A mathematical model for unsteady mixed flows in closed water pipes
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We present the formal derivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes. In the case of free surface incompressible flows, the FS-model is formally obtained, using formal asymptotic analysis, which is an extension to more classical shallow water models. In the same way, when the pipe is full, we propose the P-model, which describes the evolution of a compressible inviscid flow, close to gas dynamics equations in a nozzle. In order to cope with the transition between a free surface state and a pressured (i. e., compressible) state, we propose a mixed model, the PFS-model, taking into account changes of section and slope variation.