# Multiphase Surface Tension

nanoFluidX implementation of the multiphase surface tension model heavily relies on the work of Adami et al.

Analytically, the surface tension term in the momentum equation can be expressed as:(1)
$\frac{du}{dt}=-\alpha \left(\nabla \cdot n\right)n$
With $u$ being the velocity vector, $\alpha$ being the surface tension coefficient and the $n$ being the two-fluid interface unit normal vector. The divergence term is the curvature, as is well known from vector calculus. This term is the most complex to acquire using SPH. According to the Adami model, the SPH formulation for this term becomes:(2)
$\nabla \cdot {\phi }_{i}=d\frac{{\sum }_{j}{\phi }_{ij}\cdot \nabla {W}_{ij}n{V}_{j}}{{\sum }_{j}{r}_{ij}\nabla {W}_{ij}{V}_{j}}$

Where the indices $i$ and $j$ stand for so called ‘owner’ and ‘neighboring’ particles respectively, and the $ij$ index is a difference between the respective variables of particle $i$ and particle $j$ . The $d$ stands for the number of dimensions of the problem, the $\nabla W$ is the gradient of the kernel, r is the position vector and $V$ is the particle volume.

The surface tension model has only four options required for the setup. First, and most important, is to turn the surface tension model on. In the Simulation parameters, the option surften_model specifies the selected surface tension model. The current version (v2022) has three options: NONE, SINGLE_PHASE or ADAMI. For the SINGLE_PHASE surface tension model, refer to the section on Adhesion Model and Single Phase Surface Tension. The second important parameter is the reference curvature ref_curv [1/m] in the Domain parameters, which is the largest expected surface curvature. Third, in the Phase parameters, specify the surface tension coefficient surf_ten [N/m] for the two-phase interaction, for example if you have an oil phase and an air phase, you would specify the same surface tension coefficient for both phases. If surface tension model is set to ADAMI or SINGLE_PHASE, the reference curvature and surface tension coefficient definitions are mandatory.
Important: Enabling of surface tension models in the simulation and requirement to resolve small droplets, for example, ref_curv set to 1000, can be very computationally expensive. Unless it is of utter importance to accurately resolve small droplets, for example, Rdroplet < 1 cm, it is recommended to use a relatively high ref_curv value of ≈ 20. This will make runs much faster, while still including surface tension effects for surface fluid structures which are of the approximate size of 5 cm.

S. Adami, X. Hu und N. Adams, „A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation,“ Journal of Computational Physics, Nr. 229, pp. 5011-5021, 2010.

M. P. Allen und D. J. Tildesley, Computer simulation of liquids, New York: Oxford University Press, 1989.