NONLOCAL VIBRATION AND INSTABILITY ANALYSIS OF EMBEDDED DWCNT.pdf

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online 29 April 2014 publishedProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

A Ghorbanpour Arani, E Haghparast, Z Khoddami Maraghi and S Amirwith slip conditions consideration

Nonlocal vibration and instability analysis of embedded DWCNT conveying fluid under magnetic field

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XML Template (2014) [24.4.2014–4:06pm] [1–15]//blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/PICJ/Vol00000/140102/APPFile/SG-PICJ140102.3d (PIC) [PREPRINTER stage]

Original Article

Nonlocal vibration and instabilityanalysis of embedded DWCNTconveying fluid under magnetic fieldwith slip conditions consideration

A Ghorbanpour Arani1,2, E Haghparast1, Z Khoddami Maraghi1

and S Amir1

Abstract

In this study, vibration of double-walled carbon nanotubes (DWCNTs) conveying fluid placed in uniform magnetic field is

carried out based on nonlocal elasticity theory. DWCNT is embedded in Pasternak foundation and is simulated as a

Timoshenko beam (TB) model which includes rotary inertia and transverse shear deformation in the formulation.

Considering slip boundary conditions and van der Waals (vdW) forces between inner and the outer nanotubes, the

governing equations of motion are discretized and differential quadrature method (DQM) is applied to obtain the

frequency of DWCNTs for clamped–clamped boundary condition. The detailed parametric study is conducted, focusing

on the remarkable effects of small scale, Knudsen number, elastic medium, magnetic field, density, and velocity of

conveying fluid on the stability of DWCNT. Results indicate that considering slip boundary conditions has significant

effect on stability of DWCNTs. Also, it is found that trend of figures have good agreement with the previous researches.

Results of this investigation could be applied for optimum design of nano/micro mechanical devices for controlling

stability of DWCNTs conveying fluid under magnetic fields.

Keywords

Nonlocal vibration, DWCNTs, Timoshenko beam model, magnetic field, Knudsen number

Date received: 16 April 2013; accepted: 2 April 2014

Introduction

Nanotubes have received much attention because oftheir great importance in the development of nanode-vices. Carbon nanotubes (CNTs) have demonstratedexceptional mechanical and thermal properties andthe most important features of CNTs are their extre-mely high stiffness combined with excellent resilience.CNTs are considered as one of the most promisingreinforcement materials for high performance struc-tural and multi-functional composites with extremelyapplication potentials.1 Placing CNTs on the mag-netic substrate or adding electrons or holes, the mag-netic properties can be created for it.2 Nanotubes areusually grown on nano-particles of magnetic metal(Fe, Co), which facilitates production of electronicdevices.

In recent years, fluid-conveying micro/nano struc-tures have attracted a large number of studies in theliterature.3–6 The application of nanotubes as fluidtransport, gas storage, and drug delivery devices areconsidered recently. It is noted that most nanodevices

can be modeled as a beam.7 Therefore, investigatingthe mechanical behaviors of these structures isimportant in the design of the nanodevices. Wanget al.8 investigated single- and multi-walled TBmodels for free vibration of CNTs with various endconditions. They showed that TB model should beused for a better prediction of the frequencies. Luet al.9 used nonlocal beam elasticity theory for vibra-tional properties of CNTs and concluded that nonlo-cal parameter had a significant effect on the dynamicproperties of the beams. Vibration of short CNTsusing cylindrical shell, Euler–Bernoulli beam (EBB)and TB models was developed by GhorbanpourArani et al.10 They studied the thermal effect on the

1Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran2Institute of Nanoscience & Nanotechnology, University of Kashan,

Kashan, Iran

Corresponding author:

A Ghorbanpour Arani, University of Kashan, Bolvar-e-Ghotb Ravandi,

Kashan, 87317-5516, Iran, Republic of Islamic.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

