Vibrations of fluid-conveying inclined single-walled carbon nanotubes acted upon by a longitudinal magnetic field

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1 Appl. Phys. A (2016)122:1038 DOI /s Virations o luid-conveying inclined single-walled caron nanotues acted upon y a longitudinal magnetic ield Keivan Kiani 1 Received: 29 August 2016 / Accepted: 28 Octoer 2016 Springer-Verlag Berlin Heidelerg 2016 Astract This work deals with the inluence o the longitudinal magnetic ield on virations o inclined singlewalled caron nanotues (SWCNTs) sujected to an inside luid low. Using an equivalent continuum structure model or the SWCNT and a plug-like model or the moving inside luid low, the nonlocal longitudinal and transverse equations o motion o magnetically aected SWCNTs are otained in the context o small deormations. By application o the assumed-mode methodology, the displacements are discretized in terms o viration mode shapes, and y exploiting generalized Newmark- scheme, their corresponding time-dependent parameters are determined at each time. In the presence o the longitudinal magnetic ield, the eects o the small-scale parameter, luid low velocity, and inclination angle on oth longitudinal and transverse virations o SWCNTs are addressed. The otained results reveal that the longitudinal magnetic ield has airly no eect on the longitudinal dynamic ehavior o the nanostructure. However, maximum values o oth transverse displacement and nonlocal ending moment o the luid-conveying SWCNT would reduce as the strength o the magnetic ield grows. Such a act ecomes more highlighted or high levels o the luid low velocity. The otained results indicate that the longitudinal magnetic ield can e exploited as an eicient way to control transverse virations o SWCNTs conveying luids. & Keivan Kiani k_kiani@kntu.ac.ir; keivankiani@yahoo.com 1 Department o Civil Engineering, K.N. Toosi University o Technology, P.O. Box , Tehran, Iran 1 Introduction Currently, there is an enormous deal o theoretical and experimental works on mechanical ehavior o caron nanotues (CNTs). Experimentally oserved data are highly required; however, the technical challenge and expense o testing at the nanoscale restricts the availaility o these data. Such a act has motivated researchers to exploit theoretical models to characterize mechanical ehaviors o CNTs. Generally, there are two major categories or theoretical investigation o CNTs: atomistic-ased models and continuum mechanics. The common methods or atomistic modeling are molecular dynamics (MD), density unction theory (DFT), and tight-inding molecular dynamics (TBMD). Movement o water molecules inside the CNT s pore was modeled y MD [1, 2]. These studies oered CNTs as eicient small conveyor o water molecules and proteins. The huge computational requirements and laor costs o atomic models have encouraged some researchers to employ continuum mechanics models to examine their virations under externally applied loads. Much o these works are ocused on using Euler Bernoulli or Timoshenko eam theories to predict lexural virations o singleand multi-walled caron nanotues (SWCNTs and MWCNTs). To date, classical continuum theory (CCT) has een extensively used to study virations o CNTs sujected to moving inside luids low [3 8]. At the nanoscale, the onds o atoms play an important role in the virations o nanostructures. The CCT has no sense with respect to such an issue. To improve such a deiciency o the CCT, some advanced theories o elasticity have een estalished during the past century including couple stress theory [9 12], surace elasticity theory [13, 14], gradient elasticity theory [15 17], and nonlocal

