The Terminal Velocity of a Bubble in an Oscillating Flow

A bubble in an acoustic field experiences a net ``Bjerknes'' force from the nonlinear coupling of its radial oscillations with the oscillating buoyancy force. It is typically assumed that the bubble's net terminal velocity can be found by considering a spherical bubble with the imposed ``Bjerknes stresses''. We have analyzed the motion of such a bubble using a rigorous perturbation approach and found that one must include a term involving an effective mass flux through the bubble that arises from the time average of the second-order nonlinear terms in the kinematic boundary condition. The importance of this term is governed by the dimensionless parameter $\alpha ={R^2\omega } \mathord{\left/ {\vphantom {{R^2\omega } \nu }} \right. \kern-\nulldelimiterspace} \nu $, where $R$ is the bubble radius, $\omega $ is the driving frequency, and $\nu $ is the liquid kinematic viscosity. If $\alpha $ is large, this term is unimportant, but if $\alpha $ is small, this term is the dominant factor in determining the terminal velocity. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.