of a viscous liquid induced by a balance of the acceleration of the added mass of the liquid with the Levich drag. The equation of rectilinear motion is linear and may be integrated giving rise to exponential decay with a decay constant  where  is the kinematic viscosity of the liquid and a is the bubble radius. The problem of decay to rest of a bubble moving initially when the forces maintaining motion are inactivated and the acceleration of a bubble initially at rest to terminal velocity are considered. The equation of motion follows from the assumption that the motion of the viscous liquid is irrotational. It is an elementary example of how potential flows can be used to study the unsteady motions of a viscous liquid suitable for the instruction of undergraduate students. Another example, considered here, is the purely radial irrotational motion of a viscous liquid associated with the motions of a spherical gas bubble. This gives rise to an exact potential flow solution of the Navier-Stokes equations in which the jump of the viscous component of the normal stress is evaluated on the potential flow. The equation of motion for the liquid is almost always called the Reyleigh-Plesset equation but the viscous terms were introduced by Poritsky (1951) and not by Plesset (1949). We show that when the normal stress equation is converted into energy equation in the conventional way used for inviscid fluid, the viscous normal stress term is converted into the viscous dissipation in the liquid evaluated on potential flow.