Nonlocal vibration and instability analysis of carbon nanotubes conveying fluid considering the influences of nanoflow and non-uniform velocity profile

  • Mehdi RahmatiEmail author
  • Seyedvahid Khodaei
Research Paper


In this article, the influences of non-uniform velocity profile attributable to slip boundary condition and viscosity of fluid on the dynamic instability of carbon nanotubes (CNTs) conveying fluid are investigated. The nonlocal elasticity theory and the Euler–Bernoulli beam theory are employed to derive partial differential equation of nanotubes conveying fluid. Furthermore, a dimensionless momentum correction factor (MCF) is obtained as a function of Knudsen number (Kn) so as to insert the effects of non-uniform velocity profile into the equation of motion. In continuation, complex eigen-frequencies of the system are attained with respect to different boundary conditions, the momentum correction factor, slip boundary condition and nonlocal parameter. The results delineate that considering the effects of non-uniform velocity profile could diminish predicted critical velocity of flow. Therefore, the divergence instability occurs in the lower values of flow velocity. In addition, the MCF decreases through enhancement of Kn; hence, the effects of non-uniform velocity profile are more noticeable for liquid fluid than gas fluid.


Fluid structure interaction (FSI) Knudsen number Carbon nanotubes Nonlocal elasticity theory Non-uniform velocity profile of nanoflow 



