2016
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Effects of thermal diffusion and chemical reaction on MHD transient free convection flow past a porous vertical plate with radiation, temperature gradient dependent heat source in slip flow regime
2
2
An analytical investigation is conducted to study the unsteady free convection heat and mass transfer flow through a nonhomogeneous porous medium with variable permeability bounded by an infinite porous vertical plate in slip flow regime while taking into account the thermal radiation, chemical reaction, the Soret number, and temperature gradient dependent heat source. The flow is considered under the influence of magnetic field applied normal to the flow. Approximate solutions for velocity, temperature, and concentration fields are obtained using perturbation technique. The expressions for skinfriction, rate of heat transfer, and rate of mass transfer are also derived. The effects of various physical parameters, encountered in the problem, on the velocity field, temperature field, and concentration field are numerically shown through graphs, while the effects on skinfriction, rate of heat, and mass transfer are numerically discussed by tables.
1

83
95


S.
Mohammed Ibrahim
Department of Mathematics, Gitam Institute of Technology, GITAM University, Visakhapatnam, Andhra Pradesh – 530045, India
Department of Mathematics, Gitam Institute
Iran


K.
Suneetha
Department of Mathematics, Priyadarshini College of Engineering and Technology, Nellore, Andhra Pradesh 524004, India
Department of Mathematics, Priyadarshini
Iran
free convection
chemical reaction
The Soret number
nonhomogeneous porous medium
Temperature gradient heat source
Radiation
[[1] S. Middleman, An Introduction to Mass and Heat Transfer, John Wiley & Sons, Inc., (1998). ##[2] H. Rubin, and J. Atkinson, Environmental Fluid Mechanics, Marcel Dekker, Inc, New York, (2001). ##[3] P. S. Ghoshdastidar, Heat Transfer, Oxford University Press, UK., (2004). ##[4] P. Cheng, and W. J. Minkowycz, “Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike”, Journal of Geophysical Research, Vol. 82, No. 14, pp. 20402044, (1977). ##[5] D. A. Nield, and A. Bejan, Convection in Porous Media, Springer, Berlin, (1992). ##[6] A. J. Chamaka, “Hydromagnetic combined heat and mass transfer by natural convection from a permeable surface embedded in a fluid saturated porous medium”, International Journal Numerical Methods Heat Fluid Flow, Vol. 10, No. 5, pp. 455476, (2000). ##[7] M. Gupta, and S. Sharma, “MHD flow of viscous fluid through a porous medium bounded by an oscillating porous plate in slip flow regime”, Acta Ciencia Indica, Vol. 17M, No. 2, pp. 389 394, (1991). ##[8] J. Y. Kim, “Unsteady MHD convective heat transfer past a semiinfinite vertical porous moving plate with variable suction”, International Journal of Engineering Sciences, Vol. 38, No. 8, pp. 833845, (2000). ##[9] M. C. Raju, and S. V. K. Varma, “Unsteady MHD free convection oscillatory coquette flow through a porous medium with periodic wall temperature, imanager’s Journal on Future Engineering and Technology, Vol. 6, No. 4, pp. 712, (2011). ##[10] A. Raptis, and N. Kafousias, “Magneto hydrodynamics free convection flow and mass transfer through a porous medium bounded by an infinite vertical porous plate with constant heat flux”, Canadian Journal of Physics, Vol. 60, No. 12, pp. 17251729, (1982). ##[11] E. R. G. Eckert, and R. M. Drake, Heat and mass transfer, Tata McGrawHill, 2nd ed., New Delhi, (1979). ##[12] A. Y. Ghaly, “Radiation effects on a certain MHD free convection flow”, Chaos, solutions and Fractals, Vol. 13, No. 9, pp. 18431850, (2002). ##[13] N. Ahmed, and H. K Sarmah, “The radiation effect on a transient MHD flow mass transfer past an impulsively fixed infinite vertical plate”, International Journal of Appled Mathematics and Mechanics, Vol. 5, No. 5, pp. 8798, (2009). ##[14] A. Raptis, and C. V. Massalas, “Magnetohydrodynamic Flow Past a Plate by the Presence of Radiation,” Heat and Mass Transfer, Vol. 34, No. 23, pp. 107109, (1998) ##[15] H. Poonia, and R. C. Chaudary, “The influence of radiative heat transfer on MHD oscillating flow in a planner channel with slip condition”, International Journal of Energy and Technology, Vol. 4, No. 2, pp. 17, (2012). ##[16] N. Ahmed and H. K. Sarmah, “The Radiation Effect on a Transient MHD Flow Mass Transfer Past an Impulsively Fixed Infinite Vertical Plate,” International Journal of Applied Mathematics and Mechanics, Vol. 5, No. 5, pp. 8798, (2009). ##[17] V. Rajesh, and S. V. K. Varma, “Radiation effects on MHD flow through a porous medium with variable temperature and mass diffusion,” International Journal of Applied Mathematics and Mechanics, Vol. 6, No. 11, pp. 3957, (2010). ##[18] D. Pal, and H. Mondal, “Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity,” Meccanica, Vol. 44, No. 2, pp. 133144, (2009). ##[19] A. M. D. Samad, and M. M. Rahman, “Thermal radiation interaction with unsteady MHD flow past a vertical porous plate immersed in a porous medium”, Journal of Naval Architecture and Marine Engineering, Vol. 3, No. 1, pp. 714, (2006). ##[20] S. Ostrach, “Laminar natural convection flow and heat transfer of fluid with and without heat source in channel with wall temperature”, NACA TN, 2863, (1952). ##[21] A. A. Raptis, “Free convection and mass transfer effects on the oscillatory flow past an infinite moving vertical isothermal plate with constant suction and heat sources”, Astrophy., Space Sci., Vol. 86, No. 1, pp. 4353, (1982). ##[22] A. K. Singh, “MHD free convection and mass transfer flow with heat source and thermal diffusion”, Journal of Energy, Heat and Mass Transfer, Vol. 23, No. 2, pp. 227249, (2001). ##[23] S. S. Sexena, and G. K. Dubey, “MHD free convection heat and mass transfer flow of viscoelastic fluid embedded in a porous medium of variable permeability with radiation effect and heat source in slip flow regime”, Advances in Applied Science Research, Vol. 2, No. 5, pp. 115129, (2011). ##[24] S. Mohammed Ibrahim, and N Bhaskar Reddy, “Radiation and mass transfer effects on MHD free convection flow along a stretching surface with viscous dissipation and heat generation”, International Journal of Applied Mathematics and Mechanics, Vol. 8, No. 8, pp. 121, (2012). ##[25] S. Mohammed Ibrahim, and K. Suneetha, “Effects of heat generation and thermal radiation on steady MHD flow near a stagnation point on a linear stretching sheet in porous medium and presence of variable thermal conductivity and mass transfer, Journal of Computational and Applied Research in Mechanical Engineering, Vol. 4, No. 2, pp. 133144, (2015). ##[26] M. Ghalambaz and A. Naghrehabadi, “Effects of heat generation/absorption on natural convection of nanofluids over the vertical plate embedded in porous medium using driftflux model”, Journal of Computational and Applied Research in Mechanical Engineering, Vol. 3, No. 2, pp. 113123, (2014). ##[27] B. Madhusudhana Rao, G. Viswanatha Reddy, M. C. Raju and S. V. K. Varma, “MHD transient free convection and chemical reactive flow past a porous vertical plate with radiation and temperature gradient dependent heat source in slip flow regime”, IOSR Journal of Applied Physics, Vol. 3, No. 6, pp. 2232, (2013). ##[28] G. V. Ramana Reddy, S. Mohammed Ibrahim, and V. S. Bhagavan, “Similarity transformations of heat and mass transfer effects on steady MHD free convection dissipative fluid flow past an inclined porous surface with chemical reaction”, Journal of Naval Architecture & Marine Engineering, Vol. 11, No. 2, pp. 157166, (2014). ##[29] T. Sudhakar Reddy, M. C. Raju, and S. V. K. Varma, “Chemical reaction and radiation effects on MHD free convective flow through a porous medium bounded by a vertical surface with constant heat and mass flux, Journal of Computational and Applied Research in Mechanical Engineering Vol. 3, No. 1, pp. 5362, (2013). ##[30 ] A. Sahin, “Influence of chemical reaction on transient mhd free Convective flow over a vertical plate in slipflow Regime”, Emirates Journal for Engineering Research, Vol. 15, No. 1, pp. 2534, (2010). ##[31] E. R. G. Eckert, and R. M. Drake, Analysis of heat and mass transfer, New York, McGraw Hill, (1972). ##[32] Z. Durunkya, and W. A. Worek, “DiffusionThermo and thermaldiffusion effects in transient and steady natural convection from vertical surface”, International Journal of Heat and Mass Transfer, Vol. 35, No. 8, pp. 20602065, (1992). ##[33] N. G. Kafoussias, and E. M. Williams, “Thermaldiffusion and diffusionthermo effects on mixed free forced convection and mass transfer boundary layer flow with temperature dependent viscosity”, International Journal of Engineering Science, Vol. 33, No. 9, pp. 13691384, (1995). ## [34] M. A. Sattar, and M. S. Alam, “Thermal diffusion as well as transpiration effects on MHD free convection and mass transfer flow past an accelerated vertical porous plate”, Indian Journal of Pure and Applied Mathematics, Vol. 25, No. 6, pp. 679688, (1994). ##[35] M. S. Alam, and M. M. Rahman, Dufour and Soret effects on MHD free convective heat and mass transfer flow past a vertical porous flat plate embedded in a porous medium, Journal of Naval Architecture and Marine Engineering, Vol. 2, No. 1, pp. 5565, (2005). ##[36] J. Prakash, K. S. Balamurugan and S. V. K. Varma, “Soret and chemical reaction effects on a three dimensional MHD convective flow of dissipative fluid along an infinite vertical plate”, Journal of Computational and Applied Research in Mechanical Engineering, Vol. 4, No. 1, pp. 1942, (2015). ##[37] A. J. Ede., Advances in Heat Transfer, Academic Press, New York (1967). ##[38] K. Gerstenk, and J. F. Gross, “ Flow and heat transfer along a plane wall with periodic suction, Zeitschrift für angewandte Mathematik und Physik, Vol. 25, No. 3, pp. 399408, (1974).##]
Effects of temperature gradient magnitude on bending angle in laser forming process of aluminium alloy sheets
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Laser forming is a thermal forming process which uses laser beam irradiation to produce the desired final forms. In this article, the effect of temperature gradient across Al 6061T6 aluminum sheets on bending angle is studied. Input parameters including laser power, scan velocity, beam diameter, and sheet thickness are the effective process parameters which influence the temperature gradient. Thus, a set of 81 numerical simulations based on a full factorial design with varying parameters is carried out and temperature gradient across the sheet thickness is measured. Effects of each input parameter on temperature gradient are determined using analysis of variance. Also, an equation is derived which predicts the temperature gradient for any arbitrary input parameter. The validity of the equation is done by comparing actual and predicted results. Numerical simulation is validated by experimental tests, which show a very close agreement. Finally, the effects of temperature gradient for three different sheet thicknesses on a final bending angle are derived. Results demonstrate that increase in temperature gradient across sheet thickness leads to increase in bending angle.
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97
109


