2016
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Thermomechanical properties of polymer nanocomposites reinforced with randomly distributed silica nanoparticles Micromechanical analysis
2
2
A threedimensional micromechanicsbased analytical model is developed to study thermomechanical properties of polymer composites reinforced with randomly distributed silica nanoparticles. Two important factors in nanocomposites modeling using micromechanical models are nanoparticle arrangement in matrix and interphase effects. In order to study these cases, representative volume element (RVE) of nanocomposites is extended to c×r×h nanocells in three dimensions and consists of three phases including nanoparticles, polymer matrix and interphase between the nanoparticles and matrix. Nanoparticles are surrounded by the interphase in all composites. Effects of volume fraction, aspect ratio and size of nanoparticle on the effective thermomechanical response of the nanocomposite are studied. Also, the effects of polymer matrix properties and interphase including its elastic modulus and thickness are theoretically investigated in detail. It is revealed that when nanoparticles are randomly distributed in the matrix and interphase effects are considered, the results of present micromechanical model are in very good agreement with experimental data.
1

1
8


Reza
Ansari
Department of Mechanical Engineering, University of Guilan, Rasht, I. R. Iran
Department of Mechanical Engineering, University
Iran


Mohammad Kazem
Hassanzadeh Aghdam
Department of Mechanical Engineering, University of Guilan, Rasht, I. R. Iran
Department of Mechanical Engineering, University
Iran
Interphase
Micromechanics
Nanocomposite
Random distribution
Thermomechanical properties
[[1] B.M. Novak: Hybrid nanocomposite materialsbetween inorganic glasses and organic polymers, Adv Mater 5 (1993) 422433.##[2] H.L. Frisch, J.E. Mark: Nano composites prepared by threading polymer chains through zeolites, mesoporous silica, or Silica nanotubes, Chem Mater 8 (1996) 17351738. ##[3] B. Wetzel, F. Haupert, M.Q. Zhang: Epoxy nanocomposites with high mechanical and tribological performance, Compos Sci Technol 63 (2003) 2055–2067.##[4] H. Wang, Y. Bai, S. Liu, J. Wu, C.P. Wong: combinedeffects of silica filler and its interface in epoxy resin, ##Acta Mater 50 (2002) 4369–4377.##[5] J.N. Coleman, M. Cadek, R. Blake, V. Nicolosi,K.P. Ryan, C. Belton, A. Fonseca, J.B. Nagy, Y.K. Gun'ko, W. J. Blau: Highperformance nanotubereinforcedplastics: understanding the mechanism of strength increase, Adv Func Mater 14 (2004) 791–798.##[6] M. Izadi, M.M. Shahmardan, A. Behzadmehr, A.M.Rashidi, A. Amrollahi: Modeling of effective ##thermal conductivity and viscosity of carbonstructured nanofluid, Trans Phenom in Nano Micro scale 3 (2015) 113.##[7] G.D. Seidel, D.C. Lagoudas: A micromechanics model for the electrical conductivity of nanotubepolymer nanocomposites, J Compos Mater 43(2009) 917–941.##[8] H. Liu, L.C. Brinson: Reinforcing efficiency of nanoparticles: A simple comparison for polymer nanocomposites, Compos Sci Technol 68 (2008) 15021512.##[9] L.S. Schadler, L.C. Brinson, W.G. Sawyer: Polymer Nanocomposites: A Small Part of the Story, J Miner Metal Mater Soc 59 (2007) 5360.##[10] M. Avella, F. Bondioli, V. Cannillo, M.E. Errico, A.M. Ferrari, B. Focher, M. Malinconico, T. Manfredini, M. Montorsi: Preparation, characterisation and computational study of poly (ecaprolactone) based nanocomposites, Mater Sci Technol 20 (2004a) 1340–1344.##[11] M. Avella, F. Bondioli, V. Cannillo, S. Cosco, M.E. Errico, A.M. Ferrari, B. Focher, M. Malinconico: Properties/structure relationships in innovativePCL–SiO2 nanocomposites, Macromol Symp 218 (2004b) 201–210.##[12] L.S. Schadler: Designed Interfaces in Polymer Nanocomposites: A Fundamental Viewpoint, MRS Bulletin 32 (2007) 335340.##[13] R.A. Vaia, H.D. Wagner: Framework for Nanocomposites, Mater Today 7 (2004) 3237.##[14] J.S. Snipes, C.T. Robinson, S.C. Baxter: Effects of scale and interface on the threedimensional micromechanics of polymer nanocomposites, J Compos Mater 45 (2011) 25372546.##[15] S.C. Baxter, C.T. Robinson: Pseudopercolation: Critical volume fractions and mechanical percolation in polymer nanocomposites, Compos Sci Technol 71 (2011) 1273–1279.