oal

831

Fluid-structure interaction (FSI)

e pendundg chr, wmeowe

the mology,erties onic pro-compcationations

obtained based on local theory of continuum mechanics. He foundthat an increase in Eringens nonlocal parameter for higher wavenumbers could cause a decrease in corresponding frequencies, ascompared to local continuum mechanics theory. Eringen [9]linearized the nonlocal elasticity given in [7] and obtained a onedimensional plane wave equation. Polizzotto [10] presented a

equation of continuummechanics called the peridynamic formula-tion. The peridynamic elasticity theory could introduce inter-actions between solid elements with the aid of proper long-rangepotentials. This formulation permitted the solution of fractureproblems using the same equations corresponding to either on oroff the crack surface or crack tip. In this topic, Silling et al. [16] usedthe peridynamics formulation for analyzing deformation relationsof a bar. Askes and Aifantis [17], showed that Eringens nonlocaltheory was incapable of investigating wave propagation phenom-ena. For example, they concluded that in the range of higher wave

Corresponding author. Tel.: +98 311 391 5248; fax: +98 311 391 2628.E-mail addresses: fareedkaviani@yahoo.com (F. Kaviani), hrmirdamadi@cc.iut.

Computers and Structures 116 (2013) 7587

Contents lists available at

n

lseac.ir (H.R. Mirdamadi).are among important engineering analyses with the most interestin uid-structure interaction (FSI) community [3,4]. There are dif-ferent structural mechanics models in use for the investigation ofvibration and wave propagation in nano-size [5,6]. A model, usedfrequently for the investigation of nano-size problems in mechan-ics, is Eringens nonlocal theory of elasticity [7]. In the literature,research groups have presented diverse publications on the prob-lem of wave propagation in nano-tubes by use of Eringens nonlocaltheory of elasticity. Wang [8] showed that Eringens nonlocaltheory had dramatic inuence on dispersion relations previously

of the crack tip. To develop a nonlocal continuum theory, Di Paolaand Zingales [12] dened a nonlocal continuum model includinglong-range force between non-adjacent volume elements. Theyused a proper selection of the attenuation function in the Eringensmodel to yield stress-strain relation involving fractional deriva-tives, instead of classical derivatives or convolution integrals. DiPaola et al. [13] used the model developed by Di Paola and Zingales[12] to analyze beams and Zingales [14] used this model for wavepropagation analysis in 1/D elastic solids. To develop another non-local analysis, Silling [15] proposed a new framework for the basic1. Introduction

Carbon nano-tubes are one ofmaterials, developed by nano technrange of applications. Superior propas mechanical, thermal, and electroelectronics, nano-devices, and nanmajor reasons for their wide appli[1,2]. The investigation of pipe vibr0045-7949/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruc.2012.10.025ost important nano-that are nding a widef these materials, suchoperties used in nano-osites are among thes in various industriesand wave propagation

model of nonlocal theory, rened by assuming an attenuationfunction depending on the geodetical distance between materialparticles. He showed that the nonlocality effects would propagatealong geodetical paths from the source points to the others, in away that they would not traverse obstacles such as cracks, holes,and in general, any gap in the convexity of the domain. Further-more, Eringen [11] solved the eld equation of nonlocal elasticityfor determining the state of stress in the neighborhood of a linecrack in an elastic plate subject to a uniform shear at the surfaceSlip boundary conditionKnudsen number (Kn)

2012 Elsevier Ltd. All rights reserved.Wave propagation analysis of carbon nanslip boundary condition and strain/inerti

Fareed Kaviani, Hamid Reza Mirdamadi Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156

a r t i c l e i n f o

Article history:Received 16 May 2012Accepted 19 October 2012Available online 12 November 2012

Keywords:Wave propagationCarbon nano-tube (CNT)Strain/inertia gradient

a b s t r a c t

This article addresses wavmodeled by using size-depow interaction by slip borelations and correspondinow velocity, wave numbetions are discussed and sowave frequencies at both lhigher range.

Computers a

journal homepage: www.ell rights reserved.-tube conveying uid includinggradient theory

11, Iran

ropagation in carbon nano-tube (CNT) conveying uid. CNT structure isent strain/inertia gradient theory of continuum mechanics, CNT wall-uidary condition and Knudsen number (Kn). Complex-valued wave dispersionaracteristic equations are derived. Fluid viscosity, gyroscopic inertial force,ave frequency, and decaying ratio are among parameters that their varia-remarkable results are drawn. It was observed Kn could impress complexr and higher ranges of wave numbers, while small-size had impression at

SciVerse ScienceDirect

d Structures

vier .com/locate /compstruc

boundary condition between uid and structure into equation ofmotion of uid and nano-structure in the nano-level, by using a

tersnumbers, with increasing wave number, the corresponding phasevelocity remained constant at a non-zero value, an analytical resultagainst experimental observations for nano-tubes. They observedcloser agreement between theoretical and experimental resultsby proposing the use of gradient elasticity for the problem of wavepropagation in nano-tubes [18]. Aifantis [19] had a brief accounton the role of higher-order strain gradient in the localization ofplastic ow and showed that the higher-order gradient could pro-vide a mechanism for capturing the evolution of plastic ow inthe material softening regime. Again Aifantis [20] provided anupdate of the standard theories of dislocation dynamics, plasticityand elasticity that were modied to include scale effects in thehigher-order spatial gradients of the governing equations in amate-rial description. He noted two new features, rst the role of waveletanalysis and randomness in interpreting deformation heterogene-ity measurements and second, indentations in a stress-straingraph. There are wide applications of gradient elasticity theory forthe investigation of structural behavior in nano-size and differentresults have been obtained as contrasted with other theories formodeling nano-size. Some examples may be found in [21,22].