2 1038 Page 2 o 7 K. Kiani continuum theory (NCT) o Eringen [18 20]. The latter one has een o ocus o attention o many investigators during the past two decades due to its simplicity in application and eiciency in predicting the dynamic ehavior o nanoscale systems. This theory explains that the state o stress at each point o the continuum does depend not only on the strains o that point, ut also on the states o strains o its neighoring points. Such a relevancy is incorporated into the nonlocal model y a so-called small-scale parameter. Its magnitude can e also determined or each prolem y adjusting the predicted results y the nonlocal model with those o a reliale atomistic model. So ar, various aspects o virations o nanostructures including ree and orced virations [21 26], wave propagation [27 31], dynamic instailities [32 34], nonlinear virations [35, 36], and moving nanoparticles eects [37 40] have een studied in the context o the NCT. Additionally, there exist a lot o works in the literature regarding virations o CNTs conveying luid low using nonlocal models [41 47]. However, a practical way to control virations o CNTs ecause o movement inside luid low has not een proposed yet. Herein we are interested in ormulating luid-conveying SWCNTs in the presence o a longitudinal magnetic ield. Using nonlocal Rayleigh eam model, plug-like model or the inside luid low, and Hamilton s principle, the equations o motion descriing longitudinal and transverse virations o magnetically aected inclined SWCNTs conveying luid low are otained. By implementing the Galerkin and Newmark- approaches, oth longitudinal and transverse displacements are evaluated. The eects o the inclination angle, velocity o the luid low, and strength o the magnetic ield on dynamic ehavior o the nanostructure are addressed. The otained results suggest that application o longitudinal magnetic ield can e considered as an eicient way to control transverse virations o luid-conveying SWCNTs. 2 Longitudinal and transverse equations o motion Consider a magnetically aected inclined SWCNT which is aimed to e used as a luid conveyor. The inclination angle o the SWCNT is a, the strength o the magnetic ield is H x, and the plug-like luid low starts to move rom the let end with velocity c. Thereore, the location o its ront is expressed y x = ct. The density, cross-sectional area, and moment inertia o the luid low are represented y q, A, and I, respectively. The SWCNT is modeled y a hollow circular cylinder o wall thickness aout 0.34 nm, length l, density q, cross-sectional area A, moment inertia I, and Young s modulus E. The kinetic energy o the magnetically aected SWCNT plus that o the luid low due to the rotation o the SWCNT, T, the elastic strain energy o the SWCNT, U, and the work done y the exerted orces o the luid and magnetic ield on the SWCNT, W, are stated y: T ¼ 1 2 U ¼ 1 2 W ¼ Z l 0 Z l 0 Z l 0 where N nl q A _u 2 þ A _w 2 þ I _w 2 2 ;x þ q I _w þ cw ;x dx; ð2:1aþ N nl u ;x M nl w ;xx dx; ð2:1þ t n w ;x u þ t w ;x þ n þ z w dx; ð2:1cþ and Mnl are the nonlocal axial orce and ending moment within the magnetically aected SWCNT, respectively, t and n in order are the longitudinal and transverse interactional orces etween the luid low and the SWCNT, z ¼ ga H 2 x w ;xx denotes the Lorentz magnetic orce per unit length o the SWCNT [31, 