0(0) 1–15

! IMechE 2014

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buckling of DWCNT resting on a Pasternak founda-tion using Eringen’s nonlocal elasticity theory andconcluded that the strength of a DWCNT was directlyrelated to the Winkler and shear modules. The flex-ural vibration of the fluid-conveying SWCNT wasstudied by Chang and Lee11 who simulated SWCNTas TB model. They showed that the real component ofa higher mode frequency is larger than of lower modesfor different flow velocities. Using DQM, Ke et al.12

studied nonlocal nonlinear free vibration ofembedded DWCNTs based on the TB theory. Theyfound that an increase in the spring constant of elasticmedium leads to higher linear and nonlinear frequen-cies but lower nonlinear frequency ratio. Using thenonlocal elasticity theory, torsional buckling of aDWCNT embedded on Winkler and Pasternak foun-dations was demonstrated by Mohammadimehret al.13 They studied the effects of the surroundingelastic medium, the van der Waals forces betweenthe inner and the outer nanotubes on the critical tor-sional buckling load and showed that the shear con-stant of the Pasternak type increases the nonlocalcritical torsional buckling load. Yang et al.14 estab-lished a new analytically nonlocal TB model for ana-lysis of the wave propagation in a DWCNTs beamwith the nonlocal effects. The results show that thenonlocal effect on the wave propagations is more sig-nificant. Based on the nonlocal TB theory and trans-fer function method, the transverse vibration of theSWCNT-based micro-mass sensor is analyzed byShen et al.15 They proved that the nonlocal TBmodel is more adequate than the nonlocal EBBmodel for short SWCNT sensors. Murmu andPradhan16 investigated the thermal vibration ofSWCNT based on thermal elasticity mechanics andnonlocal elasticity theory. They studied the influenceof nonlocal small scale effects, temperature change,Winkler constant, and vibration modes of CNT onthe natural frequency. In another study, Murmuet al.17 reported an analytical approach to study theeffect of a longitudinal magnetic field on the trans-verse vibration of DWCNT based on nonlocal elasti-city theory. Results revealed that presence of alongitudinal magnetic field increases the natural fre-quencies of the DWCNT.

In recent decades, simulation and experimental testson CNT have attracted a large number of researchers.The practical investigation of fluid flow in nanoscalechannels has been facilitated by the availability oftubular carbon structures with open central pore diam-eters in the range of one to several hundred nano-meters.18 Two types of nanoscale carbon tubes withdistinct properties and methods of synthesis havebeen used experimentally.19 In the first type, singleand multi-walled CNTs with the fullerene molecularstructure were reported by Ijima.1 In the secondinstance, carbon nanopipes were produced by chemicalvapor deposition (CVD) of amorphous carbon in alu-mina templates with honeycomb pore morphology.18

In 2006, Holt et al.19 achieved a milestone in mea-suring water flow through the central pores ofDWCNT. These tubes had inner diameters less than2 nm with defect-free graphitic walls that are expectedfrom simulation to present a low friction surface totransported fluids. The practical challenges involvedin studying transport through very small CNTs areformidable. The approach used by Holt et al. was touse fabrication techniques adapted from the semicon-ductor industry. The resulting membranes had athickness in the range of 2.0–3.0 mm and pore densities40.25� 1012cm�2. Pore diameters, calibrated bypassage of gold nanoparticles, were in the range of1.3–2.0 nm.

Majumder et al.20 published the results of furtherexperiments on fluid transport through themulti-walled CNT composite membranes. A simplepressure-driven flow apparatus was used, with theaccumulated mass of transported fluids includingwater, ethanol and alkanes being measured after afixed time period. Their results demonstrate dramat-ically enhanced flow about 7 nm diameter nanotubecores compared with conventional fluid flow theory.Observed flow rates were four to five orders of mag-nitude greater than predicted by hydrodynamicsbased on macro-scale behavior. The authors concludethat the implied slip lengths (3 to 70 mm), which aremuch greater than the tube diameter, are consistentwith a nearly frictionless interface.