One of the challenging applications of nano-tube vibrations arein uid conveying problems with the most applications in targeteddrug delivery systems. Hence, there are many remarkable investi-gations on the vibration behavior of nano-tubes conveying uid. Ofthis type of research [23], we may emphasize investigation of dy-namic stability of nano-tube conveying viscous uid by usingEulerBernoulli beam theory, conducted by Khosravian andRai-Tabar [24]. They paid attention to the effect of uid viscosityon the critical velocity of uid. They showed that increasing uidviscosity is followed by an increase in the uid critical velocity.Wang and Ni [25] modied formulation obtained by [24], andshowed that the effect of viscosity could be ignored in the uidcritical velocity in nano-tube conveying uid. Ignorance of theeffect of nano-size was the major deciency of these researchworks. Later, Lee and Chang [26], included the effect of small-sizeinto equations of motion by using Eringens nonlocal elasticityand showed that increasing Eringens nonlocal parameter had theeffect of a decrease in the critical velocity of uid.

The investigation of wave propagation in nano-tube conveyinguid, based on FSI equations for viscous ow, was conducted byWang et al. [27]. They showed that increasing viscosity had theconsequence of decreasing nano-tube frequency in the range ofhigher wave numbers. Furthermore, Wang [28] showed that byusing gradient elasticity theory, the effect of small-size might beincluded into the equation of wave propagation in nano-tube con-veying uid. He showed that the use of gradient elasticity theoryhad a dramatic effect on dispersion relation. He compared the re-sults of this theory with those of Eringens nonlocal and local clas-sical theories and showed that the results of gradient elasticitytheory were more feasible than those of the other two theories.

The effect of small-size in the study of FSI problems is not limitedto the mechanical behavior of the structure part. It may inuencethe boundary condition between the interface of uid and structure;thus, it plays an important role on the structural response. The keyparameter covering this boundary effect is the Knudsen number(Kn) dened as the ratio of uid mean free path to a characteristiclength of the structure under study. This parameter is used as a sep-arator among different ow regimes. Based on Kn, there are four dif-ferent ow regimes: (1) 0 < Kn < 102 for continuum ow regime;(2) 102 < Kn < 101 for slip ow regime; (3) 101 < Kn < 10 for tran-sition ow regime; and (4)Kn > 10 for freemolecular ow regime. Inmicro/nano-size FSI problems, it could be possible thatKnwould fallbeyond 102 value. Thus, the uid ow regime might fall in either

76 F. Kaviani, H.R. Mirdamadi / Compuslip ow or transition ow regimes, or in-between [29].Thus far, the effects of small-size in the area of nano-tube con-

veying uid, were limited to the behavior of elastic structure partdimensionless parameter called VCF (velocity correction factor).They showed that Kn parameter had a considerable effect on thegas uid critical velocity. They considered the small-size effectonly on the uid part of FSI. However, they did not incorporate thiseffect for the elastic structure part of the FSI problem. In the sequel,Mirramezani and Mirdamadi [31], surveyed the higher vibrationmodes of nano-tube conveying uid with different boundary con-ditions and observed the coupled mode utter happening soonerthan that predicted by conventional continuum mechanics. How-ever, as [30], they did not bring into consideration the small-sizeeffect for the elastic behavior of nano-structure. For completingtheir work, they used Eringens nonlocal theory in the presenceof slip boundary condition in [32] for investigating the vibrationand stability of a nite length CNT. Again, they did not pay anyattention to both a wave propagation analysis of an innite lengthCNT and incorporating strain/inertia gradient nonlocal theory intotheir size-dependent elastic analysis.

In this paper, we pay attention to the analysis of wave propaga-tion phenomena in nano-tube conveying uid such that by includ-ing the size-dependent or nonlocal continuum theories, wecompensate the deciencies of analyses due to ignoring thesmall-size effect for structural behavior by Rashidi et al. [30] andMirramezani and Mirdamadi [31]. We further bring into consider-ation the gradient strain/inertia elasticity theory and its applicationfor the behavior of nano-tube structure. In addition, as mentionedpreviously, the investigation of wave propagation is one of theimportant elds of the behavior of nano-tubes. Thus, we havecontinued our studies on the investigation of the effects ofsmall-size in the level of both nano-tube structural behavior anduid-structure interface mechanism. In this study, the resultsobtained from the incorporation of the effect of small-size appearsin such a way that in the dispersion relation for higher wave num-bers, the decaying ratio (real part of complex frequency) decreasesas Kn increases, a substantial result not observed by the authors inthe literature, as far as the authors knowledge goes. Furthermore,we would use strain/inertia gradient elasticity theory, because, asmentioned previously, Wang [28] showed that in the investigationof wave propagation phenomena in the nano-size, this theory couldpresent more convincing results. The combination of these twonano-size effectswith their corresponding theories of slip boundarycondition and strain/inertia gradient theory could show us that notdetectable by the common-sense, but as observed by dispersionrelations, the imaginary and real parts of complex frequency inthe range of higher wave numbers could decrease substantially.

