Surface Tension

Immiscibility of two fluids is a result of strong cohesion forces between their molecules and depends on the nature of the fluids. The experimentally determined surface tension coefficient $σ$ expresses the ease with which the fluids can be mixed.

The surface tension force is an interfacial force, which is modeled as a volumetric force using the Continuum Surface Force (CSF) approach of Brackbill and others [580].

For more information, see Surface Tension (VOF model theory).

Strubelj and others [548] proposed an extension of the Continuum Surface Force (CSF) model for a two-fluid model. The surface tension force is split among the phases occupying the cell as:

Figure 1. EQUATION_DISPLAY

f_{σ, i} = β_{i} f_{σ}

(2323)

where the subscript $i$ indicates the $i^{t h}$ phase and $β_{i}$ denotes the splitting factor of the surface tension force. The pressure gradient within the two-fluid model is the summation of the momentum equations:

Figure 2. EQUATION_DISPLAY

\sum_{i} α_{i} \nabla p = \nabla p = \sum_{i} β_{i} f_{σ}

(2324)

At kinetic equilibrium, the pressure gradient is equal to surface tension force, thus:

Figure 3. EQUATION_DISPLAY

\sum_{i} β_{i} = 1

(2325)

Following Strubelj, the following form of surface tension force is implemented:

Figure 4. EQUATION_DISPLAY

f_{σ, i} = α_{i} σ κ \nabla α_{1}

(2326)

Artificial Viscosity

The discretization of the surface tension force term across a sharp interface between phases can lead to errors that manifest themselves as parasitic currents. In some cases, these currents negatively affect the solution and simulation convergence.

One strategy to mitigate the effects of parasitic currents is to add artificial viscosity to the phase viscosity in the vicinity of large interface. This strategy is adapted for EMP and is termed as interface momentum dissipation in general:

Figure 5. EQUATION_DISPLAY

μ_{a r t, p} = {\hat{μ}}_{p} δ

(2327)

Figure 6. EQUATION_DISPLAY

μ_{a r t, s} = {\hat{μ}}_{p} r_{s} δ

(2328)

$μ_{a r t, p}$ and $μ_{a r t, s}$ are the artificial viscosity terms that are added to the primary and secondary phases of the phase interaction.

${\hat{μ}}_{a r t, p}$ is the artificial viscosity per interaction area density. This value is the user-specified Primary Phase Artificial Viscosity Coefficient property (a constant or field function). You can define ${\hat{μ}}_{a r t, p}$ as $Δ t σ$ to ensure a similar behavior to the semi-implicit treatment that is implemented in the VOF model. See Stabilization Term for Semi-implicit Surface Tension.

$r_{s}$ is a correlation ratio with regards to the primary phase. This value is the user-specified Secondary Phase Artificial Viscosity Correlation property (a constant or field function).

The effective viscosity is augmented with the computed artificial viscosity:

Figure 7. EQUATION_DISPLAY

μ_{e f f e c t i v e, p} = μ_{e f f e c t i v e, p} + μ_{a r t, p}

(2329)

Figure 8. EQUATION_DISPLAY

μ_{e f f e c t i v e, s} = μ_{e f f e c t i v e, s} + μ_{a r t, s}

(2330)