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Viscous flow over a shrinking sheet with a second order slip flow model. (English) Zbl 1222.76028
Summary: Viscous flow over a shrinking sheet is solved analytically using a newly proposed second order slip flow model. The closed solution is an exact solution of the full governing Navier–Stokes equations. The solution has two branches in a certain range of the parameters. The effects of the two slip parameters and the mass suction parameter on the velocity distribution are presented graphically and discussed. For certain combinations of the slip parameters, the wall drag force can decrease with the increase of mass suction. These results clearly show that the second order slip flow model is necessary to predict the flow characteristics accurately.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Altan, T.; Oh, S.; Gegel, H., Metal forming fundamentals and applications, (1979), American Society of Metals Metals Park, OH
[2] Fisher, E.G., Extrusion of plastics, (1976), Wiley New York
[3] Tadmor Z, Klein I. Engineering principles of plasticating extrusion. Polymer science and engineering series. New York: Van Nostrand Reinhold; 1970.
[4] Sakiadis, B.C., Boundary-layer behavior on continuous solid surface: I. boundary-layer equations for two-dimensional and axisymmetric flow, J aiche, 7, 26-28, (1961)
[5] Sakiadis, B.C., Boundary-layer behavior on continuous solid surface: II. boundary-layer equations for two-dimensional and axisymmetric flow, J aiche, 7, 221-225, (1961)
[6] Tsou, F.K.; Sparrow, E.M.; Goldstain, R.J., Flow and heat transfer in the boundary layer on a continuous moving surface, Int J heat mass transfer, 10, 219-235, (1967)
[7] Crane, L.J., Flow past a stretching plate, Z angew math phys, 21, 4, 645, (1970)
[8] Banks, W.H.H., Similarity solutions of the boundary-layer equations for a stretching wall, J mech theor appl, 2, 375-392, (1983) · Zbl 0538.76039
[9] Dutta, B.K.; Roy, P.; Gupta, A.S., Temperature field in flow over a stretching sheet with uniform heat flux, Int commun heat mass trans, 12, 89-94, (1985)
[10] Grubka, L.J.; Bobba, K.M., Heat transfer characteristics of a continuous stretching surface with variable temperature, ASME J heat transfer, 107, 248-250, (1985)
[11] Chen, C.K.; Char, M.I., Heat transfer of a continuous stretching surface with suction and blowing, J math anal appl, 135, 568-580, (1988) · Zbl 0652.76062
[12] Ali, M.E., On thermal boundary layer on a power law stretched surface with suction or injection, Int J heat fluid flow, 16, 280-290, (1995)
[13] Elbashbeshy, E.M.A., Heat transfer over a stretching surface with variable surface heat flux, J phys D appl phys, 31, 1951-1954, (1998)
[14] Magyari, E.; Keller, B., Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface, J phys D appl phys, 32, 5, 577-585, (1999)
[15] Magyari, E.; Keller, B., Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls, Eur J mech B fluids, 19, 1, 109-122, (2000) · Zbl 0976.76021
[16] Magyari, E.; Ali, M.E.; Keller, B., Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities, Heat mass transfer, 38, 1-2, 65-74, (2001)
[17] Liao, S.J., A new branch of solutions of boundary-layer flows over a stretching flat plate, Int J heat mass transfer, 49, 12, 2529-2539, (2005) · Zbl 1189.76142
[18] Liao, S.J., A new branch of solution of boundary-layer flows over a permeable stretching plate, Int J nonlinear mech, 42, 819-830, (2007) · Zbl 1200.76046
[19] Fang, T.; Zhang, J., Flow between two stretchable disks, Int commun heat mass transfer, 35, 8, 892-895, (2008)
[20] Fang, T., Flow over a stretchable disk, Phys fluids, 19, 128105, (2007) · Zbl 1182.76239
[21] Wang, C.Y., Exact solutions of the steady state navier – stokes equations, Ann rev fluid mech, 23, 159-177, (1991)
[22] Miklavcic, M.; Wang, C.Y., Viscous flow due to a shrinking sheet, Quart appl math, 64, 2, 283-290, (2006) · Zbl 1169.76018
[23] Fang, T., Boundary layer flow over a shrinking sheet with power-law velocity, Int J heat mass transfer, 51, 25/26, 5838-5843, (2008) · Zbl 1157.76010
[24] Fang, T.; Zhang, J., Closed-form exact solutions of MHD viscous flow over a shrinking sheet, Commun nonlinear sci numer simulat, 14, 7, 2853-2857, (2009) · Zbl 1221.76142
[25] Fang, T.; Zhang, J.; Yao, S., Viscous flow over an unsteady shrinking sheet with mass transfer, Chin phys lett, 26, 1, 014703, (2009)
[26] Fang, T.; Liang, W.; Lee, C.F., A new solution branch for the Blasius equation – a shrinking sheet problem, Comput math appl, 56, 12, 3088-3095, (2008) · Zbl 1165.76324
[27] Fang T, Zhang J. Heat transfer over a shrinking sheet – an analytical solution. Acta Mech. doi:10.1007/s00707-009-0183-2; 2009.