The remainder of the paper is organized as follows: In section 2,we present the theoretical equations of strain/inertia gradient elas-ticity. In section 3, we develop the FSI equations of motion of nano-tube conveying uid by using VCF, and then, we investigate thewave propagation in an innite nano-tube conveying uid. In sec-tion 4, we present a complex-valued dispersion relation. In section5, we present numerical examples and study the dispersion rela-tions parameterized by Kn, gyroscopic force, uid viscosity, anddifferent size-effect elasticity theories in order to show the effectsof nano-sizes of different types followed by related discussions. Insection 6, nally, we draw important conclusions.

2. Non-classical theories of strain/inertia gradient and stressgradientof FSI. Those effects were ignored in the interaction between uidand structure, until Rashidi et al. [30] inserted the effect of slip

and Structures 116 (2013) 7587According to the theory of combined strain/inertia gradientsdeveloped by Askes and Aifantis [33], a combination of equation

tersof motion and the strain/inertia gradient constitutive relationcould be written as

q ui 2mui;mm Cijkluk;jl 2s uk;jlmm 1

In the above relation, q is the mass density, Cijkl are Cartesian com-ponents of elasticity tensor, and ui denotes Cartesian displacementsfor an elastic structure. The quantities s and m denote length scalesand are related to strain gradient and inertia gradient, respectively.The strain gradient length scale is related to a representative vol-ume element (RVE) size for elastostatics and inertia gradient lengthscale to a RVE size for elastodynamics problems, since generally,inertia gradient length scale tends to be larger than the straingradient length scale [17]. The constitutive relation for 1/D Euler-Bernoulli beam theory, is obtained by equating i = 3 and using thestrain-displacement relations of Euler-Bernoulli theory. Accord-ingly, for 1/D strain-exural curvature relation, we have

e z @2w

@x2

!2

where u3 = w. The only non-zero term of right hand side of Eq. (1) isfor i = j = 1. Based on strain/inertia gradient theory, Eq. (1), thestress-strain relation may be written as [33]

r E e 2s@2e@x2

! qc2m

@2e@t2

3

where r and e are, respectively, exural stress and strain in thebeam. The bending moment M, resulting from the strain-curvatureEq. (2), and stress-strain relation (3) and (3) by integrating over thecross-sectional area of the beam, is

M ZAzrdA EI @

2w@x2

2s@4w@x4

! qcI2m

@4w@x2@t2

4

In this relation, E is Youngs modulus, I is the moment of inertia ofarea cross section, and A is the area of the beam cross-section. qc isthe mass density of nano-tube per unit volume. x is a longitudinalcoordinate along the nano-tube axis, w is the tube transversedeections along the z-direction, and t is time. According toEringens nonlocal model, the stress eld at a point x in an elasticcontinuum is dependent not only on the strain eld at the pointunder consideration but also on the strains at all other points ofthe body. The most general form of constitutive equation forEringens nonlocal elasticity involves a convolution integral overthe entire region of interest. This integral contains a kernel functionthat describes the relative inuences of strain at various locationson the stress at a given location. Thus, the nonlocal stress tensorr at point x is expressed as

rN ZVKjx0 xj; srx0dx0 5

where r(x) is the classical stress tensor at point x and the kernelfunction (Kjx0 xj; s)) represents a nonlocal modulus. jx0 xj isthe distance in Euclidean norm and s is a material constant that de-pends on internal and external characteristic lengths of the struc-ture. In the stress gradient or the differential form of Eringensnonlocal model, the constitutive relation, such as that used byReddy and Pang [34], but specialized for Euler-Bernoulli beamtheory, is represented by

rN 1 2 @2

@x2

!r Ee 6

F. Kaviani, H.R. Mirdamadi / Compuwhere , and rN are respectively, a characteristic length of nano-tube as related to the stress gradient theory, and the nonlocal stresscorresponding to the local stress r. The local equation of motion ofbeam could be as follows

@2M@x2

p qcAi w 0 7

The Eringens nonlocal bending moment MN is obtained as

MN ZAzrNdA M 2 @

2M@x2

8

The bending moment M, resulting from the stress-strain relation (6)is given by

M EI @2w

@x2 2 qcAi

@2w@t2

p !

9

where Ai is the area of beam cross section of nano-tube in whichuid is moving through, and p includes two parts that are additive:First, a dynamic pressure, pD, resulting from the uid-structureinteraction and is dened as [25]

pD mf uavgnoslip 2 @2w

@x2 2mf uavgnoslip

@2w@x@t

mf @2w@t2

leAi@3w@x2@t

leAi uavgnoslip @3w

@x310

wheremf, le, and uavg-no slip are, respectively, the uid mass densityper unit length, continuum viscosity, and average ow velocitythrough the tube subject to no-slip condition. In Eq. (10), the rstterm on the right-hand side represents an inertial force correspond-ing to the centripetal or centrifugal accelerations, while the secondterm denotes the inertial force resulting from Coriolis accelerations.The third term is the inertial force due to translational transverseaccelerations. Second, the external pressure, pe, including possiblya pretension, internal pressure, and a Winkler foundation effect,are expressed as follows [25]

pe PAi T@2w@x2

Kw 11

where P is the internal pressure on the tube, T is the pretension,and K is Winkler modulus of an elastic foundation, if available.

