S. Nadeem, Anwar Hussain, M. Y. Malik, T. Hayat. Series Solutions for the Stagnation Flow of a Second Grade Fluid Over a Shrinking Sheet[J]. Applied Mathematics and Mechanics, 2009, 30(10): 1173-1180. doi: 10.3879/j.issn.1000-0887.2009.10.005
Citation: S. Nadeem, Anwar Hussain, M. Y. Malik, T. Hayat. Series Solutions for the Stagnation Flow of a Second Grade Fluid Over a Shrinking Sheet[J]. Applied Mathematics and Mechanics, 2009, 30(10): 1173-1180. doi: 10.3879/j.issn.1000-0887.2009.10.005

Series Solutions for the Stagnation Flow of a Second Grade Fluid Over a Shrinking Sheet

doi: 10.3879/j.issn.1000-0887.2009.10.005
  • Received Date: 2009-03-22
  • Rev Recd Date: 2009-07-28
  • Publish Date: 2009-10-15
  • The analytic solutions of boundary layer flows bounded by a shrinking sheet are derived. Using similarity transformations, the partial differential equations were reduced into the ordinary differen tial equations which were then solved by homotopy analysis method (HAM). Two-dimen sional and axisymmetric shrinking flow cases were discussed.
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  • [1]
    Sajid M, Ahmad I, Hayat T, et al. Unsteady flow and heat transfer of a second grade fluid over a stretching sheet[J]. Comm Nonlinear Sci Num Sim,2009, 14(1):96-108. doi: 10.1016/j.cnsns.2007.07.014
    [2]
    Cortell R. A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet[J]. Int J Nonlinear Mech, 2006,41(1):78-85. doi: 10.1016/j.ijnonlinmec.2005.04.008
    [3]
    Cortell R. Effects of viscous dissipation and work done by deformation on the MHD flow and heat transfer of a viscoelastic fluid over a stretching sheet[J]. Phy Lett A, 2006, 357(4/5):298-305. doi: 10.1016/j.physleta.2006.04.051
    [4]
    Hayat T, Sajid M. Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet[J]. Int J Heat Mass Transfer,2007, 50(1/2):75-84. doi: 10.1016/j.ijheatmasstransfer.2006.06.045
    [5]
    Bataller R C. Effects of heat source/sink, radiation and work done by deformation on flow and heat transfer of a viscoelastic fluid over a stretching sheet[J]. Computers Mathematics Applications ,2007, 53(2): 305-316. doi: 10.1016/j.camwa.2006.02.041
    [6]
    Bataller R C. Viscoelastic fluid flow and heat transfer over a stretching sheet under the effects of a non-uniform heat source, viscous dissipation and thermal radiation[J]. Int J Heat Mass Transfer, 2007, 50(15/16): 3152-3162. doi: 10.1016/j.ijheatmasstransfer.2007.01.003
    [7]
    Cortell R. MHD flow and mass transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet with chemically reactive species[J]. Chem Eng Proc, 2007, 46(8): 721-728. doi: 10.1016/j.cep.2006.09.008
    [8]
    Cortell R. Toward an understanding of the motion and mass transfer with chemically reactive species for two classes of viscoelastic fluid over a porous stretching sheet[J]. Chem Eng Proc, 2007, 46(10): 982-989. doi: 10.1016/j.cep.2007.05.022
    [9]
    Hayat T, Saif S, Abbas Z. The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet[J]. Phy Lett A, 2008, 372(30): 5037-5045. doi: 10.1016/j.physleta.2008.03.066
    [10]
    Ahmad I, Sajid M, Hayat T, et al. The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet[J]. Computers Mathematics Applications, 2008, 56(5): 1351-1357. doi: 10.1016/j.camwa.2008.03.002
    [11]