Heremans et al.21 carried out measurements of themagnetic moment and susceptibility of CNTs. Theyexperimentally illustrated that magnetic moment andsusceptibility behavior is a function of magnetic fieldstrength and temperature.

The comparatively small diameter of the inner tubein typical DWCNT samples has led to a number ofRaman studies which attempted an assignment of theinner tube chirality based on the radial breathingmode (RBM) frequencies.22–24 In these investigations,the interaction between the tube walls is often mod-eled by adding a constant to the dependence of theRBM frequency on the inverse tube diameter. In boththeoretical and experimental studies, the distancebetween the tube walls is usually assumed the sameas the interlayer distance in graphite, i.e. �3:4A0.Since the controlled experiments on nanoscale are dif-ficult, mathematical simulations are commonly usedto investigate mechanical behaviors of the micro/nanostructures.

However, to date, no report has been found in theliterature on the nonlocal vibration and stability ofDWCNT. Motivated by these considerations, thisstudy aims to develop the vibration analysis ofDWCNTs conveying fluid subjected to magneticfield based on TB theory, where DWCNT placed inPasternak foundation. DQM is presented to solve themotion equations and estimate frequency and criticalfluid velocity of the DWCNTs. The influences of non-local parameter, slip flow regime, magnetic field,

2 Proc IMechE Part C: J Mechanical Engineering Science 0(0)




Pasternak constants, density and velocity of fluid onvibration, and stability of DWCNTs are discussedin detail. Results of this investigation could be appliedto design this kind of nanodevices.

Mathematical modeling

Fundamental formulations for TB-model

A schematic diagram of a fluid-conveying embeddedDWCNT subjected to uniform longitudinal magneticfield is shown in Figure 1 in which geometrical par-ameters of inner radius R1 and outer radius R2 arealso indicated.

In TB theory, displacement fields are assumed as25

~Uiðx, z, tÞ ¼ Uiðx, tÞ þ z iðx, tÞ,

~Viðx, z, tÞ ¼ 0,

~Wiðx, z, tÞ ¼Wiðx, tÞ, ð1Þ

where ~Ui, ~Vi, and ~Wi denote the longitudinal, circum-ferential, and transverse displacements of the middlesurface, respectively. Also, i is the rotation of beamcross-section and t is time. It is noted that i ¼ 1, 2represent the inner and outer nanotubes. Using afore-mentioned equation, the strain displacement relationcan be written as

"xxi ¼@Ui

@xþ z

@ i

@x,

�xzi ¼@Wi

@xþ i ð2Þ

Energy function

The total potential energy of DWCNT is defined as12

Ui ¼1

2

Z L

0

ZAi

ð�xx"xx þ �xz�xzÞ dAidx, ð3Þ

where �xx, �xz , and Ai represent normal stress, shearstress, and the cross-sectional area of DWCNTs,respectively.

DWCNT is conveying a fully developed isentropic,incompressible, and irrotational fluid flow. It isassumed that the fluid flow moves through theDWCNT with constant velocity and it has no mag-netic property (nonmagnetic fluid); fluid velocity isnot affected by the magnetic field.26–28 Fluid velocityis not related to applied forces on DWCNT such asthe surrounding elastic medium, etc. In addition,magneto-rheological effect of DWCNT has beenignored.29 It is worth mentioning that this researchfocused on nonmagnetic liquid water flow. Liquidwater molecules have diameter D of approximately0.29 nm30 which is smaller in size than the innerradius of DWCNT. Therefore, the assumption of con-tinuity for conveying fluid through the DWCNT iscorrect.

The kinetic energies of the DWCNT and fluid floware31

Ktube¼1

2�t

Z L

0

ZAi

@ ~Ui

@t

!2

þ@ ~Wi

@t

!224

359=;dAidx

8<: i¼1,2

Kfluid¼1

2�f

Z L

0

ZAf

@W1

@tþuf

@W1

@x

� �2

þu2f

" #( )dAfdx,

ð4Þ

where �t and �f denote the density of DWCNT andfluid, respectively.