3. FSI governing equation

For inserting slip boundary conditions into equation of motion,as mentioned previously, we would insert th VCF dimensionlessnumber into the uid-structure interaction equation of motionmultiplied by an average velocity term appearing in NavierStokesequation. The VCF is dened as the ratio of the average ow veloc-ity for a slip boundary condition to the average ow velocity forno-slip boundary condition, as follows [35]

VCF uavgslipuavgnoslip

1CrKn 4

2 rvrv

Kn

1 bKn

1

12

where rv is the tangential momentum accommodation coefcient,and herein, it is considered to be 0.7. The quantity Cr(Kn) is the rar-efaction coefcient of uid and is dened by [29], as the ratio ofdynamic viscosity to bulk viscosity of uid. Furthermore, for a sec-ond-order approximation of slip boundary conditions, b = 1. Weinsert the effect of slip boundary condition into FSI equation, as sug-gested by Rashidi et al. [30], and Mirramezani and Mirdamadi [31],for the nano-tube, by using uavg-slip instead of uavg-no slip. Further-more, we add the inuence of strain/inertia gradient elasticity the-ory through the noncbending moment suggested by Eq. (4). The

and Structures 116 (2013) 7587 77classical equation of motion for a tube, ignoring the gravity effect,and using Euler-Bernoulli kinematic relation for a tube modeledas a beam, is as follows [25,36];

ters @2Mc@x2

mf uavgno slip 2 PAi Th i @2w

@x2

2mf uavgnoslip @2w

@x@t mc mf @2w

@t2 l0Ai

@3w@x2@t

l0Aixuavgnoslip@3w@x3

Kw 0 13

In this local equation, Mc is the local bending moment resistingbeam exure in a classical continuum sense for the tube. w is thetransversal displacement of the tube center-line, E is Youngs mod-ulus, I is the moment of inertia of wall area cross-section, mc is thetube mass density, Ai is the area cross-section of the tube throughwhich the uid is passing, and t is time. As it was mentioned previ-ously, by using uavg-slip instead of uavg-noslip, and bending moment ofEq. (4) instead of Mc, all in the conventional equation of motion ofbeam FSI, we could derive a generalized equation of motion fornano-tube, specialized for size-effects, by inserting the Kn effectthrough VCF dimensionless number. This relation is as follows

EI@4w@x4

2s@6w@x6

! mf VCF2 uavgnoslip

2 PAi Th i @2w@x2

2mf VCF uavgnoslip @2w

@x@t mc mf @

2w@t2

qcI2m@6w

@x4@t2

leAi@3w@x2@t

leAi uavgnoslip @3w

@x3 Kw 0 14

In addition, by substituting Eq. (9) into Eq. (13), the equation ofmotion of nano-tube, specialized for Eringens nonlocal model isderived as [30,35,37]

EI@4w@x4

1 2 @2

@x2

!mf VCF2 uavgnoslip

2 PAi Th i @2w@x2

(

2mf VCF uavgnoslip @2w

@x@t mc mf @2w

@t2 qcI2m

@6w@x4@t2

l0CrAi@3w@x2@t

l0CrAi uavgnoslip @3w

@x3 Kw

) 0 15

The viscosity of uid, according to a ow-rate moving through thenano-tube in the Knudsen layer is dened as a function of Kn [29].Polard [38] suggested the following second relation for viscosity ofuid

CrKn 11aKnleKn l0CrKn

16

In the above relation Cr is the rarefaction coefcient, l0, is bulkviscosity, a is a theoretical constant, while le is an effective viscos-ity including the effect of Kn. Assuming a constant value for a couldbe an origination for large errors in the uid ow-rate within achannel. For deriving a, Karniadakis et al. [29] used exact results ob-tained from an analysis of Boltzmann linear equation by Loyalkaand Hamoodi and experimental results of Tison [29]. They proposedthe following equation in terms of Kn:

a 2pa0tan1a1KnB 17

The values of parameters a1 and B = 0.4 were obtained from exper-imental observations. In addition, the value of a0 was obtained forvarious values of Kn, increasing from the values corresponding tothe slip ow regime to those corresponding to the free molecularregime (as Kn?1); [29]

aKn!1 a0 643p 1 4b

!18

78 F. Kaviani, H.R. Mirdamadi / CompuIn the above equation, for a slip boundary condition of second-orderapproximation, b is equal to 1.In this relation,W is the wave amplitude, k is the wave number ands is the complex frequency of wave motion. By substituting Eq. (19)into Eq. (14), the complex-valued dispersion relation for uidconveying nano-tube, by using strain/inertia elasticity, results:

