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Investigation on vibration characteristics of fluid conveying single walled carbon nanotube via DTM.

1. Introduction

Nanoscale engineering materials have superior mechanical, electrical and thermal performances than the conventional structural materials. They have attracted great interest in modern science and technology after the invention of carbon nanotubes by (Iijima 1991). Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. Nanotubes have length-to-diameter ratio of up to 132,000,000:1 (Thostenson, Ren, and Chou 2001), significantly larger than any other material. The structure of a single-walled carbon nanotube can be understood by wrapping a one-atom-thick layer of graphite called graphene into a seamless cylinder (Fleck and Hutchinson 1997; Thostenson, Ren, and Chou 2001; Qian et al. 2002). Single-walled nanotubes are the perfect candidate for miniaturizing electronics rather than the micro-electromechanical scale currently used in electronics (Wang and Varadan 2005). Carbon nanotubes are the strongest and stiffest materials in terms of tensile strength and Elastic Modulus respectively (Yang et al. 2002). Since carbon nanotubes have a low density for a solid of 1.3 to 1.4 g*[cm.sup.-3] (Pradhan and Reddy 2011). This strength results in the formation of covalent [sp.sup.2] bonds between the individual carbon atoms.

(Eringen 1983) proposed the nonlocal continuum theories for the analysis of small-sized structures. In nonlocal elasticity theory, the small-scale effects are captured by assuming that the stress at a point is a function not only of the strain at that point but also a function of the strains at all other points of the domain. The mechanical analyses of nanostructures, theoretical and mathematical modelling become an important issue when nano-engineering comes in the picture. This is due to the scale effect of the nanostructures. The influences of long range inter-atomic and intermolecular cohesive forces on the static and dynamic properties become significant as the length scales are reduced and cannot be neglected. The classical theory of elasticity being the long-wave limit of the atomic theory excludes these effects and thus this theory would fail to analyse the structure with small-scale effects accurately. Thereby size-dependent continuum-based methods (Fleck and Hutchinson 1997; Zhou and Li 2001; Yang et al. 2002) are becoming popular in modelling small-sized structures as it offers much faster and accurate solutions. (Sudak 2003) carried out buckling analysis of multi-walled carbon nanotubes. (Wang and Varadan 2005) analysed the small-scale effect of CNT and shell model. (Yakobson, Brabec, and Bernholc 1996) introduced an atomistic model for axially compressed single-walled carbon nanotube and compared it to a simple continuum shell model. (Sears and Batra 2006) proposed a comprehensive buckling analysis of single-walled and multi-walled carbon nanotubes by molecular mechanics simulations and continuum mechanics models. Reddy reported a suitable reference concerning nonlocal theories for bending, buckling and vibration analysis of nanobeams (Reddy 2007).

(Reddy et al. 2007) studied the free vibration analysis of fluid conveying single-walled carbon nanotube and showed the effect of the flowing fluid on the fundamental frequency of single-walled carbon nanotube. (Lee and Chang 2009) analysed the influences of the nonlocal effect, viscosity effect, aspect ratio effect and elastic modulus constant effect on the vibrational frequencies of fluid conveying single-walled carbon nanotube embedded in an elastic medium. (Chang 2012) analysed the thermal--mechanical vibration and instability of fluid conveying single-walled carbon nanotube embedded in an elastic medium using nonlocal elasticity theory. (Askari, Zhang, and Esmailzadeh 2013) studied the non-linear vibration instability of fluid conveying single-walled carbon nanotube using Eringen's nonlocal elasticity theory and Euler-Bernoulli beam model. They used Galerikin method to obtain the solution of the governing differential equations. (Hosseini et al. 2014) analysed the vibration and stability response of single-walled carbon nanotube conveying nano flow embedded in biological soft tissue. They showed the effect of nano- fluid flow on the fundamental frequency of single-walled carbon nanotube. (Zhen and Fang 2015) analysed the non-linear vibration of single-walled carbon nanotube conveying fluid utilizing a multidimensional Lindstedt--Poincare method. They showed the effect of the nonlocal parameter, fluid flow velocity on fundamental frequency of single-walled carbon nanotube. (Kumar and Deol 2016) presented in detail the buckling analysis of single-walled carbon nanotube using nonlocal theory and DTM. (Ravi Kumar and Palaksha Reddy 2017) carried out vibration analysis on double-walled carbon nanotubes embedded in an elastic medium using differential transform method (DTM).

