Longitudinal vibration and instabilities of carbon nanotubes conveying fluid considering size effects of nanoflow and nanostructure Soheil Oveissi a , S. Ali Eftekhari b , Davood Toghraie b,n a Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran b Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran HIGHLIGHTS  The effects of small-scale of the both nanoflow and nanostructure on the vibrational response of fluid flowing single-walled carbon nanotubes are investigated.  Critical flow velocity decreases as the wave number increases, employed.  Kn effect has considerable impact on the reduction of critical velocities especially for the air-flow flowing through the CNT. article info Article history: Received 15 March 2016 Received in revised form 28 April 2016 Accepted 9 May 2016 Available online 10 May 2016 Keywords: Longitudinal vibration Nonlocal theory Strain-inertia gradient theory Knudsen number Nano-flow Vibration instability abstract In this study, the effects of small-scale of the both nanoflow and nanostructure on the vibrational re- sponse of fluid flowing single-walled carbon nanotubes are investigated. To this purpose, two various flowing fluids, the air-nano-flow and the water nano-flow using Knudsen number, and two different continuum theories, the nonlocal theory and the strain-inertia gradient theory are studied. Nano-rod model is used to model the fluid-structure interaction, and Galerkin method of weighted residual is utilizing to solve and discretize the governing obtained equations. It is found that the critical flow ve- locity decreases as the wave number increases, excluding the first mode divergence that it has the least value among of the other instabilities if the strain-inertia gradient theory is employed. Moreover, it is observed that Kn effect has considerable impact on the reduction of critical velocities especially for the air-flow flowing through the CNT. In addition, by increasing a nonlocal parameter and Knudsen number the critical flow velocity decreases but it increases as the characteristic length related to the strain-inertia gradient theory increases. & 2016 Elsevier B.V. All rights reserved. 1. Introduction Due to ongoing development of the science and technology, the humankind requirements become different, and to satisfy not only routine needs but also inborn curiosity to know, many researches are done about unknowns. One of those is the plenty of rooms that exist at the bottom, Feynman said [1]. Today, many scientist in- terested in nanotechnology field, and chiefly carbon nanotubes discovered by Iijima [2] in 1991. Just one of the astonishing properties of CNTs is their mechanical behavior because of their high strength, geometrical structure, low mass density and linear elastic behavior. Nano-fluidic devices are from the subjects that they are studied by researchers in this field such as fluid storage, fluid transport and drug delivery [3,4]. To this end, the dynamic behavior of CNT conveying fluid should be investigated. For ex- ample, Lee and Chang [5] investigated the effect of small-size on the equations of motion using nonlocal elasticity. They found that the combination of the first and the second modes appeared above the critical flow velocity. Wang et al. [6] studied the wave propa- gation characteristics in nanotubes conveying viscous fluid based on the nonlocal continuum theory. They reported that with dif- ferent fluid viscosities, the dispersion relation is almost the same for small wave number; but for larger wave number, the wave frequency becomes higher by increasing the fluid viscosity. Rashidi et al. [7] presented one model for a single mode of coupled vi- bration of fluid conveying CNTs considering the slip boundary conditions of nanoflow. They expressed that the critical flow ve- locities could decrease if the passage fluid is a gas with nonzero Kn, in comparison with a liquid nanoflow. Ghavanloo and Fa- zelzadeh [8] investigated the vibration characteristics of nano- tubes embedded in viscous fluid by the Timoshenko beam model Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/physe Physica E http://dx.doi.org/10.1016/j.physe.2016.05.010 1386-9477/& 2016 Elsevier B.V. All rights reserved. n Corresponding author. E-mail address: Toghraee@iaukhsh.ac.ir (D. Toghraie). Physica E 83 (2016) 164–173