Vertically aligned carbon nanotubes for sensing unidirectional fluid flow

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Vertically aligned carbon nanotubes for sensing unidirectional fluid flow Keivan Kiani n a

Department of Civil Engineering, K.N. Toosi University of Technology, Valiasr Ave., P.O. Box 15875-4416, Tehran, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 10 August 2014 Received in revised form 7 December 2014 Accepted 17 January 2015

From applied mechanics points of view, potential application of ensembles of single-walled carbon nanotubes (SWCNTs) as fluid flow sensors is aimed to be examined. To this end, useful nonlocal analytical and numerical models are developed. The deflection of the ensemble of SWCNTs at the tip is introduced as a measure of its sensitivity. The influences of the length and radius of the SWCNT, intertube distance, fluid flow velocity, and distance of the ensemble from the leading edge of the rigid base on the deflection field of the ensemble are comprehensively examined. The obtained results display how calibration of an ensemble of SWCNTs can be methodically carried out in accordance with the characteristics of the ensemble and the external fluid flow. & 2015 Published by Elsevier B.V.

Keywords: Vertically aligned SWCNTs Unidirectional fluid flow Nonlocal Rayleigh beam model Nanofluidic sensor Boundary layer

1. Introduction As an innovative material, carbon nanotubes (CNTs) have revealed outstanding sensing properties [1–3]. These characteristics have accelerated their usage as gas sensors [4–6], temperature sensors [7,8], humidity sensors [9–11], pressure sensors [12,13], biosensors [14–16], and fluid flow sensors [17–20]. In fact, such possible applications for CNTs are devoted to their unprecedented physical and chemical properties [21–23]. In the present work, the latter above-mentioned application of CNTs is of interest from applied mechanics points of view. Shortly, CNTs could be exploited as fluid flow sensors since researchers draw on electro-kinetic phenomena and slip boundary conditions that suggest a detailed insight to both inside and outside nanofluidic flows. To date, vibrations of CNTs due to inside nanofluidic flow have been widely investigated in the context of the classical continuum theory (CCT) [24–30] as well as nonlocal elasticity [31–36]. Both linear vibrations and nonlinear dynamic response of CNTs have been covered. Further, nonlocal vibrations of CNTs for nanoparticle delivery have been addressed in some detail [37–43]. However, the necessity for further studies to realize their nonlinear dynamic responses due to the moving inside fluid flow or nanoparticles is still highly demanded. Concerning the influence of an outside flow on a group of CNTs as well as their ensembles and bundles, there exist a number of experimental n

Fax: þ 98 21 88779476. E-mail addresses: [email protected], [email protected]

works [18,20,44–47]. Such works propose CNTs as fluid flow sensors. However, the theoretical aspects of CNTs acted upon by moving fluid flows have not been methodically examined yet. Further, the roles of geometry properties of CNTs' ensembles and those of the moving fluid on nanomechanical sensing mechanics of such nanostructures have not been disclosed. To bridge this scientific gap, this work has been devoted to examine the static behavior of CNTs’ ensembles due to unidirectional fluid flows in the context of the nonlocal continuum theory. Application of atomistic-based methodologies to nanostructures under externally applied loads commonly requires considerable labor and time costs. To reduce such expenditures, appropriate continuum-based theories would be a good alternative. In the framework of the CCT, the information regarding the interatomic bonds is not incorporated into the equations of motion. As a result, such a theory cannot factually capture the vibration behavior of nanostructures as well as the characteristics of the propagated waves within them. To overcome such a deficiency of the CCT, some advanced theories of elasticity have been developed during the past century. One of the most well-known theories is the nonlocal continuum theory of Eringen [48–51]. Its popularity may be related to its simplicity in taking into account the atomic bond effect in its formulations through the so-called small-scale parameter. So far, such a theory has been frequently used for a diverse range of problems pertinent to vibrations of CNTs [52–58] and nanowires [59–64]. Through comparisons of the obtained results by the nonlocal models and those of suitable atomistic methods, a reasonably good agreement has been achieved in most

http://dx.doi.org/10.1016/j.physb.2015.01.033 0921-4526/& 2015 Published by Elsevier B.V.

Please cite this article as: K. Kiani, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.033i

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of the cases. Herein, for modeling the flexural behavior of CNTs' ensembles subjected to unidirectional fluid flows, an appropriate nonlocal beam model is exploited. Since seeking closed-form solutions to the governing equations of the elastic solid problems would be a difficult job in most of the cases, application of efficient numerical methodologies would be very helpful. Concerning the problem at hand, when the thickness of the boundary layer would be comparable with the length of the CNTs, the drag force would be a function of velocity profile of the moving fluid, arrangement of the CNTs on the substrate, and diameter of the CNTs. Thereby, evaluation of the drag force and the resulting deflection field would not be an easy task. For such a case, a numerical approach based on the reproducing kernel particle method (RKPM) is proposed. RKPM is one of the most powerful and efficient methods in the family of meshless methods which was initiated by a group of scientist at Northwestern University [65,66]. To date, such a numerical technique has been applied to many structural problems of CNTs [67–69,57,70–72] and reasonably good results have been reported. In this paper, novel nonlocal mathematical models are developed for predicting static response of ensembles of SWCNTs subjected to unidirectional fluid flows. The derivation of the applied nonlocal drag force on the ensemble, extraction nonlocal equations of motion of the ensemble, and suggestion of novel numerical and analytical solutions to the governing equations are among the new features of the present study. The deflection of the ensemble at its tip is considered as a crucial factor for sensing the external fluid flow. The predicted deflections of the ensemble of SWCNTs due to the fluid flow are compared with those of experimentally observed data, and a reasonably good agreement is achieved. Thereby, the proposed models can be successfully employed for detecting any fluid flow by SWCNTs' ensembles. In order to realize the sensing mechanisms of the ensembles well, the role of length and radius of the SWCNT, intertube distance, fluid flow velocity, and distance of the ensemble from the leading edge of the rigid base on the maximum static response of the ensemble due to the fluid flows are comprehensively examined.

