Rational Chebyshev Tau Method for Solving Natural Convection of Darcian Fluid About a Vertical Full Cone Embedded in Porous Media Whit a Prescribed Wall Temperature
The problem of natural convection about a cone embedded in a porous medium at local Rayleigh numbers based on the boundary layer approximation and the Darcy-s law have been studied before. Similarity solutions for a full cone with the prescribed wall temperature or surface heat flux boundary conditions which is the power function of distance from the vertex of the inverted cone give us a third-order nonlinear differential equation. In this paper, an approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev Tau (RCT) method. The operational matrices of the derivative and product of rational Chebyshev (RC) functions are presented. These matrices together with the Tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. We also present the comparison of this work with others and show that the present method is applicable.
[1] O. Coulaud, D. Funaro, O. Kavian, Laguerre spectral approximation of
elliptic problems in exterior domains, Comput. Method. Appl. Mech.
Eng. 80 (1990) 451-458.
[2] D. Funaro, O. Kavian, Approximation of some diffusion evolution
equations in unbounded domains by hermite functions, Math. Comput.
57 (1991) 597-619.
[3] D. Funaro, Computational aspects of pseudospectral laguerre approximations,
Appl. Numer. Math. 6 (1990) 447-457.
[4] B. Guo, Error estimation of hermite spectral method for nonlinear partial
differential equations, Math. Comput. 68 (227) (1999) 1067-1078.
[5] B. Guo, J. Shen, Laguerre-galerkin method for nonlinear partial differential
equations on a semi-infinite interval, Numer. Math. 86 (2000)
635-654.
[6] Y. Maday, B. Pernaud-Thomas, H. Vandeven, Reappraisal of laguerre
type spectral methods, La. Rech. Aerospatiale. 6 (1985) 13-35.
[7] J. Shen, Stable and efficient spectral methods in unbounded domains
using laguerre functions, SIAM J. Numer. Anal. 38 (4) (2000) 1113-
1133.
[8] H. Siyyam, Laguerre tau methods for solving higher order ordinary
differential equations, J. Comput. Anal. Appl. 3 (2) (2001) 173-182.
[9] B. Guo, Gegenbauer approximation and its applications to differential
equations on the whole line, J. Comput. Anal. Appl. 226 (1) (1998)
180-206.
[10] B. Guo, Jacobi spectral approximation and its applications to differential
equations on the half line, J. Comput. Math. 18 (1) (2000) 95-112.
[11] B. Guo, Jacobi approximations in certain hilbert spaces and their
applications to singular differential equations, J. Comput. Anal. Appl.
243 (2) (2000) 373-408.
[12] J. Boyd, Chebyshev and Fourier Spectral Methods, 2nd Edition, Dover,
New York, 2000.
[13] C. Christov, A complete orthogonal system of functions in l2(−∞,∞)
space, SIAM J. Appl. Math. 42 (1982) 1337-1344.
[14] J. Boyd, Spectral methods using rational basis functions on an infinite
interval, J. Comput. Phys. 69 (1) (1987) 112-142.
[15] J. Boyd, Orthogonal rational functions on a semi-infinite interval, J.
Comput. Phys. 70 (1) (1987) 63-88.
[16] B. Guo, J. Shen, Z. Wang, A rational approximation and its applications
to differential equations on the half line, J. Sci. Comput. 15 (2) (2000)
117-147.
[17] J. Boyd, C. Rangan, P. Bucksbaum, Pseudospectral methods on a semiinfinite
interval with application to the hydrogen atom: a comparison
of the mapped fourier-sine method with laguerre series and rational
chebyshev expansions, J. Comput. Phys. 188 (1) (2003) 56-74.
[18] K. Parand, M. Razzaghi, Rational chebyshev tau method for solving
volterra-s population model, Appl. Math. Comput. 149 (2004) 893-900.
[19] K. Parand, M. Razzaghi, Rational chebyshev tau method for solving
higher-order ordinary differential equations, Int. J. Comput. Math. 81
(2004) 73-80.