Amir H.
Roohi
Department of Mechanical Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, I.R. Iran
Department of Mechanical Engineering, Faculty
Iran
amir.roohi@modares.ac.ir


H.
Moslemi Naeini
Department of Mechanical Engineering, Faculty of Engineering, Tarbiat Modares University, P.O.Box 14115/143, Tehran, I.R. Iran
Department of Mechanical Engineering, Faculty
Iran


M.
Hoseinpour Gollo
Department of Mechanical Engineering, Shahid Rajaee Teacher Training University (SRTTU), Lavizan, Tehran, Iran
Department of Mechanical Engineering, Shahid
Iran


J.
Shahbazi Karami
Department of Mechanical Engineering, Shahid Rajaee Teacher Training University (SRTTU), Lavizan, Tehran, Iran
Department of Mechanical Engineering, Shahid
Iran


Sh.
Imani Shahabad
Department of Mechanical Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, I.R. Iran
Department of Mechanical Engineering, Faculty
Iran
Laser forming process
Aluminum alloy sheet
Temperature gradient
Bending angle
Process parameters
[[1] X. Zhang, Laserassisted High Precision Bending and its Applications, Ph.D. Thesis, Purdue University, (2004). ##[2] M. Geiger, and F. Vollertsen, "The Mechanisms of Laser Forming", CIRP Annals  Manufacturing Technology, Vol. 42, No. 1, pp. 301304, (1993). ##[3] W. Shichun, and Z. Jinsong, "An experimental study of laser bending for sheet metals", Journal of Materials Processing Technology, Vol. 110, No. 2, pp. 160163, (2001). ##[4] Y. Guan, S. Sun, G. Zhao, and Y. Luan, "Influence of material properties on the laserforming process of sheet metals", Journal of Materials Processing Technology, Vol. 167, No. 1, pp. 124131, (2005). ##[5] M. S. Che Jamil, M. A. Sheikh, and L. Li, "A study of the effect of laser beam geometries on laser bending of sheet metal by buckling mechanism", Journal of Optics & Laser Technology, Vol. 43, No. 1, pp. 183193, (2011). ##[6] H. Shen, and Z. Yao, "Study on mechanical properties after laser forming", Journal of Optics and Lasers in Engineering, Vol. 47, No. 1, pp. 111117, (2009). ##[7] J. Liu, S. Sun, Y. Guan, and Z. Ji, "Experimental study on negative laser bending process of steel foils", Journal of Optics and Lasers in Engineering, Vol. 48, No. 1, pp. 8388, (2010). ##[8] D. Wu, Q. Zhang, G. Ma, Y. Guo, and D. Guo, "Laser bending of brittle materials", Journal of Optics and Lasers in Engineering, Vol. 48, No. 4, pp. 405410, (2010). ##[9] F. Quadrini, A. Guglielmotti, E. A. Squeo, and V. Tagliaferri, "Laser forming of opencell aluminium foams", Journal of Materials Processing Technology, Vol. 210, No. 11, pp. 15171522, (2010) ##[10] S. M. Knupfer, and A. J. Moore, "The effects of laser forming on the mechanical and metallurgical properties of low carbon steel and aluminium alloy samples", Journal of Materials Science and Engineering: A, Vol. 527, No. 1617, pp. 43474359, (2010) ##[11] X. Y. WANG, W. X. XU, W. J. XU, Y. F. HU, Y. D. LIANG, and L. J.WANG, "Simulation and prediction in laser bending of silicon sheet", Transactions of Nonferrous Metals Society of China, Vol. 21, pp. 188193, (2011). ##[12] Y. Shi, Y. Liu, P. Yi, and J. Hu, "Effect of different heating methods on deformation of metal plate under upsetting mechanism in laser forming", Journal of Optics & Laser Technology, Vol. 44, No. 2, pp. 486491, (2012) ##[13] S. Shekhar Chakraborty, V. Racherla, and A. Kumar Nath, "Parametric study on bending and thickening in laser forming of a bowl shaped surface" Journal of Optics and Lasers in Engineering, Vol. 50, No. 11, pp. 15481558, (2012) ##[14] K. Maji, D. K. Pratihar, and A.K. Nath, "Experimental investigations and statistical analysis of pulsed laser bending of AISI 304 stainless steel sheet", Journal of Optics & Laser Technology, Vol. 49, pp. 1827, (2013). ##[15] D. P. Shidid, M. H. Gollo, M. Brandt, and M. Mahdavian, "Study of effect of process parameters on titanium sheet metal bending using Nd: YAG laser", Journal of Optics & Laser Technology, Vol. 47, pp. 242247, (2013). ##[16] J. Kim, and S. Na, "3D laserforming strategies for sheet metal by geometrical information", Journal of Optics & Laser Technology, Vol. 41, No. 6, pp. 843852, (2009). ##[17] A. H. Roohi, M. H. Gollo, and H. M. Naeini, "External forceassisted laser forming process for gaining high bending angles", Journal of Manufacturing Processes, Vol. 14, No. 3, pp. 269276, (2012). ##[18] A. H. Roohi, M. H. Naeini, M. H. Gollo, M. Soltanpour, and M. Abbaszadeh, "On the randombased closedcell metal foam modeling and its behavior in laser forming process", Journal of Optics & Laser Technology, Vol. 72, pp. 5364, (2015). ##[19] B. S. Yilbas, and S. S. Akhtar, "Laser bending of metal sheet and thermal stress analysis", Journal of Optics & Laser Technology, Vol. 61, pp. 3444, (2014). ##[20] H. Shen, M. Ran, J. Hu, and Z. Yao, "An experimental investigation of underwater pulsed laser forming", Journal of Optics and Lasers in Engineering, Vol. 62, pp. 18, (2014). ##[21] A. H. Roohi, H. M. Naeini, and M. H. Gollo, "An experimental investigation of parameters effect on laser forming of Al6061T6 sheets", Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials Design and Applications, (2015). ##[22] M. Safari, and M. Farzin, "Experimental investigation of laser forming of a saddle shape with spiral irradiating scheme", Journal of Optics & Laser Technology, Vol. 66, pp. 14615, (2015). ##[23] X. Y. Wang, J. Wang, W. J. Xu, and D. M. Guo, "Scanning path planning for laser bending of straight tube into curve tube", Journal of Optics & Laser Technology, Vol. 56, pp. 4351, (2014). ##[24] G. N. Labeas, "Development of a local threedimensional numerical simulation model for the laser forming process of aluminium components:, Journal of Materials Processing Technology, Vol. 207, No. 1–3, pp. 248257, (2008). ##[25] M. Awang, V. Mucino, Z. Feng, and S. David, "ThermoMechanical Modeling of Friction Stir Spot Welding (FSSW) Process: Use of an Explicit Adaptive Meshing Scheme", SAE 2005 World Congress & Exibition, (2005). ##[26] M. Hoseinpour Gollo, S. M. Mahdavian, H. Moslemi Naeini, "Statistical analysis of parameter effects on bending angle in laser forming process by pulsed Nd:YAG laser", Journal of Optics & Laser Technology, Vol. 43, No. 3, pp. 475482, (2011). ##[27] S. P. Edwardson, E. Abed, C. Carey, K. Edwards, G. Dearden, and K. Watkins, "Factors influencing the bend per pass in multipass laser forming", Proceedings of the 5th LANE 1, pp. 557568, (2007). ##[28] S. P. Edwardson, E. Abed, K. Bartkowiak, G. Dearden, and K. G. Watkins, "Geometrical influences on multipass laser forming", Journal of Physics D: Applied Physics, Vol. 39, No. 2, pp. 382, (2006).##]
Diffusionthermo effects on MHD free convective radiative and chemically reactive boundary layer flow through a porous medium over a vertical plate
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The main purpose of this work is to investigate the porous medium and diffusionthermo effects on unsteady combined convection magneto hydrodynamics boundary layer flow of viscous electrically conducting fluid over a vertical permeable surface embedded in a high porous medium, in the presence of first order chemical reaction and thermal radiation. The slip boundary condition is applied at the porous interface. A uniform Magnetic field is applied normal to the direction of the fluid flow. The nonlinear coupled partial differential equation are solved by perturbation method and obtained the expressions for concentration, temperature and velocity fields. The rate of mass transfer in terms of Sherwood number , the rate of heat transfer in terms of Nusselt number and the Skin friction coefficient are also derived. The Profiles of fluid flow quantities for various values of physical parameters are presented and analyzed. Profiles of fluid flow quantities for various values of physical parameters are presented and analyzed.
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111
126


J.
Prakash
Department of Mathematics, University of Botswana, Private Bag 0022, Gaborone, Botswana
Department of Mathematics, University of
Iran
prakashj@mopipi.ub.bw


P.
Durga Prasad
Departmentof Mathematics, Sri Venkateswara University, Tirupati – 517502, A. P., India
Departmentof Mathematics, Sri Venkateswara
Iran


R. V. M. S. S.
Kiran Kumar
Departmentof Mathematics, Sri Venkateswara University, Tirupati – 517502, A. P., India
Departmentof Mathematics, Sri Venkateswara
Iran