    Hayat T, Javed T, Abbas Z. Slip flow and heat transfer of a second grade fluid past a stretching sheet through a porous space[J]. Int J Heat Mass Transfer, 2008, 51(17/18): 4528-4534. doi: 10.1016/j.ijheatmasstransfer.2007.12.022
    [12]
    Abbas Z, Hayat T, Sajid M, et al. Unsteady flow of a second grade fluid film over an unsteady stretching sheet[J]. Math Computer Modelling, 2008, 48(3/4): 518-526. doi: 10.1016/j.mcm.2007.09.015
    [13]
    Khan M, Naheed E, Fetecau T, et al. Exact solutions of starting flows for second grade fluid in a porous medium[J]. Int J Nonlinear Mech, 2008, 43(9): 868-879. doi: 10.1016/j.ijnonlinmec.2008.06.002
    [14]
    Fetecau C, Hayat T, Ali N, et al. Unsteady flow of a second grade fluid between two side walls perpendicular to a plate[J]. Nonlinear Analysis: Real World Applications, 2008, 9(3): 1236-1252. doi: 10.1016/j.nonrwa.2007.02.014
    [15]
    Khan M, Ali S H, Hayat T, et al. MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium[J]. Int J Nonlinear Mech, 2008, 43(4): 302-319. doi: 10.1016/j.ijnonlinmec.2007.12.016
    [16]
    〖JP3〗Sakiadis B C. Boundary layer behaviour on continuous solid surfaces[J]. AIChE J, 1961, 7(2): 26-28.〖JP〗 doi: 10.1002/aic.690070108
    [17]
    Xu H, Liao S-J. Dual solutions of boundary layer flow over an upstream moving plate[J]. Comm Nonlinear Sci Num Sim, 2008, 13(2): 350-358. doi: 10.1016/j.cnsns.2006.04.008
    [18]
    Liao S-J. An analytic solution of unsteady boundary layer flows caused by an impulsive stretching plate[J]. Comm Nonlinear Sci Num Sim, 2006, 11(3): 326-339. doi: 10.1016/j.cnsns.2004.09.004
    [19]
    Liao S-J. A new branch of solutions of boundary layer flows over an impermeable stretching plate[J]. Int J Heat Mass Transf, 2005, 48(12):2529-2539. doi: 10.1016/j.ijheatmasstransfer.2005.01.005
    [20]
    Hayat T, Sajid M. Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid[J]. Int J Eng Sci, 2007, 45(2/8): 393-401. doi: 10.1016/j.ijengsci.2007.04.009
    [21]
    Abel M S, Mahantesh M Nandeppanavar. Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink[J]. Comm Nonlinear Sci Num Sim, 2009, 14(5): 2120-2131. doi: 10.1016/j.cnsns.2008.06.004
    [22]
    Ishak A, Nazar R, Pop I. MHD boundary-layer flow of a micropolar fluid past a wedge with constant wall heat flux[J]. Comm Nonlinear Sci Num Sim, 2009, 14(1): 109-118. doi: 10.1016/j.cnsns.2007.07.011
    [23]
    Bose S, Chakraborty S. A boundary layer analysis of electro-magneto-hydrodynamic forced convective transport over a melting slab[J]. Int J Heat Mass Transfer, 2008, 51(21/22): 5465-5474. doi: 10.1016/j.ijheatmasstransfer.2008.02.051
    [24]
    Ishak A, Nazar R, Pop I. Dual solutions in mixed convection boundary layer flow of micropolar fluids[J]. Comm Nonlinear Sci Num Sim,2009, 14(4): 1324-1333. doi: 10.1016/j.cnsns.2008.01.017
    [25]
    Hayat T, Abbas Z, Javed T, et al. Three-dimensional rotating flow induced by a shrinking sheet for suction[J]. Chaos, Solitons and Fractals,2009,39(4):1615-1626. doi: 10.1016/j.chaos.2007.06.045
    [26]
    Nadeem S, Awais M. Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity[J]. Phy Lett A, 2008, 372(30): 4965-4972. doi: 10.1016/j.physleta.2008.05.048
    [27]
    Fang T. Boundary layer flow over a shrinking sheet with power-law velocity[J]. Heat Mass Transfer, 2008, 51(25/26): 5838-5843. doi: 10.1016/j.ijheatmasstransfer.2008.04.067.
