This paper deals with the steady two-dimensional stagnation point flow of nanofluid toward an exponentially stretching sheet with nonuniform heat generation/absorption. The employed model for nanofluid includes two-component four-equation nonhomogeneous equilibrium model that incorporates the effects of Brownian diffusion and thermophoresis simultaneously. The basic partial boundary layer equations have been reduced to a two-point boundary value problem via similarity variables and solved analytically via HAM. Effects of governing parameters such as heat generation/absorption λ, stretching parameter ε, thermophoresis , Lewis number Le, Brownian motion , and Prandtl number Pr on heat transfer and concentration rates are investigated. The obtained results indicate that in contrast with heat transfer rate, concentration rate is very sensitive to the abovementioned parameters. Also, in the case of heat generation , despite concentration rate, heat transfer rate decreases. Moreover, increasing in stretching parameter leads to a gentle rise in both heat transfer and concentration rates. 1. Introduction For years, many researchers have paid much attention to viscous fluid motion near the stagnation region of a solid body, where “body” corresponds to either fixed or moving surfaces in a fluid. This multidisciplinary flow has frequent applications in high speed flows, thrust bearings, and thermal oil recovery. Hiemenz [1] developed the first investigation in this field. He applied similarity transformation to collapse two-dimensional Navier-Stokes equations to a nonlinear ordinary differential one and then presented its exact solution. Extension of this study was carried out with a similarity solution by Homann [2] to the case of axisymmetric three-dimensional stagnation point flow. After these original studies, many researchers have put their attention on this subject [3–9]. Besides stagnation point flow, stretching surfaces have a wide range of applications in engineering and several technical purposes particularly in metallurgy and polymer industry, for instance, gradual cooling of continuous stretched metal or plastic strips which have multiple applications in mass production. Crane [10] was the first to present a similarity solution in the closed analytical form for steady two-dimensional incompressible boundary layer flow caused by the stretching plate whose velocity varies linearly with the distance from a fixed point on the sheet. The combination of stretching surface and stagnation point flow was analyzed by Yao et al. [11]. Different types fluids such
References
[1]
K. Hiemenz, “Die Grenzschicht an einem in den gleichformingen Flussigkeitsstrom eingetauchten graden Kreiszylinder,” Dingler's Polytechnic Journal, vol. 326, pp. 321–324, 1911.
[2]
F. Homann, “Der Einflu？ gro？er Z？higkeit bei der Str？mung um den Zylinder und um die Kugel,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 16, no. 3, pp. 153–164, 1936.
[3]
H. A. Attia, “Homann magnetic flow and heat transfer with uniform suction or injection,” Canadian Journal of Physics, vol. 81, no. 10, pp. 1223–1230, 2003.
[4]
K. Bhattacharyya and K. Vajravelu, “Stagnation-point flow and heat transfer over an exponentially shrinking sheet,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 2728–2734, 2012.
[5]
A. Malvandi, “The unsteady flow of a nanofluid in the stagnation point region of a time-dependent rotating sphere,” THERMAL SCIENCE, 2013.
[6]
V. Kumaran, R. Tamizharasi, and K. Vajravelu, “Approximate analytic solutions of stagnation point flow in a porous medium,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 6, pp. 2677–2688, 2009.
[7]
M. A. A. Hamad and M. Ferdows, “Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: a Lie group analysis,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 132–140, 2012.
[8]
Z. Ziabakhsh, G. Domairry, and H. Bararnia, “Analytical solution of non-Newtonian micropolar fluid flow with uniform suction/blowing and heat generation,” Journal of the Taiwan Institute of Chemical Engineers, vol. 40, no. 4, pp. 443–451, 2009.
[9]
Z. Ziabakhsh, G. Domairry, and H. R. Ghazizadeh, “Analytical solution of the stagnation-point flow in a porous medium by using the homotopy analysis method,” Journal of the Taiwan Institute of Chemical Engineers, vol. 40, no. 1, pp. 91–97, 2009.
[10]
L. J. Crane, “Flow past a stretching plate,” Zeitschrift für Angewandte Mathematik und Physik, vol. 21, no. 4, pp. 645–647, 1970.
