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.