48], and g is the magnetic permeaility o the nanotue. By exploiting Hamilton s principle, the equations o motion o the magnetically aected SWCNT sujected to a moving inside luid low accounting or oth longitudinal and transverse motions are derived as: q A u N;x nl t þ n w ;x ¼ 0; ð2:2aþ q A w I w ;xx q I w þ 2c _w ;x þ c 2 w ;xx ;xx ga Hx 2 w ;xx M;xx nl ð2:2þ t w ;x n ¼ 0: By taking into account a small element o the moving luid low within the deormed SWCNT, the governing equations will e otained y adopting the Newton s second law: PA ;x þ t n w ;x þ q A g x g z w ;x þ u þ 2c _u ;x þ c 2 u ;xx ¼ 0; ð2:3aþ PA w ;x ;x þ t w ;x þ n q A g z þ g x w ;x w 2c _w ;x c 2 w ;xx ¼ 0; ð2:3þ where g x = g sin a, g z = g cos a, and g is the gravitational acceleration. The expression o t accounting or the slip oundary eect is otained as [45, 46]: t ¼ g 0 K t ðc _u Þ; ð2:4aþ 8pr v ð1 KnÞ K t ¼ 1 þ a Kn ð rv ð1 KnÞþ42 ð r v ÞKnÞ : ð2:4þ where g 0 is the ulk viscosity o the luid, r v represents the tangential moment accommodation coeicient, Kn is the Knudsen numer, =-1, and a ¼ 2a 0 p tan1 ða 1 Kn B Þ in

3 Virations o luid-conveying inclined single-walled caron nanotues acted upon y a Page 3 o which a 0 ¼ 3pð4Þ, a 1 = 4, and B = 0.4 [49]. By comining Eqs. (2.2), (2.3), and (2.4), the longitudinal and transverse equations o motion o the luid-conveying inclined SWCNT in the presence o a longitudinal magnetic ield are otainale as ollows: q A u N;x nl þ q A g z w ;x ¼ g 0 K t ðc _uþ; ð2:5aþ q A w I w ;xx q I w þ 2c _w ;x þ c 2 w ;xx ;xx þ PA w ;xx g A Hx 2 w ;xx M;xx nl 2q A g z þ g 0 K t c w;x þ q A w þ 2c _w ;x þ c 2 w ;xx q I w þ 2c _w ;x þ c 2 w ;xx ;xx ¼ q A g z : ð2:5þ It is assumed that the plug-like nanoluid lows rom the let end o the SWCNT. Thereore, Eqs. (2.5a) and (2.5) are updated as: q A u N;x nl þ q A g z w ;x 1 Hðx x Þ ¼ g 0 K t ðc _uþ 1 Hðx x Þ ; ð2:6aþ q A w I w ;xx q I w þ 2c _w ;x þ c 2 w ;xx ; xx þ PA w ;xx g A Hx 2 w ;xx M;xx nl 2q A g z þ g 0 K t c w;x 1 Hðx x Þ þ q A w þ 2c _w ;x þ c 2 w ;xx q I w þ 2c _w ;x þ c 2 w ;xx ;xx Þ 1 Hðx x Þ ¼ q A g z 1 Hðx x Þ : ð2:6þ where H represents the Heaviside step unction. The nonlocal axial orce and ending moment o the SWCNT could e related to their local ones y [22, 39, 50]: N nl M nl ð e 0aÞ 2 N;xx nl ¼ Nl ¼ E A u ;x ; ð e 0aÞ 2 M;xx nl ¼ Ml ¼E I w ;xx ; ð2:7aþ ð2:7þ where e 0 a is the small-scale parameter. By introducing Eqs. (2.6a) and (2.6) to Eqs. (2.7a) and (2.7), the nonlocal internal orces o the SWCNT sujected to oth moving inside luid low and longitudinal magnetic ield are written as: ¼ E A u ;x þðe 0 aþ 2 q A u þ q A g z w ;x ð2:8aþ g 0 K t ðc _u ÞÞ 1 Hðx x Þ Þ;x ; ¼E I w ;xx þðe 0 aþ 2 q A w I w ;xx þ PA w ;xx ga Hx 2 w ;xx 2q A g z þ g 0 K t c w;x 1 Hðx x Þ þ q A ð w þ 2c _w ;x þ c 2 w ;xx Þq I ð w þ 2c _w ;x þ c 2 w ;xx Þ ;xx Þ 1 Hðx x Þ q A g z 1 Hðx x Þ Þ; ð2:8þ N nl M nl y sustituting Eqs. (2.8a) and (2.8) into Eqs. (2.6a) and (2.6), and considering the ollowing dimensionless quantities: n ¼ x l ; u ¼ u l ; w ¼ w l ; l ¼ e 0a l ; s ¼ t l 2 l k ¼ p iiiiiiiiiii ; g 0 ¼ g 0l ; I =A C L q A p K ¼ iiiii K t; k ¼ l ; m ¼ q A ; ¼ c ; r q A C L P ¼ PA l 2 piiiiiii siiiiiiiiiii g z l ga l ; c E I z ¼ ; H 2 x ¼ H x ; C L E I siiiiiiiiii E I ; q A ð2:9þ the nonlocal dimensionless governing equations o the magnetically aected SWCNT acted upon y a nanoluidic low in terms o displacements are otained as ollows: u ;ss l 2 u ;ssnn þ m g 0 K 2 k u ;s 1 H n n u ;s 1 H n n þ m ;nn c 2 z k2 w ;n 1 H n n l 2 w ;n 1 H