Dimension reduction for compressible pipe flows including friction

We present the full derivation of a one dimensional " Saint-Venant " like equations for barotropic compressible pipe flows including friction. The one dimensional hyperbolic system is called γ−pressurized model where γ is the adiabatic constant. It is obtained through the three dimensional barotropic Navier-Stokes equations under " thin layer " assumptions as a first order approximation. Prescribing suitable boundary conditions, one can introduce a general friction law and then explicitly show its geometrical (w.r.t the hydraulic radius) and hydrodynamical (w.r.t the Oser number) dependencies in the reduced model. In particular, for linear pressure law (γ = 1), we justify the one dimensional pressurized model (called P-model) introduced by the author in the context of unsteady mixed flows in closed water pipes. For non linear pressure law (γ = 1), the γ−pressurized model describes the evolution of a compressible (almost) gravityless flow