<p>In this paper, the free vibration of a pipe conveying sub and supercritical velocity fluid is investigated. The constitutive relation is considered as a standard solid viscoelastic model. Based on Euler–Bernoulli beam theory, the transverse nonlinear dynamic equation of the pipe conveying fluid is established by using the element method. The vibration frequencies of the pipe conveying sub and supercritical velocity fluid are analyzed by the Galerkin truncation method and verified by finite difference method. It has been found that an increase in both pipe length and flow velocity results in a decrease in vibration frequency in the subcritical regime, which is entirely opposite in the supercritical regime. The effects of pipe wall thickness and the pipe flow mass ratio on the first two natural frequencies exhibit opposing trends, with the natural frequency in the supercritical regime being more sensitive to variations in these two parameters. Moreover, the standard solid constitutive model is compared with the Kelvin–Voigt model. It is observed that irrespective of the flow velocity, an increase in pipe length results in a greater vibration difference between the two models, with this effect being more pronounced at lower flow velocity. Furthermore, the vibration frequency of the standard solid constitutive model is smaller than that of the Kelvin–Voigt model in the subcritical regime, and the opposite is true in the supercritical regime.</p>