  1. Alavi SR, Rahmati M (2016) Experimental investigation on thermal performance of natural draft wet cooling towers employing an innovative wind-creator setup. Energy Convers Manag 122:504–514CrossRefGoogle Scholar
  2. Alavi SR, Rahmati M, Mortazavi SM (2017a) Fuzzy active vibration control of an orthotropic plate using piezoelectric actuators. In: 5th Iranian joint congress on fuzzy and intelligent systems (CFIS), IEEE, pp 207–212Google Scholar
  3. Alavi SR, Rahmati M, Ziaei-Rad S (2017b) A new approach to design safe-supported HDD against random excitation by using optimization of rubbers spatial parameters. Microsyst Technol 23(6):2023–2032CrossRefGoogle Scholar
  4. Alavi SR, Rahmati M, Ziaei-Rad S (2017c) Optimization of passive control performance for different hard disk drives subjected to shock excitation. J Cent South Univ 24(4):891–899CrossRefGoogle Scholar
  5. Arani A, Ghorbanpour, Roudbari MA (2014) Surface stress, initial stress and Knudsen-dependent flow velocity effects on the electro-thermo nonlocal wave propagation of SWBNNTs. Phys B 452:159–165CrossRefGoogle Scholar
  6. Arani A, Ghorbanpour E, Haghparast Z, Khoddami Maraghi, Amir S (2015) Nonlocal vibration and instability analysis of embedded DWCNT conveying fluid under magnetic field with slip conditions consideration. Proc Inst Mech Eng Part C J Mech Eng Sci 229(2):349–363CrossRefGoogle Scholar
  7. Bahaadini R, Hosseini M (2016) Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid. Comput Mater Sci 114:151–159CrossRefGoogle Scholar
  8. Dai HL, Wang L, Abdelkefi A, Ni Q (2015) On nonlinear behavior and buckling of fluid-transporting nanotubes. Int J Eng Sci 87:13–22CrossRefGoogle Scholar
  9. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710CrossRefGoogle Scholar
  10. Fattahi I, Mirdamadi HR (2017) Novel composite finite element model for piezoelectric energy harvesters based on 3D beam kinematics. Compos Struct 179:161–171CrossRefGoogle Scholar
  11. Fereidoon A, Andalib E, Mirafzal A (2016) Nonlinear vibration of viscoelastic embedded-DWCNTs integrated with piezoelectric layers-conveying viscous fluid considering surface effects. Phys E 81:205–218CrossRefGoogle Scholar
  12. Ghavanloo E, Daneshmand F, Rafiei M (2010) Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation. Phys E 42(9):2218–2224CrossRefGoogle Scholar
  13. Guo CQ, Zhang CH, Païdoussis MP (2010) Modification of equation of motion of fluid-conveying pipe for laminar and turbulent flow profiles. J Fluids Struct 26(5):793–803CrossRefGoogle Scholar
  14. Hellum AM, Mukherjee R, Andrew J, Hull (2010) Dynamics of pipes conveying fluid with non-uniform turbulent and laminar velocity profiles. J Fluids Struct 26(5):804–813CrossRefGoogle Scholar
  15. Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354(6348):56–58CrossRefGoogle Scholar
  16. Karimi M, Shahidi AR (2017a) Nonlocal, refined plate, and surface effects theories used to analyze free vibration of magnetoelectroelastic nanoplates under thermo-mechanical and shear loadings. Appl Phys A 123:304CrossRefGoogle Scholar
  17. Karimi M, Shahidi AR (2017b) Thermo-mechanical vibration, buckling, and bending of orthotropic graphene sheets based on nonlocal two-variable refined plate theory using finite difference method considering surface energy effects. Proc Inst Mech Eng Part N J Nanomater Nanoeng Nanosyst 231:111–130Google Scholar
  18. Karimi M, Shahidi AR (2018) Buckling analysis of skew magneto-electro-thermo-elastic nanoplates considering surface energy layers and utilizing the Galerkin method. Appl Phys A 124(10):681CrossRefGoogle Scholar
  19. Karimi M, Haddad HA, Shahidi AR (2015) Combining surface effects and non-local two variable refined plate theories on the shear/biaxial buckling and vibration of silver nanoplates. Micro Nano Letters 10(6):276–281CrossRefGoogle Scholar
  20. Karimi M, Shahidi AR, Ziaei-Rad S (2017) Surface layer and nonlocal parameter effects on the in-phase and out-of-phase natural frequencies of a double-layer piezoelectric nanoplate under thermo-electro-mechanical loadings. Microsyst Technol 23:4903–4915CrossRefGoogle Scholar
  21. Karimipour I, Beni YT, Zeighampour H (2018) Nonlinear size-dependent pull-in instability and stress analysis of thin plate actuator based on enhanced continuum theories including nonlinear effects and surface energy. Microsyst Technol 24(4):1811–1839CrossRefGoogle Scholar
  22. Karniadakis G, Beskok A, Aluru N (2006) Microflows and nanoflows: fundamentals and simulation, vol 29. Springer, New yorkzbMATHGoogle Scholar
  23. Khodaei S, Fatouraee N, Nabaei M (2017) Numerical simulation of Mitral valve prolapse considering the effect of left ventricle. Math Biosci 285:75–80MathSciNetCrossRefGoogle Scholar
