Physics Letters A 381 (2017) 2898–2905

Contents lists available at ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

Effects of magnetic-fluid flow on structural instability of a carbon

nanotube conveying nanoflow under a longitudinal magnetic field

Moslem Sadeghi-Goughari ∗, Soo Jeon, Hyock-Ju Kwon

Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 May 2017Received in revised form 22 June 2017Accepted 30 June 2017Available online 5 July 2017Communicated by M. Wu

Keywords:Carbon nanotubesMagnetic-fluid flowSmall size effectsFluid structure interaction (FSI)Knudsen numberNonlocal parameter

In drug delivery systems, carbon nanotubes (CNTs) can be used to deliver anticancer drugs into target site to kill metastatic cancer cells under the magnetic field guidance. Deep understanding of dynamic behavior of CNTs in drug delivery systems may enable more efficient use of the drugs while reducing systemic side effects. In this paper, we study the effect of magnetic-fluid flow on the structural instability of a CNT conveying nanoflow under a longitudinal magnetic field. The Navier–Stokes equation of magnetic-fluid flow is coupled with Euler–Bernoulli beam theory for modeling fluid structure interaction (FSI). Size effects of the magnetic fluid and the CNT are addressed through small-scale parameters including the Knudsen number (Kn) and the nonlocal parameter. Results show the positive role of magnetic properties of fluid flow on the structural stability of CNT. Specifically, magnetic force applied to the fluid flow has an effect of decreasing the structural stiffness of system while increasing the critical flow velocity. Furthermore, we discover that the nanoscale effects of CNT and fluid flow tend to amplify the influence of magnetic field on the vibrational behavior of the system.

© 2017 Elsevier B.V. All rights reserved.

1. Introduction

Discovered by Iijima [1], carbon nanotubes (CNTs) exhibit unique chemical, electrical, thermal and magnetic properties [2–5]. They are expected to make an impact on all areas of nanotechnol-ogy including nanomechanics and nanomedicine. Because of their perfect hollow cylindrical geometry, CNTs hold substantial promise as nanopipes for conveying fluid and as nanocontainers for gas storage [6]. In the field of nanomedicine, CNTs are addressed as bio-sensors for detection and elimination of cancer cells and as pharmaceutical excipients for creating versatile drug delivery sys-tems [7].

Carbon nanotubes have potential for drug delivery applications to inject small-molecule drugs and some proteins [8,9] as well as genes [10,11]. Due to the strong optical absorbance of CNTs and their propensity for endocytosis, they have also been used as a means to destroy cancerous cells hyperthermically [12] and as car-riers of biomolecules for gene therapy and gene silencing [13]. There are mainly three approaches to the delivery of pharmaceuti-cally active components using CNTs [14]. The first approach uses a bundle of CNTs as a porous absorbent to entrap active components

* Corresponding author.E-mail addresses: m8sadegh@uwaterloo.ca (M. Sadeghi-Goughari),

soojeon@uwaterloo.ca (S. Jeon), hjkwon@uwaterloo.ca (H.-J. Kwon).

http://dx.doi.org/10.1016/j.physleta.2017.06.0540375-9601/© 2017 Elsevier B.V. All rights reserved.

within a CNT mesh. CNTs can be used as a porous nanoparticu-late absorbent in the controlled delivery of erythropoietin (EPO) in mice [15]. The second approach is through functional attachment of the compound to the exterior walls of the CNTs either by co-valent bonding to the CNT wall or by hydrophobic interaction of moieties with the CNT walls. The large outer surface of CNTs fa-cilitates their functionalization and thus the drug moieties can be attached to the exterior of CNTs for subsequent delivery into cells for more specific application of CNTs [14]. The third approach in-volves the use of the inner cavities of CNTs as nanochannels to deliver drug. With endohedral delivery, small-molecule drugs are encapsulated and transported through the inner cavities of CNTs. The ability of CNTs to serve as nanochannel makes them potential nanofluidic delivery devices for pharmaceuticals, and this raises the prospect of controlled nanofluidic delivery of drug [14].

Application of driving force such as magnetic field can promote high transduction efficiency of drug delivery systems and viabil-ity after transduction. This approach not only delivers the drugs in a relatively short time, but, by varying the intensity of mag-netic field, it can also be used to tune the efficiency of delivery [11,16]. Magnetic field may help the drug delivery in different manners. It may provide a driving force to align and position CNTs or can be utilized to deliver the drug trapped inside CNT channels by guiding nanoflow process [16]. Controlled release of drugs from functionalized CNTs under magnetic guidance, espe-cially magnetic nanoparticles (NPs), is drawing attention because

M. Sadeghi-Goughari et al. / Physics Letters A 381 (2017) 2898–2905 2899

of the feasibility in cancer therapy [14]. Cai et al. [11] reported a highly efficient molecular delivery method according to the pene-tration of nickel-embedded nanotubes into cell membranes using controlled magnetic field. Vertically aligned nanotubes grown by plasma-enhanced chemical vapor deposition (CVD) with ferromag-netic nickel particles embedded at the tips were utilized as carriers for enhanced green fluorescent protein (EGFP) coding plasmids and were speared into targeted cells under the influence of a magnetic force. Leonhardt et al. [16] proposed incorporating a ferromagnetic material, together with a therapeutic agent into CNTs. This method enables the manipulation of CNTs, using an externally-controlled magnetic field, to destroy cancerous cells hyperthermically. From these studies, it is clear that magnetic field is an effective means to realize the nanofluidic delivery through CNTs. In that sense, CNT based drug delivery technologies require full understanding of the dynamics of nanofluidic systems comprising CNTs conveying nanoflow [17].