EI k4 2s k6

mf VCF2 uavgnoslip 2 PAi Th ik2

2mf VCF uavgnoslip

ks i mc mf s2 qcI2mk4s2

leA2iks leAi uavgnoslip

k3 i K 0 20where i is the imaginary basis

1

p. In the above relation, by equat-

ing viscosity equal to zero and VCF = 1, we might obtain the disper-sion relation that was obtained by Wang [28]. Further, byconsidering inertia and strain gradient length scales tending to zero,we could obtain dispersion relation for uid conveying nano-tube inthe classical continuum elasticity model. For using Eringens non-local model for the analysis of wave propagation behavior, substi-tuting Eq. (19) into Eq. (15), the corresponding complex-valueddispersion relation results

EIk4 1 2k2fmf VCF2 uavgnoslip 2 PAi Tk2

2mf VCF uavgnoslip

ks i mc mf s2 leAik2s leAi uavgnoslip

k3 i Kg 0 21

To investigate the behavior of nano-tube in the wave propagationby using the dispersion relations (20) and (21), assume s is sepa-rated into real (r) and imaginary (x) parts. r shows the decay rate(equivalent viscous damping or Neperian frequency in the units ofNeper) and x the wave frequency (the resonance frequency in theunits of Hz) of the nano-tube for a constant wave number, respec-tively. By decomposing the complex frequency into its real andimaginary parts, both equations of (20) and (21) can be separatedinto two coupled equations, as follows:

For strain/inertia model:

EIk4 2s k6 mf VCF2uavgnoslip2 PAi Th i

k2 K

2mf VCFuavgnoslipk leAik2h i

r

mc mf qcI2mk4h i

r2 mc mf qcI2mk4h i

x2 0

2mf VCFuavgnoslipk leAik2ix 2 mc mf

qcI2mk4

irx leAi uavgnoslip

k3 0 22

For Eringens nonlocal model:

EIk4 1 2k2 mf VCF2 uavgnoslip 2 PAi Th ik2n

2mf VCF uavgnoslip

kr mc mf r2 x2 leAik2r Ko 0

1 2k2f2mf VCF uavgnoslip

kx 2mc mf rx leAi uavgnoslip

k3 leAik2xg 0 23

5. Results and discussion

In this section, there are subsections in relation to parametricstudies for the effect of gyroscopic term, slip boundary condition,viscosity, and size-dependent continuum theories. In each subsec-4. Dispersion relations

The wave propagation solution of Eqs. (14) and (15) can beexpressed as follows

wx; t Weikxst 19

and Structures 116 (2013) 7587tion, the results and discussion for each parameter is furtherdivided into discussing the dispersion relations for real (decayingratio) and imaginary (wave frequency) parts of complex wave

tersfrequency. All the numerical and graphical results and their discus-sion followed, are based on Eq. (22) for nonlocal strain/inertia gra-dient and Eq. (23) for Eringens nonlocal theories.

Before presenting the numerical results in a context of paramet-ric study, some parameters appearing in the characteristic equa-tions (20) and (21) are set to xed values. This is pursued fordecreasing the number of parametric studies. For example, the val-ues of s, m, and which are respectively, the characteristic lengthsin relation to strain gradient, inertia gradient, and stress gradient(Eringens nonlocal analysis), are set to 0.0355, 0.355, and0.0355 nm, for CNT (20,20), following [28]. The geometrical prop-erties of CNT are such that its internal radius and wall thickness arerespectively, equal to 1 and 0.34 nm. Furthermore, the CNT mate-rial properties of Youngs modulus and mass density are set to 1TPa and 2.3 g/cm3. The uid passing through CNT is consideredto be acetone. Its material properties of mass density and viscosityare 0.79 g/cm3 and 0.0003 Pa s [39]. The tangential momentumaccommodation coefcient, rv, for the CNT-acetone interaction isassumed to be 0.7. In the sequel, the complex-valued dispersionequations of the elastic wave propagation in the CNT are investi-gated thoroughly.

5.1. Effect of gyroscopic term

5.1.1. Imaginary part of complex frequencyThe investigation of dispersion relation of elastic wave propaga-

tion for a given wave number would result in two different waves,propagating in two opposite directions of upstream and down-stream. The corresponding wave frequencies are called upstreamand downstream frequencies [40]. The numerical results demon-strate that these frequencies have different values when there isa uid owing through a CNT with an innite length. These wavefrequencies are the same when there is either no ow or the owis still [28]. An investigation of parametric studies shows that thereason for having two different wave frequencies of upstreamand downstream is the presence of gyroscopic term of inertiaforces. These inertia forces (corresponding to Coriolis acceleration)originate from observing the coupled motion of uid and CNT froma non-inertial reference frame attached to the CNT wall. Theseinertia forces are the product of two terms, i.e., the relative velocityof the non-inertial coordinate system and the angular velocity ofthe non-inertial coordinate system, as observed from an inertialreference frame. The angular velocity comes from a time derivativeof angular displacement, herein, ow/@x. The presence of a spatialderivative of the rst degree, capable of sign change, is the mainsource for the generation of two different upstream and down-stream traveling wave frequencies. Fig. 1 shows the effect of gyro-scopic term on the dispersion relation of upstream anddownstream wave frequencies propagating in the CNT againstthe wave number.

Furthermore, the numerical results show that the gyroscopicterm makes an increase in upstream wave frequency and adecrease in downstream wave frequency. The reason is the uni-directional ow of uid from the upstream to the downstream.Fig. 2 shows the effect of gyroscopic term on both upstream anddownstream wave frequencies.

