Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge. (English) Zbl 1416.76011

Summary: The steady two-dimensional laminar boundary layer flow of a power-law fluid past a permeable stretching wedge beneath a variable free stream is studied in this paper. Using appropriate similarity variables, the governing equations are reduced to a single third-order highly nonlinear ordinary differential equation in the dimensionless stream function, which is solved numerically using the Runge-Kutta scheme coupled with a conventional shooting procedure. The flow is governed by the wedge velocity parameter \(\lambda \), the transpiration parameter \(f_{0}\), the fluid power-law index \(n\), and the computed wall shear stress is \(f^{\prime\prime}(0)\). It is found that dual solutions exist for each value of \(f_{0}, m\) and \(n\) considered in \(\lambda - f^{\prime\prime}(0)\) parameter space. A stability analysis for this self-similar flow reveals that for each value of \(f_{0}, m\) and \(n\), lower solution branches are unstable while upper solution branches are stable. Very good agreements are found between the results of the present paper and that of P. D. Weidman et al. [Int. J. Eng. Sci. 44, No. 11–12, 730–737 (2006; Zbl 1213.76064)] for \(n = 1\) (Newtonian fluid) and \(m = 0\) (Blasius problem [H. Blasius, Schlömilch’s Z. Math. Phys. 56, 1–37 (1908; JFM 39.0803.02)]).


76A05 Non-Newtonian fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76M20 Finite difference methods applied to problems in fluid mechanics
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI


[1] Xu, H.; Liao, S.-J.; Pop, I., Series solution of unsteady boundary layer flows of non-Newtonian fluids near a forward stagnation point, J. Non-Newtonian Fluid Mech., 139, 31-43 (2006) · Zbl 1195.76070
[2] Astarita, G.; Marrucci, G., Principles of Non-Newtonian Fluid Mechanics (1974), McGraw-Hill: McGraw-Hill London · Zbl 0316.73001
[3] Schowalter, W. R., Mechanics of Non-Newtonian Fluids (1978), Pergamon Press: Pergamon Press Oxford
[4] Bird, R. B.; Armstrong, R. C.; Hassager, O., Dynamics of Polymeric Liquids. Dynamics of Polymeric Liquids, Fluid Mechanics, vol. 1 (1987), Wiley: Wiley New York
[5] Crochet, M. J.; Davies, A. R.; Walters, K., Numerical Simulation of Non-Newtonian Flow (1984), Elsevier: Elsevier Amsterdam · Zbl 0583.76002
[6] Shenoy, A. V.; Mashelkar, A. R., Thermal convection in non-Newtonian fluids, Adv. Heat Transfer, 15, 143-225 (1982)
[7] Andersson, H. I.; Irgens, F., Film flow of power law fluids, (Cheremisionoff, N. P., Encyclopedia of Fluid Mech., vol. 9 (1990), Gulf Publishing Company: Gulf Publishing Company Texas), 617-648
[8] Irvine, T. F.; Karni, J., Non-Newtonian fluid flow and heat transfer, (Kakaç, S.; Shah, R. K.; Aung, W., Handbook of Single-Phase Convective Heat Transfer (1987), Wiley: Wiley New York), 20.1-20.57, (Chapter 20)
[9] Magyari, E.; Keller, B., Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls, Eur. J. Mech. B Fluids, 19, 109-122 (2000) · Zbl 0976.76021
[10] Magyari, E.; Keller, B., A direct method to calculate the heat transfer coefficient of steady similar boundary layer flows induced by continuous moving surfaces, Int. J. Thermal Sci., 44, 245-254 (2005)
[11] Blasius, H., Grenzschiten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys., 56, 1-37 (1908) · JFM 39.0803.02
[12] Crane, L. J., Flow past a stretching plate, J. Appl. Math. Phys. (ZAMP), 21, 645-647 (1970)
[13] Afzal, N.; Varshney, I. S., The cooling of a low heat resistance stretching sheet moving through a fluid, Wärme- und Stofübertr., 14, 289-293 (1980)
[14] Kuiken, H. K., On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small, IMA J. Appl. Math., 27, 387-405 (1981) · Zbl 0472.76045
[15] Banks, W. H.H., Similarity solutions of the boundary layer equations for a stretching wall, J. Méc. Teoret. Appl., 2, 375-392 (1983) · Zbl 0538.76039
[16] Banks, W. H.H.; Zaturska, M. B., Eigensolutions in boundary-layer flow adjacent to a stretching wall, IMA J. Appl. Math., 36, 263-273 (1986) · Zbl 0619.76011
[17] Riley, N.; Weidman, P. D., Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary, SIAM J. Appl. Math., 49, 1350-1358 (1989) · Zbl 0682.76026
[18] Afzal, N., Heat transfer from a stretching surface, Int. J. Heat Mass Transfer, 36, 1128-1131 (1993) · Zbl 0764.76011
[19] Liao, S.-J.; Pop, I., On explicit analytic solutions of boundary-layer equations about flows in a porous medium or for a stretching wall, Int. J. Heat Mass Transfer, 47, 75-85 (2004)
[20] Liao, S.-J., A new branch of solutions of boundary-layer flows over a permeable stretching plate, Int. J. Non-linear Mech., 42, 819-830 (2007) · Zbl 1200.76046
[21] Fang, T., Further study on a moving-wall boundary layer problem with heat transfer, Acta Mech., 163, 183-188 (2003) · Zbl 1064.76032
[22] Fang, T., Flow and heat transfer characteristics of the boundary layers over a stretching surface with a uniform-shear free stream, Int. J. Heat Mass Transfer, 51, 2199-2213 (2008) · Zbl 1144.80313
[23] Akçay, M.; Adil Yükselen, M., Drag reduction of a non Newtonian fluid by fluid injection on a moving wall, Archive Appl. Mech., 69, 215-225 (1999) · Zbl 0933.76004
[24] Kumari, M.; Nath, G., MHD boundary-layer flow of a non-Newtonian fluid over a continuously moving surface with a parallel free stream, Acta Mech., 146, 139-150 (2001) · Zbl 1008.76099
[25] Prasad, K. V.; Pal, Dulal; Datti, P. S., MHD power-law fluid flow and heat transfer over a non-isothermal stretching sheet, Commun. Nonlinear Sci. Numer. Simul., 14, 2178-2189 (2009)
[26] Merkin, J. H., On dual solutions occurring in mixed convection in a porous medium, J. Eng. Math., 20, 171-179 (1985) · Zbl 0597.76081
[27] Ishak, A.; Nazar, R.; Pop, I., Falkner-Skan equation for flow past a moving wedge with suction or injection, J. Appl. Math. Comput., 25, 67-89 (2007) · Zbl 1290.34040
[28] Weidman, P. D.; Kubitschek, D. G.; Davis, A. M.J., The effect of transpiration on self-similar boundary layer flow over moving surfaces, Int. J. Eng. Sci., 44, 730-737 (2006) · Zbl 1213.76064
[29] S.D. Harris, D.B. Ingham, I. Pop, Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip, Transp. Porous Med., 2008, on-line.; S.D. Harris, D.B. Ingham, I. Pop, Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip, Transp. Porous Med., 2008, on-line.
[30] Falkner, V. M.; Skan, S. W., Some approximate solutions of the boundary layer equations, Philos. Mag., 12, 865-896 (1931) · Zbl 0003.17401
[31] Blasius, H., Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys., 56, 1-37 (1908) · JFM 39.0803.02
[32] Hiemenz, K., Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eigetauchten geraden Kreiszylinder, Dinglers Polytechn. J., 326, 321-324 (1911)
[33] Chiam, T. C., Solutions for the flow of a conducting power-law fluid in a transverse magnetic field and with a pressure gradient using Crocco variables, Acta Mech., 137, 225-235 (1999) · Zbl 0991.76093
[34] Andersson, H. I.; Toften, T. H., Numerical solution of the laminar boundary layer equations for power-law fluids, J. Non-Newtonian Fluid Mech., 32, 175-195 (1989) · Zbl 0672.76011
[35] Stewartson, K., The Theory of Laminar Boundary Layers in Compressible Fluids (1964), Oxford Mathematical Monographs · Zbl 0114.18705
[36] Harris, S. D.; Ingham, D. B.; Pop, I., Unsteady heat transfer in impulsive Falkner-Skan flows: constant wall heat flux case, Acta Mech., 201, 185-196 (2008) · Zbl 1155.76324
[37] Yih, K. A., Uniform suction/blowing effect on forced convection about a wedge: uniform heat flux, Acta Mech., 128, 173-181 (1998) · Zbl 0926.76108
[38] Merrill, K.; Beauchesne, M.; Paullet, J.; Weidman, P. D., Final steady flow near stagnation point on a vertical surface in a porous medium, Int. J. Heat Mass Transfer, 49, 4681-4686 (2006) · Zbl 1121.76411
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