Vibration and instability analysis of nanotubes conveying fluid has formed a significant area of nanofluidic dynamic system re-search in computational nanotechnology and nanoscience. Yoon et al. [18] investigated the effect of internal moving fluid on flow-induced structural instability of CNTs. They revealed that the in-ternal moving fluid could substantially change the vibrational fre-quencies of nanotubes especially for suspended, longer, and larger-innermost-radius ones at higher flow velocity. Wang et al. [6]considered a multi-elastic beam model to investigate the natural vibration and buckling instability of double-wall CNTs (DWCNTs) conveying fluid. They demonstrated that the spring constant of sur-rounding elastic medium and slenderness ratios of the tubes might have a significant effect on the dynamics of system. Wang and Ni [19] criticized the study performed by Khosravian and Rafii-Tabar [20] and showed that the effects of viscosity of fluid flow on the vibrational characteristics of CNTs were waived.

At nanoscales, the small size effects associated with nanostruc-tures and fluid have a crucial role to the stability behavior of the nanostructure coupled with fluid flow. In a nanoscale fluid structure interaction (FSI) problem, the theory of no-slip bound-ary conditions between the fluid flow and nanotube walls is in-valid [21]. Rashidi et al. [22] investigated the buckling instability of nanotube conveying fluid by considering the small scale effects on the flow field. They devised a velocity correction factor (VCF) to modify the FSI governing equations and demonstrated that the small size effects on flow field might change the FSI results dras-tically. Sadeghi-Goughari and Hosseini [23] developed a nanoscale FSI model by considering the non-uniformity of the flow velocity distribution due to the viscosity of fluid. They formulated the small size effects of flow field and the non-uniformity of flow velocity through Knudsen number (Kn), as a discriminant parameter, and found that ignoring the non-uniformity of the flow velocity may generate erroneous results. In addition, Jannesari et al. [24], Hos-seini et al. [25] and Askari and Esmailzadeh [26] reported various issues of the vibration properties of a nanotube conveying fluid by considering small size effects of fluid flow.

Analytical and numerical studies have been conducted by re-searchers to take account of the nonlocal scale effect of nanos-tructures on FSI problems. As one of the first studies, Lee and Chang [27] used the nonlocal elasticity theory to investigate the influence of flow velocity on the vibration frequency of the fluid-conveying single-walled CNT (SWCNT). They indicated that the frequency and mode shape of SWCNT could be significantly influ-enced by the nonlocal parameter. Zhen and Fang [28] studied the thermal and nonlocal effects on the vibration and buckling insta-bility of a SWCNT conveying fluid and showed that thermal effect reduced the influence of nonlocal parameter. Mirramezani and Mir-damadi [29] investigated the small size effects of both fluid flow and elastic structure on the vibration and instability of a nanotube

conveying fluid by using both Kn and nonlocal continuum theory. They indicated that Kn had more effect than the nonlocal parame-ter on the reduction of critical velocities of a gas nanoflow.

The magnetic properties of CNTs in a magnetic field have been of interest to many researchers due to their applications in nanoscience such as NEMS (nano-electro-mechanical-systems), MEMS (micro-electro-mechanical-systems), nanosensors, spintron-ics and nanocomposites [30,31]. CNTs exhibit different mechanical characteristics for the direction and strength of magnetic field. In practice, the dynamic response of CNTs to both transverse and longitudinal magnetic fields may be important and have been of interest to some researchers. Li et al. [32] investigated the effect of transverse magnetic fields on dynamic characteristics of multi-walled CNTs (MWCNTs) and revealed that the transverse magnetic field exerted on MWCNTs had an effect of decreasing the lowest frequency of the MWCNTs nonlinearly while keeping the highest frequency unchanged. Murmu et al. [31] reported an analytical ap-proach to investigate the influence of a longitudinal magnetic field on the vibration of a DWCNT. They demonstrated that the longitu-dinal magnetic field might increase the natural frequencies of the DWCNT. Ghorbanpour Arani et al. [33] investigated nonlocal wave propagation of interactional behavior between SWCNTs and vis-cous fluid under a longitudinal magnetic field and showed that the magnetic field might increase the wave velocity for different values of nonlocality. Hosseini and Sadeghi-Goughari [30] used a nonlo-cal beam model to study the influence of a longitudinal magnetic field on the vibration of a SWCNT conveying fluid and concluded that the longitudinal magnetic field could increase both the natural frequencies of the SWCNT and the critical flow velocities.

Although the literature is replete with the studies regarding nanotubes conveying fluid, to the authors’ knowledge, none of them considered an analytical approach to take into account for the magnetic and small size effect of fluid flow on the structural stability of magnetically sensitive CNT conveying fluid under a lon-gitudinal magnetic field. In a drug delivery system-based CNTs, the magnetic field needs to be applied in the direction of fluid flow to efficiently deliver drug to the target. Therefore, in the present study, we focused on the influence of internal moving magnetic fluid on vibration of a magnetically sensitive CNT subjected to a longitudinal magnetic field. The nonlocal beam theory and the Navier–Stokes equation are used to couple the CNT with the mag-netic fluid and to derive the FSI governing equations. Small size effects of magnetic fluid and CNT are addressed through small scale parameters including Knudsen number and nonlocal param-eter. Then, the extended Galerkin’s method is applied to solve the FSI governing equations of motion. Finally, FSI results are presented in graphical form to show the coupled effects of magnetic field and small scale parameters on the dynamics of the FSI system.