An investigation of uid velocity passing through the CNT re-veals that when the gyroscopic term is considered, an increase inthe uid velocity causes more pronounced difference between up-stream and downstream wave frequencies. This issue is depicted inFigs. 3a and 3b.

An increase in wave number causes a decrease in the differencebetween upstream and downstream wave frequencies. The evi-

F. Kaviani, H.R. Mirdamadi / Compudence is the diminishing of gyroscopic term, as compared to otherterms in FSI equation (such as structural bending stiffness and uidcentrifugal forces), for very large wave numbers. Fig. 4a illustratesupstream and downstream wave frequencies against large wavenumbers. It is observed that there is no difference between thesetwo frequencies. The parameter f is dened as f = (xupstream -xdownstream)/xdownstream. Fig. 4b shows f against the wave num-ber. It may be seen that an increase in the wave number isaccompanied by a decrease in the parameter f. This point explainsthat for the range of higher wave numbers, as seen in Fig. 4a, theeffect of gyroscopic term is smaller, as compared with other termsof FSI equation. Therefore, the relative difference between the twowave frequencies of upstream and downstream is diminished. Forhigher velocities of uid, the difference between these two up-stream and downstream wave frequencies is remarkable for higherwave numbers and it cannot be ignored. This conclusion is in a rel-ative sense.

5.1.2. Real part of complex frequencyThe gyroscopic term is effective on the variations of decaying

ratio against wave number. This is shown in Fig. 5. Higher valuesof gyroscopic termwould cause a decrease in decaying ratio as wellas a decrease in the wave number in which decaying ratio goes tozero.

5.2. Slip boundary condition

5.2.1. Imaginary part of complex frequencyAs explained in the previous sections of this article, for the

investigation of slip boundary condition, the dimensionless param-eter VCF is used which is a function of Kn. Fig. 6 shows the effect ofslip boundary condition on the two upstream and downstreamwave frequencies without considering the gyroscopic term.Increasing Kn would cause a decrease in both upstream and down-stream wave frequencies in the CNT. The reason could be an in-crease in the uid centrifugal forces (an increase in VCF followedby an increase in the centrifugal force). Consequently, the overallstructural stiffness of the CNT would result from these forces.

Considering the presence of gyroscopic term, the effect of slipboundary condition on the dispersion relation in the CNT, couldbe variant. An increase in Kn would make an increase in the up-stream wave frequency but a decrease in the downstream wavefrequency. For a better description of the reason, we would needto digress what the meaning of an increase in Kn could be from amolecular viewpoint. In addition to providing a mechanism forcomparing the geometrical dimensions of a structure againstmolecular mean free path, an increase in Kn would result an in-crease in the number of uid molecular collisions to the CNT wall.For this reason, an increase in Kn would cause the effect of slipboundary condition to show itself more pronounced. It is worthmentioning that slip boundary condition is present for any geo-metrical dimensions of a tube interacting with a uid owinginternally. However, being Kn very low, this effect might be ig-nored. Regarding this molecular interpretation, an increase in Kncould cause an increase in the number of collisions of uid mole-cules with the CNT wall. This effect would cause an increase inthe upstream wave frequency due to being opposite the directionof relative motion of uid in the CNT and the propagation of elasticwaves. The same effect could cause a decrease in the downstreamwave frequency due to being the same the direction of relative mo-tion of uid in the CNT and the propagation of elastic waves. Fig. 7shows the effect of Kn on the dispersion relation.

The effect of slip boundary condition would decrease with anincrease in the wave number. For much higher range of wave num-bers, this effect could be ignored. The reason could be that this ef-fect of slip condition would appear in those terms of FSI

and Structures 116 (2013) 7587 79characteristic equations of (22) and (23) which would include low-er powers of wave number, as compared to the other terms. Forexample, for higher range of wave numbers, since elastic bending

ters80 F. Kaviani, H.R. Mirdamadi / Compustiffness would appear with a higher power of wave number, as op-posed to centrifugal and gyroscopic terms, the structural behaviorof CNT would be more dependent on the elastic bending stiffness

Fig. 1. Upstream and downstream wave frequencies against wave number effect of gyros

Fig. 2. Effect of including and excluding gyroscopic term on the difference between upsand Structures 116 (2013) 7587than the other terms. In addition, it would also be noticed thatfor this higher wave numbers, both the variations in uid velocityand Kn would have little inuence on the wave frequency. Fig. 8

copic term (a) neglected (b) included for constant ow velocity (uavg-no slip = 10 m/s).

tream and downstream frequencies for constant ow velocity (uavg-no slip = 10 m/s).

Fig. 3a. Effect of different ow velocities on variations of wave frequency (imaginary part of s) versus wave number for upstream and downstream waves for non-viscousuid (l = 0.0).

Fig. 3b. Effect of ow velocity on difference between upstream and downstream wave frequencies for constant wave number (k = 108) and non-viscous uid (l = 0.0).

Fig. 4a. Upstream/downstream wave frequencies against higher range of wave numbers including gyroscopic term for constant ow velocity (uavg-no slip = 10 m/s) and non-viscous uid (l = 0.0).

