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We explore a computational model of an incompressible fluid with amulti-phase field in three-dimensional Euclidean space. By investigating anincompressible fluid with a two-phase field geometrically, we reformulate theexpression 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 dimensionalLie group, the volume-preserving diffeomorphism. The variational principle tothe action integral with the surface energy reproduces their Euler equation ofthe two-phase field with the surface tension. Since the surface energy ofmultiple interfaces even with singularities is not difficult to be evaluated ingeneral and the variational formulation works for every action integral, thenew 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 surfacetension of the multi-phase field. The obtained Euler equation governs theequation of motion of the multi-phase field with different surface tensioncoefficients without any difficulties for the singularities at multiplejunctions. In other words, we unify the theory of multi-phase fields whichexpress low dimensional interface geometry and the theory of the incompressiblefluid dynamics on the infinite dimensional geometry as a variational problem.We apply the equation to the contact angle problems at triple junctions. Wecomputed the fluid dynamics for a two-phase field with a wall numerically andshow the numerical computational results that for given surface tensioncoefficients, the contact angles are generated by the surface tension asresults of balances of the kinematic energy and the surface energy.

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