  24. Liu H, Lv Z (2018) Vibration and instability analysis of flow-conveying carbon nanotubes in the presence of material uncertainties. Phys A 511:85–103MathSciNetCrossRefGoogle Scholar
  25. Liu G, Wang Y (2018) Natural frequency analysis of a cantilevered piping system conveying gas–liquid two-phase slug flow. Chem Eng Res Des 136:564–580CrossRefGoogle Scholar
  26. Mirramezani M, Mirdamadi HR (2012) Effects of nonlocal elasticity and Knudsen number on fluid–structure interaction in carbon nanotube conveying fluid. Phys E 44(10):2005–2015CrossRefGoogle Scholar
  27. Mohammadi K, Rajabpour A, Ghadiri M (2018) “Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous fluid using molecular dynamics simulation. Comput Mater Sci 148:104–115CrossRefGoogle Scholar
  28. Naeeni IP, Keshavarzi A, Fattahi I (2018) Parametric study on the geometric and kinetic aspects of the slider-crank mechanism. Iran J Sci Technol Trans Mech Eng. CrossRefGoogle Scholar
  29. Oveissi S, Eftekhari SA, Toghraie D (2016) Longitudinal vibration and instabilities of carbon nanotubes conveying fluid considering size effects of nanoflow and nanostructure. Phys E 83:164–173CrossRefGoogle Scholar
  30. Païdoussis MP (1998) Fluid-structure interactions: slender structures and axial flow, vol 1. Academic press, CambridgeGoogle Scholar
  31. Païdoussis MP, Laithier BE (1976) Dynamics of Timoshenko beams conveying fluid. J Mech Eng Sci 18(4):210–220CrossRefGoogle Scholar
  32. Pollard WG, Present RD (1948) On gaseous self-diffusion in long capillary tubes. Phys Rev 73(7):762CrossRefGoogle Scholar
  33. Rafiei M, Mohebpour SR, Daneshmand F (2012) Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium. Phys E 44(7):1372–1379CrossRefGoogle Scholar
  34. Rahmati M, Alavi SR, Sedaghat A (2016a) Thermal performance of natural draft wet cooling towers under cross-wind conditions based on experimental data and regression analysis. In: 6th Conference on Thermal Power Plants (CTPP), IEEE, pp. 1–5Google Scholar
  35. Rahmati M, Alavi SR, Tavakoli MR (2016b) Experimental investigation on performance enhancement of forced draft wet cooling towers with special emphasis on the role of stage numbers. Energy Convers Manag 126:971–981CrossRefGoogle Scholar
  36. Rahmati M, Alavi SR, Ziaei-Rad S (2017) “Improving the read/write performance of hard disk drives under external excitation sources based on multi-objective optimization. Microsyst Technol 23(8):3331–3345CrossRefGoogle Scholar
  37. Rahmati M, Alavi SR, Tavakoli MR (2018a) Investigation of heat transfer in mechanical draft wet cooling towers using infrared thermal images: an experimental study. Int J Refrig 88:229–238CrossRefGoogle Scholar
  38. Rahmati M, Mirdamadi HR, Goli S (2018b) Divergence instability of pipes conveying fluid with uncertain flow velocity. Phys A 491:650–665MathSciNetCrossRefGoogle Scholar
  39. Rashidi V, Mirdamadi HR, Shirani E (2012) A novel model for vibrations of nanotubes conveying nanoflow. Comput Mater Sci 51(1):347–352CrossRefGoogle Scholar
  40. Roohi E, Darbandi M (2009) Extending the Navier–Stokes solutions to transition regime in two-dimensional micro- and nanochannel flows using information preservation scheme. Phys Fluids (1994–present) 21(8):082001CrossRefGoogle Scholar
  41. Shahali P, Rahmati M, Alavi SR, Sedaghat A (2016) Experimental study on improving operating conditions of wet cooling towers using various rib numbers of packing. Int J Refrig 65:80–91CrossRefGoogle Scholar
  42. Shokouhmand H, Meghdadi Isfahani AH, Shirani E (2010) Friction and heat transfer coefficient in micro and nano channels filled with porous media for wide range of Knudsen number. Int Commun Heat Mass Transfer 37(7):890–894CrossRefGoogle Scholar
  43. Wang Q, Varadan VK, Quek ST (2006) Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Phys Lett A 357(2):130–135CrossRefGoogle Scholar
  44. Zeighampour H, Tadi Beni Y (2014) Size-dependent vibration of fluid-conveying double-walled carbon nanotubes using couple stress shell theory. Phys E 61:28–39CrossRefGoogle Scholar
  45. Zeighampour H, Beni YT, Karimipour I (2017a) Material length scale and nonlocal effects on the wave propagation of composite laminated cylindrical micro/nanoshells. Eur Phys J Plus 132(12):503CrossRefGoogle Scholar
  46. Zeighampour H, Beni TY, Karimipour I (2017b) Wave propagation in double-walled carbon nanotube conveying fluid considering slip boundary condition and shell model based on nonlocal strain gradient theory. Microfluid Nanofluid 21(5):85CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIsfahan University of TechnologyIsfahanIran
  2. 2.Biological Fluid Dynamics Laboratory, Biomedical Engineering FacultyAmirkabir University of TechnologyTehranIran

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