2. Definition of the nanomechanical problem Consider an array of vertically aligned SWCNTs subjected to a steady fluid flow as shown in Fig. 1. The cantilevered SWCNTs have similar geometrical properties. The SWCNTs have been uniformly placed perpendicular to the y–z plane (see Fig. 1(a)), and they have been connected to a rigid base at the bottom. The length of SWCNTs, center-to-center distance of SWCNTs along the y and z directions (i.e., intertube distance), and the dimension of the square ensemble are represented by lb, d, and as, respectively. The

SWCNTs are acted upon by a steady unidirectional fluid flow of velocity profile v(x) (see Fig. 1(b)). For continuum-based modeling of the SWCNTs of the ensemble, each SWCNT is replaced by its equivalent continuum structure (ECS). The ECS associated with a SWCNT is a circular cylindrical shell of thickness 0.34 nm whose length and mean radius are identical to those of the SWCNT. By impacting the molecules of the fluids to the outer surface of the SWCNTs, a tiny interaction force would be exerted on them. Since the fluid flow moves along the z-axis, the resultant applied force on the SWCNTs of the ensemble would have only one component in acted along the z-axis. From molecular dynamics points of view, evaluation of such forces is a difficult job. Further, implementation of such methods is commonly accompanied with high costs of both time and labor. Thereby, application of suitable elasticity models would be very helpful in analyzing the problem.

3. Characteristics of the unidirectional fluid flow 3.1. Velocity profile of the fluid flow Through the thickness of the boundary layer, a parabolic variation of the fluid flow velocity is assumed, whereas outside of the boundary layer zone, the fluid velocity is constant. Therefore,

⎧ ⎛ ⎞ 2 ⎪ 2 ⎜ x ⎟ − ⎜⎛ x ⎟⎞ , x ≤ δ v (x) = ⎨ ⎜⎝ δ ⎟⎠ ⎝ δ ⎠ v peak ⎪ x≥δ ⎩1,

(1)

where vpeak denotes the peak fluid velocity expressed by vpeak = (1 + (0.722/log ( Re/6.9)) ) vave in which vave is the average velocity of the fluid, Re is the Reynolds number which is expressed by Re = ρ f vl/μ f . In this relation, ρf and μf are the density and dynamic viscosity of the fluid flow, respectively, and l is the characteristic length. Regarding the parameter l, its magnitude relies on the problem at hand. For instance, in the case of the fluid flow inside the micropipe, over the ensemble, and through the vertically aligned SWCNTs, l is set equal to the inner diameter of the micropipe, the distance of the ensemble from the leading edge, and the outer diameter of the SWCNT, respectively [45]. In Eq. (1), δ is the thickness of the boundary layer. By assuming a laminar flow, the Blasius relation [73] is employed for predicting the boundary layer thickness: δ (z) = 5 Re−0.5 z where z is the distance from the leading edge of the rigid base. 3.2. Evaluation of the exerted hydrodynamic force on SWCNTs of the ensemble The externally applied drag force on all SWCNTs of the

Fig. 1. Schematic representation of a vertically aligned ensemble of SWCNTs on a rigid base subjected to a unidirectional fluid flow: (a) top view and (b) side view.

Please cite this article as: K. Kiani, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.033i

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Fig. 2. Plots of RKPM's shape functions as well as their first and second derivatives.

4. Nonlocal static deformation of a SWCNT subjected to a steady fluid flow

ensemble is calculated by

FD =

1 C D ρ f A v2, 2

(2)

where CD is the drag coefficient, A is the total projected area of the SWCNTs of the ensemble to a plane normal to the fluid velocity vector, and FD represents the drag force on the ensemble. The drag force per unit length of the ensemble's SWCNTs, fD, is given by

fD =

1 C D ρ f d b v2, 2

(3)

where db = ∑ dbi is the sum of the outer diameters of the ensemble's SWCNTs. The drag coefficient accounting for slip boundary condition reads [45,74,75]

⎛ ⎞⎛ ⎜ 1 + 4 Kn ⎟ ⎜ 8π CD = ⎜ ⎟⎜ ⎜ 1 + 6 Kn ⎟ ⎜ Res − o ln ⎝ ⎠⎝

(

7.4 Res − o

)

