Pulsating flow induced parametric instabilities of a smart embedded micro-shell based on nonlocal piezoelasticity theory

Document Type : Article

Authors

Department of Mechanical Engineering, Faculty of Engineering, Bu-Ali Sina University, Hamedan, Iran.

Abstract

In this study, the dynamical instabilities of an embedded smart micro-shell conveying pulsating fluid flow is investigated based on nonlocal piezoelasticity theory and nonlinear cylindrical shell model. The micro-shell is surrounded by an elastic foundation which is suitable for both Winkler spring and Pasternak shear modules. The internal fluid flow is considered to be purely harmonic, irrotational, isentropic, Newtonian and incompressible and it is mathematically modeled using linear potential flow theory, time mean Navier Stokes equations and Knudsen number. For more reality of the micro-scale problem the pulsating viscous effects as well as the slip boundary condition are also taken into accounts. Employing the modified Lagrange equations of motion for open systems, the nonlinear coupled governing equations are achieved and consequently the instability boundaries are obtained using the Bolotin’s method. In the numerical results section, a comprehensive discussion is made on the dynamical instabilities of the system (such as divergence; flutter and parametric resonance). It is found that applying positive electric potential field will improve the stability of the system as an actuator or as a vibration amplitude controller in the Micro Electro Mechanical Systems.