##[16] M.J. Mahmoodi, M.M. Aghdam: Damage analysis of fiber reinforced Tialloy subjected to multiaxial loading—A micromechanical approach, Mater Sci Eng A 528 (2011) 79837990.##[17] S.R . Falahatgar, M. Salehi, M.M. Aghdam: Nonlinear viscoelastic response of unidirectional fiber reinforced composites in offaxis loading, J Reinf Plast Compos 28 (2009)1793–1812.##[18] R.P. Nimmer, R.J. Bankert, E.S. Russell, G.A. Smith, P.K. Wright: Micromechanical modeling offiber/matrix interface effects in transversely loaded SiC/Ti64 metal matrix composite, J Compos Technol Res 13 (1991) 313.##[19] R. HajAli, J. Aboudi: Nonlinear micromechanical formulation of the high fidelity generalized method of cells, Int J Solids Struct 46 (2009) 25772592.##[20] T.W. Chou, S. Nomura, M. Taya: A selfconsistent approach to the elastic stiffness of shortfiber composites, J Compos Mater 14 (1980) 178188.##[21] J.C. Halpin, S.W. Tsai: Stiffness and expansion estimates for oriented short fiber composites, J Compos Mater 3 (1969) 732–734.##[22] T. Mori, K. Tanaka: Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metal 21 (1973) 571574.##[23] J.I. Weon, H.J. Sue: Effects of clay orientation and aspect ratio on mechanical behavior of nylon6 nanocomposite, Polymer 46 (2005) 6325–6334.##[24] H.W. Wang, H.W. Zhou, R.D. Peng, J. Leon Mishnaevsky: Nanoreinforced polymer composites: 3D FEM modeling with effective interface concept, Compos Sci Technol 71 (2011) 980–988.##[25] S. Dhala, M.C. Ray:Micromechanics of piezoelectric fuzzy fiberreinforced composite, Mech Mater 81 (2015) 1–17.##[26] R.D. Peng, H.W. Zhou, H.W. Wang, J. Leon Mishnaevsky: Modeling of nanoreinforced polymer composites: Microstructure effect on Young’s modulus, Comput Mater Sci 60 (2012) 19–31.##[27] B. Mortazavi, J. Bardon, S. Ahzi: Interphase effect on the elastic and thermal conductivity response of polymer nanocomposite materials: 3D finite element study, Comput Mater Sci 69 (2013) 100–106.##[28] S. Ajori, R. Ansari, M. Mirnezhad: Mechanical properties of defective γgraphyne using molecular dynamics simulations, Mater Sci Eng: A 561 (2013) 34–39.##[29] R. Ansari, S. Rouhi, S. Ajori: Elastic properties and large deformation of twodimensional silicene nanosheets using molecular dynamics, Super Microstruct 65 (2014) 64–70.##[30] M.M. Shokrieh, R. Rafiee: Development of a full range multiscale model to obtain elastic properties of CNT/polymer composites, Iran Polymer J 21 (2012) 397402.##[31] M.J. Mahmoodi, M.M. Aghdam, M. Shakeri: The effects of interfacial debonding on the elastoplastic response of unidirectional silicon carbide–titanium composites, J Mech Eng Sci 223 (2010) 259269.##[32] Z. Wanga, J. Lua, Y. Li, S.Y. Fu, S. Jiang, X. Zhao: Studies on thermal and mechanical properties of ##PI/SiO2 nanocomposite films at low temperature, Composites A 37 (2006) 74–79.##[33] G.M. Odegard, T.C. Clancy, T.S. Gates: Modeling of the mechanical properties of nanoparticle/polymer composites, Polymer 46 (2005) 553–562.##[34] E. Kontou, G. Anthoulis:The effect of silica nanoparticles on the thermomechanical properties of polystyrene, J. Appl. Polym. Sci 105 (2007) 1723–1731.##]
Effect of Inserting Coiled Wires in Tubes on the Fluid Flow and Heat Transfer Performance of Nanofluids
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2
In the present study, numerical study of Al2O3water nanofluid flow in different coiled wire inserted tubes are performed to investigate the effects of inserting coiled wires in tubes on the fluid dynamic and heat transfer performance ofv nanofluids. The numerical simulations of nanofluids are performed using two phase mixture model. The flow regime and the wall boundary conditions are assumed to be laminar and constant heat flux respectively. The effects of inserting coiled wires in tubes on different parameters such as heat transfer coefficient, pressure drop, temperature distribution, velocity distribution and secondary flows are presented and discussed. The results show that using coiled wire in tubes leading to increase in about 13.44% but increase the Δp about 14.66% with respect to the flow without nanofluid and coiled wire. Similarly, using nanofluid leading to increase in about 5.52% but increase the Δp about 8.92%. Finally, using both of the mentioned heat transfer enhancement mechanisms leading to increase in about 17.51% but increase the value of Δp about 22.86%.
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9
16


