Spectral Investigation for Boundary Layer Flow over a Permeable Wall in the Presence of Transverse Magnetic Field
The magnetohydrodynamic (MHD) Falkner-Skan
equations appear in study of laminar boundary layers flow over
a wedge in presence of a transverse magnetic field. The partial
differential equations of boundary layer problems in presence of
a transverse magnetic field are reduced to MHD Falkner-Skan
equation by similarity solution methods. This is a nonlinear ordinary
differential equation. In this paper, we solve this equation via
spectral collocation method based on Bessel functions of the first
kind. In this approach, we reduce the solution of the nonlinear
MHD Falkner-Skan equation to a solution of a nonlinear algebraic
equations system. Then, the resulting system is solved by Newton
method. We discuss obtained solution by studying the behavior
of boundary layer flow in terms of skin friction, velocity, various
amounts of magnetic field and angle of wedge. Finally, the results
are compared with other methods mentioned in literature. We can
conclude that the presented method has better accuracy than others.
[1] T. Hayat, M. Imtiaz, A. Alsaedi, Melting heat transfer in the mhd flow of
cu–water nanofluid with viscous dissipation and joule heating, Advanced
Powder Technology.
[2] T. Hayat, S. Qayyum, M. Imtiaz, A. Alsaedi, Mhd flow and heat transfer
between coaxial rotating stretchable disks in a thermally stratified
medium, PloS one 11 (5) (2016) e0155899.
[3] V. M. Falkner, S. Skan, Some approximate solutions of the
boundary-layer equations, J. Math. Phy. 12 (1931) 865–896.
[4] V. M. Soundalgekar, H. S. Takhar, M. Singh, Velocity and temperature
field in MHD Falkner-Skan flow, J. Phys. Soci. of Japan 50 (1981)
3139–3143.
[5] G. Ashwini, A. Eswara, MHD Falkner-Skan boundary layer flow with
internal heat generation or absorption, World Aca. Sci. Engin. Tech. 65
(2012) 687–690.
[6] X. H. Su, L. C. Zheng, Approximate solutions to MHD Falkner-Skan
flow over permeable wall, Appl. Math. Mech. -Engl. Ed. 32 (2011)
401–408.
[7] A. Ishak, R. Nazar, I. Pop, MHD boundary-layer flow of a micropolar
fluid past a wedge with constant wall heat flux, Commun. Nonlinear
Sci. Num. Simul. 14 (2009) 109–118.
[8] N. Bachoka, A. Ishak, I. Pop, Boundary layer stagnation-point flow
and heat transfer over an exponentially stretching/shrinking sheet in a
nanofluid, I. J. Heat Mass Trans. 55 (2012) 8122–8128.
[9] S. Abbasbandy, T. Hayat, Solution of the MHD Falkner-Skan flow
by Homotopy analysis method, Nonlinear Anal. Real. World Appl. 14
(2009) 3591–3598.
[10] K. Parand, M. Dehghan, A. Pirkhedri, The use of sinc-collocation
method for solving Falkner-Skan boundary-layer equationl, Int. J. Num.
Meth. Fluids 69 (2004) 353–357.
[11] A. Asaithambi, A second-order finite-difference method for the
FalknerSkan equation, appl. math. comput. 156 (2004) 779–786.
[12] M. Fathizadeh, M. Madani, Y. Khan, N. Faraz, A. Yildirim, S. Tutkun,
An effective modification of the homotopy perturbation method for mhd
viscous flow over a stretching sheet, J. King Saud Uni. Sci. 2 (2013)
107113.
[13] M. M. Rashidi, The modified differential transform method and pade
approximats for solving MHD boundary-layer equationsl, Comput. Phys.
Commun. 180 (2009) 2210–2217.
[14] R. Ellahi, E. Shivanian, S. Abbasbandy, T. Hayat, R. Lewis,
Numerical study of magnetohydrodynamics generalized couette flow of
eyring-powell fluid with heat transfer and slip condition, International
Journal of Numerical Methods for Heat & Fluid Flow 26 (5).
[15] J. Shen, T. Tang, L. L. Wang, Spectral Methods Algorithms, Analysics
And Applicattions, first edition, Springer, 2001.
