Numerical Methods for Multiple Inviscid Interfaces in Creeping Flows

Abstract

We present new, highly accurate, and efficient methods for computing the motion of a large number of two-dimensional closed interfaces in a slow viscous flow. The interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex-variable theory for the biharmonic equation. The numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole-based iterative solution procedure, which requires only O(N) operations where N is the number of nodes in the discretization of the interface. The interface is described spectrally, and we use evolution equations that preserve equal arclength spacing of the marker points. We assume that the fluid on one side of the interface is inviscid and we discuss two different physical phenomena: bubble dynamics and interfacial motion driven by surface tension (viscous sintering). Applications from buoyancy-driven bubble interactions, the motion of polydispersed bubbles in an extensional flow, and the removal of void spaces through viscous sintering are considered and we present large-scale, fully resolved simulations involving O(100) closed interfaces.