IMA Journal of Applied Mathematics Advance Access published online on January 11, 2008
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxm061
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A potential multipole theory for the hydrodynamics of bubble clouds
School of Mathematics, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Email: pmhaese{at}marshwiggle.net
Received on April 13, 2007; Accepted on October 8, 2007
The occurrence of bubbles in clouds, or assemblies, is observed in a wide range of fluid flows. In an otherwise quiescent fluid, their free rise due to buoyancy may lead to transient planar clusters and ultimately to non-spherical bubble behaviour. This paper concentrates on the application of multipole methods to study the fluid dynamics of such clouds. Under the assumptions of an irrotational flow, the boundary-value problem for the fluid velocity potential is determined from the requirement that the fluid kinetic energy, in suitable integral form, is stationary with respect to a small variation in the potential. Two alternative formulations for the bubble dynamics are then presented. The first relates the trajectories of each spherical bubble to their driving forces through the Kelvin impulse. The coupled two-phase flow equations are posed in a convenient matrix form, and solved numerically to predict the transient behaviour of the assembly. To model the more general free-surface flow, a second novel approach uses a weighted residual method. Simulations on the rise of small spherical, buoyant, bubbles indicate that compact, transient, clusters are formed in a short length of time. As non-uniformities in the bubble radii strongly influence the interaction, the smaller bubbles tend to streamline around the larger ones, leading to dispersion.
Keywords: bubbles; multipoles; clouds; clusters; Kelvin impulse.