S. V. K.
Varma
Departmentof Mathematics, Sri Venkateswara University, Tirupati – 517502, A. P., India
Departmentof Mathematics, Sri Venkateswara
Iran
Diffusionthermo effect
thermal radiation
chemical reaction
Magnetic fields
[ [1] R. S. Rath, and D. N. Parida, “Magnetohydrodynamic free convection in the boundary layer due to oscillation in the wall temperature”, Wear, Vol. 78, No. 1, pp. 305314, (1982). ## [2] A. Raptis, G. Tzivanidis, and N.Kafusias, “Free convection and mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction”, Letters in Heat and Mass Transfer, Vol. 8, No. 1, pp. 417424, (1981). ##[3] B. K. Jha, and R.Prasad, “MHD freeconvection and mass transfer flow through a porous medium with heat source”, Astrophysics and Space Science, Vol. 18, No. 1, pp. 117123, (1991). ##[4] K.Yamamoto, and N. Iwamura, “Flow with convective acceleration through a porous Medium”, J. Eng. Math, Vol. 10, No. 1, pp .4154, (1976). ##[5] B. Gebhart, and L. Pera, “The nature of vertical natural convection flows resultingfrom thecombined buoyancy effects of thermal and mass diffusion”, Int. J. Heat and Mass Transfer,Vol. 14,No. 12, pp. 20252050, (1971). ##[6] D. Pal and B. Talukdar, “Perturbation Analysis of unsteady magneto hydro dynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction”, Commun Nonlinear Sci Number Simulant, Vol. 15, No. 7, pp. 18131830, (2010). ##[7] D. Nield and A. Bejan, “Convection in porous media”, 2nd edition Springer, Wiley, New York, pp. 6275, (1995). ##[8] P. Cheng, “Heat transfer in geothermal system”, Adv Heat Transfer,Vol. 4, No. 1, pp. 1105, (1978). ##[9] N. Rudraiah, “Flow through and past porous media, ”Encyclopediaof Fluid Mechanics, Gulf Publ. Vol. 5, pp. 567647, (1986). ##[10] N. Rudraiah, D. Pal, and I. N. Shivakumara, “Effect of slip and magnetic field on composite systems”. Fluid Dyn. Res., Vol. 4, No.4, pp. 255270, (1988). ##[11] A. Apelblat, “Mass transfer with a chemical reaction of the first order: AnalyticalSolutions”, The Chemical Engineering Journal Vol. 19, No. 1, pp. 1937, (1980). ##[12] U. N. Das, R. K. Deka and V. M. Soundalgekar, “Effects of Mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemicalreaction”, Forschung imIngenieurwesen, Vol. 60, No. 10, pp. 284287, (1994). ##[13] P. L. Chambre and J. D. Young, “On the diffusion of a chemically reactive species in a laminar boundary layer flow”, The Physics of Fluids, Vol. 1, pp. 4854, (1958). ##[14] W. G. England, and A. F. Emery, “Thermal radiation effects on the laminar free convection boundary layer of an absorbing gas”, Journal of Heat Trans, Vol. 91, No. 1, pp. 3744,(1969). ##[15] V. M. Soundalgekar, and H. S. Takhar, “Radiation effects on free convection flow past a semiverticalplate”, Modeling Measurement and Control, Vol. 51,pp. 3140, (1993). ##[16] A. Raptis, and C. Peridikis, “Radiation and free convection flow past a moving plate”, International Journal ofApplied Mechanicsand Engineering, Vol. 4, No. 4, pp. 817821, (1999). ## [17] F. S. Ibrahim, A. M. Elaiw and A. A. Bakr, “Effects of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semiinfinite vertical permeable moving plate with heat source and suction”, CommunicationsinNonlinearScienceandNumericalSimulation, Vol. 13, No. 6, pp. 10561066, (2008). ##[18] Z. Dursunkaya, and W. M. Worek, “Diffusion–thermo and thermaldiffusion effects in transient and steady natural convection from vertical surface”, International Journal of Heatand Mass Transfer, Vol. 35, No. 8, pp. 20602065, (1992). ##[19] M. Anghel, H. S. Takhar and I. Pop, “Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium”,StudiaUniversitatisBabesBolyai, Mathematica Vol. 45, No. 4,pp. 1121, (2000). ##[20] A. Postelnicu, “Influence of a magnetic field on heat and mass transfer by natural convectionfrom vertical surfaces in porous media considering Soret and Dufoureffects”, International Journal of Heat and Mass Transfer, Vol. 47, No. 6, pp. 14671472, (2004). ## [21] A. J. Chamkha, “Unsteady MHD convective heat and mass transfer past a semiinfinite vertical permeable moving plate with heat absorption”, International Journal of Engineering Sciences,Vol. 42, No. 2, pp. 217230, (2004). ##[22] A. J. Chamkha, “MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction”, Int. Comm. Heat Mass Transfer, Vol. 30, No. 3, pp. 413422, (2003). ##[23] R. A. Mohamed, “Doublediffusive convection radiation interaction on unsteady MHD flow over a vertical moving porous plate with heat generation and Soret effects”, Appl. Math. Sci., Vol. 3, No. 13, pp. 629651, (2009). ##[24] R. Muthucumaraswamy and B. Janakiraman, “MHD and radiation effects on moving isothermalvertical plate with variable mass diffusion”, Theo. Appl. Mech., Vol. 33, No.1, pp. 1729, (2006). ##[25] V. Rajesh and S. V. K. Varma, “Thermal diffusion and radiation effects on MHD flow past an impulsively started infinite vertical plate with variable temperature and mass diffusion”, JP J. Heat and Mass Transfer, Vol. 3,No. 1, pp. 1739, (2009). ##[26] A. G .Vijaya Kumar and S. V. K. Varma, “Thermal radiation and mass transfer effects on MHD flow past an impulsively started exponentially accelerated vertical plate with variable temperature and mass diffusion”, Far East J. Appl. Math., Vol. 55,No. 2, pp. 93115, (2011). ##[27] R. A. Raptis, C. Perdikis and A. Leontitsis, “Effects of radiation in an optically thin gray gas flowing past a vertical infinite plate in the presence of magnetic field”, Heat and Mass Transfer, Vol. 39, pp. 771773, (2003). ##[28] A. Orhan and K. Ahmad,“Radiation effect on MHD mixed convection flow about a permeable vertical plate”, Heat and Mass Transfer, Vol. 45,No. 2, pp. 239246, (2008). ##[29] S. Ahmad, “ Inclined magnetic field with radiating fluid over a porous vertical plate: Analytical study”, Journal Naval Arch. Marine Engineering, Vol. 7, No. 2, pp. 6172, (2010). ##[30] E. R. G. Eckert, and R. M. Drake, “Analysis of Heat and Mass Transfer”, M.C.GrawHill, New York, (1972). ##[31] J. Prakash, D. Bhanumathi, A. G. Vijaya Kumar “Radiation effects on unsteady MHD flow through porous medium past an impulsively started infinite vertical plate with variable temperature and mass diffusion”, Trans. Porous. Med. Vol. 96, No. 1,pp. 135151, (2013). ##[32] A. C. Cogley, W. G. Vincent and S. E. Giles, “Differential approximation to radiative heat transfer in a nongrey gas near equilibrium”, American Institute of Aeronautics and Astronautics, Vol. 6, No. 3, pp. 551553, (1968). ##[33] Y. J. Kim, “Unsteady MHD convective heat transfer past a semiinfinite vertical moving plate with variable suction”, International Journal of Engineering Sciences, Vol. 38, No. 8, pp. 833845, (2000).##]
Estimating the unknown heat flux on the wall of a heat exchanger internal tube using inverse method
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2
In the design of heat exchangers, it is necessary to determine the heat transfer rate between hot and cold fluids in order to calculate the overall heat transfer coefficient and the heat exchanger efficiency. Heat transfer rate can be determined by inverse methods. In this study, the unknown spacetime dependent heat flux imposed on the wall of a heat exchanger internal tube is estimated by applying an inverse method and simulated temperature measurements at the specified points in the flow field. It is supposed that no prior information is available on the variation of the unknown heat flux function. Variable metric method which belongs to the function estimation approach is utilized to predicate the unknown function by minimizing an objective function. Four versions of the presented inverse method, named DFP, BFGS, SR1, and Biggs, are used to solve the problem and the results obtained by each version are compared. The estimation of the heat flux depends on the location of the sensor and the uncertainties associated with temperature measurements. The influence of each factor is investigated in this paper. Results show that variable metric method is a rapid and precise technique for estimating unknown boundary conditions in inverse heat convection problems.
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127
136


