Viscous Multiphase Flow

Multiphase flow and moving boundary problems are ubiquitous in industrial applications including chemical, cosmetics, food, printing, pharmaceutical and more recently microfluidics and lab on a chip applications. However, there are also many numerical complexities in simulation of multiphase ﬂows that are not present in single ﬂuid ﬂows. With the presence of multiple phases comes the requirement of tracking the interface between the phases. Moreover, in numerous applications, such as co-extrusion, film casting, calendaring, dip and roll coating, blends mixing and compounding, and so on, one or more phases involved in the process are complex fluids and thus it is necessary to have a multiphase solver capable of handling the complex rheology of processing fluids. In Simcenter STAR-CCM+, the Conservative Level-Set method is used for this purpose.

Hydrodynamics

The hydrodynamics of viscous multiphase flow is governed by the continuity and momentum equations as given by Eqn. (1014) and Eqn. (1015). However, the non-dimensional stress tensor is modified as:

Figure 1. EQUATION_DISPLAY

T = 2 χ_{s} D + \sum_{i = 1}^{N} Φ_{i} T_{i}^{p}

(1069)

where $Φ_{i}$ is the phase indicator function for the $i$ th phase and is obtained as described in Conservative Level Set.

In the case of non-isothermal multiphase flow, the temperature distribution can be found by solving for energy conservation, see Eqn. (1030) in conjunction with hydrodynamics.

The material properties in the governing equations, are treated as mixture properties of the phases.

Mixture Properties of the N-Phase System

The mixture properties of the system such as density, viscosity and thermal conductivity, are governed according to one of the following mixing rules:

Linear mixing rule: In this case, a mixture property Ψ can be calculated according to:
Figure 2. EQUATION_DISPLAY

$Ψ = \sum_{i = 1}^{N} Ψ_{i} ϕ_{i}$

(1070)
Harmonic mixing rule: In this case, a mixture property Ψ can be calculated according to:
Figure 3. EQUATION_DISPLAY

$Ψ = \sum_{i = 1}^{N} \frac{ϕ_{i}}{Ψ_{i}}$

(1071)
User-defined field function: In this case, write your own field function for the mixture property.
Mass average mixing rule: For the particular case of the specific heat property of mixture, the mass average mixing rule is given by:
Figure 4. EQUATION_DISPLAY

$C_{p} = \frac{\sum_{i = 1}^{N} C_{p_{i}} ρ_{i} ϕ_{i}}{\sum_{i = 1}^{N} ρ_{i} ϕ_{i}}$

(1072)
Notice that, in all of the cases depicted above (except for the user-defined field function), the values of the phase indicator function $ϕ_{i}$ are forced to remain between 0 and 1 to avoid unphysical values for mixture properties in the system. The expression for this is:
Figure 5. EQUATION_DISPLAY

$ϕ_{i}^{*} = min (max (ϕ_{i}, 0), 1)$

(1073)