Surface Tension

The surface tension force is a tensile force tangential to the interface separating two fluids. It works to keep the fluid molecules at the free surface in contact with the rest of the fluid.

The interfacial surface force is modeled as a volumetric force using the continuum surface force (CSF) approach of Brackbill and others [580].

The magnitude of the surface tension force depends mainly on the nature of the fluid pair and on temperature. For a curved interface, the surface tension force $f_{σ}$ can be resolved into two components:

Figure 1. EQUATION_DISPLAY

f_{σ} = f_{σ, n} + f_{σ, t}

(2605)

where:

Figure 2. EQUATION_DISPLAY

f_{σ, n} = σ κ n

(2606)

and:

Figure 3. EQUATION_DISPLAY

f_{σ, t} = \frac{\partial σ}{\partial t} t

(2607)

where:

$σ$ is the surface tension coefficient.
$n$ is the unit vector normal to the free surface and directed from liquid to gas.
$t$ is the unit vector in the tangential direction to the free surface.
$κ$ is the mean curvature of the free surface.

The surface tension force is calculated according to the continuum surface force (CSF) model. This model uses the smooth field of the phase volume fraction $α_{i}$ to calculate a vector normal to the interface:

Figure 4. EQUATION_DISPLAY

n = \nabla α_{i}

(2608)

The curvature of the interface can therefore be expressed in terms of the divergence of the unit normal vector $n$ :

Figure 5. EQUATION_DISPLAY

κ = - \nabla\cdot \frac{\nabla α_{i}}{| \nabla α_{i} |}

(2609)

Now the normal component of the surface tension force $f_{σ, n}$ can be expressed as:

Figure 6. EQUATION_DISPLAY

f_{σ, n} = - σ \nabla\cdot (\frac{\nabla α_{i}}{| \nabla α_{i} |}) \nabla α_{i}

(2610)

When the surface tension coefficient varies along the surface, which can be due to temperature differences, the tangential part does not vanish. In this case, Marangoni or Bénard convection can develop tangential to the free surface. The tangential force is evaluated as:

Figure 7. EQUATION_DISPLAY

f_{σ, t} = {(\nabla σ)}_{t} | \nabla α_{i} |

(2611)

where ${(\nabla σ)}_{t}$ is the gradient of the surface tension coefficient in the tangential direction.

Stabilization Term for Semi-implicit Surface Tension

Surface tension problems can be difficult to simulate, for two reasons:

The surface tension force induces a jump discontinuity in the pressure, which needs to be handled in a stable manner.
Evaluating the curvature is difficult when it is based on discontinuous geometry information (such as volume fraction).

These difficulties typically manifest themselves as parasitic currents (spurious velocities) which can affect the stability of the simulation.

To enhance stability, the momentum source term due to surface tension can be formulated in a semi-implicit way. This treatment allows you to use larger time step sizes for simulations in which surface tension is significant.

From Eqn. (2606), the surface tension force term that is applied to the right-hand side of the momentum equations is $σ κ n$ , where $σ$ is the surface tension coefficient, $κ$ is the curvature of the free surface $Γ$ , and $n$ is the unit normal vector of $Γ$ .

To treat the surface tension force term semi-implicitly with respect to the velocity (instead of explicitly), a suitable temporal linearization is introduced.

From differential geometry, on the free surface $Γ$ , $n$ can be represented in terms of the Laplace-Beltrami operator (Surface Laplacian) as:

Figure 8. EQUATION_DISPLAY

\begin{matrix} κ n = \underset{̲}{Δ} x & o n & Γ \end{matrix}

(2612)

where $\underset{̲}{Δ}$ is the Laplace-Beltrami operator, and $x$ is the identity mapping on $Γ$ .

Semi-implicit expansion replaces $x$ in the identity in Eqn. (2612) to yield:

Figure 9. EQUATION_DISPLAY

{(κ n)}^{n + 1} \approx \underset{̲}{Δ} x^{n} + Δ t \underset{̲}{Δ} v^{n + 1}

(2613)

The linearization in Eqn. (2613) adds a stabilization term to the momentum equations. The additional term can be interpreted as additional (artificial) shear stress that acts tangentially to the surface. The stabilizing term scales proportionally with surface tension and time step size, and therefore counteracts spurious currents at the free surface with the appropriate amount of diffusion.