Production and deep-sea mining risers should have sufficient flexibility to avoid severe loading due to vessel motions. Free hanging catenary is the simplest and cheapest configuration because of the ease of installation and the minimal requirements for subsea infrastructure. The interaction of the inner flow with the moving riser dominates the dynamic stability of the system. In the present work, the motion equation for a long, multilayered, fiber-reinforced polymeric (FRP) flexible riser is formulated, and a numerical method for critical inner flow velocities causing buckling in risers is proposed. Taking into account the motion equation, the physics of the inner flow-induced buckling phenomena is analyzed and a detailed interpretation of the terms of the motion equation is carried out. With the aid of transfer matrices and finite elements, the dynamic buckling modes and the natural frequencies are determined. It is proved that the dynamic stability of the riser is affected by the balance of the elastic flexural restoring force, the centrifugal force of the fluid flow in the curved portions, the Coriolis force of the fluid, the inertial force of the fluid and pipe mass, and the damping due to the effect of the surrounding water.