A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid

https://doi.org/10.1016/j.mechrescom.2009.05.003Get rights and content

Abstract

By using the Euler–Bernoulli classical beam theory to model the nanotube as a continuum structure, a reevaluation of the computational modelling of carbon nanotubes conveying viscous fluid is undertaken in this paper, with some fresh insights as to if the viscosity of flowing fluid does influence the free vibration of the nanotube. It is found that during the flow of a fluid through a nanotube, modelled as a continuum beam, the effect of viscosity of flowing fluid on the vibration and instability of CNTs can be ignored.

Introduction

The discovery of carbon nanotubes (CNTs) and their subsequent syntheses in microscopic amounts promoted a very active field of research in experimental and computational condensed matter physics, materials science, nano-science and nano-technology. Studies on the elastic and vibrational properties of CNTs (see e.g., Wang and Cai, 2006, Gibson et al., 2007, and many references cited therein) are numerous in the literature, whereas the literatures on the vibration characteristic of fluid conveying CNTs are relatively limited.

Because of perfect hollow cylindrical geometry and superior mechanical strength, CNTs hold substantial promise as nanocontainers for gas storage, and nanopipes for conveying fluid (such as water or gas) (Che and Lakshmil, 1998, Evans and Bowman, 1996, Liu et al., 1998, Gao and Bando, 2002, Hummer et al., 2001). Storage and transport of fluids (or gases) inside CNTs have been the subject of several studies (Gadd and Blackford, 1997, Li et al., 2003, Skoulidas et al., 2002, Gogotsi et al., 2001). For more details on the topic of fluid inside CNTs, the interested reader is referred to the recent review articles reported by Whitby and Quirke, 2007, Mattia and Gogotsi, 2008.

In an attempt to understand the fluid–structure interactions in CNTs, Tuzun et al. (1996) developed molecular dynamic simulations of fluids flowing through CNTs. It was found that in a fluid conveying CNT system, the motion of the CNT plays a critical role in the fluid flow. For example, a fluid flowing through the CNT tends to straighten out the CNT as it flexes, and simultaneously excites longitudinal vibration modes of the CNT. Natsuki et al., 2007, Dong et al., 2008 studied the wave propagation in single- and multi-walled carbon nanotubes (SWCNTs and MWCNTs) filled with internal fluids by using an elastic shell model.

In the past years, the flow-induced instability of CNTs has been studied by modelling carbon nanotubes with the Euler–Bernoulli beam model. In related work, Yoon et al., 2005, Yoon et al., 2006 developed simple beam models for vibrating CNTs containing fluids, both for the cantilevered and supported systems. It was concluded that the resonance frequencies depend on the fluid flow velocity, and that the critical flow velocity at which structural instability of the CNTs occurs could fall within the range of practical significance. However, the effects of fluid flow velocity on the resonance frequencies would be mitigated when the CNT is embedded in a matrix material in a composite. Another interesting work by Wang et al. (2007) reported the influence of internal fluid on the coupling vibration of fluid-filled CNTs. The results described the effects of end condition, aspect ratio, surrounding elastic medium, mass density of the fluid and layer number on the coupling natural frequencies. Dong et al. (2007) further studied the wave dispersion characteristic in fluid-filled CNTs embedded in an elastic medium and some new feature obtained.

It should be mentioned that, in the papers cited in the previous paragraph and some of the other related literatures on CNTs conveying fluid (see e.g., Wang et al., 2008a, Wang, 2009, and several references cited therein), the internal fluid is assumed to be non-viscous. In 2007, Khosravian and Rafii-Tabar (2007) have reported a computational modelling of the flow of viscous fluids in carbon nanotube. In that paper, again, the Euler–Bernoulli classical beam theory was used to model the nanotube as a continuum structure. It was found that a CNT conveying a viscous fluid is more stable against vibration-induced buckling than a CNT conveying a non-viscous fluid.

Recently, however, it has become clear that the computational modelling given by Khosravian and Rafii-Tabar (2007) might overestimate the influence of viscosity of flowing fluid on the vibration characteristic of CNTs. It will be shown here that the earlier theory given by Khosravian and Rafii-Tabar (2007) should be improved to estimate the vibration of CNTs conveying fluid.

Section snippets

Theoretical basis

Consider a laminar, infinite, incompressible and viscous fluid flowing through an MWCNT, modelled as a single Euler–Bernoulli beam. The single-elastic beam model assumes that all nested individual tubes of a MWCNT remain coaxial (defined by coincident axial lines) during deformation and thus can be described by a single deflection curve. The simplified model based on such an assumption is adequate for MWCNTs of large aspect ratio. The analytical model is shown in Fig. 1. The flexural vibration

Results and discussion

Here, we consider a viscous fluid conveying CNT clamped at both ends. The Differential Quadrature Method (DQM) (Bert and Malik, 1996, Wang and Ni, 2008, Wang et al., 2008b) has been used to formulate the solutions to Eq. (9) combined with a set of specific boundary conditions. This method of numerical analysis was introduced to problems of solid mechanics by Bert and Malik (1996) and will be used here directly. For a self-excited vibration, the solution of Eq. (9) can be written in the

Acknowledgments

The authors acknowledge the funding from the National Natural Foundation of China (Nos. 10772071 and 10802031). The helpful comments made by the anonymous reviewers are also appreciated.

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