The Newtonian method is employed to obtain nonlinear mathematical model of motion of a horizontally cantilevered and inflexible pipe conveying fluid. The order magnitudes of relevant physical parameters are analyzed qualitatively to establish a foundation on the further study of the model. The method of multiple scales is used to obtain eigenfunctions of the linear free-vibration modes of the pipe. The boundary conditions yield the characteristic equations from which eigenvalues can be derived. It is found that flow velocity in the pipe may induced the 3:1, 2:1 and 1:1 internal resonances between the first and second modes such that the mechanism of flow-induced internal resonances in the pipe under consideration is explained theoretically. The 3:1 internal resonance first occurs in the system and is, thus, the most important since it corresponds to the minimum critical velocity.