Analysing the wave scattering in single-walled carbon nanotube conveying fluid based on the symplectic theory

Based on tcontinuum mechanics theory and the symplectic theory, the single-walled carbon nanotube (SWCNT) is modelled as a Timoshenko beam. The dynamics equations of fluid-conveying SWCNT are derived from Hamilton's principle. By introducing the symplectic variable into the mechanics system, the governing equation of fluid-conveying SWCNT is transformed from Lagrange system into Hamilton system, then the governing equation is employed to analyse the energy band structure of the SWCNT and the wave scattering in the beam. Moreover, the scattering matrix of the nanotube is calculated by symplectic methodology. The influences of the fluid density and velocity to SWCNT's band structure are also analysed. The results show that the shear and flexural frequencies of SWCNT are greater than those of fluid-conveying SWCNT. The analyses indicate that the shear and flexural frequencies of fluid-conveying SWCNT decrease with the fluid velocity and density increasing, because the effect of the fluid inside makes the nanotube softer. Meanwhile, it is also found that the scattering matrix is unitary matrix, pointing the power flow of the incident wave is equal to that of the reflected wave, indicating the power flow of Hamilton system is conserved. Furthermore, the results show the superiority of the symplectic elasticity theory.