Surface tension of multi-phase flow with multiple junctions governed by the variational principle

TL;DR

This paper develops a variational formulation for multi-phase incompressible fluid flows with multiple junctions, enabling accurate modeling of surface tension effects and contact angles at triple junctions.

Contribution

It extends the variational approach to multi-phase flows with multiple interfaces and junctions, providing a unified framework for surface tension and interface geometry.

Findings

Derived a novel Euler equation for multi-phase flow with multiple junctions.

Numerically demonstrated contact angle formation based on surface tension coefficients.

Unified interface geometry and fluid dynamics in a variational framework.

Abstract

We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, Zaleski and Zanetti (J. Comp. Phys. \vol{113} \yr{1994} \pages{134-147}) as a variational problem related to an infinite dimensional Lie group, the volume-preserving diffeomorphism. The variational principle to the action integral with the surface energy reproduces their Euler equation of the two-phase field with the surface tension. Since the surface energy of multiple interfaces even with singularities is not difficult to be evaluated in general and the variational formulation works for every action integral, the new formulation enables us to extend their expression to that of a multi-phase ($N$-phase, $N\ge2$) flow and to obtain a novel Euler equation with the surface tension of the multi-phase field. The obtained Euler equation governs the equation of motion of the multi-phase field with different surface tension coefficients without any difficulties for the singularities at multiple junctions. In other words, we unify the theory of multi-phase fields which express low dimensional interface geometry and the theory of the incompressible fluid dynamics on the infinite dimensional geometry as a variational problem. We apply the equation to the contact angle problems at triple junctions. We computed the fluid dynamics for a two-phase field with a wall numerically and show the numerical computational results that for given surface tension coefficients, the contact angles are generated by the surface tension as results of balances of the kinematic energy and the surface energy.