Keywords

Main Subjects


1. Gao, J. and Xu, B. Applications of nanomaterials inside cells", Nano Today, 4(1), pp. 37{51 (2009). 2. Kong, J., Franklin, N.R., Zhou, C., Chapline, M.G., Peng, S., Cho, K., and Dai, H. Nanotube molecular wires as chemical sensors", Science, 287(5453), pp. 622{625 (2000). 3. Dharap, P., Li, Z., Nagarajaiah, S., and Barrera, E.V. Nanotube _lm based on single-wall carbon nanotubes for strain sensing", Nanotechnology, 15(3), p. 379 (2004). 4. Ashley, H. and Haviland, G. Bending vibrations of a pipeline containing owing uid", J. Appl. Mech., 72(1), pp. 229{232 (1950). 5. Paidoussis, M.P., Fluid-Structure Interactions: Slender Structures and Axial Flow, 1, Academic Press, London, England (1998). 6. Amabili, M., Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, Parma, Italy (2008). 7. Reddy, J.N. and Wang, C.M., Dynamics of Fluid Conveying Beams: Governing Equations and Finite Element Models, Centre for O_shore Research and Engineering National University of Singapore (2004). 8. Pellicano, F. and Amabili, M. Dynamic instability and chaos of empty and uid-_lled circular cylindrical shells under periodic axial loads", J. Sound Vib., 293(1), pp. 227{252 (2006). 9. Sadeghi, M.H. and Karimi-Dona, M.H. Dynamic behavior of a uid conveying pipe subjected to a moving sprung mass: an FEM-state space approach", Int. J. PressVessels Pip., 88, pp. 31{123 (2011). 10. Gu, J., Ma, T., and Menglan, D. E_ect of aspect ratio on the dynamic response of a uid-conveying pipe using the Timoshenko beam model", Ocean Eng, 114, pp. 185{191 (2016). 11. Kamm, R.D. and Pedley, T.J. Flow in collapsible tubes: a brief review", J. Biomech. Eng., 111, pp. 177{179 (1989). 12. Paidoussis, M.P., Fluid-Structure Interactions: Slender Structures and Axial Flow, 2, Academic Press (2003). 13. Yan, Y., Wang, W.Q., and Zhang, L.X. Dynamical behaviors of uid-conveyed multi walled carbon nanotubes", Appl. Math. Modell., 33, pp. 1430{1440 (2009). 14. Kuang, Y.D., He, X.Q., Chen, C.Y., and Li, G.Q. Analysis of nonlinear vibrations of double-walled carbon nanotubes conveying uid", Int. J. Comput. Mater. Sci. Surf. Eng., 45, pp. 875{880 (2009). 15. Ghorbanpour Arani, A., Shajari, A.R., Amir, S., and Atabakhshian, V. Nonlinear uid-induced vibration and instability of an embedded piezoelectric polymeric microtube using nonlocal elasticity theory", J. Mech. Eng. Sci., 227(12), pp. 2870{2885 (2013). 16. Ghorbanpour Arani, A., Shajari, A.R., Atabakhshian, V., Amir, S., and Loghman, A. Nonlinear dynamical response of embedded uid-conveyed micro-tube reinforced by BNNTs", Compos. Part B-Eng., 44(1), pp. 424{432 (2013). 17. Ghorbanpour Arani, A. and Hashemian, M. Surface stress e_ects on dynamic stability of double-walled boron nitride nanotubes conveying viscose uid based on nonlocal shell theory", Sci. Iran., 20(6), pp. 2356{ 2374 (2013). 18. Ghorbanpour Arani, A., Khoddami Maraghi, Z., and Haghparast, E. The uid structure interaction e_ect on the vibration and instability of a conveyed doublewalled boron nitride nanotube", Sci. Iran., 22(2), pp. 436{447 (2015). 19. Atabakhshian, V., Shoshtari, A.R., and Karimi, M. Electro-thermal vibration of a smart coupled nanobeam system with an internal ow based on nonlocalel asticity theory", Physica B: Condensed Matter, 456, pp. 375{382 (2015). 20. Paidoussis, M.P. and Issid, N.T. Dynamic stability of pipes conveying uid", J. Sound. Vib., 33(3), pp. 267{294 (1974). 21. Panda, L.N. and Kar, R.C. Nonlinear dynamics of a pipe conveying pulsating uid with combination, principal parametric and internal resonances", Journal of Sound and Vibration, 309, pp. 375{406 (2008). V. Atabakhshian and A. Shooshtari/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 730{744 743 22. Azrar, A., Azrar, L., and Aljinaidi, A.A. Numerical modeling of dynamic and parametric instabilities of single-walled carbon nanotubes conveying pulsating and viscous uid", Compos. Struct, 125(8), pp. 127{ 143 (2015). 23. Liang, F. and Su, Y. Stability analysis of a singlewalled carbon nanotube conveying pulsating and viscous uid with nonlocal e_ect", Appl. Math. Model., 37, pp. 6821{6828 (2013). 24. Da, H.L., Wang, L., Qian, Q., and Ni, Q. Vortexinduced vibrations of pipes conveying pulsating uid", Ocean. Eng., 77, pp. 12{22 (2014). 25. Wang, L. A further study on the non-linear dynamics of simply supported pipes conveying pulsating uid", Int. J. Non. Linear Mech., 44, pp. 115{121 (2009). 