Hamed
Safikhani
Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, I. R. Iran
Department of Mechanical Engineering, Faculty
Iran
hsafikhani@araku.ac.ir


Alireza
Zare Mehrjardi
Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, I. R. Iran
Department of Mechanical Engineering, Faculty
Iran
alirezazare7351@gmail.com


Maryam
Safari
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, I. R. Iran
Department of Mechanical Engineering, Isfahan
Iran
m_safari13671@yahoo.com
CFD
Coiled wires
Mixture model
Nanofluid
Two phase model
[[1] S. Das, N. Putra, P. Thiesen, R. Roetzel: Temperature dependence of thermal conductivity enhancement for nanofluids, J. Heat Transfer 125 (2003) 567574. ##[2] S. Murshed, K. Leong, C. Yang: A combined model for the effective thermal conductivity of nanofluids, Appl. Therm. Eng 29 (2009) 24772483. ##[3] T. Teng, Y. Hung, T. Teng, H. Mo, H. Hsu: The effect of alumina/water nanofluid particle size on thermal conductivity, Appl. Therm. Eng 30 (2010) 22132218. ##[4] H. Safikhani, A. Abbassi: Effects of tube flattening on the fluid dynamic and heat transfer performance of nanofluid flow, Adv. Powder Technolog 25 (3) (2014) 11321141. ##[5] H. Safikhani, A. Abbassi, A. Khalkhali, M. Kalteh: Multiobjective optimization of nanofluid flow in flat tubes using CFD, Artificial Neural Networks and genetic algorithms, Adv. Powder Technolog 25(5) (2014) 16081617. ##[6] M. Kalteh, A. Abbassi, M. SaffarAvval, J. Harting: Eulerian–Eulerian twophase numerical simulation of nanofluid laminar forced convection in a microchannel, Int. J. Heat Fluid Flow 32 (2011) 107–116. ##[7] R. Lotfi, Y. Saboohi, A. Rashidi: Numerical study of forced convective heat transfer of Nanofluids: Comparison of different approaches, Int. Commun. Heat Mass Transfer 37 (2010) 74–78. ##[8] M. R. Salimpour, H. Gholami: Effect of inserting coiled wires on pressure drop of R404A condensation, International Journal of Refrigeration 40 (2014) 2430. ##[9] R. Kumar, K. N. Agrawal, S. N. Lal, H. K. Varma: An experimental study on condensation enhancement of R22 by the turbulence promoter. ASHRAE Trans. 111 (2005) 1825. ##[10] V. Hejazi, M. A. AkhavanBehabadi, A. Afshari: Experimental investigation of twisted tape inserts performance on condensation heat transfer enhancement and pressure drop, Int. Commun. Heat Mass Transfer 37 (2010) 13761387. ##[11] M. R. Salimpour, S. Yarmohammadi: Effect of twisted tape inserts on pressure drop during R404A condensation. Int. J. Refrigeration 35 (2012a) 263269. ##[12] M. R. Salimpour, S. Yarmohammadi: Heat transfer enhancement during R404A vapor condensation in swirling flow. Int. J. Refrigeration 35 (2012b) 20142021. ##[13] K. N. Agrawal, A. Kumar, M. A. AkavanBehabadi, H. K. Varma: Heat transfer augmentation by coiled wire inserts during forced convection condensation of R22 inside horizontal tubes. Int. J. Multiphase Flow 24 (1998) 635650. ##[14] M. A. Akhavan  Behabadi, M. R. Salimpoor, R . Kumar , K. N. Agrawal: Augmentation of forced convection condensation heat transfer inside a horizontal tube using spiral spring inserts, J. Enhanc. Heat Transfer 12 (2005) 373384. ##[15] M. A. AkhavanBehabadi, M.R. Salimpour, V. A. Pazouki: Pressure drop increase of forced convective condensation inside coiled wire inserted tube. Int. Commun. Heat Mass Transfer 35 (2008) 12201226. ##[16] S. Eiamsaard, K. Kiatkittipong: Heat transfer enhancement by multiple twisted tape inserts and TiO2/water nanofluid, Appl. Therm. Eng 70 (2014) 896924. ##[17] M. Manninen, V.Taivassalo, S. Kallio: On the mixture model for multiphase flow VTT Publications (1996). ##[18] L. Schiller, A. Naumann:A drag coefficient corre lation, Z. Ver. Deutsch. Ing 77 (1935) 318320. ##[19] B. Pak, Y. Cho Y: Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Exp. Heat Transfer 11 (1998) 151–170. ##[20] Y. Xuan, W. Roetzel: Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707. ##[21] N. Masoumi, N. Sohrabi, A. Behzadmehr: A new model for calculating the effective viscosity of nanofluids, J. Appl. Physics 42 (2009) 055501. ##[22] C. Chon, K. Kihm, S. Lee, S. Choi: Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity Enhancement, J. Appl. Physics 87 (2005) 153107 (3). ##[23] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy driven heat transfer enhancement in a two dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 36393653. ##[24] E. Ebrahimnia  Bajestan, H. Niazmand, W. Duangthongsuk, S. Wongwises: Numerical investigation of effective parameters in convective heat transfer of nanofluids flowing under a laminar flow regime, Int. J. Heat Mass Transfer 54 (2010) 4376–4388. ##[25] S. Mirmasoumi, A. Behzadmehr: Effect of nano particles mean diameter on mixed convection heat transfer of a nanofluid in a horizontal tube, Int. J. Heat Fluid Flow 29 (2008) 557566. ##[26] M. Shariat, A. Akbarinia, A. Hossein Nezhad, A. Behzadmehr, R. Laur: Numerical study of two phase laminar mixed convection nanofluid in elliptic ducts, Appl. Therm. Eng 31 (2011) 23482359.##]
Numerical Study of Bubble Separation and Motion Using Lattice Boltzmann Method
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2
In present paper acombination of threedimensional isothermal and twodimensional nonisothermal Lattice Boltzmann Method have been used to simulate the motion of bubble and effect of wetting properties of the surface on bubble separation. By combining these models, threedimensional model has been used in twodimension for decreasing the computational cost. Firstly, it has been ensured that the surface tension effect and Laplace law for twodensity ratio 50 and 1000 have been properly implemented. Secondly, by simulation of static droplet in different conditions wettability, integrity applied equations has been investigated.Thirdly, effect of governing dimensionless numbers such as Etvos number and Morton number on Reynolds number and terminal shape of bubble have been investigated.Different flow patterns in various dimensionless numbers have been obtained and by changing the dimensionless number, terminal change of bubble’s shape has been seen. Finally, the impact of wettability of surface on departure of bubble from wall under buoyancy force in different dimensionless numbers has been studied.
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17
27


Elham
Sattari
Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, I. R. Iran
Faculty of Mechanical Engineering, Babol
Iran
elham_sattari68@yahoo.com


Mojtaba
Aghajani Delavar
Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, I. R. Iran
Faculty of Mechanical Engineering, Babol
Iran
m.a.delavar@nit.ac.ir


