Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-18T14:09:20.677Z Has data issue: false hasContentIssue false
  • English
  • Français

Variational Principles for Vibrating Carbon Nanotubes Conveying Fluid, Based on the Nonlocal Beam Model

Published online by Cambridge University Press:  07 September 2015

Sarp Adali
Sarp Adali*
Affiliation:
Discipline of Mechanical Engineering, University of KwaZulu-Natal, Durban 4041, South Africa
*
*Corresponding author. Email address: [email protected] (S. Adali)
Get access

Abstract

Variational principles are derived in order to facilitate the investigation of the vibrations and stability of single and double-walled carbon nanotubes conveying a fluid, from a linear time-dependent partial differential equation governing their displacements. The nonlocal elastic theory of Euler-Bernoulli beams takes small-scale effects into account. Hamilton’s principle is obtained for double-walled nano-tubes conveying a fluid. The natural and geometric boundary conditions identified are seen to be coupled and time-dependent due to nonlocal effects.

Keywords

58E3074H4582D80Variational principlescarbon nanotubefluid flownonlocal beamvibration
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 3 , August 2015 , pp. 209 - 221
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adali, S., Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory, Phys. Lett. A 372, 5701–5705 (2008).Google Scholar
[2]Adali, S., Variational formulation for buckling of multi-walled carbon nanotubes modelled as nonlocal Timoshenko beams, J. Theoretical Appl. Mech. 50, 321–333 (2012).Google Scholar
[3]Adali, S., Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler-Bernoulli beam model, Nano Lett. 9, 1737–1741 (2009).Google Scholar
[4]Adali, S., Variational principles for multi-walled carbon nanotubes undergoing nonlinear vibrations by semi-inverse method, Micro Nano Lett. 4, 198–203 (2009).CrossRefGoogle Scholar
[5]Adali, S., Variational principles for vibrating carbon nanotubes modeled as cylindrical shells based on strain gradient nonlocal theory, J. Comp. Theoretical Nanoscience 8, 1954–1962 (2011).CrossRefGoogle Scholar
[6]Adali, S., Variational formulation for carbon nanotubes conveying fluid, Proc. 9th South African Conf. Comp. Appl. Mech., 14–16 January 2014, Somerset West, South Africa.Google Scholar
[7]Askes, H. and Aifantis, E.C., Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struct. 48, 1962–1990 (2011).CrossRefGoogle Scholar
[8]Batra, R.C. and Sears, A., Continuum models of multi-walled carbon nanotubes, Int. J. Solids Struct. 44, 7577–7596 (2007).Google Scholar
[9]Birman, V. and Bert, C.W., Non-linear beam-type vibrations of long cylindrical shells, Int. J. Nonlinear Mech. 22, 327–334 (1987).Google Scholar
[10]Challamel, N., Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Comp. Struct. 105, 351–368 (2013).CrossRefGoogle Scholar
[11]Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Dover, New York (1981).Google Scholar
[12]Chang, T.P., Thermal-mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Appl. Math. Model. 36, 1964–1973 (2012).Google Scholar
[13]Challamel, N., Wang, C.M. and Elishakoff, I., Discrete systems behave as nonlocal structural elements: Bending, buckling and vibration analysis, European J. Mech. A/Solids 44, 125–135 (2014).Google Scholar
[14]Duan, W.H., Wang, C.M. and Zhang, Y.Y., Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, J. Appl. Phys. 101, 024305 (2007).CrossRefGoogle Scholar
[15]Eringen, A.C. and Edelen, D.G.B., On nonlocal elasticity, Int. J. Eng. Sci. 10, 233–248 (1972).Google Scholar
[16]Eringen, A.C., Linear theory of nonlocal elasticity and dispersion of plane waves, Int. J. Eng. Sci. 10, 425–435 (1972).Google Scholar
[17]Faria, B., Silvestre, N. and Canongia, J.N. Lopes, Induced anisotropy of chiral carbon nanotubes under combined tension-twisting, Mechanics of Materials 58, 97–109 (2013).CrossRefGoogle Scholar
[18]Ghavanloo, E., Daneshmand, F. and Rafiei, M., Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation, Physica E 42, 2218–2224 (2010).CrossRefGoogle Scholar
[19]Hanasaki, I. and Nakatani, A., Water flow through carbon nanotube junctions as molecular convergent nozzles, Nanotech. 17, 2794–2804 (2006).CrossRefGoogle Scholar
[20]He, J.H., Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int. J. Turbo Jet Engines 14, 23–28 (1997).CrossRefGoogle Scholar
[21]He, J.H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons and Fractals 19, 847–851 (2004).Google Scholar
[22]He, J.H., Variational approach to (2 +1)-dimensional dispersive long water equations, Phys. Lett. A 335, 182–184 (2005).Google Scholar
[23]He, J.H., Variational theory for one-dimensional longitudinal beam dynamics, Phys. Lett. A 352, 276–277 (2006).Google Scholar
[24]He, J.H., Variational principle for two-dimensional incompressible inviscid flow, Phys. Lett. A 371, 39–40 (2007).Google Scholar
[25]Hummer, G., Rasaiah, J.C. and Noworyta, J.P., Water conduction through the hydrophobic channel of carbon nanotubes, Nature 414, 188–190 (2001).Google Scholar