⎞⎛ ⎞ 5 ⎟⎜ ⎟ 3 + 2ϕ 3 ⎟⎜ ⎟, 1 5 2 ⎟ ⎜ 3 − 4.5ϕ 3 + 4.5ϕ 3 − 3ϕ ⎟ ⎠⎝ ⎠

(4)

where Res − o is the Reynolds number for a fluid flow based on the Stokes–Oseen equations [76], ϕ is the correction factor for the volume fraction of the SWCNTs in a given rigid base group, and Kn denotes the Knudsen number, Kn = λ f /l f where λf and lf are the mean free path of the fluid's molecules and the characteristic length, respectively. Herein, for the air flow at the standard temperature and pressure, λ f ≈ 80 nm, and l f is set equal to the intertube distance. The first term in the parenthesis of Eq. (4) represents the incorporation of the slip boundary condition in the exerted force on the SWCNTs. As it is obvious, due to the non-zero fluid velocity at the interface of the SWCNT and the moving fluid flow, the magnitude of the effective drag force on the nanostructure would reduce. The second term in the parenthesis is the coefficient of drag force on an individual SWCNT. The last term in the parenthesis gives the correction for the volume fraction of SWCNTs in a mat group. Based on Refs. [74,75], the given Eq. (4) would be valid for Re <0.6 and Kn < 0.5.

4.1. Nonlocal governing equations and boundary conditions In order to model the flexural behavior of the ensemble of SWCNTs in which acted upon by a unidirectional fluid flow, nonlocal Rayleigh beam model is employed. Based on this theory, the only nonzero local stress and nonlocal stress within the ECS are correlated by [58,67,52,53]

^ b w, xx, σ xnl − (e0 a)2 σ xnl, xx = σ xl = − zE

(5)

where sxl/sxnl is the local/nonlocal stress field of the ECS, e0 a is the small-scale parameter of the nonlocal model, Eb is the Young's modulus of the ECS, w = w (x) is the deflection field of the SWCNTs, and z^ is the distance from the neutral axis. The study of Duan et al. [77] revealed that for the problems pertinent to the transverse vibrations of SWCNTs, the small-scale parameter relies on the chiral angle, aspect ratio, and end conditions of the nanotubes. The magnitude of this parameter is commonly considered in the range of 0–2 nm. By considering such a range for e0 a, a reasonably good agreement between the natural frequencies of SWCNTs and those of molecular dynamics is obtained [78,79]. By premultiplying both sides of Eq. (5) by z^ , and taking the integration over the cross-sectional area of the ECSs associated with all SWCNTs of the ensemble, Ab = ∑ Abi , the nonlocal bending moment, Mbnl , is related to the local bending moment, Mbl , as follows:

Mbnl − (e0 a)2Mbnl, xx = Mbl = − Eb Ib w, xx,

(6)

where Eb Ib denotes the flexural rigidity of the CNTs' ensemble such 2 that Ib = ∑ ∫ z^ dA . Furthermore, the nonlocal shear force within Abi

the ECS is defined by

Please cite this article as: K. Kiani, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.033i

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V bnl =

dMbnl = Mbnl, x. dx

Eb Ib w, xxxx = fD (x) − (e0 a)2fD, xx (x),

The static governing equation of a SWCNT due to the laterally applied drag force reads,

w (0) = w, x (0) = 0,

−Mbnl, xx = fD ,

− Eb Ib w, xx (lb ) − (e0 a)2fD (lb ) = 0,

(8)

by combining Eqs. (6) and (8), the nonlocal equation of motion takes the following form:

Eb Ib w, xxxx = fD − (e0 a)2fD, xx .

(9)

The nonlocal bending moment and shear force in terms of the deflection field are stated as

Mbnl

= − Eb Ib w, xx − (e0

a) 2 f

D,

(10a)

V bnl = − Eb Ib w, xxx − (e0 a)2fD, x .

− Eb Ib w, xxx (lb ) − (e0 a)2fD, x (lb ) = 0.

(15b)

The analytical expression for static response of a cantilevered SWCNT subjected to the fluid flow is obtained as ⎫ ⎧ x ⎞ lb lb ⎛ γ ⎪ ⎜fD (α) − (e 0 a) 2 fD, xx (α) ⎟ dα dβ dλ dγ −⎪ 0 0 λ β ⎝ ⎪ ⎪ ⎠ 1 ⎬. ⎨ w (x) = ⎡ ⎤ 2 2 Eb Ib ⎪ (e 0 a) ⎪ (e 0 a) ⎢l b fD, x (l b ) − fD (l b ) ⎥ x 2 fD, x (l b ) x 3 + ⎪ ⎪ ⎪ ⎪ ⎥⎦ 2 ⎢⎣ ⎭ ⎩ 6

∫ ∫ ∫ ∫

(16)

(10b)

The boundary conditions of a cantilevered SWCNT that stands vertically on a rigid-immovable base read

w (0) = w, x (0) = 0,

(15a)

(7)

V bnl (lb ) = 0, Mbnl (lb ) = 0.

(11)

In the following parts, in the case of low values of the boundary layer thickness to the CNT's length ratio, an analytical solution is developed. Subsequently, when the thickness of the boundary layer would be comparable with the length of CNTs, both analytical and numerical solutions are proposed to evaluate the transverse displacement of SWCNTs acted upon by the external fluid flow.