Lorentz force

In this section, Maxwell’s relations are presented. TheMaxwell’s relations according to Murmu et al.17 areexpressed as

~J ¼ r � ~h r � ~e ¼ ��@ ~h

@t, div ~h ¼ 0,

Figure 1. The schematic of DWCNT conveying fluid embedded in Pasternak medium under longitudinal magnetic field.

Ghorbanpour Arani et al. 3




~e ¼ ��@ ~D

@t� ~H

!, ~h ¼ r � ð ~D� ~HÞ: ð5Þ

where ~D is the displacement vector as ~D ¼ ðU,V,WÞ,~J as current density, ~h as distributing vector of themagnetic field, and ~e as strength vector of the electricfield.

Applying a uniform longitudinal magnetic fieldvector ~H ¼ ðHx, 0, 0Þ on DWCNT, the Lorentz forcecan be expressed as17

Fx ¼ 0

Fy ¼ �H2x

@2V

@x2þ@2V

@y2þ@2W

@y@z

� �

Fz ¼ �H2x

@2W

@x2þ@2W

@y2þ@2V

@y@z

� �: ð6Þ

For the present vibrational analysis in DWCNTs,assume that W ¼Wðx, tÞ only, so that the Lorentzforce in the z direction is written as

Fzi ¼ �H2x

@2Wi

@x2: ð7Þ

It should be noted that in the present study the effect-ive Lorentz force is a function of magnetic permeabil-ity (�) and magnetic intensity (Hx) also. It is worthmentioning that the fluid velocity is not affected bythe Lorentz force.

Elastic medium

Based on the Pasternak foundations, the effects of thesurrounding elastic medium on the nanotubes areconsidered as follows10

FElastic medium ¼ kWW2 � Gp r2W2 ð8Þ

where kw and Gp are spring and shear modulus,respectively.

vdW interaction

The vdW force on the outer tube is written as12

q1 ¼ cðW2 �W1Þ ð9Þ

where c is vdW interaction coefficient and Wiði ¼ 1, 2Þcorresponds to the transverse displacement of layers.Subscripts 1 and 2 denote the inner and outer tubes,respectively. It is well known that the interactionforces between the inner and outer tubes are equaland in opposite directions

q1R1 ¼ �q2R2 ð10aÞ

where R1 and R2 are the inner and outer radii ofDWCNT, respectively. Substituting equations (10a)

into equation (9) yields

q2 ¼ �cR1

R2ðW2 �W1Þ: ð10bÞ

The work done due to magnetic field, surroundingelastic medium, and vdW forces can be written as31

� ¼1

2

Z L

0

Fz1W1dxþ1

2

Z L

0

Fz2W2dx

þ1

2

Z L

0

q1W1dxþ1

2

Z L

0

q2W2dx

þ1

2

Z L

0

FElasticmedium W2dx: ð11Þ

Knudsen number

In the molecular phase, fluids are composed of dis-crete molecules that collide with one another and solidobjects. The continuum flow assumes fluids to be con-tinuous, not discrete molecules32 Properties such asdensity, pressure, temperature, and velocity areassumed to vary continuously from one point toanother. Therefore, Knudsen number is an importantparameter that must be used to modify flow velocity.Flow regimes in fluid structure interaction (FSI) areidentified by Knudsen number. Knudsen number is adimensionless parameter defined as the ratio of themean free path of the molecules to a characteristiclength scale which is used for identifying the variousflow regimes in FSI.33

The mean free path of liquid would be the distancebetween neighboring molecules. Liquid water is heldtogether primarily with hydrogen bonds. On the otherhand, the molecules in liquid are loosely bound, sothey are in constant contact with each other.34 In add-ition, the length scale will depend on the problemunder consideration, for example, the radius of thebody in a fluid, the diameter of a pipe or an objectimmersed in a flow, or the thickness of a boundarylayer or a shock wave.35

For micro and nanotubes, the radius of the tube isassumed as the characteristic length scale. For CNTconveying fluid, Knudsen number may be larger than10�2. Therefore, the assumption of no-slip boundaryconditions is no longer credible, and a modified modelshould be used.

