Governing Equations of the Three-Phase Flow, Phase Field Interface

The Three-Phase Flow, Phase Field interface is used to study the evolution of three immiscible fluid phases denoted by fluid A, fluid B, and fluid C. The interface solves the Navier-Stokes equations governing conservation of momentum and mass. The momentum equations solved correspond to

where Fst is the surface tension force. The physics interface assumes that the density of each phase is constant. The density may however vary between the phases. In order to accurately handle systems where density of the phases are significantly different, as well as systems with phases of similar density, the following fully compressible continuity equation is solved by the interface

In order to track the interfaces between three immiscible fluids, a ternary phase field model based on the work of Boyer and co-workers in Ref. 1 is used. The model solves the following Cahn-Hilliard equations

governing the phase field variable, φi, and a generalized chemical potential, ηi, of each phase i = A, B, C. The phase field variables vary between 0 and 1 and are a measure of the concentration of each phase. The phase field variable for phase A is one in instances containing only this phase, and zero where there is no phase A. Since the fluids are immiscible, variations in the phase field variable occur, and define the interface separating two phases. At each point the phase field variables satisfy the following constraint

The density of each phase is assumed to be constant which implies that the phase field variable corresponds to the volume fraction of the phase in question. In order to satisfy Equation 7-11, two sets of the equations shown in Equation 7-10 are solved, namely those for phase A and phase B. The phase field variable, and correspondingly the mass fraction, for fluid C is computed from Equation 7-11.

The density and viscosity of the fluid mixture used in Equation 7-3 and 7-4 are defined as:

The free energy of the three phase system is defined as a function of the phase field variables in the manner of:

Here the σij denotes the surface tension coefficient of the interface separating phase i and j, and the capillary parameters Σi are defined for each phase in the manner of:

Λ in the free energy is a function or parameter specifying the additional free bulk energy of the system. By default Λ is zero. In this case it can be seen that the free energy in Equation 7-13 represents the mixing energy, since only interfaces between two phases (where two phase field variables varies between the limiting values) contributes to the free energy.

In Equation 7-10, ε (SI unit: 1/m) is a parameter controlling the interface thickness, M0 (SI unit: m3/s) is a molecular mobility parameter, and the parameter ΣT is defined as

When using the interface it is recommended that the parameter ε is given a value in the same order as the elements in the regions passed by the fluid-fluid interfaces. The mobility determines the time scale of the Cahn-Hilliard diffusion and must be large enough to retain a constant interfacial thickness, but small enough to avoid damping the convective transport. In order to ensure that the mobility is in the correct range, it is recommended to a apply a mobility parameter such that the following holds approximately