In the present work, the vibration analysis of fluid conveying single-walled carbon nanotube (SWCNT) embedded in an elastic medium by considering the nonlocal effect, thermal effect and elastic modulus constant effect is studied using the differential transformation method. (Zhou 1986) proposed differential transformation method to solve both linear and non-linear initial value problems in electric circuit analysis. Later (Chen and Shing Huei 1999) applied this method to eigenvalue problems. (Arikoglu and Ozkol 2005) applied differential transformation method to solve the intergro--differential equation.

2. Formulation

2.1. Non-Local Formulation of SWCNT

(Eringen 1983) first introduced Nonlocal elasticity theory; the simplified constitutive relation in a differential form is given as follows:

(1 - [([e.sub.0]a).sup.2][[nabla].sup.2])[sigma] = [tau] (1)

where, [tau] is the classical, macroscopic stress tensor at a point, a is an internal characteristic length (e.g. lattice parameter, granular size, length of C-C bonds), [e.sub.0]a is a material constant, [sigma] is nonlocal stress tensor and [[nabla].sup.2] is the Laplacian operator.

For a beam type structure, in the thickness direction the nonlocal behaviour can be neglected. In the classic elastic continuum theory, the stress field at any point depends on the strain field at the same point. However, according to Eringen's nonlocal elasticity theory (Eringen 1983), the stress field at a point is dependent on the strains at all other points in the body. Thus, for a homogeneous isotropic Euler-Bernoulli beam, the nonlocal constitutive relation takes the following form: (Chang 2012).

[[sigma].sub.xx] - [([e.sub.0]a).sup.2].[[[partial derivative].sup.2][[sigma].sub.xx]/[partial derivative][x.sup.2]] = E[[epsilon].sub.xx] (2)

where, E is the modulus of elasticity.

On the basis of the theory of thermal elasticity mechanics, the axial force [N.sub.t1] can be written as (Chang 2012),

[mathematical expression not reproducible] (3)

where, [[alpha].sub.X] denotes the coefficient of thermal expansion in the direction of x-axis and v is the Poisson's ratio and T denotes the change in temperature. The carbon nanotube is arranged in a manner given below in Figure 1; it is embedded in the elastic medium. U is the fluid velocity, K is the elastic constant for Winkler foundation and L is the length of the nanotube.

If we consider the thermal effect, the differential equation of motion related to shear force of fluid conveying single-walled carbon nanotube is given by (Chang 2012),

[mathematical expression not reproducible] (4)

where Q is the shear force, w is the transverse deflection, m is the mass of nanotube per unit length and M is the mass of fluid per unit length of the beam. Q satisfies the condition for equilibrium of Euler's beam.

Q = [[partial derivative][M.sub.1]/[partial derivative]X] (5)

[M.sub.1] - [([e.sub.0]a).sup.2][[[partial derivative].sup.2][M.sub.1]/[partial derivative][x.sup.2]] = -EI[[[partial derivative].sup.2]w/[partial derivative][x.sup.2]] (6)

where [M.sub.1] is the nonlocal bending moment and Equation (6) represents the nonlocal bending equation of nanotube. By combining Equations (4)-(6), we have

[mathematical expression not reproducible] (7)

Finally, by above equation and by Equation (6), the governing differential equation is given as

[mathematical expression not reproducible] (8)

In this study, the Euler-Bernoulli beam model using stress gradient approach for the dynamic analysis of single-walled carbon nanotube with nonlocal effect is considered. Where, w = w(x, t) is the transverse beam deflection, x, t are the spatial coordinates, I is the moment of inertia of carbon nanotube and e0 is a constant appropriate to each material, a is an internal characteristic length. For the dynamic analysis, the Equation (8) can be non-dimensionalized using L (length of carbon nanotube) and by substituting, W = w/L and X = x/L, we have;

[mathematical expression not reproducible] (9)


A = [EI + [([e.sub.0]a).sup.2][N.sub.t1] - [([e.sub.0]a).sup.2]M[U.sup.2]];

B = 2[([e.sub.0]a).sup.2]MU;