[20] K. Parand, M. Razzaghi, Rational legendre approximation for solving
some physical problems on semi-infinite intervals, Phys. Scr. 69 (2004)
353-357.
[21] K. Parand, M. Shahini, Rational chebyshev pseudospectral approach for
solving thomas-fermi equation, Phys. Lett. A. 373 (2009) 210-213.
[22] K. Parand, A. Taghavi, Rational scaled generalized laguerre function
collocation method for solving the blasius equation, J. Comput. Appl.
Math. 233 (4) (2009) 980-989.
[23] K. Parand, M. Shahini, M. Dehghan, Rational legendre pseudospectral
approach for solving nonlinear differential equations of lane-emden
type, J. Comput. Phys. 228 (23) (2009) 8830-8840.
[24] K. Parand, M. Dehghan, A. Pirkhedri, Sinc-collocation method for
solving the blasius equation, Phys. Lett. A. 273 (44) (2009) 4060-4065.
[25] T. Tajvidi, M. Razzaghi, M. Dehghan, Modified rational legendre
approach to laminar viscous flow over a semi-infinite flat plate, Chaos
Soliton. Fract. 35 (2008) 59-66.
[26] D. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New
York, 2006.
[27] J. Merkin, Free convection boundary layers in a saturated porous
medium with lateral mass flux, Int. J. Heat. Mass. Tran. 21 (1978) 1499-
1504.
[28] J. Merkin, Free convection boundary layers on axi-symmetric and two
dimensional bodies of arbitrary shape in a saturated porous medium, Int.
J. Heat. Mass. Tran. 22 (1979) 1461-1462.
[29] W. Minkowycz, P. Cheng, Free convection about a vertical cylinder
embedded in a porous medium, Int. J. Heat. Mass. Tran. 19 (1976)
805-813.
[30] W. Minkowycz, P. Cheng, Local non-similar solutions for free convective
flow with uniform lateral mass flux in a porous medium, Int. J. Heat.
Mass. Tran. 9 (1982) 159-168.
[31] W. Minkowycz, P. Cheng, F. Moalem, The effect of surface mass
transfer on buoyancy induced darcian flow adjacent to a horizontal
heated surface, Int. Commun. Heat. Mass. 12 (1985) 55-65.
[32] P. Cheng, T. Le, I. Pop, Natural convection of a darcian fluid about a
cone, Int. Commun. Heat. Mass. 12 (1985) 705-717.
[33] I. Pop, P. Cheng, An integral solution for free convection of a darcian
fluid about a cone with curvature effects, Int. Commun. Heat. Mass. 13
(1986) 433-438.
[34] D. Ingham, I. Pop, Natural convection about a heated horizontal cylinder
in a porous medium, J. Fluid. Mech. 184 (1987) 157-181.
[35] K. Vafai, Handbook of porous media, Marcel Dekker, New York, 2000.
[36] A. Sohouli, D. Domairry, M. Famouri, A. Mohsenzadeh, Analytical
solution of natural convection of darcian fluid about a vertical full cone
embedded in porous media prescribed wall temperature by means of
ham, Int. Commun. Heat. Mass. 35 (2008) 1380-1384.
[37] A. Bejan, K. Khair, Heat and mass transfer by natural convection in a
porous medium, Int. J. Heat. Mass. Tran. 28 (1985) 909-918.
[38] A. Nakayama, M. Hossain, An integral treatment for combined heat and
mass transfer by natural convection in a porous medium, Int. J. Heat.
Mass. Tran. 38 (1995) 761-765.
[39] K. Yih, Coupled heat and mass transfer by free convection over a
truncated cone in porous media: Vwt/vwc or vhf/vmf, Acta. Mech. 137
(1999) 83-97.
[40] K. Yih, Uniform transpiration effect on coupled heat and mass transfer
in mixed convection about inclined surfaces in porous media: the entire
regime, Acta. Mech. 132 (1999) 229-240.
[41] C. Cheng, An integral approach for heat and mass transfer by natural
convection from truncated cones in porous media with variable wall
temperature and concentration, Int. Commun. Heat. Mass. 27 (2000)
437-548.