26. Yang, K.S., Cheng, Y.C., Liu, M.C., and Shyu, J.C. Micro pulsating heat pipes with alternate microchannel widths", Appl. Therm. Eng., 83, pp. 131{138 (2015). 27. Tubaldi, E., Amabili, V., and Pa_doussis, M.P. Fluidstructure interaction for nonlinear response of shells conveying pulsatile ow", J. Sound. Vib., 371, pp. 252{276 (2016). 28. Tubaldi, E., Amabili, M., and Paidoussis, M.P. Nonlinear dynamics of shells conveying pulsatile ow with pulse-wave propagation: Theory and numerical results for a single harmonic pulsation", J. Sound Vib., 396, pp. 217{245 (2017). 29. Ra_i-Tabar, H., Ghavanloo, E., and Fazelzadeh, S.A. Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures", Physics Reports, 638, pp. 1{97 (2016). 30. Mercan, K. and Civalek, O. DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix", Compos. Struct., 143, pp. 300{309 (2016). 31. Akgoz, B. and Civalek, O. Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory", Acta Astronaut, 119, pp. 1{12 (2016). 32. Civalek,  O. and Demir, C_ . A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal _nite element method", Appl. Math. Comput., 289, pp. 335{352 (2016). 33. Ghorbanpour Arani, A., Atabakhshian, V., Loghman, A., Shajari, A.R., and Amir, S. Nonlinear vibration of embedded SWBNNTs based on nonlocal Timoshenko beam theory using DQ method", Physica B, 407, pp. 2549{2555 (2012). 34. Alibeigi, B., Beni, Y.T., and Mehralian, F. On the thermal buckling of magneto-electro-elastic piezoelectric nanobeams", Eur. Phys. J. Plus., 133, pp. 133{ 138 (2018). 35. Institute of Electrical and Electronics Engineers, Standard on Piezoelectricity, Std, IEEE, New York (1978). 36. Ding, H.J. and Chen, W.Q. Three dimensional problems of piezoelasticity", Nova Science, New York (2001). 37. Eringen, A.C. Nonlocal polar elastic continua", INT. J. ENG. SCI., 10(1), pp. 1{16 (1972). 38. Eringen, A.C., Nonlocal Continuum Field Theories, Springer-Verlag, New York (2002). 39. Eringen, A.C. On di_erential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, pp. 4703{4710 (1983). 40. Ke, L.L., Wang, Y.Sh., and Wang, Zh.D. Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory", Compos. Struct., 94(6), pp. 2038{ 2047 (2012). 41. Han, J.H. and Lee, I. Analysis of composite plates with piezoelectric actuators for vibration control using layerwise displacement theory", Compos. Part B-Eng., 29(5), pp. 621{632 (1998). 42. Ke, L.L., Wang, Y.Sh., and Wang, Zh.D. Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory", Compos. Struct., 94(6), pp. 2038{ 2047 (2012). 43. Kurylov, Y. and Amabili, M. Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with di_erent boundary conditions", J. Sound. Vib., 329(9), pp. 1435{1449 (2010). 44. Alinia, M.M. and Ghannadpour, S. Nonlinear analysis of pressure loaded FGM plates", Compos. Struct., 88(3), pp. 354{359 (2009). 45. Yang, J., An Introduction to the Theory of Piezoelectricity, 9th Ed., Springer, Lincoln (2005). 46. Fox, R.W., Pritchard, P.J., and McDonald, A.T., Introduction to Fluid Mechanics, 4th Ed., Wiley, New York, USA (2008). 47. Paidoussis, M.P., Misra, A.K., and Chan, S.P. Dynamics and stability of coaxial cylindrical shells conveying viscous uid", J. Appl. Mech-T., ASME., 52(2), pp. 389{396 (1985). 48. Karniadakis, G., Beskok, A., and Aluru, N., Micro Flows Nano Flows: Fundamentals and Simulation, Springer-Verlag (2005). 49. Rashidi, V., Mirdamadi, H.R., and Shirani, E. A novel model for vibrations of nanotubes conveying nanoow", Comput. Mater. Sci., 51, pp. 347{352 (2012). 50. Shokouhmand, H., Isfahani, A.H.M., and Shirani, E. Friction and heat transfer coe_cient in micro and nano channels _lled with potous media for wide range of Knudsen number", Int. Comm. Heat Mass, 37, pp. 890{894 (2010). 51. Irschik, H. and Holl, H. The equations of Lagrange written for a non-material volume", Acta Mech, 153, pp. 231{248 (2002). 744 V. Atabakhshian and A. Shooshtari/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 730{744 52. Yang, J., Ke, L.L., and Kitipornchai, S. Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", Physica E, 42, pp. 1727{1735 (2010). 53. Bolotin, V.V., The Dynamic Stability of Elastic Systems, Holden-Day, Inc, San Francisco, USA (1964). 54. Amabili, M. and Graziera, R. Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass. Part ii: Shells containing or immersed in axial ow", J. Fluid. Struct., 16(1), pp. 31{51 (2002). 55. Mohammadi, K., Rajabpour, A., and Ghadiri, M. Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous uid using molecular dynamics simulation", Comp. Mater. Sci., 148, pp. 104{115 (2018).