Coupled effects of nano-size, stretching, and slip boundary conditions on nonlinear vibrations of nano-tube conveying fluid by the homotopy analysis method Mahmood Ali-Asgari, Hamid Reza Mirdamadi n , Mostafa Ghayour Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran HIGHLIGHTS  Formulating and analyzing nonlinear vibration of CNT conveying fluid.  Coupling nonlocal model and stretch- ing effect for CNT conveying fluid.  Using the homotopy analysis method for nonlinear solution of fluid con- veying pipe.  Modeling the slip boundary condi- tion by utilizing Knudsen number.  Observing divergence instability in doubly-clamped CNT conveying fluid. GRAPHICAL ABSTRACT The nonlinear response and natural frequency of system, obtained by HAM are remarkably impressed by critical flow velocity of divergence, Knudsen number, and nonlocal parameters. article info Article history: Received 10 January 2013 Received in revised form 13 March 2013 Accepted 25 March 2013 Available online 10 April 2013 Keywords: Fluid-structure interaction (FSI) Carbon nano-tube (CNT) Slip boundary condition Nonlocal theory Homotopy analysis method (HAM) Von Karman's nonlinear strains abstract In this paper, natural frequency and nonlinear response of carbon nano-tube (CNT) conveying fluid based on the coupling of nonlocal theory and von Karman's stretching have been obtained. The homotopy analysis method (HAM) has been used for solving nonlinear differential equation of system and convergence region of approach presented. Effects of mid-plane stretching, nonlocal parameter and their coupling in the model have been investigated. It has been concluded that stretching effect is significant only for higher-amplitude initial excitations and lower beam aspect ratios. Moreover, by including the slip boundary condition, the effect of nano-size flow has been revealed in the nonlinear vibration model. We have concluded that small-size effects of nano-tube and nano-flow have impressed critical velocity of fluid significantly specially for gas fluid. Analytical results obtained from HAM solution show satisfactory agreement with numerical solutions such as Runge–Kutta. Having an analytical approach, we have been able to investigate the unbounded growth of amplitude of vibrations for flow velocities near the critical value. Moreover, by employing the second-order approximation of Galerkin's method, the estimated natural frequency of the first mode is verified. The obtained results would indicate that the effects of higher mode on the first natural frequency are negligible for the doubly- clamped CNT. & 2013 Elsevier B.V. All rights reserved. 1. Introduction In recent decades, various research groups in the fields of vibrations and instability of fluid-conveying pipes specially, carbon nano-tubes based on different theories and assumptions, have investigated the effect of different factors on divergence, flutter, and critical flow velocity appearing in fluid-structure interaction (FSI) problems. The principal motivation for these studies would Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/physe Physica E 1386-9477/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2013.03.031 Abbreviations: CNT, carbon nano-tube; MWCNT, multi-walled carbon nano-tube; HAM, homotopy analysis method; FSI, fluid structure interaction; vdW, van der Waals; DSM, dynamic stiffness method n Corresponding author. Tel.: +98 311 391 5248; fax: +98 311 391 2628. E-mail addresses: mahmood.aliasgari@gmail.com (M. Ali-Asgari), hrmirdamadi@cc.iut.ac.ir, hamidreza.mirdamadi2@gmail.com (H.R. Mirdamadi), ghayour@cc.iut.ac.ir (M. Ghayour). Physica E 52 (2013) 77–85