[28] Hayat, T.; Abbas, Z.; Sajid, M., On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet, J appl mech trans ASME, 74, 6, 1165-1171, (2007)
[29] Sajid, M.; Hayat, T.; Javed, T., MHD rotating flow of a viscous fluid over a shrinking surface, Nonlinear dyn, 51, 1-2, 259-265, (2008) · Zbl 1170.76366
[30] Sajid, M.; Hayat, T., The application of homotopy analysis method for MHD viscous flow due to a shrinking sheet, Chaos, solitons & fractals, (2007) · Zbl 1197.76100
[31] Gal-el-Hak, M., The fluid mechanics of micro-devices – the freeman scholar lecture, ASME trans J fluids eng, 121, 5-33, (1999)
[32] Shidlovskiy, V.P., Introduction to the dynamics of rarefied gases, (1967), American Elsevier Publishing Company Inc. New York
[33] Pande, G.C.; Goudas, C.L., Hydromagnetic reyleigh problem for a porous wall in slip flow regime, Astrophys space sci, 243, 285-289, (1996) · Zbl 0921.76191
[34] Yoshimura, A.; Prudhomme, R.K., Wall slip corrections for Couette and parallel disc viscometers, J rheol, 32, 53-67, (1988)
[35] Martin MJ, Boyd ID. Blasius boundary layer solution with slip flow conditions. In: AIP conference proceedings, rarefied gas dynamics: 22nd international symposium, vol. 585, 2001. p. 518-23.
[36] Andersson, H.I., Slip flow past a stretching surface, Acta mech, 158, 121-125, (2002) · Zbl 1013.76020
[37] Wang, C.Y., Flow due to a stretching boundary with partial slip-an exact solution of the navier – stokes equations, Chem eng sci, 57, 17, 3745-3747, (2002)
[38] Fang, T.; Lee, C.F., A moving-wall boundary layer flow of a slightly rarefied gas free stream over a moving flat plate, Appl math lett, 18, 5, 487-495, (2005) · Zbl 1074.76042
[39] Fang, T.; Lee, C.F., Exact solutions of incompressible Couette flow with porous walls for slightly rarefied gases, Heat mass transfer, 42, 3, 255-262, (2006)
[40] Fang, T.; Zhang, J.; Yao, S., Slip MHD viscous flow over a stretching sheet – an exact solution, Commun nonlinear sci numer simulat, 14, 11, 3731-3737, (2009)
[41] Wang, C.Y., Stagnation slip flow and heat transfer on a moving plate, Chem eng sci, 61, 23, 7668-7672, (2006)
[42] Wang, C.Y., Stagnation flow on a cylinder with partial slip – an exact solution of the navier – stokes equations, IMA J appl math, 72, 3, 271-277, (2007) · Zbl 1130.76029
[43] Wang, C.Y., Analysis of viscous flow due to a stretching sheet with surface slip and suction, Nonlinear anal real world appl, 10, 1, 375-380, (2009) · Zbl 1154.76330
[44] Aziz, A., Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition, Commun nonlinear sci numer simulat, 15, 3, 573-580, (2010)
[45] Wu, L., A slip model for rarefied gas flows at arbitrary Knudsen number, Appl phys lett, 93, 253103, (2008)
[46] Fukui, S.; Kaneko, R., A database for interpolation of Poiseuille flow rates for high Knudsen number lubrication problems, ASME J tribol, 112, 78-83, (1990)
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