C = [([e.sub.0]a).sup.2](M + m);

D = [M[U.sup.2] - [N.sub.t1] - [([e.sub.0]a).sup.2]K]

2.2. DTM formulation

The DTM is a semi-analytical method based on the Taylor series expansion. In this method, certain transformation rules are applied on the governing differential equations and the boundary conditions of the system and they are transformed into a set of algebraic equations in terms of the differential transforms of the original functions (Zhou 1986). These algebraic equations give a solution which gives the desired solution of the problem. The differential transformation of the kth derivative of the function u(x) is defined as follows

[mathematical expression not reproducible] (10)

And the differential inverse transformation of U(k) is expressed as

u(x) = [[infinity].summation over (k=0)] U(k)[(x - [x.sub.0]).sup.k] (11)

In real application function, u(x) is expressed as finite series and Equation (11) can be written as:

[mathematical expression not reproducible] (12)

with the use of certain transformation rules we convert the governing differential equation and associated boundary conditions into some algebraic equations; on solving them we can get the desired results. Following transformation tables is used for this purpose.

In order to derive DTM form of Equation (9), we referred Tables 1 and 2 and the following expression can be generated:

[mathematical expression not reproducible] (13)

2.3. Application of boundary conditions

2.3.1. Simply supported at both ends

The boundary conditions for the case of simply supported single-walled carbon nanotube at both the ends are defined as

W(0) = 0, W"(0) = 0, W(L) = 0, W"(L) = 0 (14)

By using Differential Transformation these can be written as:

W(0) = 0, W(2) = [MU[([e.sub.0]a).sup.2](i[omega])L.W(1)/EI + [([e.sub.0]a).sup.2][N.sub.t1]] (15)

[mathematical expression not reproducible] (16)


F = - EI - [([e.sub.0]a).sup.2][N.sub.t1] + [([e.sub.0]a).sup.2]M[U.sup.2]

W(k), up to n terms was calculated from the Equation (13), by assuming W(1) = [c.sub.1], W(3) = [c.sub.2] and it will be substituted in Equations (15) and (16) and by solving these equations for non-trivial solution, natural frequency ([omega]) of the carbon nanotube can be calculated. The accuracy of natural frequency increases with increase in the value of n (number of iterations) and saturates at a maximum n value, i.e. n = [N.sub.max].

2.3.2. Fixed at both ends

For the single-walled carbon nanotube supported by fixed at both the ends, the boundary conditions defined as

W(0) = 0, W'(0) = 0, W(L) = 0, W'(L) = 0 (17)

By using Differential Transformation these can be written as:

W(0) = 0, W(1) = 0 (18)

[mathematical expression not reproducible] (19)

By assuming W(2) = [c.sub.1], W(3) = [c.sub.2] the Equation (13) can be calculated up to n terms and a similar procedure is followed as that of simply supported boundary condition.

3. Results and discussion

3.1. Validation

In Figures 2-4 validation of the results has been carried out with the results reported in the available literature (Chang 2012). The outer radii [R.sub.out] = 3.5 nm and thickness of the nanotube h = 0.34 nm. The mass density of single-walled carbon nanotube is 2.3 g/[cm.sup.3] with Young's modulus E of 1 TPa, the density of water is 1 g/[cm.sup.3], aspect ratio L/[R.sub.out] = 100, nonlocal parameter [e.sub.0]a/L is taken from 0 to 0.05 and Winkler elastic constant K from 0 to 0.1 MPa.

In the present study, we consider two cases of temperature region, low and high temperature regions. The coefficient of thermal expansion are [[alpha].sub.x] = -1.6 x [10.sup.-6] and 1.1 x [10.sup.-6] [K.sup.-1] for low or room temperature and high temperature region respectively. The Poisson's ratio is considered to be as 0.3.

3.1.1. Fixed-Fixed boundary condition

Figure 2 is drawn between the natural frequency (y-axis) of the beam and fluid flow velocity (x-axis) entering through it in a high-temperature region. The change in temperature is taken as 25 K, the nonlocal parameter is 0.05 and Winkler elastic constant K is taken as 0 MPa.

We can observe that the points or data obtained here are very close to the available results of Chang (2012). The natural frequencies are obtained in Hz and flow velocities in m/s.