2. FSI governing equation

Consider Newtonian, laminar, incompressible, viscous and mag-netic fluid passing through a simply supported CNT subjected to a longitudinal magnetic field (Fig. 1). Nanotubes can be modeled as single Euler–Bernoulli beams with large aspect ratio [19]. The flex-ural vibration of a nanotube subjected to an external force Fext , acting on the nanotube in the direction of the flexural displace-ment, can be expressed as [34]:

mc∂2W

∂T 2= ∂ Q

∂ X+ Fext, (1)

where mc is the CNT mass per unit length, W = W (X, T ) is the flexural displacement of the CNT at the axial coordinate X and time T , and Q is the resultant shear force acting on the wall cross section, which is equal to the first derivative of the resultant bend-ing moment with respect to X (Q = ∂M ).

∂ X

2900 M. Sadeghi-Goughari et al. / Physics Letters A 381 (2017) 2898–2905

Fig. 1. Configuration of CNT conveying magnetic fluid subjected to a longitudinal magnetic field.

The external force Fext is a combination of several factors acting on the nanotube including the forces per unit length induced by a magnetic-fluid flow (F w ) and by an external longitudinal magnetic field (F B ).

2.1. Applied force to CNT due to magnetic field

Based on the Maxwell relation, the magnetic force along the direction of the flexural displacement acting on the CNT due to a longitudinal magnetic field B = Boex , where ex denotes the unit vector along the X direction, can be stated as [30]:

F B = 1

ηB2

o A∂2W

∂ X2, (2)

where η is the magnetic field permeability, A is the cross sectional area of the nanotube and Bo denotes the magnetic intensity of the longitudinal magnetic field, measured in teslas.

2.2. Applied force to CNT due to magnetic-fluid flow

To derive the effects of a magnetic-fluid flow on the vibration and instability of CNT, the momentum-balance equation for the fluid motion is developed. The well-known Navier–Stokes equa-tions for a fluid flow in a magnetic field B can be stated as follows [35]:

ρdV

dT= −∇ P + μ∇2V + J × B, (3)

where ρ is the fluid density, μ is the dynamic viscosity of fluid, P is the fluid pressure and V = V xex + Vrer is the fluid velocity in a cylindrical coordinate system with components in X and rdirections, respectively, where r is measured with respect to the center of the nanotube. The last term in the momentum equation is called Lorentz force that applied to magnetic fluid from exter-nal magnetic field in which J is the electric current density vector given by J = σ(V × B) with σ denoting the electrical conductivity of the magnetic fluid.

Based on the formulation derived in Appendix A, the term of force applied to the nanotube due to the magnetic-fluid flow can be stated as follows:

F w = −m f

(∂2W

∂T 2+ 2V x

∂2W

∂ X∂T+ V

2x∂2W

∂ X2

)

− σ AB2o

(∂W

∂T+ V x

∂W

∂ X

), (4)

where m f is the fluid mass per unit length, V x is the average flow velocity in the flow direction and A is the cross sectional area of the internal fluid.

2.3. Small size effects of fluid flow

The flow behavior at nanoscale is substantially different from those of large scales [21]. The small size effects on fluid flow and the low Reynolds number are the most significant differences among all [36]. In the study of fluid flow at nanoscale, the Knud-sen number (Kn) i.e., the ratio of the mean free path of the fluid molecules to the characteristic length of the flow, can be used as a

discriminating factor to consider the small size effects on the flow field. According to Kn, the fluid flow can be classified in four flow regimes [21]: continuum flow regime (0 < Kn < 10−3), slip flow regime (10−3 < Kn < 10−1), transition flow regime (10−1 < Kn <

10), and free molecular flow regime (Kn > 10).Using the velocity correction factor as derived in Appendix B

and considering the slip boundary conditions between the fluid flow and the nanotube’s wall, the applied force to the nanotube due to the magnetic-fluid flow is reformulated as follows:

F w = −m f

(∂2W

∂T 2+ 2(VCF)U

∂2W

∂ X∂T+ (VCF)2U 2 ∂2W

∂ X2

)

− σ AB2o

(∂W

∂T+ (VCF)U

∂W

∂ X

), (5)

where VCF is the velocity correction factor and U is the average flow velocity through nanotube without slip boundary conditions.

2.4. Small scale effects of CNT

At nanoscales, the material properties are size-dependent and the small scale effects of nanostructure have a crucial role in the dynamics of nanosystems. Therefore, it is necessary to use nonlo-cal theories incorporating small scale effects [29]. To this end, the nonlocal elasticity theory is employed to identify the relationship between the bending moment resultant (M) and the flexural dis-placement of the Euler–Bernoulli beam theory [37]:

M − (eoa)2 ∂2M

∂ X2= −EI

∂2W

∂ X2(6)

where eo is the material constant, a is the internal characteristic length and EI is the flexural rigidity of the nanotube.