So Vavg,slip is replaced by VCF� Vavg,ðno�slipÞ in thebasic equations where it is determined as follows36

VCF �Vavg,slip

Vavg,ðno�slipÞ

¼ ð1þ aKnÞ 1þ 42� �v�v

� �Kn

1þ Kn

� �� ð12Þ

where Vavg,slip and Vavg,no slip are average flow velocitiesthrough nanotube considering slip boundary condi-tion and ignoring it. Also, �v is dependent on





tangential momentum of molecules, and for mostpractical applications �v is chosen to be 0.7 and acan be expressed as the following relation36

a ¼ a02

�tan�1 a1Kn

B� ��

ð13Þ

in which a1 ¼ 4 and B ¼ 0:04 are some experimentalparameters. The coefficient a0 is formulated as36

limKn!1

a ¼ a0 ¼ 64= 3� 1�4

b

� �� ð14Þ

where b ¼ �1.Therefore, average flow velocity of no-slip bound-

ary conditions should be used instead of the averageflow velocity of slip boundary conditions throughapplying velocity correction factor.

Nonlocal elasticity theory

According to the Eringen’s nonlocal elasticity model,the stress state at a reference point in the body isregarded to be dependent not only on the strainstate at this point but also on the strain states at allof the points throughout the body. Thus, the well-known relation of nonlocal stress tensor � in termsof classical stress is expressed as37

ð1� ðe0aÞ2r2Þ� ¼ �, ð15Þ

where the right hand of equation (15) denotes that clas-sical stress of DWCNT contains mechanical, thermal,and piezoelectric terms, e0a is small-scale parameter,and operator r2 for TB model is defined as @2=@x2.

Hamilton’s principel

The motion equations of embedded DWCNT convey-ing fluid can be derived by Hamilton’s principles asfollows31Z t

0

Z t1

t0

�ðU� Ktube � Kfluid ��Þ dt ¼ 0: ð16Þ

Integrating equation (16) by parts and setting thecoefficient of mechanical to zero leads to the followingmotion equations.

The dimensionless parameters are defined as follow

¼x

Lðwi, uiÞ ¼

ðWi,UiÞ

rii ¼

L

rien ¼

e0a

L

�Ii ¼�Ii�Air

2i

� ¼t

L

ffiffiffiffiE

�t

sUf ¼

ffiffiffiffi�fE

ruf �Hm ¼

�H2x

E

¼ � �� ¼�f�t

�i ¼KsiGAi

EAi

�Ci ¼Cl2

EAi

�Gp ¼Gp

EAi

�Kw ¼kwL

2

EAifi ¼

EAf

EAið17Þ

It should be noted that Ksiði ¼ 1; 2Þ is the shear cor-rection factor. Using above equation, the dimension-less motion equations can be written as�U1

