Using the nonlocal Euler-Bernouli beam model, this paper is carried out to investigate the vibrations and instability of a single-walled carbon nanotube (SWCNT) conveying fluid subjected to a longitudinal magnetic field. The nanobeam with clamped-clamped boundary conditions lies on the Pasternak foundation. Hamilton’s principle is applied to derive the fluid-structure interaction (FSI) governing equation and the corresponding boundary conditions. In the solution part the differential transformation method (DTM) is used to solve the differential equations of motion. The influences of nonlocal parameter, longitudinal magnetic field, Pasternak foundation on the critical divergence velocity of the nanotubes is studied.

In this work,the wave behaviour propagation of single-walled fluid conveying carbon nanotubes (SWCNT) under longitudinal magnetic fields and elastic foundations is studied, based on the nonlocal strain gradients theory(NSGT). With considertion of surface effect, the governing differential equation are obtained utilising Timoshenko beam theory and Hamilton's variational principle. The influence of small-scale parameters, fluid density, magnetic flux, surrounding elastic medium and surface effects on wave behaviour characteristics of carbon nanotubes are discussed in detail. The numerical results illustrated that with the magnetic flux increase，the phase velocities of the carbon nanotube will increase. When the fluid effects inside the carbon nanotube, the wave frequencies of the system reduces with increase in the non-local coefficient, while promotes with increase in the strain gradient coefficient. In addition, fluid density, surrounding elastic medium and surface effects have meaningful influence on the phase velocity of the system.