By substituting the terms of external force induced by the magnetic-fluid flow (Eq. (5)) and the external magnetic field (Eq. (2)) into Eq. (1), and applying the nonlocal elasticity formula-tion (Eq. (6)), the FSI equation of system can be written as follows:

EI∂4W

∂ X4+ m f (VCF)2U 2 ∂2W

∂ X2+ 2m f (VCF)U

∂2W

∂ X∂T

+ (m f + mc)∂2W

∂T 2− B2

o A

η

∂2W

∂ X2+ σ AB2

o∂W

∂T

+ σ AB2o(VCF)U

∂W

∂ X− (eoa)2

(m f (VCF)2U 2 ∂4W

∂ X4

+ 2m f (VCF)U∂4W

∂ X3∂T+ (m f + mc)

∂4W

∂ X2∂T 2− B2

o A

η

∂4W

∂ X4

+ σ AB2o

∂3W

∂ X2∂T+ σ AB2

o(VCF)U∂3W

∂ X3

)= 0. (7)

Then, the non-dimensional form of FSI equation can be derived as:

∂4 w

∂x4+ (VCF)2u2 ∂2 w

∂x2+ 2(VCF)u

√β

∂2 w

∂x∂t+ ∂2 w

∂t2− ϕ1

∂2 w

∂x2

+ ϕ2∂ w + ϕ3(VCF)u

∂ w − τ 2(

(VCF)2u2 ∂4 w4

∂t ∂ X ∂x

M. Sadeghi-Goughari et al. / Physics Letters A 381 (2017) 2898–2905 2901

+ 2(VCF)u√

β∂4 w

∂x3∂t+ ∂4 w

∂x2∂t2− ϕ1

∂4 w

∂x4+ ϕ2

∂3 w

∂x2∂t

+ ϕ3(VCF)u∂3 w

∂x3

)= 0, (8)

where we used the following dimensionless variables and parame-ters:

x = X

L, w = W

L, t = T

L2

(EI

m f + mc

) 12

,

β = m f

m f + mc, u = U L

(m f

EI

) 12

, τ = eoa

L,

ϕ1 = B2o A

η

L2

EI, ϕ2 = σ AB2

oL2

(EI(m f + mc))12

,

ϕ3 = σ AB2o

L2

(EIm f )12

, (9)

where L denoting the length of the nanotube. Equations (8) and (9) present the dynamical behavior of system as a function of di-mensionless parameters β , τ , Kn, ϕ1, ϕ2 and ϕ3. Understanding the connection between these dimensionless parameters and pa-rameters in the real system is of practical importance. Here, β is the mass ratio which represents the role of system’s mass contain-ing fluid and nanotube on the vibration of system. The small size effects of CNT are considered through the nonlocal parameter, τ , defined based on the internal (lattice parameter, C–C bond length, granular distance, etc.) and external (nanotube length) character-istic lengths of the nanotube. Kn, i.e., the ratio of the mean free path of the fluid molecules to the characteristic length of the flow, represents the small size effect of fluid flow inside CNT. At nanoscale, Kn may be larger than its value for continuum regime (Kn > 10−3). Consequently, the vibration of system is subject to the small size effects of fluid flow including the mean free path of the fluid molecules. Given that the mean free path of fluid flow is related to the temperature of the fluid, the molecular mean free path and corresponding Kn of fluid can be changed by varying the fluid temperature [22]. In this paper, the effect of magnetic proper-ties of fluid flow and CNT on the vibration of system is investigated through the dimensionless parameters ϕ1, ϕ2 and ϕ3. Here, ϕ2 and ϕ3 represent the magnetic force applied to the CNT structure due to the magnetic properties of fluid flow while ϕ1 is specifically re-lated to the force applied to the CNT structure.

For the simply supported nanotube, the kinematic and natural boundary conditions are as follows:

w = ∂2 w

∂x2= 0 at x = 0,1. (10)

3. Galerkin’s method and solving FSI equation

The main aim of the FSI analysis is to compute the complex-valued eigenvalues of the nanotube, which allow us to determine the instability conditions of the system. To this end, we employ the extended Galerkin method to approximate the partial differential equations of motion, Eq. (8) and associated boundary conditions, Eq. (10), by a finite dimensional system of coupled ordinary dif-ferential equations. Accordingly, the flexural displacement of the nanotube can be represented in a series form as [38]:

w(x, t) =N∑

n=1

qn(t)∅n(x), (11)

where N is the number of modes, qn(t) represents nth modal coordinates and ∅n(x) denotes the basis function for the nth eigen-mode.

Fig. 2. The imaginary part of the first mode eigenvalue of a simply supported CNT when τ = Kn = Bo = 0 & β = 0.1 and when τ = 0.2, Kn = 0.001, Bo = 0 for acetone flow.

For a simply supported nanotube, φn(x) can be written as [39]:

∅n(x) = sin(nπx) n = 1,2, . . . ., N, (12)

which are orthogonal to each other and satisfy the boundary con-ditions for a simply supported nanotube.

Applying the extended Galerkin procedure, a set of coupled ordinary-differential equations is obtained as:

[M]{q(t)} + [C]{q(t)

} + [K]{q(t)} = 0, (13)

where q(t) is the overall vector of modal coordinates and the dot notation refers to the time derivative. Also, [M], [C] and [K] are mass, damping and stiffness matrices of the vibrational system, re-spectively. Therefore, the dimensionless eigenvalues of nanotube can be calculated numerically from Eq. (13) as a function of Kn, τ , Bo and other parameters of the system.

4. Results and discussion

In this section, the dynamics of a system including a CNT con-veying magnetic fluid under a longitudinal magnetic field is dis-cussed. To this end, we set the nanotube’s aspect ratio ( L

2Rout),

Young’s modulus (E), outer radius (Rout) and thickness (t) to be 50, 3.4 TPa, 3 nm and 0.1 nm, respectively [40]. The magnetic per-meability of CNT is considered to be 4π × 10−7 [30]. To make the results and magnetic effects of fluid more distinguishing, we considered a liquid metal, mercury, as the fluid medium flowing through the nanotube. CNTs can transport liquid mercury when a pressure is applied on the liquid [41]. The structural instabil-ity of the system is significantly related to the dynamical behavior of fluid flow. We assume that the mercury is Newtonian fluid and consider the fluid density and electrical conductivity to be 13.56 gr

cm3 and 1.02 × 106 Sm , respectively [42].