�@2u1@�2þ 1þ ��f1ð Þ

@2u1@�2� e2n 1þ ��f1ð Þ

@4u1@�2@�2

¼ 0

ð18aÞ

�W1

� �1@2w1

@2� 1�1

@ � 1

@þ ð1þ f1 ��Þ

@2w1

@�2

þ 2ffiffiffi��

pf1uf

@2w1

@@�� e2nð1þ f1 ��Þ

@4w1

@2@�2

� 2e2nffiffiffi��

pf1uf

@4w1

@3@�þ f1u

2f

@2w1

@2� e2nf1u

2f

@4w1

@4

� �C1w2 þ �C1w1 þ e2n�C1@2w2

@2� e2n

�C1@2w1

@2

��Hm

1

@2w1

@�2þ en2

�Hm

1

@4w1

@�4¼ 0, ð18bÞ

� 1

� �I11

1

� �2@2 � 1

@2þ �1

1

1

@w1

@þ �1 � 1

� e2nð�I1 þ �� If Þ

1

1

� �2 @4 � 1

@2@�2

þ ð �I1 þ �� If Þ1

1

� �2@2 � 1

@�2¼ 0 ð18cÞ

�U2

�@2u2@2þ@2u2@�2� e2n

@4u2@2@�2

¼ 0 ð18dÞ

�W2

� �2@2w1

@2� 2�2

@ � 2

@�þ@2w2

@�2� e2n

@4w2

@2@�2

þ �Kww2 � e2n�Kw@2w2

@2� �Gp

@2w2

@2

þ e2n�Gp@4w2

@4� �C2w1 þ �C2w2 þ e2n

�C2@2w1

@2

� e2n�C2@2w2

@2�

�Hm

2

@2w2

@�2þ e2n

�Hm

2

@4w2

@�4¼ 0, ð18eÞ

� 2

� �I21

2

� �2@2 � 2

@2þ �2

1

2

@w2

@þ �2 � 2

� e2n�I2

1

2

� �2 @4 � 2

@2@�2þ �I2

1

2

� �2@2 � 2

@�2¼ 0: ð18fÞ





Solution procedure

DQ approach is used to solve the higher order equa-tions of motion. In this method, the partial derivativeof a function with respect to spatial variables at agiven discrete point is approximated as a weightedlinear sum of the function values at all discretepoints chosen in the solution domain.

According to this method, the partial derivatives ofa function f at a point xi are expressed as38

dnf

dxnx¼xi

�� ¼XNj¼1

CðnÞij f ðxj Þ, ð19Þ

where CðnÞij is the respective weighting coefficients

matrix, N is the number of grid points and f can betaken as U, W, and �. In order to avoid the ill-con-ditioning, the Lagrange interpolation basis functionsare used as38

Cð1Þij ¼

QNk¼1,k6¼i,j

ðxi � xkÞ

� QNk¼1,k 6¼j

ðxj � xkÞ ði 6¼ j Þ

PNk¼1,k6¼i

1ðxi�xkÞ

ði ¼ j Þ

8>>><>>>:i, j ¼ 1, 2, . . . . . . ,N: ð20Þ

The solution of the motion equations (equations (18a)to (18f)) can be assumed as follow38

uiðx, tÞ ¼ uiðxÞe!�, ð21aÞ

wiðx, tÞ ¼ wiðxÞe!�, ð21bÞ

iðx, tÞ ¼ iðxÞe!� ð21cÞ

where ! ¼ lffiffiffi�fE

qis the dimensionless natural fre-

quency and is the (Fundamental) natural frequency.The DQM form of mechanical clamped boundaryconditions at both ends of nanotube may be writtenin dimensionless form as

u1 ¼ w1 ¼ 1 ¼ 0 at ¼ 0

uN ¼ wN ¼ N ¼ 0 at ¼ 1:ð22Þ

Imposing the above boundary conditions into equa-tions (18a) to (18f) leads to the following constitutivematrix equation

K½ � þ ! C½ � þ !2 M½ �� db

dd

� �¼ 0 ð23Þ

where the subscript b stands for the elements relatedto the boundary points while subscript d is associatedwith the remainder elements. The K½ �, C½ �, and M½ � arethe stiffness, damping, and mass matrixes, respect-ively. For solving equation (23) and reducing it tothe standard form of Eigen value problem, it is

convenient to rewrite this equation as the followingfirst-order variable as34

_Z �¼ A½ � Zf g ð24Þ

in which the state vector Z and state matrix A½ � aredefined as

Z ¼dd_dd

� �and A½ � ¼

0½ � I½ �� M�1K�

� M�1C� � �

ð25Þ

where 0½ � and I½ � are the zero and unitary matrices,respectively. However, the frequencies obtained fromthe solution of equation (25) are complex due to thedamping. Hence, the results contain two real andimaginary parts. The real part corresponds to thesystem damping and the imaginary part representsthe system natural frequencies.