As it is seen in Eq. (16), analytical evaluation of the integral expression may be not an easy task for a general expression of the drag force. As a result, exploiting an efficient numerical scheme would be of great importance. To this end, Galerkin approach in conjunction with a meshfree method is implemented. Let premultiply both sides of Eq. (15a) by δw where δ denotes the variational sign. Then, the resulting expressions are integrated over the longitudinal domain of the SWCNT. After taking the required integration by parts,

Eb Ib =

4.2. Deformation regime of the SWCNT in the case of δ < < lb

∫0 ∫0

lb

lb lb δw, xx w, xx dx − Eb Ib ⎡⎣w, xx δw, x ⎤⎦ + Eb Ib ⎡⎣w, xxx δw ⎤⎦ 0

lb

0

δwfD (x) dx,

(17)

by introducing the boundary conditions in Eq. (15b) to (17)?, In such a case, a fairly uniform drag force per unit length is applied on the SWCNT. To obtain the deflection field, the following partial differential equation (PDE) with its boundary conditions should be solved:

Eb Ib w, xxxx = fD ,

(12a)

w (0) = w, x (0) = 0, − Eb Ib w, xx (lb ) − (e0 a)2fD = 0, − Eb Ib w, xxx (lb ) = 0.

(12b)

Through solving Eq. (12a) for w = w (x) with the given boundary conditions in Eq. (12b), one can arrive at

w (x) =

⎞ ⎤ fD ⎡ x 4 lb x 3 1 ⎛l2 ⎢ − + ⎜⎜ b − (e0 a)2⎟⎟ x2⎥. Eb Ib ⎣⎢ 24 6 2⎝2 ⎠ ⎥⎦

Eb Ib

w (l b ) =

(1 −

)

CD ρ f db lb4 v2

16Eb Ib

δw, xx w, xx dx =

∫0

lb

⎛ ⎞ δw ⎜fD − (e0 a)2fD, xx ⎟ dx ⎝ ⎠

fore, its variation would be stated by: δw (x) = ∑iNP= 1 ϕi (x) δwi where NP is the number of RKPM's particle, ϕi (x) and wi are the shape function and nodal parameter value associated with the ith RKPM's particle, respectively. By employing penalty method for enforcing the geometrical boundary conditions of the nanostructure (i.e., w (0) = w, x (0) = 0), one can arrive at the following set of linear equations:

Kij w j = fi ,

(13)

(14)

where μ = e0 a/lb denotes the dimensionless small-scale parameter. 4.3. Deformation regime of the SWCNT when δ is comparable with lb In this case, the drag force varies continuously along the length of SWCNTs according to Eq. (3). In order to determine the deflection field of the SWCNT acted upon by a steady fluid flow, the following PDE with the corresponding boundary conditions should be solved:

(18)

Now, the deflection field is discretized in the spatial domain of the problem via Galerkin-based RKPM as w (x) = ∑iNP= 1 ϕi (x) wi . There-

Kij = Eb Ib

fi = ,

lb

− (e0 a)2fD (lb ) δw, x (lb ) + (e0 a)2fD, x (lb ) δw (lb ).

Therefore, by introducing Eq. (3) to (13), the deflection at the tip of the SWCNT can be calculated as

4μ2

∫0

∫0

(19a) lb

ϕi, xx ϕ j, xx dx + α1ϕi, x (0) ϕ j, x (0) + α2 ϕi (0) ϕ j (0),

⎛ ⎞ ϕi ⎜fD − (e0 a)2fD, xx ⎟ dx ⎝ ⎠ ⎡ ⎤ + (e0 a)2 ⎢fD, x (lb ) ϕi (lb ) − fD (lb ) ϕi, x (lb ) ⎥, ⎣ ⎦

∫0

(19b)

lb

(19c)

where α1 and α2 are the penalty parameters. By introducing the dimensionless parameter ξ = x/lb to Eqs. (19a)–(19c), such equations can be rewritten in the following dimensionless form:

Kij w j = fi , Kij =

∫0

1

ϕi, ξξ ϕ j, ξξ dξ + α1ϕi, ξ (0) ϕ j, ξ (0) + α2 ϕi (0) ϕ j (0),

Please cite this article as: K. Kiani, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.033i

(20a)

(20b)

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fi =

∫0

1

⎡ ⎤ ⎛ ⎞ ϕi ⎜fD − μ2 fD, ξξ ⎟ dξ + μ2 ⎢fD, ξ (1) ϕi (1) − fD (1) ϕi, ξ (1) ⎥, ⎣ ⎦ ⎝ ⎠

(20c)

where the dimensionless quantities in Eqs. (20a)–(20c) are given in Appendix B. According to the introduced dimensionless parameters in Eqs. (B.1a)–(B.1d), Eq. (16) could be rewritten in the dimensionless form as follows:

w (ξ ) =

⎞ ⎜fD − μ2 fD, ξξ ⎟ dα dβ dλ dγ ⎝ ⎠ ⎡ ⎤ μ2 μ2 ⎢fD, ξ (1) − fD (1) ⎥ ξ 2. fD, ξ (1) ξ 3 + − 6 2⎣ ⎦ ξ

γ

1

∫0 ∫0 ∫λ ∫β

1⎛

(21)