4.1. Validation of FSI results

To describe the vibrational behavior of the system, the first mode eigenvalue Ω is computed. Generally, Ω is a complex num-ber whose imaginary and real parts can be used to characterize the dynamics and instability of the system [38].

First, the FSI computational approach is validated by the non-magnetic results published by Ni et al. [43] when the small scale effects are ignored, and by those of Mirramazani and Mirdamadi [29] when the small scale parameters are considered. It is clear from the plots shown in Fig. 2 that our computational results co-incide with those from Refs. [29,43].

Based on the vibrational behavior of the system, as the flow velocity increases, the imaginary part of the first mode eigenvalue gets reduced and the CNT becomes more flexible. When the flow velocity reaches a certain value, the imaginary part becomes zero which induces the instability of the system.

2902 M. Sadeghi-Goughari et al. / Physics Letters A 381 (2017) 2898–2905

Table 1Validation of the dimensionless critical continuum flow velocity with considering the small scale effects.

ucr

[τ ,Kn] [0,0] [0,0.0005] [0,0.001] [0.1,0] [0.2,0] [0.2,0.001]Present 3.142 3.13 3.118 2.998 2.661 2.64Ref. [29] 3.142 3.13 3.118 2.998 2.661 2.64

Fig. 3. (a) Imaginary and (b) real parts of the first mode dimensionless eigenvalue for different values of magnetic field (τ = 0.2, Kn = 0.001).

To investigate the divergent instability of the system, the critical flow velocities, i.e., the flow velocities at which the imaginary and real parts of eigenvalue reach to zero at the first mode, are com-puted and compared with those published in Ref. [29] in Table 1. We can see that our computational results coincide with the exist-ing results from Ref. [29]. Here, the non-dimensional critical flow velocity for τ , Kn = 0 is 3.142 ≈ π which is theoretically predicted for simply supported nanotube conveying fluid [38].

4.2. Effects of magnetic field

In this subsection, the effects of longitudinal magnetic field on the fundamental eigenvalue and critical flow velocity of the system are investigated. According to the strength of the magnetic field, the imaginary and real parts of the fundamental eigenvalue of the nanotube are plotted as a function of the dimensionless fluid flow velocity for τ = 0.2, Kn = 0.001 in Fig. 3. The figure shows that the magnetic field effects tend to increase the bending stiffness of the nanotube as well as the critical flow velocity of the system. For example, in the absence of a magnetic field, the system loses its stability at the dimensionless velocity equal to 2.640 while in the presents of magnetic field with the strength of Bo = 7 T, the divergent instability may happen at a higher critical flow velocity ucr = 3.220.

According to Eqs. (8) and (9), the magnetic effects occur through three additional terms which are ϕ1, ϕ2 and ϕ3. As we can see from Eq. (9), ϕ1 is specifically related to the magnetic force applied to the CNT structure while ϕ2 and ϕ3 represent the terms of force applied to the magnetic fluid. In order to understand the role of magnetic-fluid flow (ϕ2 and ϕ3) on the structural stability

Fig. 4. Considering magnetic-fluid effect in (a) Imaginary and (b) real parts of the first mode dimensionless eigenvalues for high magnetic field (Bo = 3400 T) and for (τ = 0.2, Kn = 0.001).

Table 2Critical flow velocity versus the intensity of longitudinal magnetic field for consid-ering the magnetic effects of the fluid flow (τ = 0.2, Kn = 0.001).

Bo(T ) 0 500 1000 2000 2700 3400ucr 2.640 2.6404 2.6464 2.6737 2.7566 3.0355

of the system, we investigated the situation where the magnetic term of CNT is considered to be zero (ϕ1 = 0). Then, the imaginary and real parts of dimensionless eigenvalue with and without con-sidering magnetic properties of fluid are computed and presented in Fig. 4. Fig. 4(a) shows that the magnetic force on the fluid flow has an effect of decreasing the stiffness of the system, which is op-posite to the trend in Fig. 3(a). However, as we can see in Fig. 4(b), the real part of the eigenvalue significantly decreases from 0 to around −4.35 in the under-damped regime (i.e. the range of flow velocity where the imaginary value of the eigenvalue is non-zero). As a result, the critical flow velocity increases from 2.640 to 3.0355 despite the decrease in the bending stiffness.

It should be noted that, the results of Fig. 4 are computed with a very high magnetic intensity (Bo = 3400 T) to see the effect more clearly. Table 2 lists the critical flow velocity as a function of the intensity of longitudinal magnetic field and shows that the magnetic properties of fluid have negligible effect on the dynamics of system at low magnetic field (e.g. for Bo < 100 T), and it gets more dominant at very high magnetic field. Therefore, at a low magnetic field, the magnetic effects on fluid flows passing through nanotubes may be ignored (i.e. the terms associated with ϕ2 and ϕ3 can be ignored) in the analysis of vibrational stability of nan-otubes conveying fluid flow.

4.3. Small scale effects of fluid and nanotube

In this subsection, the small scale effects of fluid and CNT on the fundamental eigenvalues and critical flow velocities of the sys-tem are investigated. Fig. 5 shows the deterioration of the imag-inary part of the first mode eigenvalue against the dimensionless flow velocity for a nanotube conveying fluid with different Knud-

M. Sadeghi-Goughari et al. / Physics Letters A 381 (2017) 2898–2905 2903

Fig. 5. Couple effect of magnetic field and slip boundary conditions of fluid flow on the imaginary part of the first mode dimensionless eigenvalue of nanotube (τ =0.1).