Numerical results and discussion

The results of this study are based on the followinggeometric and mechanical data for DWCNT25

r1¼0:77nm r2¼1:11nm h¼r2�r1 e0a¼0:5�10�9

L=r2¼20 E¼1TPa �¼2:3gr=cm3 Kn¼0:03

�¼0:34 kw¼109N=m3 GP¼10

�7N=m ð26Þ

The variation of imaginary and real components ofdimensionless frequency versus dimensionless flowvelocity has been compared with two similar worksperformed by Ke and Wang25 and Lee and Chang39

at the same conditions. Figure 2 shows good agree-ment between present work and Ke and Wang25

which have used TBT, while this trend has been fol-lowed in Lee and Chang39 with a little difference dueto Euler-Bernoulli beam model. In order for bettercomparison, the governing equations of every paperhave been dimensionless with the same groups. As canbe seen, this figure confirms the validity of theobtained results.

In following figures, imaginary and real compo-nents (Imð!ÞandReð!Þ) of dimensionless frequencyof DWCNT are shown and the effects of various par-ameters such as magnetic intensity (Hx), Knudsennumber and elastic medium on them for clamped–clamped boundary conditions are discussed in detail.It is noted that Im ð!Þ represents the resonance fre-quencies of the DWCNTs while Re ð!Þ denotes thedamping which results from the moving fluid to theinner wall of CNTs.

Figures 3 and 4 show the imaginary and real com-ponents (Imð!Þ and Reð!Þ) of dimensionless frequencyversus the dimensionless flow velocity (uf) for two dif-ferent modes. As already mentioned, Imð!Þ is the res-onance frequency and Reð!Þ is related to the damping.Generally, the system is stable when the real part of the





frequency remains zero and it is unstable when the realand imaginary parts of the frequency become positiveand zero, respectively. As can be seen, Imð!Þ generallydecreases with increasing uf. For zero resonance fre-quency, DWCNT becomes unstable due to the diver-gence via a pitchfork bifurcation and thecorresponding fluid velocity is called the critical flowvelocity. Therefore, with increasing flow velocity,DWCNT stability decreases and becomes susceptibleto buckling. In addition, utilizing this figure anddimensionless parameters which are defined in equa-tion (17), actual fluid velocity through the DWCNT is

obtained simplistically. For example, if Uf ¼ 0:01,actual fluid velocity is 208:51m=s. Therefore, theresults of this research can be used in design and man-ufacturing of nano/micro mechanical devices.

The effect of various elastic medium on imaginaryand real parts of dimensionless frequency versusdimensionless fluid velocity is illustrated in Figures 5and 6. It is obvious that existence of spring and shearfoundations enlarge the stability region of DWCNTsand increase the resonance frequency. As can be seen,Imð!Þ increases by increasing the elastic foundationstiffness and decreases as uf increases.

Figure 3. Imaginary part of frequency versus flow velocity for different vibration modes.

(a) (b)

Figure 2. (a) Comparison natural frequency between present results with that investigated by Ke and Wang25 and Lee and Chang.39

(b) Comparison damping frequency between present results with that investigated by Ke and Wang25 and Lee and Chang.39





Figures 7 and 8 depict natural and damping dimen-sionless frequency versus flow velocity for differentvalues of small scale. It is obvious that the nonlocalparameter is a significant parameter in vibration ofDWCNT. As can be seen, increasing the nonlocal

parameter decreases the frequency and critical flowvelocity. It needs to be pointed out that the zerovalue for nonlocal parameter (i.e. e0a ¼ 0) denotesthe result obtained by the classical TB model whichhas the highest frequency and critical fluid velocity.

