The inconsistency of nonlocal effect on carbon nanotube conveying fluid and a proposed solution based on local/nonlocal model

The dynamic behavior of carbon nanotube conveying fluid has long been studied employing modified continuum models, especially Eringen's nonlocal model, which, however, is recently considered as ill-posed and leads to cantilever paradox. In this paper, the inconsistency of nonlocal effect, derived from Eringen's differential nonlocal model, on nanotube conveying fluid under various boundary conditions is detected. And then, to solve this, the local/nonlocal model is introduced as a better substitute. Both governing equations of the two models are derived. Then the integro-differential governing equation derived from local/nonlocal model is converted into a differential one with two supplementary boundary conditions. The two governing equations are deciphered by wave method, since equations in such form have been demonstrated to be impossible to obtain solutions directly. Nonlocal effect is recognized to be twofold: changing the stiffness of the nanotube itself and modifying the softening effect of inside fluid on the whole nanotube-conveying-fluid system. And it is the unusual effect of the first part, derived from Eringen's differential nonlocal model, on the fundamental frequency of fluid-conveying nanotube with clamped-free supports that leaks the inconsistency. On the contrary, nonlocal effect, derived from local/nonlocal model, consistently decreases the eigen-frequencies of all considered modes under whichever of the four boundary conditions. Besides, the first phase parameter can switch down the nolocal effect disregarding the value of nonlcoal parameter, while if the first phase parameter is out of the neighborhood of 1, it appears that the first phase parameter and nonlocal parameter don't limit the modulating-nonlocal-effect ability of each other.