n n ;nn k 2 u ;nn ¼ m g 0 K 2 k2 1 H n n l 2 1 H n n ; ð2:10aþ ;nn w ;ss l 2 w ;ssnn k 2 w ;ssnn l 2 w ;ssnnnn þ m w ;ss þ 2k w ;ns þ 2 k 2 w ;nn 1 H n n l 2 w ;ss þ 2k w ;ns þ 2 k 2 w ;nn 1 H n n m ;nn k 2 w ;ss þ 2k w ;ns þ 2 k 2 w ;nn ;nn 1 H n n l 2 w ;ss þ 2k w ;ns þ 2 k 2 w ;nn ;nn 1 H n n ;nn þ w ;nnnn H x 2 w ;nn l 2 w ;nnnn þ P w ;nn l 2 P ;nn w ;nn þ 2 P ;n w ;nnn þ P w ;nnnn m k 2 2c 2 z þ g 0K 2 w ;n l 2 w ;nnn ¼ m k 2 c 2 z 1 H n n l 2 1 H n n : ;nn ð2:10þ qiiii qiii qiiii I where r ¼ I A, r ¼ E A, and C L ¼ q. Additionally, the nonlocal axial orce and the nonlocal ending moment o the SWCNT as a unction o the dimensionless displacements are stated y: N nl ¼ E A u ;n þ l 2 u ;ss þ m k 2 c 2 z k w ;n g 0 K 2 k 1 u ;s ; ;n ð2:11aþ

4 1038 Page 4 o 7 K. Kiani M nl ¼ E I w ;nn þ l 2 w ;ss k 2 w ;ssnn þ P H x 2 w;nn l þ m w ;ss þ 2k w ;sn þ 2 k 2 w ;nn k 2 w ;ssnn þ 2k w ;snnn þ 2 k 2 : w ;nnnn k 2 2c 2 z þ g 0K 2 w ;n c 2 z 1 H n n ð2:11þ Regarding the initial conditions o the nanostructure, the SWCNT is assumed to e initially at rest. For a simply supported SWCNT with immovale ends, the oundary conditions take the ollowing orm: u ð0; sþ ¼ u ð1; sþ ¼ 0; ð2:12aþ w ð0; sþ ¼ w ð1; sþ ¼ 0; w ;nn ð0; sþ ¼ w ;nn ð1; sþ ¼ 0: ð2:12þ 3 Application o the Galerkin approach to the prolem Finding an analytical solution to the nonlocal equations o motion o the prolem at hand is a very diicult jo. It is mainly ecause o the existence o coupling terms in the governing equations. Thereore, development o numerical methodologies as an alternative approach would e o great eneicial. Herein, a numerical method ased on Galerkin approach is proposed. To this end, oth sides o Eqs. (2.10a) and (2.10) are premultiplied y du and d w, respectively, where d denotes the variational sign, and then the resulting relations are integrated over the dimensionless longitudinal coordinate o the SWCNT, namely [0,1]. Ater successul integration y parts, the dimensionless displacements in terms o admissile mode shapes are expressed y: u ðn; sþ ¼ XN m u i ðþsin s ðipnþ; w ðn; sþ ¼ XN m i¼1 i¼1 K wu K ww w i ðþsin s ðipnþ; ð3:1þ where u i ðþand s w i ðþare s the unknown time-dependent parameters associated with the ith mode and N m is the numer o viration modes. Sustitution o Eq. (3.1) into the inally otained relation would result in: M uu M uw u C M wu M ww þ uu C uw u w ;ss C wu C ww w ;s K þ uu K uw ð3:2þ u ¼ ; w where the sumatrices can e readily calculated. To solve the set o 2N m ordinary dierential equations in Eq. (3.2) in the time domain, the generalized Newmark- approach is used. u w 4 Results and discussion Consider an ECS pertinent to the SWCNT with the ollowing geometry and mechanical data: E = 1012 Pa, q = 2500 kg/m 3, m = 0.2, t = 0.34 nm, r m = 2 nm, k = 40, r 1 = r m - t /2, and r 2 = r m? t /2 where t and r m denote the thickness and the mean radius o the ECS, respectively, and r 1 and r 2 in order are the inner and outer radii o the ECS. The properties o the luid low are considered as: q = 1000 kg/m 3, g 0 = Pa s, r v = 0.8, and l = 0.3 nm. We consider the ollowing normalized parameters in analyzing o the prolem: u N ¼ 8u= m g 0 K 2, w N ¼ 384 we I = 5q A gl 3, NN ¼ 2 N N = E A m g 0 K 2, M N ¼ 8 M = q A gl 2, and p c N ¼ ck= p iiiiiiiiiiii E =q. The maximum dynamic amplitude actors (MDAFs) o the normalized transverse displacement and nonlocal ending moment o the magnetically aected SWCNT conveying luid low are expressed y MDAF w ¼ maxw N ðn; sþg and MDAF M ¼ maxm N