Korosh
Sedighi
Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, I. R. Iran
Faculty of Mechanical Engineering, Babol
Iran
Etvos Number
Inamuro Model
Lattice Boltzmann method
Morton Number
Two Phase Flow
Wettability
[[1] AK. Gunstensen, DH. Rothman, S. Zaleski, G Zanetti.:Lattice Boltzmann model of immiscible fluids, Physical Review A 43 (1991) 43204327. ##[2] X. Shan, H. Chen: Lattice Boltzmann model for simulating flows with multiple phases and components, Physical Review E 47 (1993) 18151819. ##[3] M. R. Swift, W. Osborn, J. M. Yeomans: Lattice Boltzmann simulation of nonideal fluids, Physical Review Letters 75(1995) 830833. ##[4] T. Inamuro, T. Ogata, F. Ogino: Numerical simulation of bubble flows by the lattice Boltzmann method, Future Generation Computer Systems 20(2004) 959964. ##[5] T. Inamuro, T. Ogata, S. Tajima, N. Konishi: A lattice Boltzmann method for incompressible twophase flows with large density differences, Journal of Computational Physics 198 (2004) 628644. ##[6] T. Inamuro, S. Tajima, F. Ogino: Lattice Boltzmann simulation of droplet collision dynamics, International journal of heat and mass transfer 47(2004) 46494657. ##[7] B. Sakakibara, T. Inamuro: Lattice Boltzmann simulation of collision dynamics of two unequalsize droplets, International journal of heat and mass transfer 51(2008) 32073216. ##[8] T. Inamuro: Lattice boltzmann methods for viscouse fluid flows and twophase fluid flows, Computational Fluid Dynamics. Springer: India (2008) 316. ##[9] Y.Y. Yan, Y.Q. Zu : LBM simulation of interfacial behaviour of bubbles flow at low Reynolds number in a square microchannel, Computational Methods in Multiphase Flow V. New Forest: WIT Press, 2009. ##[10] M. Yoshino, Y. Mizutani: Lattice Boltzmann simulation of liquid–gas flows through solid bodies in a square duct, Mathematics and computers in simulation 72(2006) 264269. ##[11] Y. Yan, Y. Zu: A lattice Boltzmann method for incompressible twophase flows on partial wetting surface with large density ratio, Journal of Computational Physics 227 (2007) 763775. ##[12] Y.Y. Yan,Y.Q. Zu: Numerical modelling of bubble coalescence and droplet separation, in Computational Methods in Multiphase Flow IV (2007) 227237. ##[13] Y. Tanaka, Y. Washio, M. Yoshino, T. Hirata: Numerical simulation of dynamic behavior of droplet on solid surface by the twophase lattice Boltzmann method, Computers & Fluids 40 (2011) 6878. ##[14] Y. Tanaka, M. Yoshino, T. Hirata, Lattice Boltzmann simulation of nucleate pool boiling in saturated liquid, Communications in Computational Physics 9(2011)13471361. ##[15] T. Lee: Effects of incompressibility on the elimination of parasitic currents in the lattice Boltzmann equation method for binary fluids, Computers & Mathematics with Applications 58(2009) 987994. ##[16] T. Lee, C.L. Lin: A stable discretization of the lattice Boltzmann equation for simulation of incompressible twophase flows at high density ratio, Journal of Computational Physics206(2005) 1647. ##[17] T. Lee, L. Liu: Lattice Boltzmann simulations of micronscale drop impact on dry surfaces, Journal of Computational Physics 229(2010) 80458063. ##[18] H. Zheng, C. Shu, Y.T. Chew: A lattice Boltzmann model for multiphase flows with large density ratio, Journal of Computational Physics 218 (2006) 353371. ##[19] E. Sattari, M. A. Delavar, E. Fattahi, K. Sedighi: Numerical investigation the effects of working parameters on nucleate pool boilin, International Communications in Heat and Mass Transfer 59(2014)106113. ##[20] A. Briant, A. Wagner, J. Yeomans: Lattice Boltzman simulations of contact line motion. I. Liquidgas systems, Physical Review E 69(2004)031602. ##[21] S. M. Tilehboni, K. Sedighi, M. Farhadi, E. Fattahi: Lattice Boltzmann Simulation of Deformation and Breakup of a Droplet under Gravity Force Using Interparticle Potential Model, International Journal of EngineeringTransactions A: Basics 26(2013)781794. ##[22] T. Seta, K. Okui: Effects of truncation error of derivative approximation for twophase lattice Boltzmann method, Journal of Fluid Science and Technology 2(2007)139151. ##[23] H. Huang, D. T. Thorne, M. G. Schaap, M. C. Sukop: Proposed approximation for contact angles in ShanandChentype multicomponent multiphase lattice Boltzmann models, Physical Review E 76(2007)066701 16.##]
MHD Boundary Layer Flow and Heat Transfer of Newtonian Nanofluids over a Stretching Sheet with Variable Velocity and Temperature Distribution
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2
Laminar boundary layer flow and heat transfer of Newtonian nanofluid over a stretching sheet with the sheet velocity distribution of the form (UW=cXβ) and the wall temperature distribution of the form (TW=T∞+aXr ) for the steady magnetohydrodynamic (MHD) is studied numerically. The governing momentum and energy equations are transformed to the local nonsimilarity equations using the appropriate transformations. The set of ODEs are solved using Keller–Box implicit finitedifference method. The effects of several parameters, such as magnetic parameter, volume fraction of different nanoparticles (Ag, Cu, CuO, Al2O3 and TiO2), velocity parameter, Prandtl number and temperature parameter on the velocity and temperature distributions, local Nusselt number and skin friction coefficient are examined. The analysis reveals that the temperature profile increases with increasing magnetic parameter and volume fraction of nanofluid. Furthermore, it is found that the thermal boundary layer increases and momentum boundary layer decreases with the use of water based nanofluids as compared to pure water. At constant volume fraction of nanoparticles, it is also illustrated that the role of magnetic parameter on dimensionless temperature becomes more effective in lower value.
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28
40


Payman
Elyasi
Mechanical Engineering Department,Faculty of Mechanical Engineering, Shahrekord University, Shahrekord, I. R. Iran
Mechanical Engineering Department,Faculty
Iran
paymanelyasi@gmail.com


