On dynamics of nanotubes conveying nanoflow

In this study, a size-dependent Timoshenko beam model is used for free vibration and instability analysis of a nanotube conveying nanoflow. To capture the size effects, nonlocal strain gradient theory and Knudsen number are applied. The extended Hamilton's principle is employed to obtain the size-dependent governing equations of motion and associated boundary conditions. The Galerkin approach is utilized to convert the partial differential equation into a set of ordinary differential equations. The resulting eigenvalue problem is solved for cantilever Timoshenko nanotubes. Some numerical instances are presented to study the effects of various parameters such as strain gradient length scale, small length scale, length-diameter ratio, nanotube's thickness, Knudsen number and gravity on the eigenfrequencies, critical flutter velocities and instability of the system. The results reveal that the natural frequencies and critical flutter velocities are closer to the ones from Euler-Bernoulli beam model just for long nanotubes and low mode numbers. Furthermore, it is shown that by increasing the length-diameter ratio, the critical flutter velocity increases. Moreover, by increasing the strain gradient length scale and decreasing the small length scale, the critical flutter velocity and stability region increase.