Under consideration for publication in J . Fluid Mech . 1 Two - Phase Modeling of a Fluid

We analyse and improve a recently-proposed two-phase ow model for the statistical evolution of two-uid mixing. A hyperbolic equation for the volume fraction, whose characteristic speed is the average interface velocity v , plays a central role. We propose a new model for v in terms of the volume fraction and uid velocities, which can be interpreted as a constitutive law for two-uid mixing. In the incompressible limit, the two-phase equations admit a self-similar solution for an arbitrary scaling of lengths. We show that the constitutive law for v can be expressed directly in terms of the volume fraction, and thus it is an experimentally measurable quantity. For incompressible Rayleigh-Taylor mixing, we examine the self-similar solution based on a simple zero-parameter model for v. It is shown that the present approach gives improved agreement with experimental data for the growth rate of a Rayleigh-Taylor mixing layer. Closure of the two-phase ow model requires boundary conditions for the surfaces that separate the two-phase and single-phase regions, i.e., the edges of the mixing layer. We propose boundary conditions for Rayleigh-Taylor mixing based on the inertial, drag, and buoyant forces on the furthest penetrating structures which deene these edges. Our analysis indicates that the compatibility of the boundary conditions with the two-phase ow model is an important consideration. The closure assumptions introduced here and their consequences in relation to experimental data are compared to the work of others.