Ali
Shateri
Mechanical Engineering Department,Faculty of Mechanical Engineering, Shahrekord University, Shahrekord, I. R. Iran
Mechanical Engineering Department,Faculty
Iran
Boundary Layer Flow
MHD
Nanofluid
Stretching Sheet
[[1] U.S. Choi: Enhancing thermal conductivity of fluids with nanoparticle Developments and Applications of NonNewtonian Flows 231 (1995) 99105. ##[2] S. Choi, Z. Zhang, W. Yu, F. Lockwood, E. Grulke: Anomalously thermal conductivity enhancement in nanotube suspensions, Journal of Applied Physics Letters 79 (2001) 2252–2254. ##[3] S. Z. Heris, M. N. Esfahany, S. Gh. Etemad :Experimental Investigation of Convective Heat Transfer of Al2O3/Water Nanofluid in Circular Tube, International Journal of Heat and Fluid Flow 28 (2006) 203–210. ##[4] M. Hojjat, S. Etemad, R. Bagheri: Laminar heat transfer of nonNewtonian nanofluids in a circular tube, Korean Journal of Chemical Engineering 27 (2010) 1391–1396. ##[5] B.C. Pak, Y. Cho: Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Experimental Heat Transfer 11 (1998) 151170. ##[6] Y. Xuan, Q. Li: Investigation on convective heat transfer and flow features of nanofluids, Journal of Heat Transfer 125 (2003) 151155. ##[7] A. Ahuja: Augmentation of heat transport in laminar flow of polystyrene suspensions, Journal of Applied Physics 46 (1975) 34083425. ##[8] J. Buongiorno: Convective transport in nanofluids, Journal of Heat Transfer 128 (2006) 240250. ##[9] MA Fadzilah, R Nazar, M. Arifin, I. Pop: MHD boundarylayer flow and heat transfer over a stretching sheet with induced magnetic field. Journal of Heat Mass Transfer 47 (2011) 155–162. ##[10] A. Ishak, R. Naza, I. Pop: MHD boundarylayer flow due to a moving extensible surface, Journal of Engineering Mathematics 62 (2008) 23–33. ##[11] A.V. Kuznetsov, D.A Nield: Natural convective boundarylayer flow of a nanofluid past a vertical plate, International Journal of Thermal Sciences 49 (2010) 243247. ##[12] N. Bachok, A.Ishak, I.Pop: Boundarylayer flow of nanofluids over a moving surface in a flowing fluid, International Journal of Thermal Sciences 49 (2010) 16631668. ##[13] W.Ibrahim, B. Shanker: Unsteady MHD boundarylayer flow and heat transfer due to stretching sheet in the presence of heat source or sink, International Journal of Computers & Fluids 70 (2012) 2128. ##[14] A. Ishak, R. Naza, I. Pop: Heat transfer over a stretching surface with variable heat flux in microplar fluids. Physics Letters A 5 (2008) 559–61. ##[15] A. Noghrehabadi, R. Pourrajab, M. Ghalambaz: Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature, International Journal of Thermal Sciences 54 (2012) 253261 ##[16] M. Narayana, P. Sibanda: Laminar flow of a nanoliquid film over an unsteady stretching sheet, International Journal of Heat and Mass Transfer 55 (2012) 75527560. ##[17] A. Aziz, W.A. Khan: Natural convective boundary layer flow of a nanofluid past a convectively heated vertical plate, International Journal of Thermal Sciences 52 (2012) 8390. ##[18] L. Zheng, C. Zhang, X. Zhang, J. Zhang: Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium, Journal of the Franklin Institute 350 (2013) 990–1007. ##[19] P. Rana, R. Bhargava: Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: A numerical study, Communications in Nonlinear Science and Numerical Simulation 17 (2012) 212–226. ##[20] A. Mahdy: Unsteady mixed convection boundary layer flow and heat transfer of nanofluids due to stretching sheet, Nuclear Engineering and Design 249 (2012) 248– 255. ##[21] K.V. Prasad, P.S. Pal Dulal, Datti: MHD powerlaw fluid flow and heat transfer over a nonisothermal stretching sheet, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 2178–2189. ## [22] H. Xu, S. Liao: Laminar flow and heat transfer in the boundarylayer of nonNewtonian fluids over a stretching flat sheet, Computers and Mathematics with Applications 57 (2009) 1425_1431. ##[23] N. Masoumi, N. Sohrabi, A.A. Behzadmehr: New model for calculating the effective viscosity of nanofluids, Journal of Physics D: Applied Physics 42 (2009) 055501–055506. ##[24] C.H. Chon, K.D. Kihm, S.P. Lee, S.U. Choi: Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement, Journal of Applied Physics Letters 87 (2005) 153107–153110. ##[25] H.A. Mintsa, G. Roy, C.T. Nguyen, D. Doucet: temperature dependent thermal conductivity data for waterbased nanofluids, International Journal of Thermal Sciences 48 (2009) 363–371. ##[26] C.T Nguyen., F. s Desgrange, G. Roy, T. Galanis, S. Boucher, H.A.n Mintsa: Temperature and particlesize dependent viscosity data for waterbased nanofluids–hysteresis phenomenon, International Journal of Heat Fluid Flow 28 (2009) 1492–1506. ##[27] Kh. Khanafer, K. Vafai: A critical synthesis of thermophysical characteristics of nanofluids, International Journal of Heat Mass Transfer 54 (2011) 4410–4428. ##[28] D.A.G. Bruggeman: Berechnung verschiedenerphysikalischer konstanten von heterogenen substanzen, I. Dielektrizitatskonstanten und leitfahigkeiten der mischkorper aus isotropen substanzen, Annals of Physics 24 (1935) 636– 679. ##[29] J.C. Maxwell Garnett : Colours in metal glasses and in metallic films, Philos. Trans. R. Soc. Lond. A 203 (1904). 385–420. ##[30] H.C. Brinkman: The viscosity of concentrated suspensions and solutions. Journal of Chemical Physics 20 (1952) 571–581. ##[31] S.E.B. Maiga, S.J. Palm, C.T.Nguyen, G. Roy, N. Galanis: Heat transfer enhancement by using nanofluids in forced convection flow, International Journal of Heat and Fluid Flow 26 (2005) 530–546. ##[32] E. AbuNada: Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step, International Journal of Heat Fluid Flow 29 (2008) 242–249. ##[33] A. Akbarinia, A. Behzadmehr: Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes, Journal of Applied Thermal Engineering 27 (2007) 1327–1337. ##[34] T. Cebeci, J. Cousteix: Modeling and Computation of BoundaryLayer Flows, Second Edition, Horizons Publishing Inc., Long Beach, CaliforniaSpringerVerlag, (2005). ##[35] A. Ishak, R. Nazar, I. Pop: Boundary layer flow and heat transfer over an unsteady stretching vertical surface Meccanica 44 (2009) 369–375. ##[36] M.E.Ali: Heat transfer characteristics of a continuous stretching surface, Journal of Heat Mass Transfer 29 (1904) 227–234. ##[37] L.J. Grubka, K.M. Bobba: Heat transfer characteristics of a continuous, stretching surface with variable temperature, ASME Journal of Heat Transfer 107 (1985) 248–250.##]
Nanofluid Thermal Conductivity Prediction Model Based on Artificial Neural Network
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2
Heat transfer fluids have inherently low thermal conductivity that greatly limits the heat exchange efficiency. While the effectiveness of extending surfaces and redesigning heat exchange equipments to increase the heat transfer rate has reached a limit, many research activities have been carried out attempting to improve the thermal transport properties of the fluids by adding more thermally conductive solids into liquids. In this study, new model to predict nanofluid thermal conductivity based on Artificial Neural Network. A twolayer perceptron feedforward neural network and backpropagation LevenbergMarquardt (BPLM) training algorithm were used to predict the thermal conductivity of the nanofluid. To avoid the preprocess of network and investigate the final efficiency of it, 70% data are used for network training, while the remaining 30% data are used for network test and validation. Fe2O3 nanoparticles dispersed in waster/glycol liquid was used as working fluid in experiments. Volume fraction, temperature, nano particles and base fluid thermal conductivities are used as inputs to the network. The results show that ANN modeling is capable of predicting nanofluid thermal conductivity with good precision. The use of nanotechnology to enhance and improve the heat transfer fluid and the cost is exorbitant.It can play a major role in various industries, particularly industries that are involved in that heat.
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Ali
Hosseinian naeini
Department of Chemical Engineering, Islamic Azad University,Central Tehran Branch, Tehran, I. R. Iran
Department of Chemical Engineering, Islamic
Iran