M. Sh.
Mazidi
Optimization and Development of Energy Technologies Division, Research Institute of Petroleum Industry (RIPI), Tehran, Iran
Optimization and Development of Energy Technologie
Iran


M.
Alizadeh
School of Mechanical Engineering, Iran University of Science and Technology (IUST), Tehran, Iran
School of Mechanical Engineering, Iran University
Iran


L.
Nourpour
School of Mechanical Engineering, Iran University of Science and Technology (IUST), Tehran, Iran
School of Mechanical Engineering, Iran University
Iran


V.
Shojaee Shal
School of Mechanical Engineering, Iran University of Science and Technology (IUST), Tehran, Iran
School of Mechanical Engineering, Iran University
Iran
Inverse heat transfer
Variable metric method
Heat exchanger
Unknown heat flux
[[1] J. C. Bokar, and M. N. Ozisik, “An inverse analysis for estimation of timevarying inlet temperature in laminar flow inside a parallel plate duct”, International Journal of Heat and Mass Transfer, Vol. 38, No. 1, pp. 3945, (1995). ##[2] J. V. Beck, and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis”, Measurement Science and Technology, Vol. 9, No. 6, pp. 839847, (1998). ##[3] M. N. Ozisik, and H. R. B. Orlande, Inverse Heat Transfer: Fundamentals and Applications, 1st ed., Taylor & Francis, New York, (2000). ##[4] O. M. Alifanov, Inverse Heat Transfer Problems, 1st ed., SpringerVerlag, Berlin, (1994). ##[5] L. Luksan, and E. Spedicato, “Variable metric methods for unconstrained optimization and nonlinear least squares”, Journal of Computational and Applied Mathematics, Vol. 124, No. 12, pp. 6195, (2000). ##[6] C. H. Huang, and M. N. Ozisik, “Inverse problem of determining unknown wall heat flux in laminar flow through a parallel plate duct”, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology, Vol. 21, No. 1, pp. 5570, (1992). ##[7] M. J. Colaco, and H. R. B. Orlande, “Inverse forced convection problem of simultaneous estimation of two boundary heat flux in irregularly shaped channels”, Numerical Heat Transfer, Part A: Applications, Vol. 39, No. 7, pp. 737 ##760, (2001). ##[8] P. Ding, and W. Q. Tao, “Estimation of unknown boundary heat flux in laminar circular pipe flow using functional optimization approach: effects of Reynolds numbers,” Journal of Heat Transfer, Vol. 131, No. 2, pp. 19, (2009). ##[9] C. H. Huang, I. C. Yuan, and H. Ay, “A threedimensional inverse problem in imaging the local heat transfer coefficients for plate finnedtube heat exchangers”, International Journal of Heat and Mass Transfer, Vol. 46, No. 19, pp. 36293638, (2003). ##[10] W. L. Chen, and Y. C. Yang, “An inverse problem in determining the heat transfer rate around two in line cylinders placed in a cross stream”, Energy Conversion and Management, Vol. 48, No. 7, pp. 19962005, (2007). ##[11] F. Bozzoli, L. Cattani, C. Corradi, M. Mordacci, and S. Rainieri, “Inverse estimation of the local heat transfer coefficient in curved tubes: a numerical validation”, Journal of Physics: Conference Series, Vol. 501, No. 012002, (2014). ##[12] J. H. Noha, W. G. Kima, K. U. Chab, and S. J. Yooka, “Inverse heat transfer analysis of multilayered tube using thermal resistance network and Kalman filter”, International Journal of Heat and Mass Transfer, Vol. 89, pp.10161023, (2015). ##[13] S .S. Rao, Optimization: Theory and Applications, 2nd ed., Wiley Eastern Limited, New Delhi, (1984). ##[14] M. C. Biggs, “A note on minimization algorithms which make use of nonquadratic properties of the objective function”, IMA Journal of Applied Mathematics, Vol. 12, No. 3, pp. 337338, (1973).##]
Optimizing naturally driven air flow in a vertical pipe by changing the intensity and location of the wall heat flux
2
2
Heat transfer from the internal surfaces of a vertical pipe to the adjacent air gives rise to the air flow establishment within the pipe. With the aim of optimizing the convective air flow rate in a vertical pipe, the details of the flow and thermal fields were investigated in the present study. Conservation equations of mass, momentum, and energy were solved numerically using simple implicit forwardmarching finite difference scheme for a twodimensional axissymmetric flow. In order to evaluate and optimize the air flow rate passing through the pipe, the position and intensity of the wall heat flux were altered when the total employed heat transfer rate was constant. Based on the results of the numerical analysis, relatively more air flow rate was achieved when more intensified heat flux was employed at the lowest part of the vertical pipe. This finding was then validated using a simple experimental setup. The results of the present study could be useful in the design and application of buoyancyassisted natural ventilation systems.
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137
145