F. Kaviani, H.R. Mirdamadi / Computers and Structures 116 (2013) 7587 81

Fig. 4b. Variation of f = (xupstream xdownstream)/xdownstream versus wave number for various ow velocities and non-viscous uid (l = 0.0).

Fig. 5. Variations of decaying ratio versus wave number with and without gyroscopic effect for constant ow velocity (uavg-no slip = 10 m/s) and non-viscous uid (l = 0.0).

Fig. 6. Effect of different Kns on upstream and downstream frequencies of CNT without gyroscopic term for constant ow velocity (uavg-no slip = 10 m/s) and non-viscous uid(l = 0.0).

82 F. Kaviani, H.R. Mirdamadi / Computers and Structures 116 (2013) 7587

Fig. 7. Effect of different Kns on upstream and downstream frequencies of CNT with gyroscopic term for constant ow velocity (uavg-no slip = 10 m/s) and non-viscous uid(l=0.0).

Fig. 8. Effect of different Kns on upstream/downstream frequencies against higher range of wave numbers for constant ow velocity (uavg-no slip = 10 m/s) and non-viscousuid (l = 0.0).

Fig. 9. Effect of different Kns on variations of decaying ratio versus wave number for constant ow velocity (uavg-no slip = 10 m/s) and non-viscous uid (l = 0.0).

F. Kaviani, H.R. Mirdamadi / Computers and Structures 116 (2013) 7587 83

shows that Kn could have little effect on the wave frequency forhigher wave numbers.

5.2.2. Real part of complex frequencyOf parameters effective on the decaying ratio, the slip boundary

condition might be recalled. The variations of decaying ratio withthe wave number could be such that for a non-viscous uid, an in-crease in wave number would cause an early increase in decayingratio and later on, a decrease in that parameter, and nally, becom-ing zero for a specic wave number. Higher values of Kn wouldcause an increase in the decaying ratio as well as an increase inthe wave number in which decaying ratio would be zero. Fig. 9 de-picts this effect.

5.3. Viscosity

5.3.1. Imaginary part of complex frequencyOf other inuential and debatable factors in investigating the

wave propagation in CNTs conveying uid would be the effect ofuid viscosity. The uid viscosity could have little effect on the

wave frequency, which would be dened as the imaginary partof complex frequency. In the literature, some authors have ob-served and reported a major effect of uid viscosity on the wavefrequency. However, they have not distinguished between imagi-nary part of complex frequency and complex frequency itself. Itwould be the real part of complex frequency, which might takeeffect from viscosity. Fig. 10 illustrates the non-effectiveness of vis-cosity on the imaginary part of complex frequency, which could bea representative for the structural stiffness of CNT. Merely, theimaginary part would be really a wave frequency in Hz or rad/s,not the real part which is in the units of Neper (Np).

5.3.2. Real part of complex frequencyThe uid viscosity is effective on the variations of decaying ratio

against the wave number, as shown in Fig. 11. The viscosity wouldcause that the decaying ratio could not remain constant at a zerovalue, but would decrease below zero as the wave number wouldincrease. Increasing Kn could cause a decrease in the effect ofviscosity. The reason might be that as Kn would increase, the colli-sions of uid molecules with the CNT wall could increase while the

Fig. 10. Effect of different uid viscosity on variations of upstream/downstream wave frequencies versus wave number for constant ow velocity (uavg-no slip = 10 m/s).

84 F. Kaviani, H.R. Mirdamadi / Computers and Structures 116 (2013) 7587Fig. 11. Variations of decaying ratio versus wave number for different Kns but constant gyroscopic term.ow velocity (uavg-no slip = 10 m/s) and constant uid viscosity (l = 0.0003), including

er fo

F. Kaviani, H.R. Mirdamadi / Computers and Structures 116 (2013) 7587 85collisions among molecules themselves would decrease, as ex-plained earlier. Accordingly, the uid viscosity could decrease withan increase in Kn [35]. This point might be more claried by inves-tigating Eq. (16).

5.4. Size-dependent continuum theories

5.4.1. Imaginary part of complex frequencySmall-size effects in the structural behavior of CNTs might be

described by non-classical theories of continuum mechanics. Thissmall-size effect could manifest in the higher range of wavenumbers where upstream and downstream wave frequencieswould be nearly the same. Thus, this parameter could have thesame effect on both frequencies. Fig. 12 shows this effect. Asmight be observed, strain/inertia gradient theory might be moresensitive to both ranges of lower and higher wave numbers,while Eringens nonlocal theory would seem to be sensitive only

Fig. 12. Variations of upstream/downstream wave frequencies versus wave numbtheories and constant ow velocity (uavg-no slip = 10 m/s).to the range of higher wave numbers. Accordingly, the strain/inertia gradient theory could present a wider and more completepicture of system behavior as contrasted to the Eringens non-local theory.

