WS, Michael Siegel: A fast mesh-free boundary integral method for two-phase flow with soluble surfactant

Speaker
Michael Siegel,New Jersey Institute of Technology

Abstract
We present an accurate and efficient boundary integral (BI) method to simulate the deformation of drops and bubbles in Stokes flow with soluble surfactant.
Soluble surfactant advects and diffuses in bulk fluids while adsorbing and desorbing from interfaces. Since the fluid velocity is coupled to the surfactant concentration, the advection-diffusion equation governing the bulk surfactant concentration C is nonlinear, precluding the Green’s function formulation necessary for a BI method.
However, in the physically representative large Peclet number limit an analytical reduction of the surfactant dynamics surprisingly permits a Green’s function formulation for C as an Abel-type time convolution integral at each interface point. A challenge in developing a practical numerical method based on this formulation is the fast evaluation of the time convolution, since the kernel depends on the time history of quantities at the interface, which is only found during the time-stepping process. To address this, we develop a novel, causal version of the Fast Multipole Method that is mesh-free in the bulk phase. The resulting method provides an accurate solution to the fully coupled moving interface problem with soluble surfactant. The approach extends naturally to a broader class of advection-diffusion problems in the high Peclet number regime.