  The mathematical up-scaling of gas-liquid bubbly flows was carried out under
the framework of the volume averaging theory. A two-fluid model and its
associated closure problem were deduced. The closure problem was solved for a
case study: a liquid flow past an assembly of homogeneous and monodisperse
distorted bubbles. To this end, a cylindrical unit cells with concentric
bubbles was used as approach of a representative elementary volume. The
following conditions were evaluated: 0 < Re <= 675, 0 < {\epsilon}g <= 0.2,
7.15 x 10-9 <= We <= 3.96 and Mo = 2.55 x 10-11. The interfacial force closure
was analyzed in terms of the ratio between the interfacial force for bubbly
flow and that existing for a single bubble with the equivalent Re (in an
unbounded liquid). It was found that, for creeping regime (0.001 <= Re <= 0.1),
the ratio between drag coefficients increases as {\epsilon}g increases
independently of Re. For intermediate and inertial regimes (1 <= Re <= 650),
the ratio also increases as {\epsilon}g increases; however, two different
trends were observed with respect to Re: the ratio decreases for the
intermediate regime (1 <= Re <= 50) and increases for the inertial regime (50 <
Re <= 650). A transition point around Re = 50 was found, which could be related
to the boundary layer formation. Similar trends with respect to existing data
from direct numerical simulations of bubbly flows were obtained, showing that
this approach represents a good alternative, in terms of computing efforts, to
obtain closure laws for two-phase flow modeling.