Fig. 13. Variation of decaying ratio versus wave number for constant uid viscosity (l =theories.5.4.2. Real part of complex frequencyThe non-classical continuum theories could also be effective on

the variations of decaying ratio against the wave number. The ef-fect of small-size might be only for a viscous uid because for anon-viscous uid in lower range of wave numbers, before the ef-fects of non-classical theories and their dependence on the sizewould be disclosed, the decaying ratio would reach a constant va-lue of zero. Furthermore, for a viscous uid, Eringens nonlocal the-ory would have little effect on the variations of decaying ratioagainst the wave number and could predict results similar to thosefor classical continuum theory; however, the strain/inertia gradi-ent theory would make the decaying ratio decrease. Fig. 13 showsthese variations. The effect of small-size on the decaying ratio for anon-viscous uid could not be predicted by using the Eringensnonlocal and strain/inertia gradient theories. These theories wouldseem to be disabled and limited in this eld. Nevertheless, as Fig. 9might illustrate, the effect of small-size in investigating the decay-

r both viscous and non-viscous uid ow by different size-dependent continuuming ratio for a non-viscous uid could be predicted by using the slipboundary condition and the dimensionless number VCF, such thatfor Kn = 0, the effect of small-size could be ignorable, while byincreasing Kn, this effect would become apparent.

0.0003) and constant ow velocity (uavg-no slip = 10 m/s) but different size-dependent

tle effect on the variations of decaying ratio against wave num-ber and could predict results similar to the classical continuum

ters6. Conclusions

This researchwork investigated the elasticwavepropagation in aCNT of innite length conveying viscous uid and enforcing slipboundary condition between the CNT and passing uid, by usingboth the strain/inertia gradient theory and Eringens nonlocal con-tinuum theory for the structural part, and Kn-dependent ow foruid part of size-dependent FSI problem. For deriving an interactionequation between the CNT structure and passing internal uid (FSI),the Euler-Bernoulli beam was used. The effect of viscosity, as pro-posed by [25], was added to the terms showing the effect of uidon the CNT structure. Noticing that in the nano-scale, the character-istic length of CNT would be comparable with the uid molecularmean free path, the uid-CNT boundary condition would begoverned by a slip condition. The effect of slip boundary conditionsbetween a CNT and uid was modeled in FSI problem by using thedimensionless number of VCF, dened as the ratio of mean owvelocityofuidpassing throughtheCNT ina slipboundaryconditionto themeanuidvelocityof the sameuid inano-slipboundarycon-dition. For solving the elastic wave propagation, the linear FSI equa-tion in a general form was derived and the non-separable(interwoven time and space-dependent) eigenfunctions of linearpartial differential equation of transverse motion were substituted,representing a travelingwave phenomenon. In this type of solution,therewere two transformations from temporal and spatial indepen-dent variables, respectively to the complex frequency-domain andwave number-domain. The complex frequency was divided intothe real (decaying ratio) and the imaginary (wave frequency) parts.The coupled dispersion equations were divided into two categoriesbased on these two parts of complex frequency. By studying numer-ical results obtained from these coupled dispersion equations, wecould investigate the behavior of this FSI system, which would havebeen paid little attention so far. These numerical and parametricstudies on gyroscopic inertia force, slip boundary condition, viscos-ity, and size-dependent continuum theories disclosed to us thefollowing remarkable results:

Gyroscopic effects

The gyroscopic term could affect on the dispersion relation ofboth upstream and downstream wave frequencies propagatingin a CNT against wave number.

The gyroscopic term could make an increase in the upstreamwave frequency and a decrease in the downstream wavefrequency.

When the gyroscopic term is considered, an increase in the uidvelocity causes more pronounced difference between theupstream and downstream wave frequencies.

An increase in the wave number would cause a decrease in thedifference between upstream and downstream wave frequen-cies.

The higher values of gyroscopic term could cause the decreasein the decaying ratio as well as in the wave number in whichthe decaying ratio goes to zero.

Slip boundary condition effects

The effect of slip boundary condition on the two upstream anddownstream wave frequencies without considering the gyro-scopic term would be observed like this: An increase in Knwould cause a decrease in both upstream and downstreamwave frequencies.

In the presence of gyroscopic term, an increase in Kn couldmake an increase in the upstream wave frequency, but adecrease in the downstream wave frequency.

86 F. Kaviani, H.R. Mirdamadi / Compu The effect of slip boundary condition could decrease with anincrease in the wave number, and for much higher range ofwave numbers, it could be ignorable.theory; however, the strain/inertia gradient theory would makethe decaying ratio decrease.

The non-classical continuum theories would seem to be dis-abled and limited to predict the size-dependent effects on thedecaying ratio for a non-viscous uid ow. The effect ofsmall-size in investigating the decaying ratio for a non-viscousuid could be predicted by using the slip boundary conditionand the dimensionless number VCF, such that for Kn = 0 theeffect of small-size could be ignorable, while by increasing Kn,this effect would become apparent.

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Wave propagation analysis of carbon nano-tube conveying fluid including slip boundary condition and strain/inertial gradient theory1 Introduction2 Non-classical theories of strain/inertia gradient and stress gradient3 FSI governing equation4 Dispersion relations5 Results and discussion5.1 Effect of gyroscopic term5.1.1 Imaginary part of complex frequency5.1.2 Real part of complex frequency

5.2 Slip boundary condition5.2.1 Imaginary part of complex frequency5.2.2 Real part of complex frequency

5.3 Viscosity5.3.1 Imaginary part of complex frequency5.3.2 Real part of complex frequency

5.4 Size-dependent continuum theories5.4.1 Imaginary part of complex frequency5.4.2 Real part of complex frequency

6 ConclusionsReferences