[42] C. Cheng, Soret and dufour effects on natural convection heat and mass
transfer from a vertical cone in a porous medium, Int. Commun. Heat.
Mass. 36 (2009) 1020-1024.
[43] S. Rahman, K. Mahgoub, A. Nafees, Natural convective mass transfer
from upward pointing conical surfaces in porous media, Chem. Eng.
Commun. 194 (2007) 280-290.
[44] I. Pop, T. Na, Naturnal convection of a darcian fluid about a wavy cone,
Int. Commun. Heat. Mass. 21 (1994) 891-899.
[45] I. Pop, T. Na, Natural convection over a frustum of a wavy cone in a
porous medium, Mech. Res. Commun. 22 (1995) 181-190.
[46] O. D. Makinde, On mhd boundary-layer flow and mass transfer past a
vertical plate in a porous medium with constant heat flux, Int. J. Numer.
Meth. Heat. Fluid. Flow. 19 (2009) 546-554.
[47] O. D. M. A. Ogulu, Unsteady hydromagnetic free convection flow of
a dissipative and radiating fluid past a vertical plate with constant heat
flux, Chem. Eng. Comm. 196 (2009) 454-462.
[48] P. S. O. D. Makinde, Magnetohydrodynamic mixed-convective flow and
heat and mass transfer past a vertical plate in a porous medium with
constant wall suction, J. Heat. Transfer. 130 (2008) 1-8.
[49] C. Lanczos, Applied Analysis, Springer-Verlag, 1987.
[50] D. Gottlieb, M. Hussaini, S. Orszg, In: R. Voigt and D. Gottlieb and and
M. Y. Hussaini (Eds.) Theory and applications of spectral methods in
spectral methods for partial differential equations, Philadelphia: SIAM,
1984.
[51] C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral methods in
fluid dynamic, Englewood Cliffs (NJ): Prentice-Hall, 1986.
[1] O. Coulaud, D. Funaro, O. Kavian, Laguerre spectral approximation of
elliptic problems in exterior domains, Comput. Method. Appl. Mech.
Eng. 80 (1990) 451-458.
[2] D. Funaro, O. Kavian, Approximation of some diffusion evolution
equations in unbounded domains by hermite functions, Math. Comput.
57 (1991) 597-619.
[3] D. Funaro, Computational aspects of pseudospectral laguerre approximations,
Appl. Numer. Math. 6 (1990) 447-457.
[4] B. Guo, Error estimation of hermite spectral method for nonlinear partial
differential equations, Math. Comput. 68 (227) (1999) 1067-1078.
[5] B. Guo, J. Shen, Laguerre-galerkin method for nonlinear partial differential
equations on a semi-infinite interval, Numer. Math. 86 (2000)
635-654.
[6] Y. Maday, B. Pernaud-Thomas, H. Vandeven, Reappraisal of laguerre
type spectral methods, La. Rech. Aerospatiale. 6 (1985) 13-35.
[7] J. Shen, Stable and efficient spectral methods in unbounded domains
using laguerre functions, SIAM J. Numer. Anal. 38 (4) (2000) 1113-
1133.
[8] H. Siyyam, Laguerre tau methods for solving higher order ordinary
differential equations, J. Comput. Anal. Appl. 3 (2) (2001) 173-182.
[9] B. Guo, Gegenbauer approximation and its applications to differential
equations on the whole line, J. Comput. Anal. Appl. 226 (1) (1998)
180-206.
[10] B. Guo, Jacobi spectral approximation and its applications to differential
equations on the half line, J. Comput. Math. 18 (1) (2000) 95-112.
[11] B. Guo, Jacobi approximations in certain hilbert spaces and their
applications to singular differential equations, J. Comput. Anal. Appl.
243 (2) (2000) 373-408.
[12] J. Boyd, Chebyshev and Fourier Spectral Methods, 2nd Edition, Dover,
New York, 2000.
[13] C. Christov, A complete orthogonal system of functions in l2(−∞,∞)
space, SIAM J. Appl. Math. 42 (1982) 1337-1344.
