Modeling time delay in clusters of interacting bubbles Free

Modeling the dynamics of large clusters of interacting bubbles requires that the effects of fluid compressibility be taken into account. Compressibility manifests itself through radiation damping and bubble‐bubble interactions due to time delays associated with the finite sound speed. The time delays convert the dynamical equations for interacting bubbles in an incompressible fluid from a system of nonlinear ordinary differential equations to one of delay differential equations (DDEs). Special care must be taken when integrating DDEs numerically to maintain acceptable bounds on errors. The dynamical equations determined to be most suitable for solving as DDEs were obtained using Hamiltonian mechanics [Ilinskii et al., J. Acoust. Soc. Am. 121, 786 (2007)]. These first‐order differential equations were augmented to include time delays in the bubble interaction terms and then solved numerically using a sixth‐order Runge-Kutta method with a continuous interpolant (DDE_SOLVER). The same equations were also solved without the time delays but with correction terms that account for mutual radiation damping. Comparison of the results reveals the importance of time delay in bubble‐bubble interactions as a function of the size and density of a bubble cluster. [Work supported by the ARL:UT McKinney Fellowship in Acoustics and NIH Grant No. DK070618.]