Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory

• Nonlocal strain gradient theory is used to model fluid-conveying tubes.
• An analytic model for the wave propagation analysis is formulated.
• Dispersion relation between wave frequency and wave number is investigated.
• Dispersion relation is increased with material length scale parameter increasing.
• Dispersion relation is increased with nonlocal parameter decreasing.

The governing equation of wave motion of fluid-conveying viscoelastic single-walled carbon nanotubes is formulated on the basis of the nonlocal strain gradient theory and the Kelvin–Voigt viscoelastic model. Based on the formulated equation of wave motion, the closed-form dispersion relation between the wave frequency (or phase velocity) and the wave number is derived. It is found that, the effects of nonlocal parameters and small scale material parameters on the dispersion relation between the phase velocity and the wave number are significant at high wave numbers, however, may be ignored at low wave numbers. The upstream phase velocities decrease as increasing flow velocity, whereas the downstream phase velocities firstly increase as increasing flow velocity and then decrease as increasing flow velocity. The effect of damping coefficient on the phase velocity of both upstream and downstream waves is negligible at low wave numbers, however, is remarkable at high wave numbers.