{"title":"Numerical Solutions of Boundary Layer Flow over an Exponentially Stretching\/Shrinking Sheet with Generalized Slip Velocity","authors":"Ezad Hafidz Hafidzuddin, Roslinda Nazar, Norihan M. Arifin, Ioan Pop","volume":100,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":244,"pagesEnd":250,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10001145","abstract":"
In this paper, the problem of steady laminar boundary
\r\nlayer flow and heat transfer over a permeable exponentially
\r\nstretching\/shrinking sheet with generalized slip velocity is
\r\nconsidered. The similarity transformations are used to transform the
\r\ngoverning nonlinear partial differential equations to a system of
\r\nnonlinear ordinary differential equations. The transformed equations
\r\nare then solved numerically using the bvp4c function in MATLAB.
\r\nDual solutions are found for a certain range of the suction and
\r\nstretching\/shrinking parameters. The effects of the suction parameter,
\r\nstretching\/shrinking parameter, velocity slip parameter, critical shear
\r\nrate and Prandtl number on the skin friction and heat transfer
\r\ncoefficients as well as the velocity and temperature profiles are
\r\npresented and discussed.<\/p>\r\n","references":"[1]\tB.C. Sakiadis, \u201cBoundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow,\u201d AIChE J., vol. 7, 1961, pp. 26\u201328.\r\n[2]\tF.K. Tsou, E.M. Sparrow, and R.J. Goldstein, \u201cFlow and heat transfer in the boundary layer on a continuous moving surfaces,\u201d Int. J. Heat Mass Transf.,vol. 10, 1967, pp. 219\u2013235.\r\n[3]\tL.J. Crane, \u201cFlow past a stretching plate,\u201d ZeitschriftF\u00fcrAngew. Math. Und Phys.,vol. 21, 1970, pp. 645\u2013647.\r\n[4]\tC.Y. Wang, \u201cLiquid film on an unsteady stretching sheet,\u201d Quart. Appl. Math.,vol. 48, 1990, pp. 601\u2013610.\r\n[5]\tM. Miklavcic, and C.Y. Wang, \u201cViscous flow due to a shrinking sheet,\u201d Quart. Appl. Math.,vol. 64, 2006, pp. 283\u2013290.\r\n[6]\tE. Magyari, and B. Keller, \u201cHeat and mass transfer in the boundary layers on an exponentially stretching continuous surface,\u201d J. Phys. D Appl. Phys.,vol. 32, 1999, pp. 577\u2013585.\r\n[7]\tE.M.A. Elbashbeshy, \u201cHeat transfer over an exponentially stretching continuous surface with suction,\u201d Arch. Mech.,vol. 53, 2001, pp. 643\u2013651.\r\n[8]\tK. Bhattacharyya, \u201cBoundary layer flow and heat transfer over an exponentially shrinking sheet,\u201d Chinese Phys. Lett.,vol. 28, 2011, pp. 074701.\r\n[9]\tN. Najib, N. Bachok, N.M. Arifin, and A. Ishak, \u201cBoundary layer stagnation point flow and heat transfer past a permeable exponentially shrinking cylinder,\u201d Int. J. Math. Model. Methods Appl. Sci.,vol. 8, 2014, pp. 121\u2013126.\r\n[10]\tG.S. Beavers, and D.D. Joseph, \u201cBoundary conditions at a naturally permeable wall,\u201d J. Fluid Mech.,vol. 30, 1967, pp. 197\u2013207.\r\n[11]\tC.Y. Wang, \u201cStagnation flow with slip: Exact solutions of the Navier-Stokes equations,\u201d Z. Angew. Math. Phys.,vol. 54, 2003, pp. 184\u2013189.\r\n[12]\tP.D. Ariel, \u201cAxisymmetric flow due to a stretching sheet with partial slip,\u201d Comput. Math. with Appl.,vol. 54, 2007, pp. 1169\u20131183.\r\n[13]\tK. Bhattacharyya, S. Mukhopadhyay, and G.C. Layek, \u201cSlip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet,\u201d Int. J. Heat Mass Transf.,vol. 54, 2011, pp. 308\u2013313.\r\n[14]\tF. Aman, A. Ishak, and I. Pop, \u201cMagnetohydrodynamic stagnation-point flow towards a stretching\/shrinking sheet with slip effects,\u201d Int. Commun. Heat Mass Transf.,vol. 47, 2013,pp. 68\u201372. [15]\tP.A. Thompson, and S. M. Troian, \u201cA general boundary conditon for liquid flow at solid surfaces,\u201d Nature,vol. 389, 1997, pp. 360\u2013362.\r\n[16]\tM. Sajid, K. Mahmood, and Z. Abbas, \u201cAxisymmetric stagnation-point flow with a general slip boundary condition over a lubricated surface,\u201d Chinese Phys. Lett.vol. 29, 2012, pp. 24702.\r\n[17]\tA. Aziz, \u201cA similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition,\u201d Commun. Nonlinear Sci. Numer. Simul.,vol. 14, 2009, pp. 1064\u20131068.\r\n[18]\tJ. Kierzenka, and L.F. Shampine, \u201cA BVP solver based on residual control and the Maltab PSE,\u201d C. Trans. Math. Softw.,vol. 27,2001, pp. 299\u2013316.\r\n[19]\tL.F. Shampine, I. Gladwell, and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, 2003.\r\n[20]\tJ.H. Merkin, \u201cOn dual solutions occurring in mixed convection in a porous medium,\u201d J. Eng. Math.,vol. 20, 1986, pp. 171\u2013179.\r\n[21]\tP.D. Weidman, D.G. Kubitschek, and A.M.J. Davis, \u201cThe effect of transpiration on self-similar boundary layer flow over moving surfaces,\u201d Int. J. Eng. Sci.,vol. 44, 2006, pp. 730\u2013737.\r\n[22]\tA.V. Ro\u015fca, and I. Pop, \u201cFlow and heat transfer over a vertical permeable stretching\/shrinking sheet with a second order slip,\u201d Int. J. Heat Mass Transf.,vol. 60, 2013, pp. 355\u2013364.\r\n","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 100, 2015"}