  We present the derivation of a new unidirectional model for We present the
derivation of a new unidirectional model for unsteady mixed flows in non
uniform closed water pipes. We introduce a local reference frame to take into
account the local perturbation caused by the changes of section and slope. Then
an asymptotic analysis is performed to obtain a model for free surface flows
and another one for pressurized flows. By coupling these models through the
transition points by the use of a common set of variables and a suitable
pressure law, we obtain a simple formulation called PFS-model close to the
shallow water equations with source terms. It takes into account the changes of
section and the slope variation in a continuous way through transition points.
Transition point between the two types of flows is treated as a free boundary
associated to a discontinuity of the gradient of pressure. The numerical
simulation is performed by making use of a Roe-like finite volume scheme that
we adapted to take into account geometrical source terms in the convection
matrix. Finally some numerical tests are presented.

Analysis of PDEs;Classical Physics