Jafar
Baghbani Arani
Chemical Engineering Department, Kashan University, Kashan, I. R. Iran
Chemical Engineering Department, Kashan University
Iran
jafar.baghbani@gmail.com


Afsaneh
Narooei
Department of Material Engineering, University of Sistan and Baluchestan, Zahedan, I. R. Iran
Department of Material Engineering, University
Iran
a.narooei91@gmail.com


Reza
Aghayari
Daneshestan Institute Of Higher Education, Saveh, Iran
Daneshestan Institute Of Higher Education,
Iran


Heydar
Maddah
Department of Chemistry, Sciences Faculty, Arak Branch, Islamic Azad University, Arak, I. R. Iran
Department of Chemistry, Sciences Faculty,
Iran
Nanofluid
Neural Network
thermal conductivity
[[1] B. Wang, L. Zhou, X. Peng: A Fractal Model for Predicting the Effective Thermal Conductivity of Liquid with Suspension of Nanoparticles, Int. J. Heat Mass Tran 46 (2003) 26652672. ##[2] P. Keblinski, S. R. Phillpot, S. U. S. Choi, J. A. Eastman: Mechanisms of Heat Flow in Suspensions of NanoSized Particles (Nanofluids), Int. J. Heat Mass Tran 45 (2002) 855863. ##[3] H. Masuda, A. Ebata, K. Teramae, N. Hishinuma: Alteration of Thermal Conductivity and Viscosity of Liquid by Dispersing UltraFine Particles (Dispersion of γAl2O3, SiO2, and TiO2) (1993). ##[4] S. U. S. Choi: Enhancing Thermal Conductivity of Fluids with Nanoparticles, Developments and Applications of NonNewtonian Flows (1995), D. A. Siginer, H. P. Wang: The American Society of Mechanical Engineers, New York, FEDVol. 231 / MD(66) 99105. UltraFine Particles),” Netsu Bussei 4(4) 227233. ##[5] R. Chein, J. Chuang: Experimental Microchannel Heat Sink Performance Studies using Nanofluids, Int. J. Therm. Sci 46(2007) 57 66. ##[6] J. Lee, I. Mudawar: Assessment of the Effectiveness of Nanofluids for SinglePhase and TwoPhase Heat Transfer in MicroChannels, Int. J. Heat Mass Tran 50 (2007) 452463. ##[7] J. A. Eastman, S. U. S. Choi, S. Li, W. Yu, L. J. Thompson: Anomalously Increased Effective Thermal Conductivities of Ethylene GlycolBased Nanofluids Containing Copper Nanoparticles, Appl. Phys. Lett 78 (2001) 718720. ##[8] W. Yu, D. M. France, J. L.Routbort, S. U. S. Choi: Review and Comparison of Nanofluid Thermal Conductivity and Heat Transfer Enhancements, Heat Transfer Eng. 29 (2008) 432460. ##[9] J. M. Romano, J. C. Parker, Q. B. Ford: Application Opportunities for Nanoparticles Made from the Condensation of Physical Vapors, Adv. Pm. Part. (1997) 1213. ##[10] C. H. Chon, K. D. Kihm, S. P. Lee, S. U. S. Choi: Empirical Correlation Finding the Role of Temperature and Particle Size for Nanofluid (Al2O3), (2005). ##[11] S. K. Das, N. Putra, P. Thiesen, W. Roetzel: Temperature Dependence of Thermal Conductivity Enhancement for Nanofluids, J. Heat Transfer 125 (2003) 567574. ##[12] C. H. Li, G. P. Peterson: Experimental Investigation of Temperature and Volume Fraction Variations on the Effective Thermal Conductivity of Nanoparticle Suspensions (Nanofluids), J. Appl. Phys. 99 (2006) 084314. ##[13] B. C. Pak, Y. I. Cho: Hydrodynamic and Heat Transfer Study of Dispersed Fluids with Submicron Metallic Oxide Particles, Exp. Heat Transfer 11(1998) 151 170. ##[14] K. S. Hwang, S. P. Jang, S. U. S. Choi: Flow and Convective Heat Transfer Characteristics of WaterBased Al2O3 (2009). ##[15] S. Z. Heris, M. N. Esfahany, S. Etemad: Experimental Investigation of Convective Heat Transfer of Al Nanofluids in Fully Developed Laminar Flow Regime, Int. J. Heat Mass Tran. 52 (2007) 193199. ##[16] R.L. Hamilton, O.K. Crosser: Thermal conductivity of heterogeneous two component systems, I & EC Fundamentals 1 (1962) 182–191. ##[17] F.J. Wasp: Solid–Liquid Flow Slurry Pipeline Transportation, Trans. Tech. Pub., Berlin (1977). ##[18] J.C. MaxwellGarnett: Colours in metal glasses and in metallic films, Philos. Trans. Roy. Soc. A 203 (1904) 385–420. ##[19] D.A.G. Bruggeman: Berechnung Verschiedener Physikalischer Konstanten von Heterogenen Substanzen, I. Dielektrizitatskonstanten und Leitfahigkeiten der Mischkorper aus Isotropen Substanzen, Annalen der Physik. Leipzig 24 (1935) 636–679. ##[20] B.X. Wang, L.P. Zhou, X.F. Peng: A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles, Int. J. Heat Mass Transfer 46 (2003) 2665–2672. ##[21] W. Yu, S.U.S. Choi: The role of interfacial layer in the enhanced thermal conductivity of nanofluids: A renovated Maxwell model, J. Nanoparticles Res. (2003) 167–171. ##[22] L. Xue, P. Keblinski, S.R. Phillpot, S.U.S. Choi, A.J. Eastman: Effect of liquid layering at the liquid–solid interface on thermal transport, Int. J. Heat Mass Transfer 47 (2004) 4277–4284. ##[23] D.H. Kumar, H.E. Patel, V.R.R. Kumar, T. Sundararajan, T. Pradeep, S.K. Das: Model for conduction in nanofluids, Phys. Rev. Lett. 93 (2004) 1443011– 1443014. ##[24] R. Prasher, P. Bhattacharya, P.E. Phelan: Brownianmotionbased convectiveconductive model for the effective thermal conductivity of nanofluid, ASME J.Heat Transfer 128 (2006) 588–595. ##[25] H.E. Patel, T. Pradeep, T. Sundarrajan, A. Dasgupta, N. Dasgupta, S.K. Das: A microconvection model for thermal conductivity of nanofluid, Pramana–J. Phys. 65 (2005) 863–869. ##[26] S.K. Das, et al: Reply, Phys. Rev. Lett. 95 (2005) 019402.##]
Investigation of two phase unsteady nanofluid flow and heat transfer between moving parallel plates in the presence of the magnetic field using GM
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2
In this paper, unsteady two phase simulation of nanofluid flow and heat transfer between moving parallel plates, in presence of the magnetic field is studied. The significant effects of thermophoresis and Brownian motion have been contained in the model of nanofluid flow. The three governing equations are solved simultaneously via Galerkin method (GM). Comparison with other works indicates that this method is very applicable to solve these problems. The semi analytical analysis is accomplished for different governing parameters in the equations e.g. the squeeze number, Eckert number and Hartmann number. The results showed that skin friction coefficient value increases with increasing Hartmann number and squeeze number in a constant Reynolds number. Also, it is shown that the Nusselt number is an incrementing function of Hartmann number while Eckert number is a reducing function of squeeze number .This type of results can help the engineers to make better and researchers to investigate faster and easier.
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53