M.
Rahimi
Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran
Faculty of Engineering, University of Mohaghegh
Iran


M.
KhalafiSalout
Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran
Faculty of Engineering, University of Mohaghegh
Iran
Natural ventilation
vertical pipe
wall heat flux
[[1] K. T. Lee, and W. M. Yan, “Laminar natural convection between partially heated vertical parallel plates”,Wärme und Stoffübertragungb, Vol. 29, No. 3, pp. 145151, (1994). ##[2] K. T. Lee, “Laminar natural convection heat and mass transfer in vertical rectangular ducts”,Int. J. heat and mass transfer, Vol. 42, No. 24, pp. 45234534, (1999). ##[3] H. B. Awbi, “Design considerations for naturally ventilated buildings”, Renewable Energy, Vol. 5, No. 58, pp. 1081–1090, (1994). ##[4] N. K. Bansal, R. Mathur, and M. S. Bhanduri, “Solar chimney for enhanced stack ventilation”, Building and Environment, Vol. 28, No. 3, pp. 373377, (1993). ##[5] N. K. Bansal, R. Mathur, and M. S. Bhandari, “A study of solar chimney assisted wind tower system for natural ventilation in buildings”, Building and Environment, Vol. 29, No. 4, pp. 495500, (1994). ##[6] K. S. Ong, “A mathematical model of solar chimney”, Renewable Energy, Vol. 28, No. 7, pp. 10471060, (2003). ##[7] N. Hatami, and M. Bahadorinejad, “Experimental determination of natural convection heat transfer coefficient in a vertical flatplate solar air heater”, Solar Energy, Vol. 82, No. 10, pp. 903910, (2008). ##[8] J. Arce, M. J. Jimenez, J. D. Guzman, M. R. Heras, G. Alvarez, and J. Xaman, “Experimental study for natural ventilation on a solar chimney”, Renewable Energy, Vol. 34, No. 12, pp. 29282934, (2009). ##[9] M. Rahimi, and M. M. Bayat, “An experimental study of naturally driven heated air flow in a vertical pipe”, Energy and Building, Vol. 43, No. 1, pp. 126129, (2011). ##[10] U. Frisch, Turbulence, Cambridge University Press, London, (1995). ##[11] P. Bradshaw, Turbulence, SpringerVerlag, Berlin, (1978). ##[12] O. T. Hanna, O. C. Sandal, and P. R. Mazet, “Heat and mass transfer in turbulent flow under condition of drag reduction”, American institute of Chemical Engineering Journal, Vol. 27, pp. 693697, (1981). ##[13] B. Carnahan, H. A. Luther and J. D. Wilkes, Applied Numerical methods, Wiley, pp. 298301, (1969). ##[14] Y. A. Cengel and J. M. Cimbala, Fluid Mechanics, Fundamentals and Applications, 3nded., McGrawHill, pp. 381–394, (2010).##]
Incompressible laminar flow computations by an upwind leastsquares meshless method
2
2
In this paper, the laminar incompressible flow equations are solved by an upwind leastsquares meshless method. Due to the difficulties in generating quality meshes, particularly in complex geometries, a meshless method is increasingly used as a new numerical tool. The meshless methods only use clouds of nodes to influence the domain of every node. Thus, they do not require the nodes to be connected to form a mesh and decrease the difficulty of meshing, particularly around complex geometries. In the literature, it has been shown that the generation of points in a domain by the advancing front technique is an order of magnitude faster than the unstructured mesh for a 3D configuration. The Navier–Stokes solver is based on the artificial compressibility approach and the numerical methodology is based on the higherorder characteristicbased (CB) discretization. The main objective of this research is to use the CB scheme in order to prevent instabilities. Using this inherent upwind technique for estimating convection variables at the midpoint, no artificial viscosity is required at high Reynolds number. The Taylor leastsquares method was used for the calculation of spatial derivatives with normalized Gaussian weight functions. An explicit fourstage RungeKutta scheme with modified coefficients was used for the discretized equations. To accelerate convergence, local time stepping was used in any explicit iteration for steady state test cases and the residual smoothing techniques were used to converge acceleration. The capabilities of the developed 2D incompressible NavierStokes code with the proposed meshless method were demonstrated by flow computations in a liddriven cavity at four Reynolds numbers. The obtained results using the new proposed scheme indicated a good agreement with the standard benchmark solutions in the literature. It was found that using the third order accuracy for the proposed method could be more efficient than its second order accuracy discretization in terms of computational time.
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147
160


