cylinder-motion-2-augl 3. tex

Irrotational motions of bubbles under the action of acceleration of added mass and viscous drag

J. Wang, T. Funada and D. D. Joseph

Abstract

We present elementary examples of how irrotational flows can be used to study the unsteady motions of bubbles in a viscous liquid suitable for the instruction of undergraduate students. 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 viscous drag. The equation of rectilinear motion is linear and may be integrated giving rise to exponential decay with a decay constant 18 v t/a² where v 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. We also discuss the motion of a massless cylindrical bubble under the combined action of Kutta-Joukowski lift, acceleration of added mass and viscous drag. The cylinder moves with a constant speed along a circular path if the liquid is inviscid, whereas it goes to rest in a spiral fashion when the viscous drag is added.