Knudsen number

Viscosity parameter
Size-dependent continuum theory
a b s t r a c t
In this article, we reappraise the well-known equation of motion for a pipe conveying viscous fluid. We
utilize prominent principles of fluid mechanics such as Navier–Stokes’ equation as well as several benchmark
references in the field of fluid–structure interaction (FSI) to reveal that the viscosity of the fluid flow
should not appear explicitly in the equation of motion of pipe conveying fluid. Based on this result, we
could develop an innovative model for one dimensional coupled vibrations of carbon nano-tubes (CNTs)
conveying fluid using slip velocity of the fluid flow on the CNT walls as well as utilizing size-dependent
continuum theories to consider the size effects of nano-flow and nano-structure. Therefore, this innovative
coupled FSI equation suggests that CNTs conveying nano-flow remain stable for higher velocities. In
the other words, the critical average velocity of the fluid flow at which the divergence instability occurs,
should be greater in comparison with the critical velocity predicted by the models used plug flow and
classical continuum theories.
2013 Elsevier B.V. All rights reserved.
1. Introduction
Carbon nano-tubes (CNTs) are becoming the most promising
material for nano-electronics, nano-devices and nano-composites
because of their enormous application such as nano-pipettes, actuators,
reactors, fluid filtration devices, biomimetic selective transport
of ions, targeted drug delivery devices, scanning molecule
microscopy, and scanning ion conductance microscopy [1–4]. In
this regard, a remarkable number of studies have been accomplished
to disclose the vibrational behavior of such nano-structures
conveying fluid. Tuzun et al. [5], Amabili et al. [6], Yoon et al. [7],
Natsuki et al. [8], Wang et al. [9], Xia et al. [10] and Wang and Qiao
[11] made important contributions in this practical area. In this research,
we would undertake a reevaluation for computational
modeling of carbon nano-tubes conveying viscous fluid with some
fresh insights as well as we try to develop an innovative one
dimensional (1D) coupled fluid–structure interaction (FSI) equation
by considering slip condition on the nano-tube wall. Khosravian
and Rafii Tabar [12] studied the flow of viscous fluid through a

carbon nano-tube and established a new equation of motion of
pipe conveying fluid by considering the viscosity effect. They found
that a nano-tube conveying a viscous fluid was more stable against
vibration-induced buckling than a nano-tube conveying a non-viscous
fluid. Wang and Ni [13] reappraised the computational modeling
of carbon nano-tube conveying viscous fluid represented by
Khosravian and Rafii Tabar [12] and then corrected the FSI equation
and disclosed that the effect of viscosity of fluid flow on the
vibration and instability of CNTs could be ignored. Lee and Chang
[14] analyzed the influences of nonlocal effect, viscosity effect, aspect
ratio, and elastic medium constant on the fundamental frequency
of a single-walled carbon nano-tube (SWCNT) conveying
viscous fluid embedded in an elastic medium. They revealed that
the frequency increased as the values of the viscosity parameter increased.
Soltani et al. [15] developed a transverse vibrational model
for a viscous fluid-conveying SWCNT embedded in biological soft
tissue. Their investigation determined that the structural instability
and the associated critical flow velocity could be affected by
the viscosity of the fluid and the nonlocal parameter. Khoddami
et al. [16] studied electro-thermo nonlinear vibration and instability
of embedded double-walled Boron Nitride nano-tubes
(DWBNNTs) conveying viscous fluid based on nonlocal piezoelasticity
theory. They reported that increasing the small scale parameter
decreased the real and imaginary parts of frequency and
critical fluid velocity. Furthermore, they concluded that the effect
of fluid viscosity on the vibration and instability of DWBNNTs
might be ignored. In many recent studies various size-dependent
continuum theories have been developed for vibration and stability
analysis of CNTs conveying fluid. Lee and Chang [17], Zhen
0927-0256/$ – see front matter 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.commatsci.2013.04.047
⇑ Corresponding author. Tel.: +98 311 391 5248; fax: +98 311 391 2628.
E-mail addresses: m.mirramezani@me.iut.ac.ir (M. Mirramezani), hrmirdamadi@
cc.iut.ac.ir (H.R. Mirdamadi), ghayour@cc.iut.ac.ir (M. Ghayour).
Computational Materials Science 77 (2013) 161–171
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Computational Materials Science
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