[11]
S. Yao, T. Fang, and Y. Zhong, “Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 752–760, 2011.
[12]
K. R. Rajagopal, T. Y. Na, and A. S. Gupta, “Flow of a viscoelastic fluid over a stretching sheet,” Rheologica Acta, vol. 23, no. 2, pp. 213–215, 1984.
[13]
R. Nazar, N. Amin, D. Filip, and I. Pop, “Stagnation point flow of a micropolar fluid towards a stretching sheet,” International Journal of Non-Linear Mechanics, vol. 39, no. 7, pp. 1227–1235, 2004.
[14]
M. A. A. Hamad, “Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field,” International Communications in Heat and Mass Transfer, vol. 38, no. 4, pp. 487–492, 2011.
[15]
Z. Ziabakhsh, G. Domairry, M. Mozaffari, and M. Mahbobifar, “Analytical solution of heat transfer over an unsteady stretching permeable surface with prescribed wall temperature,” Journal of the Taiwan Institute of Chemical Engineers, vol. 41, no. 2, pp. 169–177, 2010.
[16]
S. Nadeem, A. Hussain, and M. Khan, “HAM solutions for boundary layer flow in the region of the stagnation point towards a stretching sheet,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 475–481, 2010.
[17]
M. M. Nandeppanavar, K. Vajravelu, M. Subhas Abel, S. Ravi, and H. Jyoti, “Heat transfer in a liquid film over an unsteady stretching sheet,” International Journal of Heat and Mass Transfer, vol. 55, no. 4, pp. 1316–1324, 2012.
[18]
A. Malvandi, F. Hedayati, and D. D. Ganji, “Thermodynamic optimization of fluid flow over an isothermal moving plate,” Alexandria Engineering Journal, 2013.
[19]
A. K. Singh, “Heat source and radiation effects on magneto-convection flow of a viscoelastic fluid past a stretching sheet: analysis with Kummer's functions,” International Communications in Heat and Mass Transfer, vol. 35, no. 5, pp. 637–642, 2008.
[20]
S. Nadeem and C. Lee, “Boundary layer flow of nanofluid over an exponentially stretching surface,” Nanoscale Research Letters, vol. 7, article 94, pp. 1–15, 2012.
[21]
M. R. H. Nobari and A. Malvandi, “Torsion and curvature effects on fluid flow in a helical annulus,” International Journal of Non-Linear Mechanics, vol. 57, pp. 90–101, 2013.
[22]
S. U. S. Choi, “Enhancing thermal conductivity of fluids with nanoparticles,” in Developments and Applications of Non-Newtonian Flows, D. A. Siginer and H. P. Wang, Eds., pp. 99–105, ASME, 1995.
[23]
H. Masuda, A. Ebata, K. Teramae, and N. Hishinuma, “Alteration of thermalconductivity and viscosity of liquid by dispersing ultra-fine particles. Dispersion of Al2O3, SiO2 and TiO2 ultra-fine particles,” Netsu Bussei, vol. 7, no. 4, pp. 227–233, 1993.
[24]
J. Buongiorno, “Convective transport in nanofluids,” Journal of Heat Transfer, vol. 128, no. 3, pp. 240–250, 2006.
[25]
A. V. Kuznetsov and D. A. Nield, “Natural convective boundary-layer flow of a nanofluid past a vertical plate,” International Journal of Thermal Sciences, vol. 49, no. 2, pp. 243–247, 2010.
[26]
W. A. Khan and I. Pop, “Boundary-layer flow of a nanofluid past a stretching sheet,” International Journal of Heat and Mass Transfer, vol. 53, no. 11-12, pp. 2477–2483, 2010.
[27]
N. Bachok, A. Ishak, and I. Pop, “Boundary-layer flow of nanofluids over a moving surface in a flowing fluid,” International Journal of Thermal Sciences, vol. 49, no. 9, pp. 1663–1668, 2010.
[28]
A. Alsaedi, M. Awais, and T. Hayat, “Effects of heat generation/absorption on stagnation point flow of nanofluid over a surface with convective boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4210–4223, 2012.
[29]
P. Rana and R. Bhargava, “Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: a numerical study,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 212–226, 2012.