[16] J. P. Boyd, Chebyshev and Fourier spectral methods. 2nd ed, New York
Dover, New York, 2000.
[17] K. Parand, J. Rad, M. Nikarya, A new numerical algorithm based on
the first kind of modified bessel function to solve population growth in
a closed system, International Journal of Computer Mathematics 91 (6)
(2014) 1239–1254.
[18] K. Parand, M. Nikarya, Solving the unsteady isothermal gas through
a micro-nano porous medium via bessel function collocation method,
Journal of Computational and Theoretical Nanoscience 11 (1) (2014)
131–136.
[19] W. W. Bell, Special Functions For Scientists And Engineers, Published
simultaneously in Canada by D. Van Nostrand Company, (Canada), Ltd,
1967.
[20] G. Ben-yu, Spectral methods and their applications, World Scientific,
Shelton Street, Covent Garden, London WC2H 9HE, 1998.
[21] J. D. C. G. W. Bluman, Similarity Methods for Differential Equations,
Springer, New York, 1974.
[22] L. Dresner, Similarity solutions of nonlinear partial differential
equations, Longman Group, London, 1983.
[23] F. M. White, Fluid Mechanics, Seventh Edition, McGraw-Hill, New
York, 2009.
[24] H. Schlichting, K. Gersten, Boundary layer theory. 6th ed, Oxford, New
York, 1979.
[25] T. Y. Na, Computational Method in Engineering Boundray Value
Problem, Academic Pressl, New Yorkl, 1979.
[26] L. Rosenhead, Laminar Boundary layers, Clarendon Pressl,
McGraw-Hill, 1963.
[27] T. Hayat, Q. Hussain, T. Javed, The modified decomposition method
and pad`e aapproximants for theMHD flow over a non-linear stretching
sheet, Commun. Nonlinear Sci. Num. Simul. 10 (2009) 966–973.
[28] G. Watson, A treatise on the theory of Bessel Functions, 2nd edition,
Cambridge University Press, Cambridge (England), 1967.
[1] T. Hayat, M. Imtiaz, A. Alsaedi, Melting heat transfer in the mhd flow of
cu–water nanofluid with viscous dissipation and joule heating, Advanced
Powder Technology.
[2] T. Hayat, S. Qayyum, M. Imtiaz, A. Alsaedi, Mhd flow and heat transfer
between coaxial rotating stretchable disks in a thermally stratified
medium, PloS one 11 (5) (2016) e0155899.
[3] V. M. Falkner, S. Skan, Some approximate solutions of the
boundary-layer equations, J. Math. Phy. 12 (1931) 865–896.
[4] V. M. Soundalgekar, H. S. Takhar, M. Singh, Velocity and temperature
field in MHD Falkner-Skan flow, J. Phys. Soci. of Japan 50 (1981)
3139–3143.
[5] G. Ashwini, A. Eswara, MHD Falkner-Skan boundary layer flow with
internal heat generation or absorption, World Aca. Sci. Engin. Tech. 65
(2012) 687–690.
[6] X. H. Su, L. C. Zheng, Approximate solutions to MHD Falkner-Skan
flow over permeable wall, Appl. Math. Mech. -Engl. Ed. 32 (2011)
401–408.
[7] A. Ishak, R. Nazar, I. Pop, MHD boundary-layer flow of a micropolar
fluid past a wedge with constant wall heat flux, Commun. Nonlinear
Sci. Num. Simul. 14 (2009) 109–118.
[8] N. Bachoka, A. Ishak, I. Pop, Boundary layer stagnation-point flow
and heat transfer over an exponentially stretching/shrinking sheet in a
nanofluid, I. J. Heat Mass Trans. 55 (2012) 8122–8128.
[9] S. Abbasbandy, T. Hayat, Solution of the MHD Falkner-Skan flow
by Homotopy analysis method, Nonlinear Anal. Real. World Appl. 14
(2009) 3591–3598.
[10] K. Parand, M. Dehghan, A. Pirkhedri, The use of sinc-collocation
method for solving Falkner-Skan boundary-layer equationl, Int. J. Num.
Meth. Fluids 69 (2004) 353–357.
[11] A. Asaithambi, A second-order finite-difference method for the
FalknerSkan equation, appl. math. comput. 156 (2004) 779–786.