5. Results and discussion 5.1. Comparison of the obtained results with those of other works In this part, the capability of the proposed numerical model in capturing the experimentally observed data is investigated. To this end, consider an ensemble of SWCNTs of length 50 μm, average diameter 46 nm, mean intertube distance 150 nm covers an area of dimensions 5 μm  5 μm. The distance of the SWCNT's ensemble from the leading edge is about 0.5 mm. For different levels of the fluid flow, the static responses of the SWCNTs’ ensemble acted upon by the air and argon flows are experimentally reported at the tip [45]. Such data are specified by triangular markers as shown in Fig. 3(a) and (b). Further, the predicted results for the air and argon flows are, respectively, presented by the solid and dashed lines. As it is seen in these figures, there is a reasonably good agreement between the experimentally observed data and the obtained results of the numerical model for both air and argon flows. According to Fig. 3(a), for the case of the ensemble of SWCNTs subjected to the air flow, the maximum discrepancy between the predicted results by the present model and those of lab data is about 17 percent. However, in the case of the SWCNT ensemble under the argon fluid flow, the proposed model can capture all experimental data with relative error lower than 8 percent (see Fig. 3(b)). It should be emphasized that the accuracy of the proposed model assumes that the experimental data is accurate. However, no uncertainty of the experimental data is available. The agreement between the proposed model and experiment may

5

vary according to the actual magnitude of the experimental uncertainty. It is noted that the argon is about 1.37 times heavier than the air. Interestingly, a close survey of the plotted results in Fig. 3 (a) and (b) displays that the deflection of the SWCNT subjected to the argon is about 1.37 times greater than that acted upon by the moving air flow. To understand the main reason of this fact, the plots of the thickness of the boundary layer as well as exerted drag force on SWCNTs in terms of the fluid flow velocity for both air and argon flows are given in Fig. 4(a) and (b). As it is seen in Fig. 4(a), the plots of the boundary layer thickness of both air and argon flows are fairly close to each other for different levels of the fluid velocity. Furthermore, the thickness of the boundary layer of the argon is somewhat lesser than that of the air flow. As a result, the drag coefficients of both the air and argon flows are almost the same. Fig. 4(b) shows that the exerted hydrodynamic force on the SWCNTs' ensemble due the moving argon flow is about 1.37 times greater than that because of the air flow. This matter is strongly related to this fact that the density of the argon flow is about 1.37 times greater than that of the air flow. The recently obtained result would guide us to the possible usage of any fluid flow, with respect to a reference one like as air or argon, through a simple density scaling. Additionally, such a crucial issue would provide a new horizon for application of SWCNTs as nanomechanical sensors of any external fluid flow. 5.2. Numerical studies In this part, the effects of the geometry data of the SWCNTs' ensemble as well as the characteristics of the fluid flow on the static response of the ensemble due to the fluid flow are of interest. The maximum static response of the nanostructure (i.e., the deflection at the tip) is considered as a measure of its sensitivity. In the following investigations, an ensemble of SWCNTs subjected to the air flow with the given properties in the previous part is taken into account. In all carried out calculations, the small-scale parameter is set equal to 2 nm. Since the length of the SWCNTs is very high (about 50 μm), the dimensionless small-scale parameter would be a tiny value (i.e., μ ¼ 4  10  5). Thereby, it is expected that the choice of the small-scale parameter for the problem at hand would have a negligible effect on the flexural behavior of the ensemble of SWCNTs.

Fig. 3. Comparison of the predicted deflections at the tip of SWCNT's ensemble via the proposed model and the experimentally observed data of Deck et al. [45] for different levels of the fluid velocity: (a) air and (b) argon; ((▵) Deck et al. [45]; the solid and dashed plots present the predicted results by the proposed model for the air and argon flows, respectively).

Please cite this article as: K. Kiani, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.033i

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K. Kiani / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Fig. 4. The predicted hydrodynamic force on the SWCNT as well as the thickness of the boundary layer as a function of the fluid velocity for two fluid flows: ((—) air and (—) argon).

5.2.1. Convergence check To ensure regarding the accuracy of the proposed model, a convergence check study is performed. In the case of vave ¼ 20 (m/ s), the deflection at the tip of the ensemble of CNTs as a function of number of RKPM's particles has been demonstrated in Fig. 5. By an increase of the particles of RKPM, the predicted deflection of the nanostructure at the tip increases. For higher levels of NP, such an increase is followed with a lower rate and the deflection converges to a specific value. For example, the discrepancies between the predicted deflection at the tip for NP¼ 17, 19, 21, and 50 and that for NP ¼200 are about 6.3, 5.5, 4.9, and 1.7 percent, respectively. In the remainder of this paper, 51 particles with equal distances and 50 similar computational cells with 5 Gauss points in each cell are used. The shape functions of RKPM have been numerically calculated at the Gauss points based on exponential window function, linear base function, and the dilation parameter is set equal to 3.2 inter-particle distance. 5.2.2. Effect of length of SWCNTs Fig. 6 displays the influence of the length of the SWCNTs' ensemble on the maximum static response of the nanostructure for various levels of the velocity of the fluid flow. As it seen, the deflection of the SWCNTs' ensemble at the tip would magnify with the length of the SWCNT. It implies that the sensitivity of the