[14] J. Boyd, Spectral methods using rational basis functions on an infinite
interval, J. Comput. Phys. 69 (1) (1987) 112-142.
[15] J. Boyd, Orthogonal rational functions on a semi-infinite interval, J.
Comput. Phys. 70 (1) (1987) 63-88.
[16] B. Guo, J. Shen, Z. Wang, A rational approximation and its applications
to differential equations on the half line, J. Sci. Comput. 15 (2) (2000)
117-147.
[17] J. Boyd, C. Rangan, P. Bucksbaum, Pseudospectral methods on a semiinfinite
interval with application to the hydrogen atom: a comparison
of the mapped fourier-sine method with laguerre series and rational
chebyshev expansions, J. Comput. Phys. 188 (1) (2003) 56-74.
[18] K. Parand, M. Razzaghi, Rational chebyshev tau method for solving
volterra-s population model, Appl. Math. Comput. 149 (2004) 893-900.
[19] K. Parand, M. Razzaghi, Rational chebyshev tau method for solving
higher-order ordinary differential equations, Int. J. Comput. Math. 81
(2004) 73-80.
[20] K. Parand, M. Razzaghi, Rational legendre approximation for solving
some physical problems on semi-infinite intervals, Phys. Scr. 69 (2004)
353-357.
[21] K. Parand, M. Shahini, Rational chebyshev pseudospectral approach for
solving thomas-fermi equation, Phys. Lett. A. 373 (2009) 210-213.
[22] K. Parand, A. Taghavi, Rational scaled generalized laguerre function
collocation method for solving the blasius equation, J. Comput. Appl.
Math. 233 (4) (2009) 980-989.
[23] K. Parand, M. Shahini, M. Dehghan, Rational legendre pseudospectral
approach for solving nonlinear differential equations of lane-emden
type, J. Comput. Phys. 228 (23) (2009) 8830-8840.
[24] K. Parand, M. Dehghan, A. Pirkhedri, Sinc-collocation method for
solving the blasius equation, Phys. Lett. A. 273 (44) (2009) 4060-4065.
[25] T. Tajvidi, M. Razzaghi, M. Dehghan, Modified rational legendre
approach to laminar viscous flow over a semi-infinite flat plate, Chaos
Soliton. Fract. 35 (2008) 59-66.
[26] D. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New
York, 2006.
[27] J. Merkin, Free convection boundary layers in a saturated porous
medium with lateral mass flux, Int. J. Heat. Mass. Tran. 21 (1978) 1499-
1504.
[28] J. Merkin, Free convection boundary layers on axi-symmetric and two
dimensional bodies of arbitrary shape in a saturated porous medium, Int.
J. Heat. Mass. Tran. 22 (1979) 1461-1462.
[29] W. Minkowycz, P. Cheng, Free convection about a vertical cylinder
embedded in a porous medium, Int. J. Heat. Mass. Tran. 19 (1976)
805-813.
[30] W. Minkowycz, P. Cheng, Local non-similar solutions for free convective
flow with uniform lateral mass flux in a porous medium, Int. J. Heat.
Mass. Tran. 9 (1982) 159-168.
[31] W. Minkowycz, P. Cheng, F. Moalem, The effect of surface mass
transfer on buoyancy induced darcian flow adjacent to a horizontal
heated surface, Int. Commun. Heat. Mass. 12 (1985) 55-65.
[32] P. Cheng, T. Le, I. Pop, Natural convection of a darcian fluid about a
cone, Int. Commun. Heat. Mass. 12 (1985) 705-717.
[33] I. Pop, P. Cheng, An integral solution for free convection of a darcian
fluid about a cone with curvature effects, Int. Commun. Heat. Mass. 13
(1986) 433-438.
[34] D. Ingham, I. Pop, Natural convection about a heated horizontal cylinder
in a porous medium, J. Fluid. Mech. 184 (1987) 157-181.
[35] K. Vafai, Handbook of porous media, Marcel Dekker, New York, 2000.