ðn; sþg. In Fig. 1a c, the time history plots o the longitudinal and transverse displacements as well as nonlocal axial orce and ending moment are provided or three levels o the strength o the magnetic ield as well as small-scale parameter. As it is seen, the maximum values o longitudinal displacement and nonlocal axial orce would decrease as small-scale parameter increases. Such a act is more ovious or nonlocal axial orce. However, the maximum values o transverse displacement as well as nonlocal ending moment magniy with the small-scale parameter. Such an increase is more apparent or the plots o the nonlocal ending moment. Generally, variation o the strength o the magnetic ield has airly no eect on the variation o longitudinal displacement and nonlocal axial orce. Nevertheless, oth transverse displacement and nonlocal ending moment would reduce as the magnetic ield strength magniies. Because o this act, in the remaining o this paper, the role o the magnetic ield on only transverse displacement and ending moment o the SWCNT will e investigated. Figure 2a c displays the eect o the longitudinal magnetic ield on the maximum values o transverse displacement and nonlocal ending moment o the SWCNT acted upon y a moving inside low. The plotted results are given or three levels o the luid low velocity as well as inclination angle. According to Fig. 2, irrespective o the velocity o the moving inside low and the inclination angle o the SWCNT, oth transverse displacement and nonlocal ending moment would decrease as the strength o the longitudinal magnetic ield increases. Such a act is ollowed with a higher rate or higher levels o the luid low

5 Virations o luid-conveying inclined single-walled caron nanotues acted upon y a Page 5 o Fig. 1 Time history o normalized dynamic displacements, nonlocal axial orce, and nonlocal ending moment at the midspan point o the SWCNT or dierent levels o strength o magnetic ield as well as small-scale parameter: a H x ¼ 0:1, H x ¼ 1, c H x ¼ 2; ((dotted lines) e 0 a = 0 nm, (dashed lines) e 0 a = 1 nm, (lines) e 0 a = 2 nm; a = 0, k = 30, c N = 0.2) (a) () (c) Fig. 2 Plots o MDAF w and MDAFM as a unction o H x or various levels o the luid low velocity and inclination angle: a c N = 0.05, c N = 0.1, c c N = 0.15; ((dotted lines) a = 20, (dashed lines) a = 50, (lines) a = 80; k = 40, e 0 a = 1 nm) velocity and lower inclination angle. These evidences indicate that the longitudinal magnetic ield can e used as an eicient tool to control transverse virations o SWCNTs conveying luid. In another crucial investigation, the role o the luid low velocity on the maximum dynamic transverse displacement and nonlocal ending moment within the luid-conveying SWCNTs is o concern. To this end, the plots o MDAF w and MDAF M as a unction o c N are demonstrated in Fig. 3a c. The results have een provided or three levels o the magnetic ield strength as well as inclination angle. Irrespective o the luid low velocity and the strength o the applied longitudinal magnetic ield, oth maximum values o transverse displacement and nonlocal ending moment would lessen as the inclination angle would magniy. This issue is mainly related to this act that the interactional normal orce etween the moving luid and the nanotue would reduce as the inclination angle grows. Additionally, or all levels o the luid low velocity and inclination angle, oth maximum transverse displacements and nonlocal ending moment would decrease as the inluence o the longitudinal magnetic ield ecomes