Above figure (Figure 3) is drawn in low temperature region by taking same boundary condition on the same axes as Figure 2. Here the change in temperature is taken as15 K, nonlocal parameter value and Winkler constant are same as 0.05 and 0 MPa, respectively.

The results through DTM are close to the available results (Chang 2012). In Figure 2, it shows that the natural frequency decreases with increase in flow velocity similarly as in Figure 3, the values of natural frequencies at different fluid velocities are much higher than the natural frequencies in the high-temperature region (Figure 2).

In Figure 4 the comparison has been carried out between the results through DTM and available results (Chang 2012) with a temperature change of 15 K, ea/L at 0.05 and K at 0.1 MPa in low-temperature region. In both the methods a very good agreement has been obtained. We observe that natural frequency increases with the increase of Winkler constant.

After the validation of the results, we have observed that the application of the differential transformation method (DTM) gives the values very close to the available results in the literature (Chang 2012).

3.2. New results and discussion

3.2.1. (Simply supported boundary condition) Effect of temperature. In Figure 5 by keeping the value of the nonlocal parameter and elastic force constant at 0.05 and 0 MPa, respectively and it is drawn in the low-temperature region, where the coefficient of thermal expansion has negative values. The natural frequency and critical flow velocity are also related to the temperature change. There is an increase in the temperature change (from 0 to 35 K) which tends to increase the natural frequencies of the single-walled carbon nanotube as well as critical flow velocity in the low-temperature region, when the flow velocity is lower than the critical flow velocity. Moreover, the natural frequency and critical flow velocity are 1.2 x [10.sup.8] Hz and 215 m/s which are much lower than the fixed-fixed boundary condition (Figure 2) at T = 15 K. Also in comparison with the Figure 3, we can observe that the natural frequency and critical flow velocity are much lower at T = 0 K.

Figure 6 is drawn with the value of the nonlocal parameter and elastic force constant as 0.05 and 0 MPa, respectively, in the high-temperature region, which means the coefficient of thermal expansion has positive values. When the flow velocity is lower than the critical flow velocity, the increase of the temperature change (from 0 to 35 K) tends to decrease the natural frequencies of the SWCNT (from 1 x [10.sup.8] to 0.4 x [10.sup.8] Hz) and critical flow velocity (from 180 to 70 m/s). Moreover, the natural frequency and critical flow velocity are much lesser when compared with Figure 2. Effect ofthe Nonlocalparameter. (Eringen 1983) nonlocal elasticity theory allows for the small-scale effect that becomes significant when dealing with micro and nanostructures. Figure 7 is drawn in a high-temperature region which represents the variation of the fundamental frequency of single-walled carbon nanotube with flow velocity for different values of [e.sub.0]a/L. We observe that the nonlocal parameter ([e.sub.0]/L) increases and natural frequency decreases at T = 35 K and K = 0 MPa. Also the nonlocal parameter ([e.sub.0]/L) increases from 0 to 0.1 at zero flow velocity, natural frequency decreases from 0.46 x [10.sup.8] to 0.35 x [10.sup.8] Hz and critical flow velocity reduces from 80 to 60 m/s. The variation of [e.sub.0]a/L from 0 to 0.05 has a significant effect on the natural frequency at zero flow velocity but does not have very significant effect on critical flow velocity. When dealing with [e.sub.0]a/L = 0 the nonlocal beam theory reduces to local beam theory. A curve is drawn at T = 25 K and with [e.sub.0]a/L = 0, which corresponds to a local beam theory. We clearly observe that in local beam theory the reduction of the natural frequencies and critical flow velocities happens when the nonlocal parameter is introduced.

We can observe that the nonlocal parameter ([e.sub.0]/L) increases and natural frequency decreases again. At zero flow velocity, nonlocal parameter ([e.sub.0]/L) is increased from 0 to 0.05, but does not increase the natural frequency so much. Similarly, the variation in [e.sub.0]a/L from 0 to 0.05 does not change the critical flow velocity, same as in the high-temperature region. If we compare above graph at [e.sub.0]a/L = 0.05 from fixed-fixed condition (Figure 4) we notice that the natural frequency of later case has a higher value (1.84 x [10.sup.8] Hz) than former (1 x [10.sup.8] Hz) and critical flow velocity has a higher value (210 m/s) than former (170 m/s). The reason is only that the fixed end gives an extra stiffness to the beam.