M. Y.
Hashemi
Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz, 5375171379, Iran
Department of Mechanical Engineering, Azarbaijan
Iran
Incompressible laminar artificial compressibility
Leastsquares meshless method
Characteristic based scheme
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Li, “Numerical simulation of three dimensional non Newtonian free surface flows in injection molding using ALE finite element method”, Finite Elements in Analysis and Design, Vol. 46, No. 7, pp. 551562, (2010). ##[6] R. W. Lewis, K. Ravindran and A. S. Usmani, “Finite element solution of incompressible flows using an explicit segregated approach”, Archives of Computational Methods in Engineering, Vol. 2, No. 4, pp. 6993, (1999). ##[7] J. H. Kent, “Air Conditioning Modelling by Computational Fluid Dynamics”, Architectural Science Review, Vol. 37, No. 3, pp. 103113, (1994). ##[8] S. Wang and D. Zhub, “Application of CFD in retrofitting airconditioning systems in industrial buildings”, Energy and Buildings, Vol. 35, No. 9, pp. 893902, (2003). ##[9] S. W. Hwang, D. H. Kim, J. K. Min and J. H. Jeong, “CFD analysis of fin tube heat exchanger with a pair of delta winglet vortex generators”, Journal of Mechanical Science and Technology, Vol. 26, No. 9, pp. 29492958, (2012). ##[10] M. Yataghene and J. Legrand, “A 3DCFD model thermal analysis within as craped surface heat exchanger”, Computers & Fluids, Vol. 71, pp. 380399, (2013). ##[11] S. Aradag, U. Olgun, F. Aktrk and B. Basibyk, “CFD analysis of cooling of electronic equipment as an undergraduate design project”, International Journal of Hydrogen Energy, Vol. 20, No. 1, pp. 103113, (2012). ##[12] H. SadatHosseini H, Pablo Carrica, F. Stern, N. Umeda, H. Hashimoto, S. Yamamura and A. Mastuda,“CFD, systembased and EFD study of ship dynamic instability events: Surfriding, periodic motion, and broaching”, Ocean Engineering, Vol. 38, No. 1, pp. 88110, (2011). ##[13] J. F. Thompson, F. C. Thomas and C. W. Mastin, “Automated numerical generation of body fitted curvilinear coordinate system for field containing any number of arbitrary 2D bodies”, Journal of Computational Physics, Vol. 15, pp. 299319, (1974). ##[14] N. P. Weatherill, “A method for generating irregular computational grids in multiply connected planar domains”, International Journal for Numerical Methods in Fluids, Vol. 8, pp. 181197, (1998). ##[15] R. Lohner and E. Onate, “An advancing front point generation technique”, Communications in Numerical Methods in Engineering, Vol. 14, No. 12, pp. 10971108, (1998). ##[16] R. Lohner and E. Onate, “A general advancing front technique for filling space with arbitrary objects”, International Journal for Numerical Methods in Engineering, Vol. 61, No. 12, pp. 19771991, (2004). ##[17] P. W. Randles and L. D. Libersky, “Smoothed particle hydrodynamics: some recent improvements and applications”, Computer Methods in Applied Mechanics and Engineering, Vol. 139, No. 1, pp. 375 408, (1996). ##[18] C. S. Chew CS, K. S. Yeo and C. Shu,“A generalized finitedifference (GFD) ALE scheme for incompressible flows around moving solid bodies on hybrid meshfreeCartesian grids”, Journal of Computational Physics, Vol. 218, No. 2, pp. 510548, (2006). ##[19] H. Ding, C. Shu, K. S. Yeo and D. Xu, “Development of leastsquarebased twodimensional finitedifference schemes and their application to simulate natural convection in a cavity”, Computers & Fluids, Vol. 33, No. 1, pp. 137154, (2004). ##[20] S. N. Atluri and T. Zhu, “A new meshless local PetrovGalerkin (MLPG) approach”, Computational Mechanics, Vol. 22, No. 2, pp. 117127, (1998). ##[21] J. S. Chen, C. T. Wux, S. Yoon and Y. You, “A stabilized conforming nodal integration for Galerkin meshfree methods”, International Journal for Numerical Methods in Engineering, Vol. 50, No. 2, pp. 435466, (2001). ##[22] M. Polner, L. Pesch and J. J. W. van der Vegt, “Construction of stabilization operators for Galerkin leastsquares discretizations of compressible and incompressible flows”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, pp. 24312448, (2007). ##[23] C. Shu, and B. E. Richards, “Application of generalized differential quadrature to solve twodimensional incompressible NavierStokes equations”, International Journal for Numerical Methods in Fluids, Vol. 15, No. 7, pp. 791798, (1992). ##[24] H. Q. Chen and C. Shu, “An efficient implicit meshfree method to solve two dimensional compressible Euler equations”, International Journal of Modern Physics, Vol. 16, No.3, pp. 439454, (2005). ##[25] W. K. Liu, S. Jun and Y. F. Zhang, “Reproducing kernel particle methods”, International Journal for Numerical Methods in Fluids, Vol. 20, No. 89, pp. 10811106, (1995). ##[26] M. Najafi, A. Arefmanesh, and V. Enjilela, “Meshless local PetrovGalerkin methodhigher Reynolds numbers fluid flow applications”, Engineering Analysis with Boundary Elements, Vol. 36, No. 11, pp. 16711685, (2012). ##[27] Y. L. Wu, G. R. Liu and Y. T. Gu, “Application of meshless local PetrovGalerkin (MLPG) approach to simulation of incompressible flow”, Numerical Heat Transfer, Vol. 48, No. 5, pp. 459475, (2005). ##[28] R. Lohner, C. Sacco, E. Onate and S. Idelsohn, “A finite point method for compressible flow”, International Journal for Numerical Methods in Engineering, Vol. 53, pp. 17651779, (2002). ##[29] E. Ortega, E. Onate, S. Idelsohn and R. Flores, “A meshless finite point method for threedimensional analysis of compressible flow problems involving moving boundaries and adaptivity”, International Journal for Numerical Methods in Fluids, Vol. 73, No. 4, pp. 323343, (2013). ##[30] Z. Ma, H. Chen and C. Zhou, “A study of point moving adaptivity in gridless method”, Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 9261937, (2008). ##[31] Y. Hashemi, and A. Jahangirian, “Implicit fully meshless method for compressible viscous flow calculations”, Journal of Computational and Applied Mathematics, Vol. 235, No. 16, pp. 46874700, (2011). ##[32] M. Y. Hashemi, and A. Jahangirian,“An efficient implicit meshless method for compressible flow calculations”, International Journal for Numerical Methods in Fluids, Vol. 67, No. 6, pp. 754770, (2011). ##[33] X. K. Zhang, K. C. Kwon and S. K. Youn, “The leastsquares meshfree method for the steady incompressible viscous flow”, Journal of Computational Physics, Vol. 206, pp. 182207, (2005). ##[34] X. Su, S.Yamamoto