Efficient numerical methods for multiple surfactant-coated bubbles in a two-dimensional stokes flow

We present efficient and highly accurate numerical methods to compute the deformation of surfactant-coated, two-dimensional bubbles in a slow viscous flow. Surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion. In this paper, we restrict our attention to the case of a dilute insoluble surfactant. The convection–diffusion equation for the surfactant concentration on the interface is coupled with the Stokes equations in the fluid domain through a boundary condition based on the Laplace-Young condition. The Stokes equations are first recast as an integral equation and then solved using a fast-multipole accelerated iterative procedure. The computational cost per time-step is only O(N log N) operations, with N being the number of discretization points on the interface. The bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points. This equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient, implicit treatment of the stiffest terms in the dynamics. Several phenomenologically different examples are presented.