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Abstract. In this paper, we propose a thermal buckling analysis of a functionally graded (FG)

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Purpose This study aims to study the transverse vibration and instabilities of the fluid-conveying single-walled carbon nanotubes (CNTs). To this purpose, the Euler–Bernoulli beam model is used. Also, the surface effects, small-size effects of the both fluid and structure and two different elastic mediums viscoelastic and Pasternak elastic are investigated. Design/methodology/approach To consider the nano-scale for the CNT, the strain-inertia gradient theory is used and to solve the governing equation of motion for the system, the Galerkin’s method is used. The effect of the flow velocity, aspect ratio, characteristic lengths of the mentioned theory, effects of Knudsen number and effects of the Winkler, the Pasternak elastic and the viscoelastic medium on the frequencies and stabilities of the system are studied. The effects of the above parameters on the vibrational behavior are investigated both separately and simultaneously. Findings The results show that the critical flow velocity value is increased as the aspect ratio, characteristic lengths, Winkler modulus, shear and damping factors increase. Also, the critical flow velocity is increased by considering the surface effects. In addition, the consequence of increase in the nano-flow-size effects (Knudsen number) is decreasing the critical flow velocity. Moreover, it can be observed that the effect of the shear factor on increasing the critical flow velocity is different from the rest of parameters. Originality/value Use of Timoshenko and modified couple stress theories and taking into account Von-Karman expressions for investigating the nonlinear vibrations of triple-walled CNTs buried within Pasternak foundation.