Fig. 5. Convergence check of the proposed numerical model (vave ¼ 20 m/s).

nanostructure for detecting the fluid flow increases with its length. Further, the variation of the SWCNT's tip deflection as a function of its length is more obvious for higher levels of the fluid flow velocity. 5.2.3. Effect of radius of SWCNTs In Fig. 7, the effect of the radius of the SWCNT on the deflection of SWCNTs' ensemble is studied for different levels of the fluid flow velocity. It is revealed from Fig. 1 that the variation of the radius of the SWCNT at higher levels of the fluid velocity has a more impact on the variation of the deflection of the SWCNTs' tip. According to Fig. 7, the deflection of the SWCNTs' tip would lessen with the radius of the SWCNT up to 20 nm. Thereafter, the deflection would increase as the SWCNT's radius magnifies. To explain such a behavior, the variations of various parts of the drag coefficient as a function of the SWCNT's radius are also investigated which are not included in the paper. Such investigations reveal that the coefficient associated with the slip boundary conditions does not alter in terms of SWCNT's radius; however, the coefficient corresponding to the volumetric fraction of SWCNTs generally increases with the radius of SWCNTs. Further, the coefficient pertinent to the drag on a single SWCNT would reduce as

Fig. 6. Role of the length of the SWCNTs' ensemble on its deflection at the tip for different levels of the fluid flow velocity: ((⋯) vave ¼5 m/s, (—) vave ¼ 10 m/s, (—) vave ¼20 m/s).

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Fig. 7. Role of the mean radius of the SWCNT on its deflection at the tip for different levels of the fluid flow velocity: ((⋯) vave ¼ 5 m/s, (—) vave ¼ 10 m/s, (—) vave ¼ 20 m/s).

the SWCNT's radius magnifies. Thereby, a combination effect of the coefficient of the volumetric fraction of SWCNTs as well as that of drag force on the single SWCNT is corresponding to such a static behavior of SWCNTs’ ensemble acted upon by a moving fluid flow. 5.2.4. Effect of intertube distance Another important study has been carried out to determine the role of the intertube distance on the sensitivity of the SWCNTs’ ensemble for detecting a gas flow. Investigations on the influence of the intertube distance can be followed for two cases. In the first study, the dimensions of the ensemble are kept fixed, and the intertube distance is variable. Fig. 8(a) shows the variation of the deflection SWCNTs’ tip in terms of the intertube distance.

7

According to the plotted results in Fig. 8(a), the deflection of the SWCNTs' tip would generally magnify as the intertube distance increases. This matter is also more obvious for those ensembles which are subjected to fluid flows with a higher velocity. Close scrutiny reveals that the main cause of such a fact is the reduction of the flexural rigidity of the SWCNTs’ ensemble. Further calculations show that although the exerted drag force on the ensemble would decrease, but due to the decrease of the number of SWCNTs within the ensemble because of the increase of the intertube distance, the deflection of the SWCNTs' tip due to the fluid flow would increase. In the second case, the number of SWCNTs of the ensemble remains unchanged, and the intertube distance varies. The plotted results of this case have been provided in Fig. 8(b). As it is obvious from this figure, the deflection of the SWCNTs’ tip would decrease as the intertube distance increases. In contrast to the previous case, the flexural rigidity of the ensemble of SWCNTs would magnify with the intertube distance. On the other hand, the exerted drag force on the ensemble would also lessen due to the reduction of the drag coefficient. As a result, the reduction of the drag coefficient and the magnification of the flexural rigidity are the main reason for the reduction of the sensitivity of the SWCNTs’ ensemble in detecting the fluid flow. 5.2.5. Effect of distance from the leading edge Another interesting issue in sensing behavior of SWCNTs' ensemble is its distance from the leading edge of the rigid base. In Fig. 9, the plots of the deflection of the SWCNTs' tip as a function of the distance from the leading edge are provided for three levels of the air velocity. Fig. 9 displays that the deflection of the SWCNTs' tip would generally decrease as the distance of the ensemble from the leading edge increases. Furthermore, such a reduction in the deflection is more obvious for higher levels of the fluid flow velocities. In finding out the reason of such a behavior, a detailed survey has been conducted to determine the variations of both drag coefficient and force in terms of z. Such explorations reveal that drag coefficient of the air flow would generally increase with z for all levels of the flow velocity (this issue is mainly attributed to

Fig. 8. Role of the intertube distance of the SWCNTs' ensemble on its deflection at the tip for different levels of the fluid flow velocity: (a) the dimensions of the ensemble are kept fixed and (b) the number of SWCNTs in the ensemble is kept fixed; ((…) vave ¼ 5 m/s, (—) vave ¼10 m/s, (—) vave ¼20 m/s).

