  In the present study we investigate analytically the process of velocity and
energy relaxation in two-phase flows. We begin our exposition by considering
the so-called six equations two-phase model [Ishii1975, Rovarch2006]. This
model assumes each phase to possess its own velocity and energy variables.
Despite recent advances, the six equations model remains computationally
expensive for many practical applications. Moreover, its advection operator may
be non-hyperbolic which poses additional theoretical difficulties to construct
robust numerical schemes |Ghidaglia et al, 2001]. In order to simplify this
system, we complete momentum and energy conservation equations by relaxation
terms. When relaxation characteristic time tends to zero, velocities and
energies are constrained to tend to common values for both phases. As a result,
we obtain a simple two-phase model which was recently proposed for simulation
of violent aerated flows [Dias et al, 2010]. The preservation of invariant
regions and incompressible limit of the simplified model are also discussed.
Finally, several numerical results are presented.

Classical Physics;Analysis of PDEs;Numerical Analysis;Computational Physics;Fluid Dynamics