Computing interfacial flows of viscous fluids

We present a novel technique for solving for the flow of a viscous multi-fluid system in which interfaces are present. The Navier-Stokes equations are used in their exact form, without the need for phase-field approximations. A Boussinesq-type approach is invoked, in which the component fluids of differing densities are represented by a single fluid in which the density changes smoothly but rapidly across each narrow interfacial zone. This formulation is illustrated in detail for classical planar Rayleigh-Taylor flow containing two horizontal layers of fluid, in which the upper fluid has greater density. The interface between the fluids is therefore unstable and overturns as time progresses. The results of this new approach are compared against the predictions of classical Boussinesq theory and a recent Extended Boussinesq model proposed by the authors. Finally, a comparison with the predictions of a smoothed-particle-hydrodynamics model gives strong support for this new technique.