Boundary element approach for the slow viscous migration of spherical bubbles

This paper examines the slow viscous migration of a collection of N ≥ 1 spherical bubbles immersed in a bounded Newtonian liquid under the action of prescribed uniform gravity field and/or arbitrary ambient Stokes flow. The liquid domain is either open or closed with fixed boundary(ies) where the ambient Stokes flow vanishes. The incurred translational velocity of each bubble is obtained by resorting to a well-posed boundary formulation which requires to invert 3N boundary-integral equations. Depending upon the selected Green tensor, these integral equations holds on the cluster's surface plus the boundaries or solely on the cluster's surface. The advocated numerical strategy resorts to quadratic triangular curvilinear boundary elements on each encoutered surface and enables one to accurately compute at a reasonable cpu time cost each bubble velocity. A special attention is paid, both theoretically and numerically, to the case of N-bubble cluster located near a solid and motionless plane wall with numerical results given and discussed for a few clusters subject to gravity effects and ambient linear or quadratic shear flows.