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The interaction between a flowing fluid and the transmission mains is in relation with the applied pressure level as well as the sought structure lifetime. The numerical approach developed in this study is based on a global modeling of the fluid and the deformation of the duct under transient conditions. The viscous tangential stress is expressed using a mixed polynomial development representing the variation of the radial profile of the fluid speed. The resolution of the obtained system of equations is carried out using the method of the characteristics coupled with the finite differences method. In the case of a Newtonian fluid, the results were compared to the experimental data of Holmboe and the theoretical ones of Zielke. This comparison allowed confirming the validity of the adopted calculation scheme. An extension of calculations in a similar way was carried out for the same installation in the case of a linear viscoelastic fluid. [ABSTRACT FROM AUTHOR]