    [28]
    Hayat T, Javed T, Sajid M. Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface[J]. Phy Lett A, 2008, 372(18): 3264-3273. doi: 10.1016/j.physleta.2008.01.069
    [29]
    Hayat T, Abbas Z, Ali N. MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species[J]. Phy Lett A, 2008, 372(26): 4698-4704. doi: 10.1016/j.physleta.2008.05.006
    [30]
    Wang C-Y. Stagnation flow towards a shrinking sheet[J]. Int J Nonlinear Mech, 2008, 43(5): 377-382. doi: 10.1016/j.ijnonlinmec.2007.12.021
    [31]
    Liao S-J. Beyond Perturbation Introduction to Homotopy Analysis Method[M]. Boca Raton: Chapman & Hall/CRC Press, 2003.
    [32]
    Abbasbandy S. The application of homotopy analysis method to nonlinear equations arising in heat transfer[J]. Phy Lett A, 2006, 360(1): 109-113. doi: 10.1016/j.physleta.2006.07.065
    [33]
    Abbasbandy S. Homotopy analysis method for heat radiation equations[J]. Int Comm Heat Mass Transfer, 2007, 34(3): 380-387. doi: 10.1016/j.icheatmasstransfer.2006.12.001
    [34]
    Abbasbandy S, Tan Y, Liao S-J. Newton-homotopy analysis method for nonlinear equations[J]. Applied Mathematics Computation, 2007, 188(2): 1794-1800. doi: 10.1016/j.amc.2006.11.136
    [35]
    Abbasbandy S. Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method[J]. Chem Eng J, 2008, 136(2/3): 144-150. doi: 10.1016/j.cej.2007.03.022
    [36]
    Abbasbandy S. Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method[J]. Applied Math Modelling, 2008, 32(12): 2706-2714. doi: 10.1016/j.apm.2007.09.019
    [37]
    Tan Y, Abbasbandy S. Homotopy analysis method for quadratic Ricati differential equation[J]. Comm Nonlinear Sci Num Sim, 2008, 13(3): 539-546. doi: 10.1016/j.cnsns.2006.06.006
    [38]
    Alomari A K, Noorani M S M, Nazar R. Adaptation of homotopy analysis method for the numeric-analytic solution of Chen system[J]. Comm Nonlinear Sci Num Sim, 2009,14(5): 2336-2346. doi: 10.1016/j.cnsns.2008.06.011.
    [39]
    Sajid M, Hayat T, Asghar S. Comparison of the HAM and HPM solutions of thin film flow of a non-Newtonian fluids on a moving belt[J]. Nonlinear Dynam, 2007, 50(1/2): 27-35. doi: 10.1007/s11071-006-9140-y
    [40]
    Sajid M, Awais M, Nadeem S, et al. The influence of slip condition on thin film flow of a fourth grade fluid by the homotopy analysis method[J]. Computers Math Applications, 2008, 56(8): 2019-2026. doi: 10.1016/j.camwa.2008.04.022
    [41]
    Chowdhury M S H, Hashim I, Abdulaziz O. Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems[J]. Comm Nonlinear Sci Num Sim, 2009, 14(2): 371-378. doi: 10.1016/j.cnsns.2007.09.005
    [42]
    Sajid M, Hayat T. Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations[J]. Nonlinear Analysis: Real World Applications, 2008, 9(5): 2296-2301. doi: 10.1016/j.nonrwa.2007.08.007
    [43]
    Bataineh A S, Noorani M S M, Hashim I. Modified homotopy analysis method for solving systems of second-order BVPs[J]. Comm Nonlinear Sci Num Sim, 2009, 14(2): 430-442. doi: 10.1016/j.cnsns.2007.09.012
    [44]
    Sajid M, Hayat T. The application of homotopy analysis method to thin film flows of a third order fluid[J]. Chaos,
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