Nima
Hedayati
Babol University of Technology, Department of Mechanical Engineering, Babol, I. R. Iran
Babol University of Technology, Department
Iran
nima.hedayati883@gmail.com


Abbas
Ramiar
Babol University of Technology, Department of Mechanical Engineering, Babol, I. R. Iran
Babol University of Technology, Department
Iran
aaramiar@stu.nit.ac.ir
Brownian
Eckert number
Galerkin method (GM)
Hartmann number
Nanofluid
Thermophoresis
[[1] M.Sheikholeslami , H.R.Ashorynejad, D.D. Ganji, A. Kolahdooz: Investigation of Rotating MHD Viscous Flow and Heat Transfer between Stretching and Porous Surfaces Using Analytical Method, Hindawi Publishing Corporation Mathematical Problems in Engineering (2011). ##[2] M. Sheikholeslami, H.R. Ashorynejad, D.D.Ganji, Yıldırım A: Homotopy perturbation method for threedimensional problem of condensation film on inclined rotating disk, Scientia Iranica B 19 (2012) 437–442. ## [3] D.D.Ganji, H.B.Rokni, M.G.Sfahani, S.S.Ganji : Approximate traveling wave solutions for coupled shallow water. Advances in Engineering Software 41 (2010) 956–961. ##[4] M. Keimanesh, M.M.Rashidi, A.J. Chamkha,R.Jafari: Study of a third grade non Newtonian fluid flow between two parallel plates using the multistep differential transform method, Computers and Mathematics with Applications 62 (2011) 2871–2891. ##[5] M.Hatami, Kh.Hosseinzadeh, G. Domairry, M.T.Behnamfar: Numerical study of MHD twophase Couette flow analysis for fluidparticle suspension between moving parallel plates. Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 22382245. ##[6] K.Khanafer , K.Vafai , M. Lightstone: Buoyancy driven heat transfer enhancement in a twodimensional enclosure utilizing nanofluids. International Journal of Heat and Mass Transfer 46 (2003) 3639–3653. ##[7] E.AbuNada ,Z. Masoud , A.Hijazi:Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids, International Communications in Heat and Mass Transfer 35 (2008) 657–665. ##[8] M.M. Rashidi ,S.Abelman, N. Freidooni Mehr: Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid, International Journal of Heat and Mass Transfer 62 (2013) 515–525. ##[9] M. Sheikholeslami ,Sh. Abelman, D.D.Ganji: Numerical simulation of MHD nanofluid flow and heat transfer considering viscous dissipation, International Journal of Heat and Mass Transfer 79 (2014) 212–222. ##[10] M.Sheikholeslami, M.GorjiBandpy ,D.D Ganji: MHD free convection in an eccentric semiannulus filled with nanofluid, Journal of the Taiwan Institute of Chemical Engineers 45(2014)1204–16. ##[11] M. Sheikholeslami, M.GorjiBandpy, D.D.Ganji S.Soleimani: MHD natural convection in a nanofluid filled inclined enclosure with sinusoidal wall using CVFEM, Neural Comput & Applic 24 (2014) 873–882. ##[12] A.Malvandi ,D.D. Ganji: Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field, International Journal of Thermal Sciences 84 (2014) 196206. ##[13] M. Hatami , D.D.Ganji: Heat transfer and nanofluid flow in suction and blowing process between parallel disks in presence of variable magnetic, Field Journal of Molecular Liquids190 (2014) 159–168. ##[14] H.R. Ashorynejad, A.A. Mohamad, M.Sheikholeslami: Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method. International Journal of Thermal Sciences 64 (2013) 240250. ##[15] D.A.Nield, A.V.Kuznetsov: Thermal instability in a porous medium layer saturated by a nanofluid, International, Journal of Heat and Mass Transfer 52 (2009) 5796–5801. ##[16] W.A. Khan: Pop I. Boundarylayer flow of a nanofluid past a stretching sheet, International Journal of Heat and Mass Transfer 53 (2010) 2477–2483. ##[17] M.Sheikholeslami, M.GorjiBandpy, D.D.Ganji S.Soleimani: Thermal management for free convection of nanofluid using two phase model, Journal of Molecular Liquids 194(2014) 179–87 ##[18] M .Mahmood, S. Asghar, M.A.Hossain:Squeezed flow and heat transfer over a porous surface for viscous fluid, Heat Mass Transfer 44 (2007) 165–173 ##[19] G.Domairry, A .Aziz: Approximate analysis of MHD squeeze flow between two parallel disks with suction or injection by homotopy perturbation method, Hindawi Publishing Corporation Mathematical Problems in Engineering (2009). ##[20] M.Mustafa, T.Hayat, S.Obaidat: On heat and mass transfer in the unsteady squeezing flow between parallel plates, Meccanica 47(2012)1581–1589.##]
The effect of small scale on the vibrational response of nanocolumn based on differential quadrature method
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The present paper deals with the dynamic behavior of nanocolumn subjected to follower force using the nonlocal elasticity theory. The nonlocal elasticity theory is used to analyze the mechanical behavior of nanoscale materials. The used method of solution is the Differential Quadrature Method (DQM). It is shown that the nonlocal effect plays an important role in the vibrational behavior of nanocolumns. The results can provide useful guidance for the study and design of the next generation of nanodevices and could be useful in biomedical and bioengineering applications as well as in other fields related with the nanotechnology.
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58


