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In this paper, we first construct a model for free surface flows that takes into account
the air entrainment by a system of four partial differential equations. We derive it by
taking averaged values of gas and fluid velocities on the cross surface flow in the Euler
equations (incompressible for the fluid and compressible for the gas). The obtained system
is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of
this system to finally construct a two-layer kinetic scheme in which a special treatment
for the “missing” boundary condition is performed. Several numerical tests on closed water
pipes are performed and the impact of the loss of hyperbolicity is discussed and
illustrated. Finally, we make a numerical study of the order of the kinetic method in the
case where the system is mainly non hyperbolic. This provides a useful stability result
when the spatial mesh size goes to zero.
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