Figure 4. Real part of frequency versus flow velocity for different vibration modes.

Figure 5. Effect of elastic medium on the natural frequency of DWCNT.





Figure 6. Effect of elastic medium on the damping frequency of DWCNT.

Figure 7. Natural frequency versus flow velocity for different values of nonlocal parameter.





The effect of magnetic field on dimensionless fre-quency of DWCNT is shown in Figures 9 and 10. Asalready mentioned, applying magnetic field in axialdirection generates the force in radial direction callLorentz force. It is found that increasing magneticintensity (Hx) increases frequency and critical

flow velocity. Regarding Lorentz force effect, it hasbeen concluded that the magnetic field is basicallyan effective factor on increasing resonance frequencyleading to stability of DWCNT.

Finally, the effect of Knudsen number on naturaland damping dimensionless frequency of DWCNT is

Figure 9. Effect of magnetic field on natural frequency of DWCNT.

Figure 8. Damping frequency versus flow velocity for different values of nonlocal parameter.





illustrated in Figures 11 and 12. Knudsen number isdefined based on various flow regimes. Here, the slipflow regime is considered. As shown in these figures,continuum fluid (Kn ¼ 0) predicts the highest fre-quency zone, and considering fluid with higherKnudsen number results in shifting the curves to the

lower frequency zone. Therefore, critical flow velocityof DWCNT decreases with increasing Kn.

Figures 13 and 14 demonstrate the imaginary andreal components of frequency versus the flow velocityfor different values of aspect ratio. Since the flow fluidin slender pipe can be assumed as a steady state flow

Figure 10. Effect of magnetic field on damping frequency of DWCNT.

Figure 11. The effect of Knudsen number on imaginary component of frequency.





for l=r2 4 15,4 results have been obtained for thevalues of aspect ratio more than 20. It is clear thatthe frequency of the DWCNT increases with decreas-ing l=r2 and simultaneously the critical fluid velocityincreases as aspect ratio decreases. It is worth notingthat the low aspect ratio is better for vibration of

DWCNT conveying fluid for optimum design ofnano/micro devices.

Different values for inner and outer radii ofDWCNT were reported in the literature. Figure 15illustrates natural frequency versus fluid velocity fordifferent values of inner and outer radius of DWCNT

Figure 12. The effect of Knudsen number on real component of frequency.

Figure 13. The effect of aspect ratio on imaginary component of frequency.





corresponding to the literatures.13,17,25,39–42 It is evi-dent from this figure that the increasing fluid velocitycauses to decrease the natural frequency. In addition,critical fluid velocity is different while trend of figure isconstant for all cases.

Conclusion

In this paper, nonlocal vibration analysis ofDWCNTs subjected to magnetic field was investi-gated, where DWCNT embedded in Pasternak

Figure 15. Natural frequency for different values of inner and outer radius of DWCNT corresponding to the literatures.13,17,25,39–42

Figure 14. The effect of aspect ratio on real component of frequency.





medium is considering Winkler and shear constants.Also, the effects of vdW forces between the inner andouter nanotubes were taken into account. Based onnonlocal TB theory, fundamental equations wereevaluated and solved by DQ approach. Regardingslip boundary conditions for fluid in DWCNT, itcan be observed that Knudsen number have signifi-cant effects on the mechanical behavior of theDWCNT. Also, it has been found that the magnitudeof fundamental frequency is strongly dependent onthe imposed magnetic field so that increasing the mag-netic intensity significantly increases the critical flowvelocity. The findings of the present study may beused in advanced applications of this kind of nano/micro mechanical devices.

Funding

This work was supported by University of Kashan [grant

number 65475/54]. Iranian Nanotechnology DevelopmentCommittee provided financial support.

Conflict of Interest

None declared.

Acknowledgement

The authors would like to thank the reviewers for their

comments and suggestions to improve the clarity of thisarticle.

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