Fig. 6. Couple effect of magnetic field and small scale of nanotube on the imaginary part of the first mode dimensionless eigenvalues of nanotube (Kn = 0.001).

sen numbers. Similarly, Fig. 6 shows the role of nonlocal parameter on the dynamics and instability of the nanotube conveying mag-netic fluid. According to the Figs. 5 and 6, the small scale effects of system, i.e., the nonlocal parameter and Knudsen number, tend to decrease the critical flow velocity as well as the bending stiffness of the nanotube conveying magnetic fluid.

It can be seen from Figs. 5 and 6 that the magnetic field may affect the role of nonlocal parameter and Knudsen number on the vibrations of the nanotube. The difference of imaginary parts of the first mode dimensionless eigenvalue for different values of τ , Kn and Bo is used as a discriminating factor to identify the role of each parameter (Table 3). The comparison between the column 2 and 3 shows that the presence of magnetic field tends to reduce the influence of the nonlocal parameter, τ , on the vibrational be-havior of the system. Similarly, if we compare the column 6 with 7, we can see that the magnetic field diminishes the effect of the

Knudsen number. On the other hand, if we compare the column 4 with 5, we can see that the small scale effects of CNT (i.e. τ ) tend to amplify the effect of the magnetic field on the dynamics of sys-tem. The same thing happens for the Knudsen number as we can see from the comparison between the column 8 and 9.

5. Conclusion

In this study, the vibrational behavior of a CNT conveying mag-netic fluid subjected to a longitudinal magnetic field was investi-gated. The influence of the magnetic fluid and the small size effects of fluid flow and CNT on the structural instability of CNT were considered. It was demonstrated that the magnetic field had a sig-nificant effect on the vibration characteristics of a CNT conveying magnetic fluid. Specifically, the study showed that the magnetic field effects led to increasing the bending stiffness of the nanotube as well as the critical flow velocity of the system, which makes the system more stable.

Additionally, we discovered that the magnetic field effect on the fluid flow tended to decrease the stiffness of the system while increase the critical flow velocity, thus making the system more stable. This behavior is noticeable for high magnetic intensity so it may be ignored at a low intensity magnetic field.

Finally, our results suggested that the stability of a nanotube conveying fluid was highly sensitive to the small size effects for both fluid flow and CNT. The small scale effects of the system led to the decrease of the critical flow velocity as well as the bend-ing stiffness of the nanotube conveying magnetic fluid. Also, we showed that the magnetic field affected the impact of small scale parameters on the vibrations of the CNT. The presence of mag-netic field led to the reduction of the influence of the nonlocal parameters and Knudsen number on the vibrational behavior of the system.

Appendix A

In order to obtain the applied force to the CNT due to the fluid flow, a number of simplifications are made to the fluid flow sys-tem:

(i) Fluid flow is fully developed in the flow direction, so V =V x(r)ex + Vr(X, T )er

(ii) For the fluid flow inside the nanotube with high aspect ratio, the slender body theory is applied to determine the velocity component in the vertical direction of flow [19]:

Vr(X, T ) = dW

dT, (A.1)

d = ∂ + V x∂

. (A.2)

dT ∂T ∂ X

Table 3The coupled effects of magnetic field with small scale parameters, i.e. τ and Kn.

u Kn = 0.001 τ = 0.1

Im(Ω)[τ = 0]−Im(Ω)[τ = 0.1]

Im(Ω)[Bo = 14]−Im(Ω)[Bo = 0]

Im(Ω)[Kn = 0]−Im(Ω)[Kn = 0.01]

Im(Ω)[Bo = 14]−Im(Ω)[Bo = 0]

Bo = 0 Bo = 14 τ = 0 τ = 0.1 Bo = 0 Bo = 14 Kn = 0 Kn = 0.1

0 0.4537 0.2890 5.4154 5.5801 0 0 5.5801 5.58010.5 0.4681 0.2984 5.4560 5.6257 0.0266 0.0192 5.6250 5.63241 0.5116 0.3267 5.5839 5.7688 0.1092 0.0771 5.7658 5.79791.5 0.5854 0.3732 5.8207 6.0329 0.2589 0.1752 6.0253 6.10902 0.6966 0.4368 6.2194 6.4793 0.5116 0.3161 6.4619 6.65752.5 0.8879 0.5156 6.9286 7.3010 1.0556 0.5055 7.2569 7.80693 – 0.6070 – – – 0.7563 – –3.5 – 0.7073 – – – 1.1028 – –4 – 0.81559 – – – 1.6824 – –4.5 – 0.99712 – – – – – –

2904 M. Sadeghi-Goughari et al. / Physics Letters A 381 (2017) 2898–2905

The force applied to the CNT induced by magnetic-fluid flow is directly related to the fluid pressure gradient in the direction of flexural displacement of nanotube. The Navier–Stokes relation of the flow in r-direction is developed as follows:

ρdVr

dT= −∂ P

∂r+ μ

∂2 Vr

∂ X2− σ B2

o Vr . (A.3)

By substituting Eq. (A.1) into Eq. (A.3) and using Eq. (A.2), the fluid pressure gradient in r-direction is obtained as:

∂ P

∂r= −ρ

(∂2W

∂T 2+ 2V x

∂2W

∂ X∂T+ V

2x∂2W

∂ X2

)

+ μ

(∂3W

∂ X2∂T+ V x

∂3W

∂ X3

)− σ B2

o

(∂W

∂T+ V x

∂W

∂ X

).

