Abstract The non-linear equations of motion of a flexible pipe conveying unsteadily flowing fluid are derived from the continuity and momentum equations of unsteady flow. These partial differential equations are fully coupled through equilibrium of contact forces, the normal compatibility of velocity at the fluid– pipe interfaces, and the conservation of mass and momentum of the transient fluid. Poisson coupling between the pipe wall and fluid is also incorporated in the model. A combination of the finite difference method and the method of characteristics is employed to extract displacements, hydrodynamic pressure and flow velocities from the equations. A numerical example of a pipeline conveying fluid with a pulsating flow is given and discussed.