[26]Khosravian, N. and Rafii-Tabar, H., Computational modelling of the flow of viscous fluids in carbon nanotubes, J. Phys. D: Appl. Phys. 40, 7046–7052 (2007).Google Scholar
[27]Khosravian, N. and Rafii-Tabar, H., Computational modelling of a non-viscous fluid flow in a multi-walled carbon nanotube modelled as a Timoshenko beam, Nanotech. 19, 275703 (2008).Google Scholar
[28]Kiani, K., Vibration behaviour of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model, Appl. Math. Model. 37, 1836–1850 (2013).CrossRefGoogle Scholar
[29]Kucuk, I., Sadek, I.S. and Adali, S., Variational principles for multi-walled carbon nanotubes undergoing vibrations based on nonlocal Timoshenko beam theory, J. Nanomaterials, 461252 (2010).Google Scholar
[30]Lee, H.-L. and Chang, W.-J., Vibration analysis of a viscous fluid conveying single-walled carbon nanotube embedded in an elastic medium, Physica E 41, 529–532 (2009).Google Scholar
[31]Liang, Y.J. and Han, Q., Prediction of nonlocal scale parameter for carbon nanotubes, Science China – Physics, Mechanics and Astronomy 55, 1670–1678 (2012).Google Scholar
[32]Liu, H.M., Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method, Chaos, Solitons and Fractals 23, 573–576 (2005).Google Scholar
[33]Mao, Z. and Sinnott, S.B., A computational study of molecular diffusion and dynamics flow through CNTs, J. Physical Chem. B 104, 4618–4624 (2000).Google Scholar
[34]Mattia, D. and Gogotsi, Y., Review: Static and dynamic behaviour of liquids inside carbon nanotubes, Microfluidics and Nanofluidics 5, 289–305 (2008).Google Scholar
[35]Narendar, S., Mahapatra, D.R. and Gopalakrishnan, S., Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation, Int. J. Eng. Science 49, 509–522 (2011).Google Scholar
[36]Rafiei, M., Mohebpour, S.R. and Daneshmand, F., Small-scale effect on the vibration of nonuniform carbon nanotubes conveying fluid and embedded in viscoelastic medium, Physica E 44, 1372–1379 (2012).Google Scholar
[37]Rinaldi, S., Prabhakar, S., Vengallatore, S. and Paidoussis, M.P., Dynamics of microscale pipes containing internal fluid flow: Damping, frequency shift, and stability, J. Sound Vib. 329, 1081–1088 (2010).Google Scholar
[38]Silvestre, N., On the accuracy of shell models for torsional buckling of carbon nanotubes, European J. Mech.- A/Solids 32, 103–108 (2012).CrossRefGoogle Scholar
[39]Supple, S. and Quirke, N., Rapid imbibition of fluids in CNTs, Phys. Rev. Lett. 90, 21450 (2003).CrossRefGoogle Scholar
[40]Vaziri, A., Mechanics of highly deformed elastic shells, Thin-Walled Structures 47, 692–700 (2009).Google Scholar
[41]Wang, L., Wave propagation of fluid-conveying single-walled carbon nanotubes via gradient elasticity theory, Comp. Mat. Sci. 49, 761–766 (2010).Google Scholar
[42]Wang, L., Size-dependent vibration characteristics of fluid-conveying microtubes, J. Fluids Struct. 26, 675–684 (2010).Google Scholar
[43]Wang, L., Guo, W. and Hu, H., Flexural wave dispersion in multi-walled carbon nanotubes conveying fluids, Acta Mechanica Solida Sinica 22, 623–629 (2009).Google Scholar
[44]Wang, Y.-Z., Li, F.M. and Kishimoto, K., Wave propagation characteristics in fluid-conveying double-walled nanotubes with scale effects, Comp. Mat. Sci. 48, 413–418 (2010).CrossRefGoogle Scholar
[45]Wang, L. and Ni, Q., A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid, Mech. Res. Comm. 44, 833–837 (2009).Google Scholar
[46]Wang, L. and Ni, Q., On vibration and instability of carbon nanotubes conveying fluid, Comp. Mat. Sci. 43, 399–402 (2008).Google Scholar
[47]Wang, L., Ni, Q. and Li, M., Buckling instability of double-wall carbon nanotubes conveying fluid, Comp. Mat. Sci. 44, 821–825 (2008).Google Scholar
[48]Wang, Q. and Wang, C.M., The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology 18, 075702 (2007).Google Scholar
[49]Whitby, M. and Quirke, N., Fluid flow in carbon nanotubes and nanopipes, Nature Nanotech. 2, 87–94 (2007).CrossRefGoogle ScholarPubMed
[50]Whitham, G.B., Linear and Nonlinear Waves, Wiley, New York (2011).Google Scholar
[51]Yan, Y., He, X.Q., Zhang, L.X. and Wang, Q., Flow-induced instability of double-walled carbon nanotubes based on an elastic shell model, J. Appl. Phys. 102, 044307 (2007).Google Scholar
[52]Yan, Y., He, X.Q., Zhang, L.X. and Wang, C.M., Dynamic behaviour of triple-walled carbon nanotubes conveying fluid, J. Sound Vib. 319, 1003–1018 (2009).Google Scholar
[53]Yan, Y., Wang, W.Q. and Zhang, L.X., Dynamical behaviours of fluid-conveyed multi-walled carbon nanotubes, Appl. Math. Model. 33, 1430–1440 (2009).Google Scholar
[54]Yoon, J., Ru, C.Q. and Mioduchowski, A., Vibration and instability of CNTs conveying fluid, Comp. Sci. Tech. 65, 1326–1336 (2005).Google Scholar
[55]Yoon, J., Ru, C.Q. and Mioduchowski, A., Flow-induced flutter instability of cantilever CNTs, Int. J. Solids Struct. 43, 3337–3349 (2006).Google Scholar
[56]Zeighampour, H. and Beni, Y.T., Size-dependent vibration of fluid-conveying double-walled carbon nanotubes using couple stress shell theory, Physica E 61, 28–39 (2014).Google Scholar