The problem of stability of fluid conveying carbon nanotubes clamped at one end and pinned at the other end and subjected to an axial magnetic field is investigated in this paper. Non-local continuum mechanics formulation is utilized to derive the governing fourth-order partial differential equations, which takes into consideration the small length scale effects and the axial magnetic field. Galerkin’s technique is used to find the solution of the governing equation for the case of clamped-pinned boundary. Closed-form expressions for the critical flow velocity above which the system becomes unstable, of the fluid conveying carbon nanotubes, are obtained and numerical results for different values of axial magnetic field parameter are presented in this paper for use in industrial dynamic design of such devices. The results obtained from these simple and approximate expressions are compared with those existing in literature, wherever available and an excellent agreement is found between them. Along with extensive results on critical velocities new and interesting results are also reported for varying values of nonlocal length parameter. From the results presented in this paper, it is observed that the non-local length parameter along with axial magnetic field parameter are having considerable influence on the critical velocities of the fluid conveying nanotubes. © 2017, Polish Academy of Sciences. All rights reserved.