Equations of motion are obtained for spatially-curvilinear elastic pipelines containing nonstationary flow of a viscous incompressible fluid. The influence of such factors as the rotational inertia and transverse shear strain of the pipe, the friction of the fluid on the internal pipeline surface, the pressure of the flow, is taken into account (the flow is characterized by parameters averaged over the cross-section). The problem is solved in a linear formulation under the assumption of nondeformability of the pipeline cross section