in domains initially inside the interface. Here, inside refers to domains where <0.5 and outside refers to domains where >0.5.

For the initialization to work it is crucial that there are two Initial Value features and one Initial Interface feature. One of the Initial Value feature should use Domain initially: Inside interface and the other Domain initially:

Outside interface. The Initial Interface feature should have all interior

Important

boundaries where the interface is initially present as selection. If the selection of the Initial interface feature is empty, the initialization fails.

Variables For Geometric Properties of the Interface

Geometric properties of the interface are often needed. The unit normal to the interface is given by

These variables are available in the physics interface as the interface normal and mean curvature.

It is only possible to compute the curvature explicitly when using

second-order or higher-order elements.

Note

Reference for the Level Set Interface

1. 1. E. Olsson and G. Kreiss, “A Conservative Level Set Method for Two Phase Flow,” J. Comput. Phys., vol. 210, pp. 225–246, 2005.

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T h e o r y f o r t h e P h a s e F i e l d I n t e r f a c e

The Phase Field Interface theory is described in this section:

•About the Phase Field Method

•The Equations for the Phase Field Method

•Conservative and Non-Conservative Forms

•Additional Sources of Free Energy

•Variables and Expressions

•Reference For the Phase Field Interface

About the Phase Field Method

The phase field method offers an attractive alternative to more established methods for solving multiphase flow problems. Instead of directly tracking the interface between two fluids, the interfacial layer is governed by a phase field variable, . The surface tension force is added to the Navier-Stokes equations as a body force by multiplying the chemical potential of the system by the gradient of the phase field variable.

The evolution of the phase field variable is governed by the Cahn-Hilliard equation, which is a 4th-order PDE. The Phase Field interface decomposes the Cahn-Hilliard equation into two second-order PDEs.

For the level set method, the fluid interface is simply convected with the flow field. The Cahn-Hilliard equation, on the other hand, does not only convect the fluid interface, but it also ensures that the total energy of the system diminishes correctly. The phase field method thus includes more physics than the level set method.

The free energy of a system of two immiscible fluids consists of mixing, bulk distortion, and anchoring energy. For simple two-phase flows, only the mixing energy is retained, which results in a rather simple expression for the free energy.

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The Equations for the Phase Field Method

The free energy is a functional of a dimensionless phase field parameter, :

where is a measure of the interfacial thickness. The following equation describes the evolution of the phase field parameter:

where ftot is the total free energy density of the system (SI unit: J/m3), and u is the velocity field for the advection (SI unit: m/s). The right-hand side of Equation 7-6

strives to minimize the total free energy with a relaxation time controlled by the mobility (SI unit: m3·s/kg).

The free energy density of an isothermal mixture of two immiscible fluids is the sum of the mixing energy and elastic energy. The mixing energy assumes the Ginzburg-Landau form:

where is the dimensionless phase field variable, defined such that the volume fraction of the components of the fluid are (1+ )/2 and (1 )/2. The quantity is the mixing energy density (SI unit: N) and is a capillary width that scales with the thickness of the interface (SI unit: m). These two parameters are related to the surface tension coefficient, (SI unit: N/m), through the equation

The PDE governing the phase field variable is the Cahn-Hilliard equation:

where G is the chemical potential (SI unit: Pa) and is the mobility (SI unit: m3·s/kg). 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 so that the convective terms are not overly damped. In COMSOL Multiphysics the mobility is

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determined by a mobility tuning parameter that is a function of the interface thickness2. The chemical potential is:

The Cahn-Hilliard equation forces to take a value of 1 or 1 except in a very thin region on the fluid-fluid interface. The Phase Field interface breaks Equation 7-8 up into two second-order PDEs:

Conservative and Non-Conservative Forms

If the velocity field is divergence free, use the conservative formulation:

+ u = -----t 2

Using the conservative phase field form the exact numerical conservation of the integral of is obtained. However, the non-conservative form is better suited for numerical calculations and usually converges more easily. The non-conservative form, which is the default form, only conserves the integral of the phase field function approximately, but this is sufficient for most applications.

Additional Sources of Free Energy

In some cases, the expression for the free energy may include other sources. It is possible to incorporate these by modifying Equation 7-11:

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