Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) Journal o Theoretical and Applied Vibration and Acoustics I S A V journal homepage: http://tava.isav.ir Transverse vibration and instability o luid conveying triple-walled carbon nanotubes based on strain-inertia gradient theory Soheil Oveissi a, Hassan Nahvi b* a Department o Mechanical Engineering, Islamic Azad University, Khomeinishahr Branch, 8475/9, Khomeinishahr, Isahan, Iran b Department o Mechanical Engineering, Isahan University o Technology, Isahan 84568, Iran K E Y W O R D S Triple-walled carbon nanotube Knudsen number Strain-inertia gradient theory Galerkin s method A B S T R A C T In this paper, the transverse vibration o a triple-walled carbon nanotube (TWCNT) conveying luid low is studied based on the strain/inertia gradient theory with van der Waals interaction taken into consideration. The nanotube is modelled using Euler-Bernoulli beam model and the Galerkin s method is employed to obtain the CNT complex valued Eigen-requencies. The eects o the luid low thorough the innermost tube and the van der Waals orce interaction between any two walls on the instability o the nanotube are studied. In addition, the eects o the nano-low size, the characteristic lengths and the aspect ratio on the critical low velocities are investigated. Results indicate that due to the luid low the nanotube natural requencies decrease. By considering the size eect o the luid low, requencies decrease more rapidly causing reduction o the stability region. Moreover, it is shown that the length o the nanotube can play an important role in the vibration response.. 05 Iranian Society o Acoustics and Vibration, All rights reserved. Introduction Carbon nanotubes with their excellent properties have applications in ields such as nanotubes conveying luid [], nano luidic devices [], drug delivery devices [], micromechanical oscillators, sensors, etc. [4]. To this end, many researchers ocused on the interesting dynamic characteristics o luid-structure systems in small-scale [5, 6]. Due to the act that molecular dyn0amics simulations (MDS) are complex and time consuming, using the classical continuum mechanics theories can be an eective and useul way to study mechanical behaviour o both single-walled and multi-walled carbon nanotubes. Raiei et al. [7] discussed the eects o taper ratio and small-scale parameter on the vibration o non-uniorm carbon nanotubes. They reported that dimensionless requencies obtained rom nonlocal theory are less than those obtained rom local ones. Wang [8] utilized nonlocal elasticity theory integrated with surace elasticity theory to model the luid conveying nanotubes with inner and outer surace layers. He showed that the * Corresponding Author: Hassan Nahvi, Email: hasnahvi@yahoo.com
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) predicted undamental requency is generally higher than that predicted by the Euler-Bernoulli beam model without surace eects. Yoon et al. [9] investigated transverse sound wave propagation in multi-walled carbon nanotubes (MWCNTs) based on the multiple-elastic beam model. Yan et al. [0] studied the dynamical characteristics o the luid-conveying MWCNTs by using the classical continuum theory. They reported that the van der Waals (vdw) interaction has no inluence on the biurcation o the system. Rashidi et al. [] presented a model or a single mode coupled vibration o luid conveying CNTs considering the slip boundary conditions o nano low. They ound that the critical low velocities decrease i the passage luid is a gas with nonzero Kn in contrast to a liquid nano low. In this research, the eect o small size or both luid low and solid structure is considered on the o vibration study o CNT. Also, or the irst time, by using the Euler-Bernoulli beam theory a model is proposed or the coupled vibrations o triple-walled carbon nanotubes conveying luid low taking into account the small-size eects o both low luid and solid structure utilizing Knudsen number and strain-inertia gradient theory. In addition, the inluences o small-size eects and aspect ratio on the natural requencies are investigated. For the numerical solution, the governing luid-structure equation is discretized by using the Galerkin method. It has been shown that the size-eect has signiicant inluence on the non-dimensional critical low velocities. Moreover, it is ound that the length and CNT radius have eect on the natural requencies and critical low velocities.. Strain-inertia gradient theory The theory o combined strain-inertia gradient is developed by Askes and Aiantis []. They showed that the combination o equation o motion and the strain-inertia gradient constitutive relation can be expressed as u&& i lmu&& i, mm Cijkl uk, jl ls uk, jlmm () ρ( ) ( ) where ρ is the mass density o nanostructure, C ijkl are Cartesian components o