[30]
W. Daungthongsuk and S. Wongwises, “A critical review of convective heat transfer of nanofluids,” Renewable and Sustainable Energy Reviews, vol. 11, no. 5, pp. 797–817, 2007.
[31]
X.-Q. Wang and A. S. Mujumdar, “Heat transfer characteristics of nanofluids: a review,” International Journal of Thermal Sciences, vol. 46, no. 1, pp. 1–19, 2007.
[32]
S. Kaka？ and A. Pramuanjaroenkij, “Review of convective heat transfer enhancement with nanofluids,” International Journal of Heat and Mass Transfer, vol. 52, no. 13-14, pp. 3187–3196, 2009.
[33]
S. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004.
[34]
B. Wu and H. Zhong, “Summation of perturbation solutions to nonlinear oscillations,” Acta Mechanica, vol. 154, no. 1–4, pp. 121–127, 2002.
[35]
G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988.
[36]
J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.
[37]
F. Hedayati, D. Ganji, S. Hamidi, and A. Malvandi, “An analytical study on a model describing heat conduction in rectangular radial fin with temperature-dependent thermal conductivity,” International Journal of Thermophysics, vol. 33, no. 6, pp. 1042–1054, 2012.
[38]
A. Malvandi, D. D. Ganji, F. Hedayati, M. H. Kaffash, and M. Jamshidi, “Series solution of entropy generation toward an isothermal flat plate,” Thermal Science, vol. 16, no. 5, pp. 1289–1295, 2012.
[39]
S. Liao, “On the relationship between the homotopy analysis method and Euler transform,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 6, pp. 1421–1431, 2010.
[40]
S. Liao, “Homotopy analysis method: a new analytical technique for nonlinear problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 2, pp. 95–100, 1997.
[41]
S.-I. Liao, “A short review on the homotopy analysis method in fluid mechanics,” Journal of Hydrodynamics, vol. 22, no. 5, pp. 839–841, 2010.
[42]
D. G. Domairry, A. Mohsenzadeh, and M. Famouri, “The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery-Hamel flow,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 1, pp. 85–95, 2009.
[43]
G. Domairry and M. Fazeli, “Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 489–499, 2009.
[44]
G. Domairry and N. Nadim, “Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation,” International Communications in Heat and Mass Transfer, vol. 35, no. 1, pp. 93–102, 2008.
[45]
S.-J. Liao, “An explicit, totally analytic approximate solution for Blasius' viscous flow problems,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 759–778, 1999.
[46]
S. Liao, “An optimal homotopy-analysis approach for strongly nonlinear differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 2003–2016, 2010.
[47]
M. Hassani, M. Mohammad Tabar, H. Nemati, G. Domairry, and F. Noori, “An analytical solution for boundary layer flow of a nanofluid past a stretching sheet,” International Journal of Thermal Sciences, vol. 50, no. 11, pp. 2256–2263, 2011.
[48]
N. Bachok, A. Ishak, and I. Pop, “On the stagnation-point flow towards a stretching sheet with homogeneous-heterogeneous reactions effects,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4296–4302, 2011.
[49]
G. Domairry and Z. Ziabakhsh, “Solution of boundary layer flow and heat transfer of an electrically conducting micropolar fluid in a non-Darcian porous medium,” Meccanica, vol. 47, no. 1, pp. 195–202, 2012.
[50]
A. A. Joneidi, G. Domairry, and M. Babaelahi, “Analytical treatment of MHD free convective flow and mass transfer over a stretching sheet with chemical reaction,” Journal of the Taiwan Institute of Chemical Engineers, vol. 41, no. 1, pp. 35–43, 2010.
[51]
Z. Ziabakhsh, G. Domairry, H. Bararnia, and H. Babazadeh, “Analytical solution of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium,” Journal of the Taiwan Institute of Chemical Engineers, vol. 41, no. 1, pp. 22–28, 2010.
[52]
K. Yabushita, M. Yamashita, and K. Tsuboi, “An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method,” Journal of Physics A, vol. 40, no. 29, pp. 8403–8416, 2007.
[53]
Z. Niu and C. Wang, “A one-step optimal homotopy analysis method for nonlinear differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 2026–2036, 2010.
