A coupled immersed boundary and immersed interface method for interfacial flows with soluble surfactant

In this paper, we develop a coupled immersed boundary (IB) and immersed interface method (IIM) to simulate the interfacial flow problems with soluble surfactant. That is, a coupled system of surface-bulk convection-diffusion equations must be solved not only on a moving interface but also in an evolving irregular domain. Based on the immersed interface framework, we first begin with the numerical development for the diffusion equation in a fixed irregular domain and then extend the scheme to solve the convection-diffusion equation in an evolving domain. The fluid motion governed by the incompressible Navier-Stokes equations is solved by using traditional immersed boundary method while the bulk surfactant is solved by the proposed immersed interface method. A series of numerical tests for the present scheme have been conducted to illustrate the accuracy and applicability of the method. We first perform the accuracy and efficiency tests for the present IIM solver. We then check the convergence of the surface and bulk surfactant and the fluid variables in the interfacial flow problems. We further run a series of numerical simulations for a suspended droplet under shear flow with presence of soluble surfactant to study the effects of the dimensionless Biot number and the bulk Peclet number on the droplet deformation in details.