elasticity tensor and u i denotes the Cartesian displacements or an elastic structure. Also l m and l s are the two length scales related to the inertia and strain gradients respectively, which represent volume element sizes o electrostatics and electrodynamics problems. According to this theory, the CNT conveying luid remains more stable compared to the classical theories []. Setting i and using the linearized strain-displacement relation o the Euler- Bernoulli beam theory yields the constitutive relation or -D Euler-Bernoulli beam model. The bending moment is M A where σ denotes the lexural or axial stress. According to the strain-inertia gradient theory, the stress-strain relation may be written as zσ da () ε σ σ ε + ρ x ε t xx E ls clm in which ε is the bending strain that or -D strain-lexural curvature can be written as, () 6
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) ε w x ε xx z Finally, by integrating over the beam cross-sectional area, Eq. () gives; w w w M EI l Il x x x t 4 4 s ρ 4 c m (4) (5). Size eect o nano low To investigate the small-scale eect o nano low, the velocity correction actor (VCF) is inserted into the luid-structure interaction equation o motion multiplied by the low velocity term. By considering the VCF parameter, the luid slip-boundary condition can be inserted into the equation o motion. VCF parameter is deined as ollows []. where V slip and ( no slip) VCF Vslip σ v Kn 4 + V Cr ( no slip ) ( kn ) σ v bkn V are the low velocities with and without slip boundary conditions respectively. Also σ v and b are the tangential momentum accommodation coeicient and the general slip coeicient; these two parameters are considered to be 0.7 and - respectively. Parameter Cr is the rareaction coeicient which is deined as ollows by using the Polard relation or viscous luids [4]. where α is a constant obtained as, Cr( Kn) + αkn (7) α α α π B 0[tan ( Kn )] (8) in which α0 64 π ( (4 b)), α 0.4 and B 0.4 are practical parameters. In Eq. (6), the eective parameter is Knudsen number (Kn) that presents the ratio o the ree path o the luid molecules to a characteristic length o the low geometry. The range o Kn can vary rom 0.00 to 0.0 or a liquid nano low [].The values o the parameters in Eqs. (6)-(8) are extracted rom []. 4. Governing equations Flexural vibration o the Euler-Bernoulli beam model is studied using the equations o motion o the CNTs. The external orce can be considered as an incompressible, laminar, ininite and viscous luid lowing through the CNT. On the other hand, the TWCNTs consist o three single SWCNTs with the van der Waals interaction between any two tubes, as shown in Fig. (a). The luid lows inside the innermost tube o the CNT. Based on the strain-inertia gradient theory, the (6) 64
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) innovative governing -D coupled luid-structure interaction equations o motion or a TWCNT conveying luid are derived as W W W W W W EI l m m m VCF U m VCF U Il x x t x t x t x 4 6 W W W EI ( l ) + m 6 c x x t 4 6 W W W EI( l ) + m 6 c x x t 4 6 6 ( ) + ( 6 c + ) ( ) ( ) + + + ρc m 4 where W, W and W denote the lexural displacements o the innermost, middle and outermost CNT walls respectively. Also x is the axial coordinate, E is the beam s Young modulus and U is the low velocity. In addition, I, I and I are the moments o inertia related to the three walls o CNT and m is the luid mass per unit length. Parameters m, m and m are the mass per unit length o inner, middle and outer CNTs respectively. In Eq. (9),, and denote the acting Van der waals interaction orces exerted on the inner, middle and outer CNT walls due to the van der Waals interaction among any two walls. These orces can be expressed as, i ij i j j, i j c c ( x,0) c ( W W ) (0) or i,, where c ij is the van der Waals coeicient or the related CNT wall that can be obtained as [0], in which 00πεσ 0πεσ c R E E R a 9a 6 7 ij i 4 ij 4 ij j π n n ij ( i + j ) 0 l [ Kij cos θ ] E R R K ij c (9) () dθ () 4RjRi ( R + R ) The equation o motion may be rendered dimensionless. For this purpose, the ollowing dimensionless variables are deined j i () 65
