An immersed boundary method for mass transport applications in multiphase systems with discontinuous species concentration fields

We present a numerical approach to address mass transport problems in multiphase systems, where a diffusing species concentration may exhibit discontinuities across phase boundaries. The approach employs a fixed structured grid, non-conforming with the probably complex or even evolving phase interfaces, in the same spirit as in numerous works in the literature focused on the dynamics of multiphase flows containing solid particles, immiscible fluids, elastic embedded structures, etc. The distinctive feature of the proposition is that in the transport equation, solved over the entire domain, the discontinuities are captured by including a distribution of source-dipoles along the phase boundaries. Moreover, the magnitude of the discontinuities and the source-dipole field strength do not need to be predetermined but are found as parts of the solution by a compatibility condition on the composite concentration field. A numerical implementation based on the finite element method is presented and examples are discussed demonstrating the validity of the approach. In addition, several types of multiphase mass transport problems are discussed, and simple examples are also presented, where it could find application.