## THERMAL SCIENCE

International Scientific Journal

### AN EFFICIENT SPECTRAL SOLUTION FOR UNSTEADY BOUNDARY-LAYER FLOW AND HEAT TRANSFER DUE TO A STRETCHING SHEET

**ABSTRACT**

In this paper, an efficient spectral collocation method based on the shifted Legendre polynomials is applied to study the unsteady boundary-layer flow and heat transfer due to a stretching sheet. A similarity transformation is used to reduce the governing unsteady boundary-layer equations to a system of non-linear ordinary differential equations. Then, the shifted Legendre polynomials and their operational matrix of derivative are used for producing an analytical approximate solution of this system of non-linear ordinary differential equations. The main advantage of the proposed method is that the need for guessing and correcting the initial values during the solution procedure is eliminated and a stable solution with good accuracy can be obtained by using the given boundary conditions in the problem. A very good agreement is observed between the obtained results by the proposed spectral collocation method and those of previously published ones.

**KEYWORDS**

PAPER SUBMITTED: 2015-03-29

PAPER REVISED: 2015-06-10

PAPER ACCEPTED: 2015-06-24

PUBLISHED ONLINE: 2015-07-03

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**5**, PAGES [2167 - 2176]

- C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral Methods in Fluid Dynam- ics, Springer, 1988.
- E. Babolian, M. M. Hosseini, A Modified Spectral Method for Numerical Solution of Ordinary Dierential Equations with Non-analytic Solution, Appl. Math. Comput., 132 (2002), 341351.
- F. Mohammadi, M. M. Hosseini, Syed Tauseef Mohyud-Din. Legendre wavelet galerkin method for solving ordinary dierential equations with non-analytic solu- tion. Int. J. Syst. Sci. 42 (4) (2011) 579-585.
- H. A. Khater, R. S. Temsah, M. M. Hassan. A Chebyshev spectral collocation method for solving Burgers-type equations. J. Comput. Appl. Math. 222 (2) (2008) 333-350.
- M. Kamrani, and S. M. Hosseini. Spectral collocation method for stochastic Burgers equation driven by additive noise, Math. Comput. Simul. 82 (9) (2012) 1630-1644.
- M. R. Malik, T. A. Zang, M.Y Hussaini. A spectral collocation method for the Navier- Stokes equations. J. Comput. Phys. 61 (1) (1985) 64-88.
- A. Karageorghis, T. N. Phillips, A. R. Davies. Spectral collocation methods for the primary two-point oundary value problem in modelling viscoelastic ows. Int. J. Nu- mer. Methods. Eng. 26 (4) (1988) 805-813.
- H. Chen, Y. Su, B. D. Shizgal. A direct spectral collocation Poisson solver in polar and cylindrical coordinates. J. Comput. Phys. 160 (2) (2000) 453-469.
- Y. Chen, T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput. 79 (269) (2010) 147-167.
- S. Nemati, P. M. Lima, Y. Ordokhani. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials. J. Comput. Appl. Math. 242 (2013) 53-69.
- C. D. Pruett, C. L. Streett, A spectral collocation method for compressible, non- similar boundary layers. Int. J. Numer. Methods. Fluids. 13 (6) (1991) 713-737.
- M. R. Malik, Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86 (2) (1990) 376-413.
- B. Bialecki, A. Karageorghis, Spectral Chebyshev-Fourier collocation for the Helmholtz and variable coecient equations in a disk. J. Comput. Phys. 227 (19) (2008) 8588-8603.
- M. T. Darvishi, S. Kheybari, F. Khani, Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation. Commun. Non- linear. Sci. Numer. Simul. 13 (10) (2008) 2091-2103.
- A. Shidfar, R. Pourgholi, Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials, Appl. Math. Comput 175 (2) (2006): 1366-1374.
- H. Khalil, R. Ali Khan, A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation, Comput. Math. Appl 67 (10) (2014) 1938-1953.
- Rong-Yeu Chang, Maw-Ling Wang, Shifted Legendre function approximation of dif- ferential equations; application to crystallization processes. Comput. Chem. Eng. 8 (2) (1984) 117-125.
- A. Saadatmandi, M. Dehghan. A new operational matrix for solving fractional-order dierential equations. Comput. Math. Appl. 59 (3) (2010) 1326-1336.
- L. J. Crane, Flow past a stretching plate, J. Appl. Math. Phys., 21 (1970) 645-647.
- P. Carragher, L. J. Crane, Heat transfer on a continuous stretching sheet, J. Appl. Math. Mech., 62 (1982) 564-565.
- B. K. Dutta, P. Roy, A. S. Gupta, Temperature field in ow over a stretching surface with uniform heat ux, Int. Comm. Heat Mass Transfer, 12 (1985) 89-94.
- L. J. Grubka, K. M. Bobba, Heat transfer characteristic of a continuous stretching surface with variable temperature, J. Heat Transf., 107 (1985) 248-250.
- E. M. A. Elbashbeshy, Heat transfer over a stretching surface with variable surface heat ux, J. Phys. D Appl. Phys., 31 (1998) 1951-1954.
- E. M. A. Elbashbeshy, M. A. A. Bazid, Heat transfer over an unsteady stretching surface, Heat Mass Tran., 41 (2004) 1-4.
- S. Sharidan, T. Mahmood, I. Pop, Similarity solutions for the unsteady boundary layer ow and heat transfer due to a stretching sheet, Int. J. Appl. Mech. Eng., 11 (2006) 647-654.
- M. M. Rashidi, M. Keimanesh, Using Dierential Transform Method and Pade Ap- proximant for Solving MHD Flow in a Laminar Liquid Film from a Horizontal Stretch- ing Surface, Mathematical Problems in Engineering, Volume 2010 (2010).
- M. M. Rashidi, E. Erfani, The Modified Dierential Transform Method for Investi- gating Nano Boundary-Layers over Stretching Surfaces, Int. J. Numer. Methods Heat Fluid Flow 21 (7) (2011) 864-883.
- M. M. Rashidi, N. Freidoonimehr, A. Hosseini, O. Anwar Beg, T. K. Hung, Homo- topy Simulation of Nano uid Dynamics from a Non-Linearly Stretching Isothermal Permeable Sheet with Transpiration, Meccanica 49 (2) (2014) 469-482.
- M. M. Rashidi, S. A. Mohimanian Pour, Analytic approximate solutions for unsteady boundary-layer ow and heat transfer due to a stretching sheet by homotopy analysis method, Nonlinear Anal. Model. Control. 15 (1) (2010) 83-95.
- W. Ibrahim, B. Shanker, Unsteady Boundary Layer Flow and Heat Transfer Due to a Stretching Sheet by Quasilinearization Technique, World Journal of Mechanics, 1 (6) (2011) 288-293.