On hydrodynamic phase field models for binary fluid mixtures

Abstract

Two classes of thermodynamically consistent hydrodynamic phase field models have been presented for binary fluid mixtures of incompressible viscous fluids with possibly different densities and viscosities. One is derived based on the mass-averaged velocity and quasi-incompressible while the other is derived based on the volume-averaged velocity and incompressible. To describe the same binary fluid mixture of two viscous fluid components, which model is more appropriate? To answer this question, we conduct a comparative study in this paper. First, we visit their derivation, conservation and energy dissipation properties and show that the quasi-incompressible model conserves both mass and linear momentum while the incompressible one does not. We then show that the quasi-incompressible model is sensitive to the density difference of the fluid components while the incompressible model is not in a linear stability analysis. Second, we conduct a numerical investigation on coarsening or coalescent dynamics of protuberances using the two models. We find that they can predict quite different transient dynamics depending on the initial conditions and the density difference although they predict essentially the same quasi-steady results in some cases. This study thus cast a doubt on the applicability of the incompressible model to describe dynamics of binary mixtures of two incompressible viscous fluids especially when the two fluid components have a large density difference.