We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve gamma. The one-dimensional model is obtained via singular perturbation of the Navier-Stokes system as epsilon, the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow in powers of epsilon is computed. The first term in the expansion depends only on the tangential injection along the central curve gamma of the pipe and the velocity as well as the pressure drop are in the tangential direction. The second term contains the effects of the curvature (flexion) of gamma in the direction of the tangent while the effects of torsion appear in the direction of the normal and the binormal to gamma. The boundary layers at the ends of the pipe are studied. The error estimate is proved.