and K. Nakahashi, “Analysis of a meshless solver for high Reynolds number flow”, International Journal for Numerical Methods in Fluids, Vol. 72, No. 5, pp. 505527, (2013). ##[35] A. J. Chorin, “A numerical method for solving incompressible viscous flow problems”, Journal of Computational Physics, Vol. 2, No. 1, pp. 1226, (1967). ##[36] J. Farmer, L. Martinelli and A. Jameson, “Fast multigrid method for solving incompressible hydrodynamic problems with free surface”, American Institute of Aeronautics and Astronautics Journal, Vol. 32, pp. 11751182, (1994). ##[37] V. Esfahanian and P. Akbarzadeh, “The Jameson’s numerical method for solving the incompressible viscous and in viscid flows by means of artificial compressibility and preconditioning method”, Applied Mathematics and Computation, Vol. 206, pp. 651661, (2008). ##[38] C. Liu, X. Zheng and C. H. Sung, “Preconditioned multigrid methods for unsteady incompressible flows”, Journal of Computational Physics, Vol. 139, No. 1, pp. 3557, (1998). ##[39] Y. Kallinderis and H. T. Ahn, “Incompressible NavierStokes method with general hybrid meshes”, Journal of Computational Physics, Vol. 210, No. 1, pp. 75108, (2005). ##[40] D. Drikakis, P. A. Govatsos and D. E. Papantonis, “A characteristic based method for incompressible flows”, International Journal for Numerical Methods in Fluids, Vol. 19, No. 8, pp. 667685, (1994). ##[41] D. Drikakis, O. P. Iliev and D. P. Vassileva, “A nonlinear multigrid method for the threedimensional incompressible NavierStokes equations”, Journal of Computational Physics, Vol. 146, No. 1, pp. 301321, (1998). ##[42] K. Siong and C. Y. Zhao, “Numerical study of steady/unsteady flow and heat transfer in porous media using a characteristicsbased matrixfree implicit FV method on unstructured grids”, International Journal of Heat and Fluid Flow, Vol. 25, pp. 10151033, (2004). ##[43] E. Shapiro and D. Drikakis, “Artificial compressibility, characteristicsbased schemes for variable density, incompressible, multispecies flows. Part I. Derivation of different formulations and constant density limit”, Journal of Computational Physics, Vol. 210, No. 2, pp. 584607, (2005). ##[44] E. Shapiro and D. Drikakis, “Artificial compressibility, characteristicsbased schemes for variable density, incompressible, multispecies flows. Part II. Multi grid implementation and numerical tests”, Journal of Computational Physics, Vol. 210, pp. 608631, (2005). ##[45] M. Y. Hashemi and K. Zamzamian,“A multidimensional characteristicbased method for making incompressible flow calculations on unstructured grids”, Journal of Computational and Applied Mathematics, Vol. 259(B), pp.795805, (2014). ##[46] S. Sridar and N. Balakrishnan , “An upwind finite difference scheme for meshless solvers”, Journal of Computational Physics, Vol. 189, No. 1, pp.129, (2003). ##[47] C. Praveen C. and S. M. Deshpande, “Kinetic meshless method for compressible flows”, International Journal for Numerical Methods in Fluids, Vol. 55, No. 11, pp.10591089, (2007). ##[48] H. Luo, J. D. Baum and R. Lohner, “Hybrid buildingblock and gridless method for compressible flows”, International Journal for Numerical Methods in Fluids, Vol. 59, No. 4, pp. 459474, (2009). ##[49] A. Katz, “Meshless methods for computational fluid dynamics”, .2009, PhD Thesis, Stanford University, (2009). ##[50] A. Katz and A. Jameson, “Meshless scheme based on alignment constraints”, American Institute of Aeronautics and Astronautics Journal, Vol. 48, No. 11, pp. 25012511, (2010). ##[51] G. May and Jameson, “Unstructured algorithms for inviscid and viscousflows embedded in a unified solver architecture Flo3xx”, AIAA 43rd Aerospace Sciences Meeting 2005; 0318, Reno, Nevada, (2005). ##[52] C. Hirsch, Numerical Computation of Internaland External Flows (Volume 1:Fundamentals of Computational FluidDynamics), 2ndedition, Elsevier, Burlington, pp. 357360, (2007). ##[53] K. Zamzamian and S. E. Razavi, “Multidimensional up winding for incompressible flows based on characteristics”, Journal of Computational Physics, Vol. 227, No. 19, pp. 86998713, (2008). ##[54] Y. Zhao and B. Zhang, “A highorder characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grid”, Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 5, pp. 733756, (2000). ##[55] C. H. Tai and Y. Zhao, “Parallel unsteady incompressible viscous flow computations using an unstructured multigrid method”, Journal of Computational Physics, Vol. 192, No. 1, pp. 277311, (2003). ##[56] R. Lohner, C. Sacco, E. Onate, S. A. Idelsohn,“Finite point method for compressible flow”, International Journal for Numerical Methods in Engineering, Vol. 53, pp.1765–1779, (2002). ##[57] D. J. Mavriplis, A. Jameson, and L. Martinelli,“Multigrid solution of the NavierStokes equations on the triangular meshes”, AIAA paper, 27th Sciences Meeting, Reno, Nevada, USA, January 912, (1989). ##[58] U. Ghia, K. N. Ghia, and C. T. Shin, “HighRe solutions for incompressible flow using the NavierStokes equations and a multigrid method”, Journal of Computational Physics, Vol. 48, pp.387411, (1982).##]
A study on the use of perturbation technique for analyzing the nonlinear forced response of piezoelectric microcantilevers
2
2
In this paper, a comparison is made between direct and indirect perturbation approaches to solve the nonlinear vibration equations of a piezoelectrically actuated cantilever microbeam. In this comparison, the equation of motion is considered according to EulerBernoulli theory with considering the nonlinear geometric and inertia terms resulted from shortening effect. In the direct perturbation approach, the multiple scales method is directly applied to the partial differential equation of motion. In the indirect approach, the multiple scales perturbation technique is applied to the discretized equation of motion. It is shown that, if the equation of motion is discretized using one nonuniform microbeam mode shape as a comparison function, then the results of indirect perturbation approach will be identical to those of the direct perturbation approach. Moreover, it is observed that discretization using one uniform microbeam mode shape as a comparison function results in a different output. The concept of nonuniform microbeam mode shape is the linear mode shape of the microbeam by considering the geometric and inertia effects of the piezoelectric layer.
1