[36] A. Sohouli, D. Domairry, M. Famouri, A. Mohsenzadeh, Analytical
solution of natural convection of darcian fluid about a vertical full cone
embedded in porous media prescribed wall temperature by means of
ham, Int. Commun. Heat. Mass. 35 (2008) 1380-1384.
[37] A. Bejan, K. Khair, Heat and mass transfer by natural convection in a
porous medium, Int. J. Heat. Mass. Tran. 28 (1985) 909-918.
[38] A. Nakayama, M. Hossain, An integral treatment for combined heat and
mass transfer by natural convection in a porous medium, Int. J. Heat.
Mass. Tran. 38 (1995) 761-765.
[39] K. Yih, Coupled heat and mass transfer by free convection over a
truncated cone in porous media: Vwt/vwc or vhf/vmf, Acta. Mech. 137
(1999) 83-97.
[40] K. Yih, Uniform transpiration effect on coupled heat and mass transfer
in mixed convection about inclined surfaces in porous media: the entire
regime, Acta. Mech. 132 (1999) 229-240.
[41] C. Cheng, An integral approach for heat and mass transfer by natural
convection from truncated cones in porous media with variable wall
temperature and concentration, Int. Commun. Heat. Mass. 27 (2000)
437-548.
[42] C. Cheng, Soret and dufour effects on natural convection heat and mass
transfer from a vertical cone in a porous medium, Int. Commun. Heat.
Mass. 36 (2009) 1020-1024.
[43] S. Rahman, K. Mahgoub, A. Nafees, Natural convective mass transfer
from upward pointing conical surfaces in porous media, Chem. Eng.
Commun. 194 (2007) 280-290.
[44] I. Pop, T. Na, Naturnal convection of a darcian fluid about a wavy cone,
Int. Commun. Heat. Mass. 21 (1994) 891-899.
[45] I. Pop, T. Na, Natural convection over a frustum of a wavy cone in a
porous medium, Mech. Res. Commun. 22 (1995) 181-190.
[46] O. D. Makinde, On mhd boundary-layer flow and mass transfer past a
vertical plate in a porous medium with constant heat flux, Int. J. Numer.
Meth. Heat. Fluid. Flow. 19 (2009) 546-554.
[47] O. D. M. A. Ogulu, Unsteady hydromagnetic free convection flow of
a dissipative and radiating fluid past a vertical plate with constant heat
flux, Chem. Eng. Comm. 196 (2009) 454-462.
[48] P. S. O. D. Makinde, Magnetohydrodynamic mixed-convective flow and
heat and mass transfer past a vertical plate in a porous medium with
constant wall suction, J. Heat. Transfer. 130 (2008) 1-8.
[49] C. Lanczos, Applied Analysis, Springer-Verlag, 1987.
[50] D. Gottlieb, M. Hussaini, S. Orszg, In: R. Voigt and D. Gottlieb and and
M. Y. Hussaini (Eds.) Theory and applications of spectral methods in
spectral methods for partial differential equations, Philadelphia: SIAM,
1984.
[51] C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral methods in
fluid dynamic, Englewood Cliffs (NJ): Prentice-Hall, 1986.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51941", author = "Kourosh Parand and Zahra Delafkar and Fatemeh Baharifard", title = "Rational Chebyshev Tau Method for Solving Natural Convection of Darcian Fluid About a Vertical Full Cone Embedded in Porous Media Whit a Prescribed Wall Temperature", abstract = "The problem of natural convection about a cone embedded in a porous medium at local Rayleigh numbers based on the boundary layer approximation and the Darcy-s law have been studied before. Similarity solutions for a full cone with the prescribed wall temperature or surface heat flux boundary conditions which is the power function of distance from the vertex of the inverted cone give us a third-order nonlinear differential equation. In this paper, an approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev Tau (RCT) method. The operational matrices of the derivative and product of rational Chebyshev (RC) functions are presented. These matrices together with the Tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. We also present the comparison of this work with others and show that the present method is applicable.
", keywords = "Tau method, semi-infinite, nonlinear ODE, rational Chebyshev, porous media.", volume = "5", number = "8", pages = "1172-6", }
