Phase Field Thin-Film Flow Equations
The phase field method, used to track the interface, adds the following equations:
(6-9)
(6-10)
where the quantity λ (SI unit: N) is the mixing energy density and ε (SI unit: m) is a capillary width that scales with the thickness of the interface. These two parameters are related to the surface tension coefficient, σ (SI unit: N/m), through the equation
and γ is the mobility parameter which is related to ε through γ = χε2 where χ is the mobility tuning parameter (set to 1 by default). The volume fractions of Fluid 1 and Fluid 2 are computed as
where the min and max operators are used so that the volume fractions have a lower limit of 0 and an upper limit of 1. Let ρ1 and ρ2 be the constant densities of Fluid 1 and Fluid 2, respectively. When Density averaging is set to Volume average, the density is defined as,
switching to Heaviside function, the density is defined as,
where H is a smooth step function and lρ is a mixing parameter defining the size of the transition zone. When the Harmonic volume average is selected,
Similarly, the dynamic viscosity can be defined by setting Viscosity averaging to Volume average,
Heaviside function,
Harmonic volume average,
Mass average,
or Harmonic mass average,
where μ1 and μ2 are the dynamic viscosities of Fluid 1 and Fluid 2, respectively.
The mean curvature (SI unit: 1/m) can be computed by entering the following expression:
where G is the chemical potential defined as:
Details of the theory for the phase field method are found in Ref. 2.
Force Terms
The forces applied on the right-hand side of the momentum equation are due to surface tension and a user-defined volume force.
The Surface Tension Force for the Phase Field Method
The surface tension force for the phase field method is implemented as a body force
where G is the chemical potential (SI unit: J/m3) defined in The Equations for the Phase Field Method and f/∂ϕ is a user-defined source of free energy.
The User-Defined Volume Force
When using a Phase Field in Fluids interface, a force arising due to a user-defined source of free energy is computed according to:
This force is added when a ϕ-derivative of the external free energy has been defined in the External Free Energy section of the Fluid Properties feature.