Coupled pulsation and translation of two gas bubbles in a liquid

Abstract

We present and analyse a model for the spherical pulsations and translational motions of a pair of interacting gas bubbles in an incompressible liquid. The model is derived rigorously in the context of potential flow theory and contains all terms up to and including fourth order in the inverse separation distance between the bubbles. We use this model to study the cases of both weak and moderate applied acoustic forcing. For weak acoustic forcing, the radial pulsations of the bubbles are weakly coupled, which allows us to obtain a nonlinear time-averaged model for the relative distance between the bubbles. The two parameters of the time-averaged model classify four different dynamical regimes of relative translational motion, two of which correspond to the attraction and repulsion of classical secondary Bjerknes theory. Also predicted is a pattern in which the bubbles exhibit stable, time-periodic translational oscillations along the line connecting their centres, and another pattern in which there is an unstable separation distance such that bubble pairs can either attract or repel each other depending on whether their initial separation distance is smaller or larger than this value. Moreover, it is shown that the full governing equations possess the dynamics predicted by the time-averaged model. We also study the case of moderateamplitude acoustic forcing, in which the bubble pulsations are more strongly coupled to each other and bubble translation also affects the radial pulsations. Here, radial harmonics and nonlinear phase shifting play a significant role, as bubble pairs near resonances are observed to translate in patterns opposite to those predicted by classical secondary Bjerknes theory. In this work, dynamical systems techniques and the method of averaging are the primary mathematical methods that are employed.

Model for the dynamics of two interacting axisymmetric spherical bubbles undergoing small shape oscillations

“…(18), (22), (25), and (29) using a standard backward differentiation routine 35 are presented. The equilibrium radii of the bubbles were assumed to be on the order of a few microns.…”

“…The translational (secondary Bjerknes) force acting between the bubbles is attractive as a result of their in-phase pulsation. 18,37 The monopole mode decays with time due to viscous damping, while the quadrupole and octupole modes increase as the bubbles move closer together. The dynamics is qualitatively consistent with Fig.…”

“…[15][16][17][18][19][20] In addition, shock-wave induced bubble collapse and jetting near a liquid-solid interface have been studied via direct solution of the Navier-Stokes equations. 21, 22 Blake et al studied bubble deformation near a rigid boundary 23 and a free surface.…”

“…Viscous dissipation is represented at leading order by the term 4 _ R ei =R ei in Eq. (18). 31 The terms of orders e and el 2 are…”

Ultrasound contrast agents : dynamics of coated bubbles

“…Assuming potential flow, the interaction with the wall can be modeled through the "method of images" by placing a virtual bubble with identical source strength (the image bubble) in the mirrored position of the wall plane. The models predict a decrease of the resonance frequency and translatory oscillations near the wall [29,32,[105][106][107][108][109]. Other phenomena that possibly occur are shape oscillations [38] or asymmetrical collapse of the bubbles leading to jet formation [110,111].…”