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) m u n + c x W W W EI t ξ, η, η, η, τ, LU, L L L L m m L EI m m I m I β, β, β, m + mc ( m ) + mc I ( m + mc ) I c L c L c L c L c L c L c, c, c, c, c, c ψ 4 4 4 4 4 4 EI EI EI EI EI EI ρ I ρ I ρ I l l c c c m s, ψ, ψ, λ, m λs L ( m + mc ) L ( m ) ( ) c L m c L L where L is the CNT length. Using Eq. (4) the dimensionless equations o motion can be expressed as η η η η η η ξ ξ τ ξ τ ξ τ ξ 4 6 6 λ + + ( VCF ) u 6 n + β ( VCF ) un + ψλm 4 c ( η η ) c ( η η ) 0 4 6 6 η η η η λ + β 6 + ψ λm c 4 ( η η) c( η η) 0 ξ ξ τ τ ξ 4 6 6 η η η η λ + β 6 + ψ λm c( η η 4 ) c( η η) 0 ξ ξ τ τ ξ (4) (5) 5. Approximate solution and discretization To solve Eqs. (5), the Galerkin s approximate method is used. The solution o the dimensionless dierential equations are approximated as N N N (6) η ( ξ, τ ) q ( τ ) φ ( ξ), η ( ξ, τ ) q ( τ ) φ ( ξ), η ( ξ, τ ) q ( τ ) φ ( ξ) r r r+ N r r+ N r r r r where qr ( τ ), qr+ N ( τ ) and qr+ N ( τ ) are the generalized coordinate o the discretized innermost tube conveying luid, middle and outermost tubes respectively, and φr ( ξ ) are the dimensionless eigen-unctions o the CNTs that should satisy the essential and natural boundary conditions. In this work, the boundary conditions are considered as simply supported ends or pinned-pinned which can be written as, η 0 and η ξ 0 at ξ 0, (7) For this boundary condition, considering simple harmonic motion, the comparison unctions and generalized coordinate are considered as, φr ( ξ ) sin( rπξ ) (8) 66
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) qr ( τ ) Ar exp( srτ ) (9) where A r are the dimensionless constant amplitude o the r th -generalized coordinate and denotes the r th mode complex-valued Eigen-requency. To discretize equation o motion by Galerkin weighted-residual technique, or the irst mode, the wave number is set to and the residual is obtained by substituting Eqs. (6) - (9) into Eq. (5). In this approximate method, the main governing dierential equations o motion are used to compute the residue. For this purpose, one generalized coordinate is chosen. In the dimensionless equations (5), the approximate unction η( ξ, τ ) is substituted by sin( πξ ) q ( τ ) to calculate the residual. Then, this residual is multiplied by a weighting unction (test unction). Herein, the comparison and weighting unctions are the same and are selected as the irst mode Eigenunction ( sin( πξ ) ). The resulting weighted residual is then integrated over the domain o the structure and then it is set to zero. This means that the error in the subspace spanned by the trial unctions is nulliied and only the residual error orthogonal to this subspace would remain. This residual could have components in the complementary subspace o the exact solution not spanned by the inormation subspace. The trial eigen-unctions play the role o the basis unctions that span the complementary space o ininite-dimensional space spanning the exact solution. For example, the resulting discretized equation or the innermost tube conveying luid is obtained as ollows 4 6 4 ( π + λs π ( VCF ) unπ ) q( τ ) + ( ψλmπ + ) q&& ( τ ) 0 (0) The coeicient o q ( ) τ denotes the sum o elastic and geometric stiness parameters and the coeicient o q && ( τ ) represents the equivalent mass parameter. This process is continued or the next modes; or example, to study the two irst modes, the approximate unction is considered as sin( πξ ) q ( τ ) + sin( πξ ) q ( τ ). 6. Results and discussion The material and geometric properties o the CNT and the luid are shown in Table. Results are extracted rom the dimensionless parameters and using a MATLAB code. Water is considered as the luid passing through the CNT. The eects o the aspect ratio, Knudsen number and characteristic lengths on the critical low velocities are studied. Table. Material and geometric properties o CNT and luid [0] Young s modulus (E) Tpa Fluid viscosity ( µ ). 0 Pa.s s r CNT mass density( ρ ) c. 0 kgm - Fluid mass density( ρ ) 0 kgm - Geometric parameters R.9 nm R.4 nm R.58 nm h 0.4 nm van der Waals parameters ε.968 mev σ.407 A o o a.4 A 67
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) In Table, the parameters R, R and R are the internal radii o the inner, middle and outer CNTs respectively. These parameters are shown in Fig.. Fig.. (a) A luid conveying TWCNT. (b) Cross sectional view o a circular TWCNT 6.. Eects o aspect ratio In this subsection, the eects o aspect ratio on the critical low velocity and the vibration behavior o CNTs conveying luid low are studied using size dependent continuum theory. The critical velocity is a velocity in which the imaginary part o Eigen requency (natural requency) reaches zero while its real part (damping) is non-zero and thereore the system is instable. According to Paidoussis [5], by equating the equivalent stiness o a system (K eq ) to zero, the critical value o luid low velocity will be obtained. In this case, the divergence instability occurs in the system. Fig. shows the variation o dimensionless critical low velocity against CNT thickness or the our aspect ratio values o,, 5 and 0. It can be seen that by increasing the CNT thickness the critical low velocity decreases. Moreover, the eect o aspect ratio on the critical low velocity increases as the CNT thickness decreases. Fig. illustrates the imaginary parts o dimensionless undamental Eigen-requencies or the nanotube versus dimensionless Fig.. Dimensionless critical low velocity against dimensionless CNT thickness or dierent aspect ratios 68
