An immersed boundary method for two-fluid mixtures

We present an Immersed Boundary method for interactions between elastic boundaries and mixtures of two fluids. Each fluid has its own velocity field and volume-fraction. A penalty method is used to enforce the condition that both fluids' velocities agree with that of the elastic boundaries. The method is applied to several problems: Taylor's swimming sheet problem for a mixture of two viscous fluids, peristaltic pumping of a mixture of two viscous fluids, with and without immersed particles, and peristaltic pumping of a mixture of a viscous fluid and a viscoelastic fluid. The swimming sheet and peristalsis problems have received much attention recently in the context of a single viscoelastic fluid. Numerical results demonstrate that the method converges and show its capability to handle a number of flow problems of substantial current interest. They illustrate that for each of these problems, the relative motion between the two fluids changes the observed behaviors profoundly compared to the single fluid case.