161
172


M.
Zamanian
Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, P.O. Box 1571914911, Tehran, Iran
Department of Mechanical Engineering, Faculty
Iran


M.
Hadilu
Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, P.O. Box 1571914911, Tehran, Iran
Department of Mechanical Engineering, Faculty
Iran


B.
Firouzi
Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, P.O. Box 1571914911, Tehran, Iran
Department of Mechanical Engineering, Faculty
Iran
Direct perturbation
Indirect perturbation
Microcantilever
Nonlinear Vibration
Shortening effect
Piezoelectric
[[1] M. I. Younis, MEMS Linear And Nonlinear statics and dynamics, Springer, New York , pp. 6870, (2011) . ##[2] S. N. Mahmoodi and N. Jalili, “Nonlinear vibrations and frequency response analysis of piezoelectrically driven microcantilevers”, Int. J. Nonlinear Mech, Vol. 42, No. 4, pp. 577 587, (2007). ##[3] S. N. Mahmoodi, M. Afshari and N. Jalili, “Nonlinear vibrations of piezoelectric microcantilevers for biologicallyinduced surface stress sensing”, Commun Nonlinear Sci Numer Simul, Vol. 13, No. 9, pp. 19641977, (2007). ##[4] S. N. Mahmoodi and N. Jalili, Coupled “FlexuralTorsional Nonlinear Vibrations of Piezoelectrically Actuated Microcantilevers With Application to Friction Force Microscopy”, J Vib Acoust., Vol. 130 No. 6, 061003(10pp), (2008). ##[5] S. N. Mahmoodi, N. Jalili and M. Ahmadian, “Subharmonics analysis of nonlinear flexural vibrations of piezoelectrically actuated microcantilevers”, Nonlinear Dyn., Vol. 59, No. 3, 397409, (2010). ##[6] A. Shooshtari, , S.M. Hoseini, , S.N. Mahmoodi, and H. Kalhori, “Analytical solution for nonlinear free vibrations of viscoelastic microcantilevers covered with a piezoelectric layer”, Smart Mater. Struct., Vol. 21, No. 7, pp. 075015 (10pp) ,(2012). ##[7] S. M Hosseini, A. Shooshtari , H. Kalhori and S. N. Mahmoodi “Nonlinearforced vibrations of piezoelectrically actuated viscoelastic cantilevers”, Nonlinear Dyn., Vol. 78, No. 1, pp. 571583, (2014). ##[8] S. M. Hoseini, H. Kalhori, A. Shooshtari and S. N. Mahmoodi, “Analytical solution for nonlinear forced response of a viscoelastic piezoelectric cantilever beam resting on a nonlinear elastic foundation to an external harmonic excitation”, Composites Part B ;Vol. 67, No. 1, pp. 464471, (2014). ##[9] M. H. Korayem and R. Ghaderi, “Vibration response of a piezoelectrically actuated microcantilever subjected to tip–sample interaction”, Sci. Iran. Vol. 20, No. 1, pp. 195206, (2013). ##[10] A. J. Dick, B. Balachandran, D. L. DeVoe and J. Mote, “Parametric identification of piezoelectric microscale resonators”, J. Micromech. Microeng., Vol. 16, No. 8, pp. 15931601, (2006). ##[11] H. Raeisifard, M. Zamanian, M. Nikkhah Bahrami, A. YousefiKomad and H. Raeisi Fard, “On the nonlinear primary resonances of a piezoelectric laminated micro system under electrostatic control voltage”. J Sound Vib .,Vol. 333, No. 21, pp. 5494–5510, (2014). ##[12] H. Li, S. Preidikman, , B. Balachandran, and J. Mote, “Nonlinear free and forced oscillations of piezoelectric microresonators”, J. Micromech. Microeng., Vol. 16, No. 2, pp. 356367, (2006). ##[13] M. Zamanian, S. E. Khadem and S. N. Mahmoodi, “Analysis of nonlinear vibrations of a microresonator under piezoelectric and electrostatic actuations”, Proc. IMechE Part C: J. Mech. Eng. Sci., Vol. 223, No. 2, pp. 329344, (2009). ##[14] M. C. Rachael and N. Mahmoodi, “Parameter sensitivity analysis of nonlinear piezoelectric probe in tapping mode atomic force microscopy for measurement improvement”, J. applied physics,Vol. 115, No.7, 074501(9pp), (2014). ##[15] M. C. Rachael and N. Mahmoodi, “Nonlinear forced response of piezoelectric microcantilevers with application to tapping mode atomic force microscopy” Proc.of SPIE 9057, Vol. 9057, No. 1, 905722(13pp), (2014). ##[16] M. Zamanian, H. Rezaei, M. Hadilu and S. A. A. Hosseini, A comprehensive analysis on the discretization method of the equation of motion in piezoelectrically actuated microbeam Smart Structures and Systems, Vol. 16, No. 5, pp. 891918 , (2015) ##[17] A. H. Nayfeh, Introduction to perturbation technique, Wiley, New York , pp.338400, (1993).##]