(A.4)

While the fluid viscosity has a crucial role in determining the axial fluid velocity profile inside the nanotube, the effects of viscosity terms in the applied force to the nanotube due to the fluid flow on the vibrational equation of nanotubes can be ignored according to Wang and Ni [19]. The term of force applied to the nanotube due to the magnetic-fluid flow can be stated as follows:

F w = −m f

(∂2W

∂T 2+ 2V x

∂2W

∂ X∂T+ V

2x∂2W

∂ X2

)

− σ AB2o

(∂W

∂T+ V x

∂W

∂ X

). (A.5)

Appendix B

According to Rashidi et al. [22], an average velocity correction factor (VCF) can be used to consider the small size effect on fluid flow.

VCF = V x

U= (1 + aKn)

(4

(2 − σv

σv

)(Kn

1 + Kn

)+ 1

), (B.1)

where V x and U are average flow velocities through nanotube with and without slip boundary conditions, respectively. The value, σv =0.7, is tangential momentum accommodation coefficient and a is a function of Kn and is defined as [22]:

a = ao2

π

[tan−1(a1KnB)]

(B.2)

where a1 and B are empirical parameters and are equal to 4 and 0.4, respectively and ao is given by [22]:

ao = 64

3π(1 − 4b )

. (B.3)

For the second-order term of slip boundary conditions, b = 1.By applying the velocity correction factor, the applied force to

the nanotube due to the magnetic-fluid flow is reformulated as follows:

F w = −m f

(∂2W

∂T 2+ 2(VCF)U

∂2W

∂ X∂T+ (VCF)2U 2 ∂2W

∂ X2

)

− σ AB2o

(∂W

∂T+ (VCF)U

∂W

∂ X

). (B.4)

References

[1] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (6348) (1991) 56–58.

[2] T.W. Ebbesen, Carbon Nanotubes: Preparation and Properties, CRC Press, 1996.[3] E.V. Dirote, Trends in Nanotechnology Research, Nova Science Publishers, 2004.[4] J. Che, T. Cagin, W.A. Goddard III, Thermal conductivity of carbon nanotubes,

Nanotechnology 11 (2) (2000) 65.[5] H. Ajiki, T. Ando, Magnetic properties of carbon nanotubes, J. Phys. Soc. Jpn.

62 (7) (1993) 2470–2480.[6] L. Wang, Q. Ni, M. Li, Buckling instability of double-wall carbon nanotubes con-

veying fluid, Comput. Mater. Sci. 44 (2) (2008) 821–825.[7] S. Sawano, T. Arie, S. Akita, Carbon nanotube resonator in liquid, Nano Lett.

10 (9) (2010) 3395–3398.[8] D. Pantarotto, C.D. Partidos, J. Hoebeke, F. Brown, E.D. Kramer, J.P. Briand, S.

Muller, M. Prato, A. Bianco, Immunization with peptide-functionalized carbon nanotubes enhances virus-specific neutralizing antibody responses, Chem. Biol. 10 (10) (2003) 961–966.

[9] A. Bianco, J. Hoebeke, S. Godefroy, O. Chaloin, D. Pantarotto, J.P. Briand, S. Muller, M. Prato, C.D. Partidos, Cationic carbon nanotubes bind to CpG oligodeoxynucleotides and enhance their immunostimulatory properties, J. Am. Chem. Soc. 127 (1) (2005) 58–59.

[10] R. Singh, et al., Binding and condensation of plasmid DNA onto functionalized carbon nanotubes: toward the construction of nanotube-based gene delivery vectors, J. Am. Chem. Soc. 127 (12) (2005) 4388–4396.

[11] D. Cai, et al., Highly efficient molecular delivery into mammalian cells using carbon nanotube spearing, Nat. Methods 2 (6) (2005) 449–454.

[12] N.W.S. Kam, M. O’Connell, J.A. Wisdom, H. Dai, Carbon nanotubes as multifunc-tional biological transporters and near-infrared agents for selective cancer cell destruction, Proc. Natl. Acad. Sci. USA 102 (33) (2005) 11600–11605.

[13] Z. Yinghuai, A.T. Peng, K. Carpenter, J.A. Maguire, N.S. Hosmane, M. Takagaki, Substituted carborane-appended water-soluble single-wall carbon nanotubes: new approach to boron neutron capture therapy drug delivery, J. Am. Chem. Soc. 127 (27) (2005) 9875–9880.

[14] M. Foldvari, M. Bagonluri, Carbon nanotubes as functional excipients for nanomedicines: II. Drug delivery and biocompatibility issues, Nanomedicine 4 (3) (2008) 183–200.

[15] N. Venkatesan, J. Yoshimitsu, Y. Ito, N. Shibata, K. Takada, Liquid filled nanopar-ticles as a drug delivery tool for protein therapeutics, Biomaterials 26 (34) (2005) 7154–7163.

[16] A. Leonhardt, I. Mönch, A. Meye, S. Hampel, B. Büchner, Synthesis of ferro-magnetic filled carbon nanotubes and their biomedical application, J. Adv. Sci. Technol. 49 (2006) 74–78.

[17] N. Khosravian, H. Rafii-Tabar, Computational modelling of a non-viscous fluid flow in a multi-walled carbon nanotube modelled as a Timoshenko beam, Nan-otechnology 19 (27) (2008) 275703.

[18] J. Yoon, C.Q. Ru, A. Mioduchowski, Vibration and instability of carbon nanotubes conveying fluid, Compos. Sci. Technol. 65 (9 spec. iss.) (2005) 1326–1336.