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) low velocity or the aspect ratio values o and, and the dimensionless length scale values o 0.055 and 0.. According to Askes and Aiantis [], or the length scales l m and l s large ranges are possible; lm 0ls is used here or CNT (0, 0). It can be observed that or a certain aspect ratio, the critical low velocity increases as the characteristic length decreases. For example, by using Table., or aspect ratios equal to and, approximately.0% and 0.58% reductions occur in the magnitude o the critical low velocities respectively. It means that the divergence instability occurs at a lower critical low velocity. Fig.. First dimensionless natural requency against dimensionless low velocity or dierent aspect ratios and characteristic lengths 6.. Eect o Kn o nano low The irst and second dimensionless natural requencies against dimensionless low velocity or three values o Kn are shown in Fig. 4 and 5 respectively. By comparing the values o the critical low velocities or Kn equal to 0, 0.00 and 0.0, the divergence and coupled mode lutter phenomena occur at a lower critical low velocity as Knudsen number increases. As shown in Figs. 4 and 5, approximately 4.% and 7.% reductions occur or increasing the Knudsen number rom 0 to 0.00 and rom 0.00 to 0.0 respectively. Moreover, as the low velocity increases, the eect o small-size o nano low on the natural requencies increases. In order to validate the numerical results, the results o Paidoussis [5] are considered. Since Paidoussis investigated the plug low theory, the Knudsen number is set equal to zero or VCF is substituted by one in Eq. (5). From Figs. 4 and 5 it can be observed that by increasing the low velocity rom zero to its critical value, the natural requencies approach to zero; as a result, the system s stiness disappears and divergence mode occurs. As shown in Fig. 4 the irst mode divergence is equal to π which is expected rom the Paidoussis observations [5]. 69
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) Fig 4. First dimensionless natural requency versus low velocity or three values o Knudsen number (Kn0, 0.00 and 0.0) Fig 5. Second dimensionless natural requency versus low velocity or three values o Knudsen number (Kn0, 0.00 and 0.0) 6.. Simulations o the divergence and lutter instabilities Fig. 6 shows the variations o the dimensionless natural requencies versus the dimensionless low velocities or the irst three modes. The Knudsen number is set to be 0.0 and the characteristic length related to the strain gradient is equal to 0.055. Also, the aspect ratio is considered to be L/Rout000. It should be noted that the van der Waals orce is the coeicient 70
S. Oveissi and H. Nahvi / Journal o Theoretical and Applied Vibration and Acoustics () 6-7 (05) o the W term in Eqs. (9) and (0). Hence, by adding these orces, it is expected that the equivalent stiness be increased. By setting this stiness term equal to zero, it can be seen that the critical low velocity value increases. It means that considering the van der Waals orce makes the system more stable. Moreover, the dimensionless natural requencies o three modes decrease as the dimensionless low velocity increases. By increasing the low velocity, the irst natural requency reaches to zero at the critical low velocity value o.576 and the system becomes unstable, known as the irst divergence instability. Next, by increasing the low velocity to 4.8, the system becomes stable again. At this point, the requencies o the irst and second modes are coupled which is called the lutter instability. The same phenomenon is occurred at the critical low velocity equal to 7.9 in which the third divergence and second lutter instabilities occur. 7. Conclusions Fig 6. Dimensionless low velocity dependence o the lowest three modes o a TWCNT The ree vibration and instabilities o TWCNT conveying luid low were investigated considering the small-size eects o both luid low and the structure by using strain-inertia gradient theory. To study the vibration response, the Euler- Bernoulli beam model is used and to calculate the governing equation o luid-structure interaction, the Galerkin weighted residual method is employed. The variations o the dimensionless critical low velocities and dimensionless requencies or dierent aspect ratios and characteristic lengths in the mentioned theory were examined. It is shown that the low inside CNTs decreases the system natural requencies. It means that the luid low can destabilize the luid-structure system. Also it is shown that by increasing the Knudsen number o the passing nano low, instability occurs at a lower low velocity value. The results indicate that the critical low velocity increases as the thickness or outer radius o CNT decreases and it decreases as the CNT length increases. 7
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