  The modeling of multi-phase flow is very challenging, given the range of
scales as well as the diversity of flow regimes that one encounters in this
context. We revisit the discrete equation method (DEM) for two-phase flow in
the absence of heat conduction and mass transfer. We analyze the resulting
probability coefficients and prove their local convexity, rigorously
establishing that our version of DEM can model different flow regimes ranging
from the disperse to stratified (or separated) flow. Moreover, we reformulate
the underlying mesoscopic model in terms of an one-parameter family of PDEs
that interpolates between different flow regimes. We also propose two sets of
procedures to enforce relaxation to equilibrium. We perform several numerical
tests to show the flexibility of the proposed formulation, as well as to
interpret different model components. The one-parameter family of PDEs provides
an unified framework for modeling mean quantities for a multiphase flow, while
at the same time identifying two key parameters that model the inherent
uncertainty in terms of the underlying microstructure.