Amin
Pourasghar
Young Researchers and Elite club, Islamic Azad University, Central Tehran Branch, Tehran, I. R. Iran
Young Researchers and Elite club, Islamic
Iran
aminpourasghar@yahoo.com


Ali
Ghorbanpour Arani
Faculty of Mechanical Engineering, University of Kashan, Kashan, I. R. IranInstitute of Nanoscience and Nanotechnology, University of Kashan, Kashan, I. R. Iran
Faculty of Mechanical Engineering, University
Iran
a_ghorbanpour@yahoo.com


Saeed
Kamarian
Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, I. R. Iran
Young Researchers and Elite Club, Kermanshah
Iran
kamarian.saeed@yahoo.com
differential quadrature method (DQM)
nanocolumn
nonlocal elasticity
Vibration
[[1] AC . Eringen: Linear theory of nonlocal elasticity and dispersion of planewaves, International Journal of Engineering Science 10 (1972) 233–48. ##[2] S. Iijima: Helical microtubules of graphitic carbon, Nature 354 (1991) 56–8. ##[3] CR. Martin: Membranebased synthesis of nanomaterials, Chemical Mater 8 (1996) 1739–46. ##[4] KE. Drexler: Nanosystems: molecular machinery, manufacturing and computation, New York: Wiley (1992). ##[5] J. Han, A. Globus, R. Jaffe , G. Deardorff : Molecular dynamics simulation of carbon nanotubebased gear, Nanotechnology 8 (1997) 95–102. ##[6] A. Fennimore , TD. Yuzvinsky, WQ. Han, MS. Fuhrer, J. Cumings, A. Zettl: Rotational actuators based on carbon nanotubes, Nature (2003) 408–24. ##[7] B. Bourlon, DC. Glattli, C. Miko, L. Forro, A. Bachtold: Carbon nanotube based bearing for rotational motions, Nano Letter 4 (2004) 709–12. ##[8] V. Saji, H. Choe, K. Young: Nanotechnology in biomedical applicationsa review, International Journal of Nano Biomater 3 (2010) 119–39. ##[9] AC. Eringen: On differentialequations of nonlocal elasticity and solutions of screw dislocation and surfacewaves, Journal of Apply Physics 54 (1983) 4703–10. ##[10] AC. Eringen: Nonlocal polar elastic continua, International Journal of Engineering Science 10 (1972) 1–16. ##[11] J. Peddieson, GR. Buchanan , RP. McNitt: Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41 (2003) 305–12. ##[12] JN. Reddy: Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45 (2007) 288–307. ##[13] J. Loya, J. LopezPuente, R. Zaera, J. FernandezSaez: Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model, Journal of Apply Physics 105 (2009) 044309. ##[14] K. Kiani: Free longitudinal vibration of tapered nanowires in the context of nonlocal continuum theory via a perturbation technique, Physica E 43 (2010) 387–97. ##[15] T. Murmu, S. Adhikari: Non local effects in the longitudinal vibration of doublenanorod systems, Physica E 43 (2010) 415–22. ##[16] L.L. Ke, Y.S. Wang, Z.D. Wang: Nonlocal elastic plate theories, Proceedings of the Royal Society of London A 463 (2008) 3225–40. ##[17] T. Murmu, SC. Pradhan: Smallscale effect on the free inplane vibration of nanoplates by nonlocal continuum model, Physica E 41 (2009) 1628–33. ##[18] R. Ansari, A. Shahabodini, H. Rouhi: A thicknessindependent nonlocal shell model for describing the stability behavior of carbon nanotubes under compression, Composite Structures 100 (2013) 323–31. ##[19] HT. Thai: A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science 52 (2012) 56–64 ##[20] JN. Reddy, SD. Pang: Nonlocal continuum theories of beams for the analysis of carbon nanotubes, J Appl Phys 103 (2008) 1–16. ##[21] M.A. De Rosa, N.M. Auciello, M. Lippiello: Dynamic stability analysis and DQM for beams with variable crosssection, Mechanics Research Communications 35 (2008) 187–192. ##[22] A.A. Mahmoud, A. Ramadan, M.M. Nassar: Free vibration of nonuniform column using DQM, Mechanics Research Communications 38 (2011) 443–448 ##[23] C. Shu: Differential quadrature and its application in engineering, Springer, Berlin, (2000). ##[24] C. Shu, BE. Richards: Application of generalized differential quadrature to solve twodimensional incompressible Navier Stockes equations, International Journal for Numerical Methods in Fluids 15 ( 1992) 791798. ##[25] R. Bellman, BG. Kashef, J. Casti: Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computation Physics 10 (1972) 4&52. ##[26] C.W. Bert, M. Malik: Differential quadrature method in computational mechanics, A review, Applied Mechanics Reviews 9 (1996) 128.##]