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In this contribution, we introduce a versatile formalism to derive unifiedtwo-phase models describing both the separated and disperse regimes. It relieson the stationary action principle and interface geometric variables. The mainideas are introduced on a simplified case where all the scales and phases havethe same velocity and that does not take into account large-scale capillaryforces. The derivation tools yield a proper mathematical framework throughhyperbolicity and signed entropy evolution. The formalism encompasses ahierarchy of small-scale reduced-order models based on a statisticaldescription at a mesoscopic kinetic level and is naturally able to include thedescription of a disperse phase with polydispersity in size. This hierarchyincludes both a cloud of spherical droplets and non-spherical dropletsexperiencing a dynamical behaviour through incompressible oscillations. Theassociated small-scale variables are moments of a number density functionresulting from the Geometric Method Of Moments (GeoMOM). This method selectsmoments as small-scale geometric variables compatible with the structure anddynamics of the interface; they are defined independently of the flow topologyand, therefore, this model pursues the goal of unifying the modelling of afully-coupled two-scale flow. It is particularly showed that the resultingdynamics provides closures for the interface area density equation obtainedfrom the averaging approach. The extension to mass transfer from one scale tothe other including capillary phenomena, as well as the extension to multiplevelocities are possible and proposed in complementary works.

Two-scale modelling of two-phase flows based on the Stationary Action Principle and a Geometric Method Of Moments