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Appendix A. A brief introduction to one-dimensional RKPM According to the works of Liu and his coworkers [65,66,80,81], an arbitrary field u(x) can be approximated continuously by

ua (x) =

∫Ω ϕa⁎ (x; x − s) u (s) ds,

(A.1)

where Ω is the one-dimensional spatial domain, and ϕa⁎ represents the modified kernel function stated as

ϕa⁎ (x; x − s) = ϕa (x − s) C (x; x − s),

ϕa (x − s) =

1 ⎛x − s⎞ ⎟, ϕ⎜ a ⎝ a ⎠ (A.2)

where a is the dilation parameter, ϕa is the window function, and C is the correction function which is expressed in terms of base function vector, HT , as N

C (x ; x − s ) =

∑ bi (x)(x − s)i = HT (x − s) b (x), i=0

Fig. 9. Role of the distance of the SWCNTs' ensemble from the leading edge on its deflection at the tip for different levels of the fluid flow velocity: ((⋯) vave ¼ 5 m/s, (—) vave ¼ 10 m/s, (—) vave ¼ 20 m/s).

an increase in the term associated with the drag on an individual SWCNT whereas the other terms of the drag coefficient do not depend on z). Further, the increase of the distance from the leading edge would lead to an increase of the thickness of the boundary layer. Such an increase would lead to the decrease of the fluid flow velocity. Thereby, the drag force which is a monotonically increasing function of velocity would surely reduce. Finally, in spite of the increase of the drag coefficient, the drag force lessens due to the reduction of the fluid velocity at the vicinity of the SWCNTs' ensemble.

6. Concluding remarks The possible use of SWCNTs' ensembles as nanomechanical fluid flow sensors is rationally investigated by a simple model. Using nonlocal Rayleigh beam theory, analytical and numerical models are developed when the thickness of the boundary layer is negligible and comparable with respect to the length of the SWCNT. The deflection field of the ensemble of SWCNTs is then determined analytically and numerically. The capabilities of the proposed nonlocal models in capturing the deflection field of the ensemble due to the applied drag force of the moving fluid flow are investigated. The undertaken comparison studies reveal the efficacy of the proposed models in predicting static behavior of ensembles subjected to fluid flows. The maximum deflection of the SWCNTs' ensemble is considered as a measure for sensing the fluid flow. Subsequently, the sensitivity analysis of the SWCNTs' ensembles as fluid flow sensors is performed through various parametric studies. The influences of length and radius of the SWCNT, intertube distance, and distance from the leading edge on the maximum deflection of the ensemble are noted.

Acknowledgments The financial support of the Iran National Science Foundation (INSF) is gratefully acknowledged. The author would also like to express his gratitude to the anonymous reviewers for their fruitful comments in which lead to improvement of the present paper.

(A.3)

where

H T = [1, x, x2 , …, x N ], bT (x) = [b0 (x), b1 (x), …, b N (x)],

(A.4)

by substituting Eq. (A.3) into Eq. (A.2) and then substituting the resulting statement into Eq. (A.1), it is obtainable: N

ua (x) =

∞

∑∑ k = 0 n= 0

( − 1)n bk (x) mn + k (x) u(n) (x), n!

u(0) (x)

u(n)

∂ nu/∂x n,

where = = u (x), tions, mn(x), are defined by

mn (x) =

(A.5)

n > 1, and the moment func-

∫Ω (x − s)n ϕa (x − s) ds.

(A.6)

The satisfaction of the Nth order completeness condition for ua (x) yields ∑kN = 0 bk (x) mk (x) = 1 and ∑kN = 0 bk (x) mn + k (x) = 0, n ≥ 1. By solving these equations for bk(x),

b (x) = M−1(x) H (0),

(A.7)

where the moment matrix, M , is defined as

⎛ m0 (x) m1 (x) ⎜ ⎜ m (x) m2 (x) M(x) = ⎜ 1 ⋮ ⎜⎜ ⋮ ⎝m N (x) MN + 1 (x)

m N (x) ⎞ ⎟ … m N + 1 (x) ⎟ ⎟. ⋱ ⋮ ⎟⎟ … m2N (x) ⎠ …

(A.8)

By introducing Eqs. (A.7), (A.2) and (A.3) to (A.1), the modified kernel function is calculated as

ϕa⁎ (x; x − s) = H T (x − s) M−1(x) H (0) ϕa (x − s).

(A.9)

Using the trapezoidal rule to evaluate the integral in Eq. (A.1), ua (x) can be discretized as NP

ua (x) =

∑

ϕ I (x) u I ,

I=1

(A.10)

where NP is the number of RKPM's particles, uI and ϕ I (x) in order are the nodal parameter value and the shape function of the Ith RKPM's particle such that

ϕ I (x) = ϕ⁎ (x; x − x I ) Δx I ,

(A.11)

where Δx I is the length pertinent to the Ith particle. The first and the second derivatives of the shape functions of RKPM are calculated as

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K. Kiani / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

⎛H T (x − s) M−1(x) ϕ (x; x − x )+⎞ I a, x ⎜ ⎟ ⎜ T 1 − ϕ I, x (x) = H,x (x − s) M (x) ϕa (x; x − x I )+ ⎟ H (0) Δx I , ⎜ ⎟ ⎜ T ⎟ −1 ⎝H (x − s) M,x (x) ϕa (x; x − x I ) ⎠

(A12a)

⎛ H T (x − s) M−1(x) ϕ (x; x − x ) + H T (x − s) I ,xx a, xx ⎜ ⎜ −1(x) ϕ (x; x − x )+ M I a ⎜ ⎜ T T −1 ( ) ( ) ( x s x x − ϕ H M ,xx a ; x − x I ) + 2H,x (x − s) ⎜ ϕ I, xx (x) = ⎜ −1 ⎜ M (x) ϕa, x (x; x − x I )+ ⎜ T −1 T ⎜ 2H,x (x − s) M,x (x) ϕa (x; x − x I ) + 2H (x − s) ⎜ −1 ⎝ M,x (x) ϕa, x (x; x − x I )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

H (0) Δx I .