6 1038 Page 6 o 7 K. Kiani (a) () (c) Fig. 3 Plots o MDAF w and MDAFM as a unction o c N or various levels o the strength o magnetic ield and inclination angle: a H x ¼ 0:1, H x ¼ 1, c H x ¼ 2; ((dotted lines) a = 20, (dashed lines) a = 50, (lines) a = 80; k = 40, e 0 a = 1 nm) highlighted. Such a act is relevant to the increase in the lexural stiness o the SWCNT due to the application o the longitudinal magnetic ield. 5 Conclusions Virations o magnetically aected inclined SWCNTs acted upon y moving inside luids low are o high interest. Using a plug-like model or the luid low and a nonlocal continuum-ased model or the SWCNT, the nonlocal equations o motion o luid-conveying SWCNTs in the presence o a longitudinal magnetic ield are developed. By employing an eicient numerical scheme, the unknown elastic ields o the continuum under study are evaluated. Then, eect o the strength o the longitudinal magnetic ield on the virations o luid-conveying SWCNTs is studied in some detail. The otained results show that the longitudinal magnetic ield can e eiciently exploited to control the transverse virations o inclined SWCNTs conveying luid low. Reerences 1. A. Kalra, S. Garde, G. Hummer, Proc. Natl. Acad. Sci. 100, (2003) 2. F. Zhu, K. Schulten, Biophys. J. 85, (2003) 3. J. Yoon, C.Q. Ru, A. Mioduchowski, Int. J. Solids Struct. 43, (2006) 4. N. Khosravian, H. Raii-Taar, J. Phys. D Appl. Phys. 40, (2007) 5. L. Wang, Q. Ni, M. Li, Q. Qian, Phys. E 40, (2008) 6. W.J. Chang, H.L. Lee, Phys. Lett. A 373, (2009) 7. Y. Yan, X.Q. He, L.X. Zhang, C.M. Wang, J. Sound Vi. 319, (2009) 8. Y. Yan, W.Q. Wang, L.X. Zhang, Appl. Math. Model. 34, (2010) 9. R.D. Mindlin, H.F. Tiersten, Arch. Ration. Mech. Anal. 11, (1962) 10. R.D. Mindlin, Exp. Mech. 3, 1 7 (1963) 11. R.D. Mindlin, Arch. Ration. Mech. Anal. 16, (1964) 12. R.A. Toupin, Arch. Ration. Mech. Anal. 17, (1964) 13. M.E. Gurtin, A.I. Murdoch, Int. J. Solids Struct. 14, (1978) 14. A.I. Murdoch, J. Mech. Phys. Solids 24, (1976) 15. R.D. Mindlin, N.N. Eshel, Int. J. Solids Struct. 4, (1968) 16. N. Triantayllidis, E.C. Aiantis, J. Elast. 16, (1986) 17. E.C. Aiantis, Int. J. Plast 3, (1987) 18. E.C. Eringen, Int. J. Eng. Sci. 5, (1967) 19. A.C. Eringen, Int. J. Eng. Sci. 10, 1 16 (1972) 20. A.C. Eringen, D.G.B. Edelen, Int. J. Eng. Sci. 10, (1972) 21. Y.Q. Zhang, G.R. Liu, X.Y. Xie, Phys. Rev. B 71, (2005) 22. Q. Wang, J. Appl. Phys. 98, (2005) 23. J.N. Reddy, S.D. Pang, J. Appl. Phys. 103, (2008) 24. K. Kiani, Phys. E 60, (2014) 25. K. Kiani, Appl. Math. Comput. 234, (2014) 26. K. Kiani, Int. J. Mech. Sci. 68, (2013) 27. Q. Wang, V.K. Varadan, Smart Mater. Struct. 16, 178 (2007) 28. Y.G. Hu, K.M. Liew, Q. Wang, X.Q. He, B.I. Yakoson, J. Mech. Phys. Solids 56, (2008) 29. H. Heireche, A. Tounsi, A. Benzair, M. Maachou, E.A. Bedia, Phys. E 40, (2008) 30. K. Kiani, Phys. E 45, (2012) 31. K. Kiani, Meccanica 50, (2015) 32. X.Q. He, C.M. Wang, Y. Yan, L.X. Zhang, G.H. Nie, Arch. Appl. Mech. 78, (2008) 33. K. Kiani, Curr. Appl. Phys. 13, (2013) 34. K. Kiani, Compos. Struct. 107, (2014) 35. L.L. Ke, Y. Xiang, J. Yang, S. Kitiporncha, Comput. Mater. Sci. 47, (2009) 36. J. Yang, L.L. Ke, S. Kitipornchai, Phys. E 42, (2010) 37. M. Simsek, Phys. E 43, (2010) 38. K. Kiani, Q. Wang, Eur. J. Mech. A. Solids 31, (2012) 39. K. Kiani, Nonlinear Dyn. 76, (2014) 40. K. Kiani, Compos. Struct. 116, (2014) 41. H.L. Lee, W.J. Chang, J. Appl. Phys. 103, (2008) 42. L. Wang, Phys. E 41, (2009)

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