Nonlinear forced vibrations of a slightly curved nanotube conveying fluid based on the nonlocal strain gradient elasticity theory

Advances in miniaturization of medical and engineering equipment made nanotubes and nanopipes to be very important components for these devices. A nonlinear mechanical behaviour of nonlocal strain gradient of a slightly curved tube conveying pressurized fluid under thermal loading subjected to forced vibration is investigated in this study. The microtube’s viscoelasticity of the material is assumed using the Kelvin–Voigt model. First the effects of scale due to fluid and solid are considered. Then using Hamilton’s principle, and the nonlocal strain gradient elasticity, the nonlinear size-dependent governing partial integro-differential equation (PDE) is derived. Two different methods are used to solve this problem. These are; (1) finite difference method (FDM), is used to solve the PDE, and (2) the Eigenfunction expansion methods was combined using Runge–Kutta and Heun schemes to solve the resulting ODE in time. The results of pipe’s midpoint displacement and frequency are almost indistinguishable with Runge–Kutta and Heun schemes. However, comparing FDM with RK, the displacement is within 16% while frequency is within 2% respectively. Results show that particularly the effect of initial curvature have profound effects on the resonance of the system. For the linear analysis, the slip, nonlocal and thermal parameters degraded the natural frequency of the nanotube. For forced vibration, when initial curvature is zero, one distinct resonant frequency was obtained. However, for slightly curved pipe, two distinct resonant frequencies were obtained for flow velocity between 3.7 and 4.5 respectively. Slightly curved nanopipes with slip boundary condition behave very differently from those without slip boundary condition. There are no comparable results in the study of micropipes conveying fluids in the oil and gas industry.