[12] M. Fathizadeh, M. Madani, Y. Khan, N. Faraz, A. Yildirim, S. Tutkun,
An effective modification of the homotopy perturbation method for mhd
viscous flow over a stretching sheet, J. King Saud Uni. Sci. 2 (2013)
107113.
[13] M. M. Rashidi, The modified differential transform method and pade
approximats for solving MHD boundary-layer equationsl, Comput. Phys.
Commun. 180 (2009) 2210–2217.
[14] R. Ellahi, E. Shivanian, S. Abbasbandy, T. Hayat, R. Lewis,
Numerical study of magnetohydrodynamics generalized couette flow of
eyring-powell fluid with heat transfer and slip condition, International
Journal of Numerical Methods for Heat & Fluid Flow 26 (5).
[15] J. Shen, T. Tang, L. L. Wang, Spectral Methods Algorithms, Analysics
And Applicattions, first edition, Springer, 2001.
[16] J. P. Boyd, Chebyshev and Fourier spectral methods. 2nd ed, New York
Dover, New York, 2000.
[17] K. Parand, J. Rad, M. Nikarya, A new numerical algorithm based on
the first kind of modified bessel function to solve population growth in
a closed system, International Journal of Computer Mathematics 91 (6)
(2014) 1239–1254.
[18] K. Parand, M. Nikarya, Solving the unsteady isothermal gas through
a micro-nano porous medium via bessel function collocation method,
Journal of Computational and Theoretical Nanoscience 11 (1) (2014)
131–136.
[19] W. W. Bell, Special Functions For Scientists And Engineers, Published
simultaneously in Canada by D. Van Nostrand Company, (Canada), Ltd,
1967.
[20] G. Ben-yu, Spectral methods and their applications, World Scientific,
Shelton Street, Covent Garden, London WC2H 9HE, 1998.
[21] J. D. C. G. W. Bluman, Similarity Methods for Differential Equations,
Springer, New York, 1974.
[22] L. Dresner, Similarity solutions of nonlinear partial differential
equations, Longman Group, London, 1983.
[23] F. M. White, Fluid Mechanics, Seventh Edition, McGraw-Hill, New
York, 2009.
[24] H. Schlichting, K. Gersten, Boundary layer theory. 6th ed, Oxford, New
York, 1979.
[25] T. Y. Na, Computational Method in Engineering Boundray Value
Problem, Academic Pressl, New Yorkl, 1979.
[26] L. Rosenhead, Laminar Boundary layers, Clarendon Pressl,
McGraw-Hill, 1963.
[27] T. Hayat, Q. Hussain, T. Javed, The modified decomposition method
and pad`e aapproximants for theMHD flow over a non-linear stretching
sheet, Commun. Nonlinear Sci. Num. Simul. 10 (2009) 966–973.
[28] G. Watson, A treatise on the theory of Bessel Functions, 2nd edition,
Cambridge University Press, Cambridge (England), 1967.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:74499", author = "Saeed Sarabadan and Mehran Nikarya and Kouroah Parand", title = "Spectral Investigation for Boundary Layer Flow over a Permeable Wall in the Presence of Transverse Magnetic Field", abstract = "The magnetohydrodynamic (MHD) Falkner-Skan
equations appear in study of laminar boundary layers flow over
a wedge in presence of a transverse magnetic field. The partial
differential equations of boundary layer problems in presence of
a transverse magnetic field are reduced to MHD Falkner-Skan
equation by similarity solution methods. This is a nonlinear ordinary
differential equation. In this paper, we solve this equation via
spectral collocation method based on Bessel functions of the first
kind. In this approach, we reduce the solution of the nonlinear
MHD Falkner-Skan equation to a solution of a nonlinear algebraic
equations system. Then, the resulting system is solved by Newton
method. We discuss obtained solution by studying the behavior
of boundary layer flow in terms of skin friction, velocity, various
amounts of magnetic field and angle of wedge. Finally, the results
are compared with other methods mentioned in literature. We can
conclude that the presented method has better accuracy than others.", keywords = "MHD Falkner-Skan, nonlinear ODE, spectral
collocation method, Bessel functions, skin friction, velocity.", volume = "10", number = "7", pages = "1354-7", }
