A level-set method for interfacial flows with surfactant

A level-set method for the simulation of fluid interfaces with insoluble surfactant is presented in two-dimensions. The method can be straightforwardly extended to three-dimensions and to soluble surfactants. The method couples a semi-implicit discretization for solving the surfactant transport equation recently developed by Xu and Zhao [J. Xu, H. Zhao. An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput. 19 (2003) 573–594] with the immersed interface method originally developed by LeVeque and Li and [R. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31 (1994) 1019–1044] for solving the fluid flow equations and the Laplace–Young boundary conditions across the interfaces. Novel techniques are developed to accurately conserve component mass and surfactant mass during the evolution. Convergence of the method is demonstrated numerically. The method is applied to study the effects of surfactant on single drops, drop–drop interactions and interactions among multiple drops in Stokes flow under a steady applied shear. Due to Marangoni forces and to non-uniform Capillary forces, the presence of surfactant results in larger drop deformations and more complex drop–drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the level-set method has been used to simulate fluid interfaces with surfactant.