## THERMAL SCIENCE

International Scientific Journal

### NUMERICAL STUDY OF HEAT TRANSFER OF A MICROPOLAR FLUID THROUGH A POROUS MEDIUM WITH RADIATION

**ABSTRACT**

An efficient Spectral Collocation method based on the shifted Legendre polynomials was applied to get solution of heat transfer of a micropolar fluid through a porous medium with radiation. A similarity transformation is applied to convert the governing 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 approximate solution for this system of non-linear 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: 2015-09-18

PAPER REVISED: 2017-04-16

PAPER ACCEPTED: 2017-04-19

PUBLISHED ONLINE: 2017-05-06

**THERMAL SCIENCE** YEAR

**2018**, VOLUME

**22**, ISSUE

**1**, PAGES [557 - 565]

- C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral Methods in Fluid Dynamics, Springer, 1988.
- E. Babolian, M. M. Hosseini, A Modified Spectral Method for Numerical Solution of Ordinary Differential 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 differential equations with non-analytic solution. Int. J. Syst. Sci. 42 (4) (2011) 579-585.
- 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.
- 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. 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 flows. Int. J. Numer. 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, nonsimilar 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 coefficient 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. Nonlinear. Sci. Numer. Simul. 13 (10) (2008) 2091-2103.
- 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 differential 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 differential equations. Comput. Math. Appl. 59 (3) (2010) 1326-1336.
- I. K. Vafai, C. L. Tien, Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Trans., 24 (1981) 195-203.
- A. Raptis, Boundary layer flow of a micropolar fluid through a porous medium. J. Porous Media, 3 (1) (2000) 95-97.
- E.M. Abo-Eldahab, M.S. El Gendy, Convective heat transfer past a continuously moving plate embedded in a non-Darcian porous medium in the presence of a magnetic field. Canadian Journal of Physics, 79 (7) (2001), 1031-1038.
- E. M. Abo-Eldahab, A. F. Ghonaim, Radiation effect on heat transfer of a micropolar fluid through a porous medium. Applied Mathematics and Computation, (2005) 169 (1), 500-510.
- M. M. Rashidi, S. Abbasbandy, Analytic approximate solutions for heat transfer of a micropolar fluid through a porous medium with radiation. Communications in Nonlinear Science and Numerical Simulation, (2011) 16 (4), 1874-1889.
- M. M. Rashidi, S. A. Mohimanian Pour, A novel analytical solution of heat transfer of a micropolar fluid through a porous medium with radiation by DTM-Pad´e. Heat Transfer-Asian Research, (2010) 39 (8), 575-589.
- A. J. Willson, Boundary layers in micropolar liquids. Camb. Phil. Soc. 67 (1970) 469-476.
- D. A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 1999.
- F. Mohammadi, M. M. Hosseini, A. Dehgahn, F. M. Ghaini, Numerical Solutions of Falkner-Skan Equation with Heat Transfer, Studies in Nonlinear Sciences, (2012) 3 (3) 86-93.
- F. Mohammadi, M. M. Rashidi, Spectral Collocation Solution of MHD Stagnation- Point Flow in Porous Media with Heat Transfer, Journal of Applied Fluid Mechanics (to appear).
- F. Mohammadi, M. M. Rashidi, An Efficient Spectral Solution for Unsteady Boundary Layer Flow and Heat Transfer Due to a Stretching Sheet, Thermal Science (to appear).