A Class of Conservative Phase Field Models for Multiphase Fluid Flows

Hydrodynamic phase field models for multiphase fluids formulated using volume fractions of incompressible fluid components do not normally conserve mass. In this paper, we formulate phase field theories for mixtures of multiple incompressible fluids, using volume fractions, to ensure conservation of mass and momentum for the fluid mixture as well as the total volume for each fluid phase. In this formulation, the mass-average velocity is nonsolenoidal when the densities of incompressible fluid components in the mixture are not equal, making it a bona fide compressible model subject to an internal constraint. Derivation of mass conservation and energy dissipation in phase field models based on both Allen–Cahn dynamics and Cahn–Hilliard dynamics are presented. One salient feature of the phase field models is that the hydrostatic pressure is coupled with the transport of the volume fractions making the momentum transport and the volume fraction transport fully coupled in light of the mass conservation. Near equilibrium dynamics are explored using a linear analysis. In the case of binary fluid mixtures, one potential growth mode is identified in all the models for a class of free energy, which has been adopted for multiphase fluids. The growth is either absent for all waves or of a longwave feature.