[19] L. Wang, Q. Ni, A reappraisal of the computational modelling of carbon nan-otubes conveying viscous fluid, Mech. Res. Commun. 36 (7) (2009) 833–837.

[20] N. Khosravian, H. Rafii-Tabar, Computational modelling of the flow of viscous fluids in carbon nanotubes, J. Phys. D, Appl. Phys. 40 (22) (2007) 7046.

[21] A. Beskok, G.E. Karniadakis, Report: a model for flows in channels, pipes, and ducts at micro and nano scales, Microscale Thermophys. Eng. 3 (1) (1999) 43–77.

[22] V. Rashidi, H.R. Mirdamadi, E. Shirani, A novel model for vibrations of nan-otubes conveying nanoflow, Comput. Mater. Sci. 51 (1) (2012) 347–352.

[23] M. Sadeghi-Goughari, M. Hosseini, The effects of non-uniform flow velocity on vibrations of single-walled carbon nanotube conveying fluid, J. Mech. Sci. Tech-nol. 29 (2) (2015) 723–732.

[24] H. Jannesari, M.D. Emami, H. Karimpour, Investigating the effect of viscosity and nonlocal effects on the stability of SWCNT conveying flowing fluid using nonlinear shell model, Phys. Lett. A 376 (12) (2012) 1137–1145.

[25] M. Hosseini, M. Sadeghi-Goughari, S.A. Atashipour, M. Eftekhari, Vibration anal-ysis of single-walled carbon nanotubes conveying nanoflow embedded in a viscoelastic medium using modified nonlocal beam model, Arch. Mech. 66 (4) (2014) 217–244.

[26] H. Askari, E. Esmailzadeh, Forced vibration of fluid conveying carbon nanotubes considering thermal effect and nonlinear foundations, Composites, Part B, Eng. 113 (2017) 31–43.

[27] H.L. Lee, W.J. Chang, Free transverse vibration of the fluid-conveying single-walled carbon nanotube using nonlocal elastic theory, J. Appl. Phys. 103 (2) (2008).

[28] Y. Zhen, B. Fang, Thermal–mechanical and nonlocal elastic vibration of single-walled carbon nanotubes conveying fluid, Comput. Mater. Sci. 49 (2) (2010) 276–282.

[29] M. Mirramezani, H.R. Mirdamadi, Effects of nonlocal elasticity and Knudsen number on fluid–structure interaction in carbon nanotube conveying fluid, Physica E, Low-Dimens. Syst. Nanostruct. 44 (10) (2012) 2005–2015.

M. Sadeghi-Goughari et al. / Physics Letters A 381 (2017) 2898–2905 2905

[30] M. Hosseini, M. Sadeghi-Goughari, Vibration and instability analysis of nan-otubes conveying fluid subjected to a longitudinal magnetic field, Appl. Math. Model. 40 (4) (Feb. 2016) 2560–2576.

[31] T. Murmu, M.A. McCarthy, S. Adhikari, Vibration response of double-walled car-bon nanotubes subjected to an externally applied longitudinal magnetic field: a nonlocal elasticity approach, J. Sound Vib. 331 (23) (2012) 5069–5086.

[32] S. Li, H.J. Xie, X. Wang, Dynamic characteristics of multi-walled carbon nan-otubes under a transverse magnetic field, Bull. Mater. Sci. 34 (1) (2011) 45–52.

[33] A.G. Arani, M.A. Roudbari, S. Amir, Longitudinal magnetic field effect on wave propagation of fluid-conveyed SWCNT using Knudsen number and surface con-siderations, Appl. Math. Model. 40 (3) (2016) 2025–2038.

[34] M.P. Paidoussis, B.E. Laithier, Dynamics of Timoshenko beams conveying fluid, J. Mech. Eng. Sci. 18 (4) (1976) 210–220.

[35] H.I. Andersson, An exact solution of the Navier–Stokes equations for magneto-hydrodynamic flow, Acta Mech. 113 (1) (1995) 241–244.

[36] G. Zhang, M. Liu, X. Zhang, T. Ye, Y. Chen, L. Wang, Continuum-based slip model and its validity for micro-channel flows, Chin. Sci. Bull. 51 (9) (2006) 1130–1137.

[37] J.N. Reddy, S.D. Pang, Nonlocal continuum theories of beams for the analysis of carbon nanotubes, J. Appl. Phys. 103 (2) (2008).

[38] M.P. Paidoussis, Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 1, Academic Press, 1998.

[39] W. Weaver Jr., S.P. Timoshenko, D.H. Young, Vibration Problems in Engineering, John Wiley & Sons, 1990.

[40] P. Soltani, M.M. Taherian, A. Farshidianfar, Vibration and instability of a viscous-fluid-conveying single-walled carbon nanotube embedded in a visco-elastic medium, J. Phys. D, Appl. Phys. 43 (42) (2010) 425401.

[41] H.W. Zhang, Z.Q. Zhang, L. Wang, Y.G. Zheng, J.B. Wang, Z.K. Wang, Pressure control model for transport of liquid mercury in carbon nanotubes, Appl. Phys. Lett. 90 (14) (2007) 144105.

[42] C.K. Chiang, C.R. Fincher Jr., Y.W. Park, A.J. Heeger, H. Shirakawa, E.J. Louis, S.C. Gau, A.G. MacDiarmid, Electrical conductivity in doped polyacetylene, Phys. Rev. Lett. 39 (17) (1977) 1098–1101.

[43] Q. Ni, Z.L. Zhang, L. Wang, Application of the differential transformation method to vibration analysis of pipes conveying fluid, Appl. Math. Comput. 217 (16) (2011) 7028–7038.