(

(A12b)

Appendix B. The dimensionless quantities in RKPM formulations

α1 =

α1lb , Eb Ib

α2 =

α2 lb3 , Eb Ib

fD (ξ) = K CD (ξ) v 2 (ξ), K =

fD, ξ

2 ρ f db lb3 v peak

, 2Eb Ib ⎛ ⎞ = K ⎜CD, ξ (ξ) v 2 (ξ) + 2CD (ξ) v (ξ) v, ξ (ξ) ⎟, ⎜ ⎟ ⎝ ⎠

⎛C ⎞ 2 D, ξξ (ξ) v (ξ) + 4C D, ξ (ξ) v (ξ) v, ξ (ξ)+ ⎟, fD, ξξ = K ⎜ ⎜ 2C (ξ ) v 2 (ξ ) + 2C (ξ ) v (ξ ) v (ξ ) ⎟ , ξξ D ,ξ ⎝ D ⎠ ⎧ ⎛1 ξ ⎞ ⎪2 ⎜ − ⎟, ξ ≤ δ v, ξ = ⎨ ⎝ δ , δ2⎠ ⎪ ξ≥δ ⎩ 0,

v, ξξ

⎧ 2 ⎪− , ξ ≤ δ = ⎨ δ2 , ⎪ ξ≥δ ⎩ 0,

(B.1a)

δ =

δ , lb

(B.1b)

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ 5/3 1 + 4 Kn ⎟ ⎜ 3 + 2ϕ ⎟ C D , ξ (ξ ) = ⎜ ⎜ 1 + 6 Kn ⎟ ⎜ 3 − 4.5ϕ1/3 + 4.5ϕ5/3 − 3ϕ2 ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎛ ⎜ 8π × ⎜ ⎜ ⎜ Res − o ln ⎝

(

(

7.4 Res − o

⎞ ⎞⎛ ⎟ ⎟⎜ ⎛ ⎞ 7.4 ⎟ ⎜ ln ⎜ ⎟ − 1⎟ Res − o, ξ , 2 ⎟⎜ ⎟ ⎝ Res − o ⎠ ⎟ ⎟⎜ ⎠ ⎠⎝

))

⎛⎛ ⎜⎜ ⎜⎜ 8π × ⎜⎜ ⎛ ⎜ ⎜⎜ ⎜ Re ⎜ ⎝ s − o ln ⎝⎝ ⎛ ⎜ 8π ⎜ +⎜ ⎜⎜ ⎛⎜ Re s − o ln ⎝⎝

()

wi , lb

⎛ ⎞⎛ ⎞ 5 ⎜ ⎟⎜ ⎟ 3 + 2ϕ 3 ⎜ 1 + 4Kn ⎟ ⎜ ⎟ C D, ξξ (ξ) = ⎜ ⎟⎜ ⎟ 1 5 ⎜ 1 + 6Kn ⎟ ⎜ 3 − 4.5ϕ 3 + 4.5ϕ 3 − 3ϕ 2 ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

(

Herein, linear base function (i.e., HT (x) = [1, x]), and the ex⎛ 2⎞ ponential window function (i.e., ϕa (x) = exp ⎜ − αx ⎟ ; α = 0.3) is ⎝ ⎠ used to construct the shape functions of RKPM. In the case of using 11 RKPM's particles with uniform distribution over the dimensionless spatial domain of our interest, namely [0, 1], the RKPM's shape functions as well as their first and second derivatives have been plotted in Fig. 2.

wi =

9

(B.1c)

Re s − o =

ρ f vlv peak

Re s − o, ξ =

Re s − o, ξξ =

μf

)

⎞ ⎞ ⎞⎛ ⎟ ⎟ ⎟⎜ ⎛ ⎞ ⎟ 7.4 ⎟ ⎟⎜ ln ⎜ ⎟ − 1⎟ Re s − o, ξξ ⎟ , ⎜ ⎟ ⎞2 ⎟ ⎜ ⎝ Re s − o ⎠ ⎟ ⎟⎟ ⎟ ⎟⎜ ⎟ ⎠ ⎠⎝ ⎠ ⎠

,

ρ f v, ξξ lv peak μf

7.4 Res − o

)

,

ρ f v, ξ lv peak μf

7.4 Res − o

⎞ ⎞⎛ ⎟ ⎟⎜ ⎛ ⎞ ⎛ 7.4 ⎞2 7.4 2 ⎟ ⎟⎜ ⎜ ⎟ ⎟ ⎜2 ln ⎜⎝ Re s − o ⎟⎠ − 3 ln ⎜ Re s − o ⎟ + 2⎟ ( Re s − o, ξ ) 3 ⎞ ⎟⎜ ⎝ ⎠ ⎟⎟ ⎟ ⎟⎜ ⎠ ⎠⎝ ⎠

. (B.1d)

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