4. Conclusion

This paper presents an analytical model for studying the effects of temperature change, the nonlocal parameter on the natural frequency of single-walled carbon nanotube conveying fluid for various boundary conditions. Based on the theories of thermal elasticity mechanics and nonlocal elasticity, an elastic Bernoulli-Euler beam model is developed for thermal-mechanical vibration instability of a SWCNT embedded in an elastic medium.

Several results are presented on the variation of the fundamental natural frequency of single-walled carbon nanotube with flow velocity for various parameter values. It is found that at low or room temperature, the fundamental natural frequency and critical flow velocity for the single-walled carbon nanotube increase as the temperature change increases, on the contrary, while at high temperature the fundamental natural frequency and critical flow velocity for the single walled carbon nanotube decrease as the temperature change increases. From this analysis, it can be seen that the solutions obtained for a SWCNT can be helpful in investigating more complicated nanotube structures with nonlocal effects and design of nanotubes in a transverse magnetic field which may be used in space elevator applications in future.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

Ravi Kumar Bikramsingh has completed his BE (Aeronautical Engineering) degree from The aeronautical society of India, in the year 2009. He started his research as MTech (By Research) Scholar in Indian Institute of Technology, Kharagpur (INDIA) in 2011 and graduated his Master degree in 2013. He continues research in nanostructures for aerospace applications in SASTRA University. Presently, he is working as an assistant professor in Aerospace Engineering Department under the School of Mechanical Engineering, SASTRA University. His research interests include Nanomaterials, Composite structures, Aero-thermodynamics, Smart Materials, and Aerospace Propulsion.

Hariharan Sankara Subramanian is an automobile engineer by training with PhD from Indian Institute of Technology, Delhi. He is presently a member of School of Mechanical Engineering, SASTRA teaching faculty in the capacity of a senior assistant professor. His research interests include vehicle safety, vehicle dynamics, automobile engineering, injury biomechanics, and CAE-related optimization studies.


Ravi Kumar Bikramsingh

Hariharan Sankara Subramanian


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Ravi Kumar Bikramsingh [iD] and Hariharan Sankara Subramanian [iD]

School of Mechanical Engineering, SASTRA University, Thanjavur, India


Vibration; DTM; nonlocal; SWCNT; critical flow velocity


Received 10 May 2017

Accepted 6 July 2017
Table 1. Differential transformations for mathematical equations.

Original function                      Transformed function

y(x) = u(x) [+ or -] v(x)              Y(k) = U(k) [+ or -] V(k)
y(x) = [lambda]u(x)                    Y(k) = [lambda]U(k)
y([chi]) = [[d.sup.n]u(x)/[dx.sup.n]]  Y(k) = (k + 1)(k + 2)....(k + n)
                                       U(k + n)

Table 2. Differential transformations for the boundary conditions (BC).

At x= 0
Original BC       Transformed  Original

                  BC           BC
f(0) = 0          F(0) = 0     f(1) = 0
[df(0)/dx] = 0    F(1) = 0     [df(1)/dx] = 0
[[d.sup.2]f(0)    F(2) = 0     [[d.sup.2]f(1)/[dx.sup.2]] = 0
/[dx.sup.2]] = 0
[[d.sup.3]f(0)    F(3) = 0     [[d.sup.3]f(1)/[dx.sup.3]] = 0
/[dx.sup.3]] = 0

At x= 0           At x = 1
Original BC       Transformed BC

f(0) = 0          [[infinity].summation over (k = 0)] F(k)= 0
[df(0)/dx] = 0    [[infinity].summation over (k = 0)] k.F(k) = 0
[[d.sup.2]f(0)    [[infinity].summation over (k = 0)] k(k - 1)F(k) = 0
/[dx.sup.2]] = 0
[[d.sup.3]f(0)    [[infinity].summation over (k = 0)] k(k - 1)(k - 2)
/[dx.sup.3]] = 0  F(k) = 0
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Author:Bikramsingh, Ravi Kumar; Subramanian, Hariharan Sankara
Publication:Australian Journal of Mechanical Engineering
Date:Oct 1, 2018
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