Numerical solution of falkner-skan equation by iterative transformation method
Abstract
In this paper, we study the nonlinear boundary-layer equation of Falkner-Skan defined on a semi-infinite domain. An iterative finite difference (IFD) scheme is proposed to numerically solve such nonlinear ordinary differential equation. A computational iterative scheme is developed based on Newton-Kantorovich quasilinearization. At every iteration, the obtained linearized differential equation is numerically solved using the standard finite difference method. Numerical experiments show the accuracy and efficiency of the method compared to existing solvers. The computation is performed for different parameter values, including the special case of Blasius problem.
Keyword : Falkner-Skan equation, Blasius equation, quasi-linearization, iterative nite dierence method
How to Cite
Temimi, H., & Ben-Romdhane, M. (2018). Numerical solution of falkner-skan equation by iterative transformation method. Mathematical Modelling and Analysis, 23(1), 139-151. https://doi.org/10.3846/mma.2018.009
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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N.S. Asaithambi. A numerical method for the solution of the Falkner-Skan equation. Applied Mathematics and Computation, 81(2):259-264, 1997. https://doi.org/10.1016/S0096-3003(95)00325-8
R.E. Bellman and R.E. Kalaba. Quasilinearization and Nonlinear Boundary-Value Problems. American Elsevier Publishing Company, New York, 1965.
H. Blasius. Grenzschichten in Flussigkeiten mit kleiner Reibung. Zeitschrift fur angewandte Mathematik und Physik, 56:1-37, 1908.
T. Cebeci and H. Keller. Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equations. Journal of Computational Physics, 7(2):289-300, 1971. https://doi.org/10.1016/0021-9991(71)90090-8
R. Cortell. Numerical solutions of the classical Blasius at-plate problem. Applied Mathematics and Computation, 170(1):706-710, 2005. https://doi.org/10.1016/j.amc.2004.12.037
V.M. Falkner and S.W. Skan. LXXXV. solutions of the boundary-layer equations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 12(80):865-896, 1931. https://doi.org/10.1080/14786443109461870
T. Fang, W. Liang and C.F. Lee. A new solution branch for the Blasius equation - A shrinking sheet problem. Computers & Mathematics with Applications, 56(12):3088-3095, 2008. https://doi.org/10.1016/j.camwa.2008.07.027
T. Fang and J. Zhang. An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching. International Journal of Non-Linear Mechanics, 43(9):1000-1006, 2008. https://doi.org/10.1016/j.ijnonlinmec.2008.05.006
R. Fazio. The iterative transformation method for the Sakiadis problem. Computers & Fluids, 106(SupplementC):196-200, 2015. https://doi.org/10.1016/j.compuid.2014.10.007
D. Hartree. On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer. Mathematical Proceedings of the Cambridge Philosophical Society, 33(2):223-239, 1937. https://doi.org/10.1017/S0305004100019575
K. Hiemenz. Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder. Polytechnisches Journal, 326:391-393, 1911.
F. Homann. Der Einuss grosser Zahigkeit bei der Stromung um den Zylinder und um die Kugel. Zeitschrift fur Angewandte Mathematik und Mechanik, 16(3):153-164, 1936. https://doi.org/10.1002/zamm.19360160304
L. Howarth. On the solution of the laminar boundary layer equations. Proceedings of the Royal Society of London A : Mathematical, Physical and Engineering Sciences, 164(919):547-579, 1938. https://doi.org/10.1098/rspa.1938.0037
S. Liao. A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite at plate. Journal of Fluid Mechanics, 385:101-128, 1999. https://doi.org/10.1017/S0022112099004292
S. Liao. Beyond Perturbation. Introduction to the Homotopy Analysis Method. Chapman Hall/CRC, Boca Raton, 2003. https://doi.org/10.1201/9780203491164
C.-S. Liu and J.-R Chang. The Lie-group shooting method for multiple-solutions of Falkner-Skan equation under suction-injection conditions. International Journal of Non-Linear Mechanics, 43(9):844-851, 2008. https://doi.org/10.1016/j.ijnonlinmec.2008.05.005
S. Mukhopadhyay, I.C. Mondal and A.J. Chamkha. Casson fluid flow and heat transfer past a symmetric wedge. Heat Transfer-Asian Research, 42(8):665-675, 2013. https://doi.org/10.1002/htj.21065
K. Pohlhausen. Zur naherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht. Journal of Applied Mathematics and Mechanics (ZAMM), 1(4):252-268, 1921. https://doi.org/10.1002/zamm.19210010402
C.S.K. Raju, M.M. Hoque and T. Sivasankar. Radiative ow of Casson uid over a moving wedge filled with gyrotactic microorganisms. Advanced Powder Technology, 28(2):575-583, 2017. https://doi.org/10.1016/j.apt.2016.10.026
C.S.K. Raju, S.M. Ibrahim, S. Anuradha and P. Priyadharshini. Bio-convection on the nonlinear radiative flow of a Carreau fluid over a moving wedge with suction or injection. The European Physical Journal Plus, 131(11):409-409, 2016. https://doi.org/10.1140/epjp/i2016-16409-7
C.S.K. Raju and N. Sandeep. A comparative study on heat and mass transfer of the Blasius and Falkner-Skan flow of a bio-convective Casson fluid past a wedge. The European Physical Journal Plus, 131(11):405, 2016. https://doi.org/10.1140/epjp/i2016-16405-y
C.S.K. Raju and N. Sandeep. Nonlinear radiative magnetohydrodynamic Falkner-Skan flow of Casson fluid over a wedge. Alexandria Engineering Journal, 55(3):2045-2054, 2016. https://doi.org/10.1016/j.aej.2016.07.006
C.S.K. Raju and N. Sandeep. MHD slip flow of a dissipative Casson fluid over a moving geometry with heat source/sink: A numerical study. Acta Astronautica, 133:436-443, 2017. https://doi.org/10.1016/j.actaastro.2016.11.004
P.L. Sachdev, R.B. Kudenatti and N.M. Bujurke. Exact analytic solution of a boundary value problem for the Falkner-Skan equation. Studies in Applied Mathematics, 120(1):1-16, 2008. https://doi.org/10.1111/j.1467-9590.2007.00386.x
A.A. Salama. Higher-order method for solving free boundary-value problems. Numerical Heat Transfer, Part B: Fundamentals, 45(4):385-394, 2004. https://doi.org/10.1080/10407790490278002
I. Sher and A. Yakhot. New approach to solution of the Falkner-Skan equation. American Institute of Aeronautics and Astronautics, 39(1):965-967, 2001. https://doi.org/10.2514/2.1403
H. Temimi and M. Ben-Romdhane. An iterative finite difference method for solving Bratu's problem. Journal of Computational and Applied Mathematics, 292(SupplementC):76-82, 2016. https://doi.org/10.1016/j.cam.2015.06.023
K.A. Yih. The effect of uniform suction/blowing on heat transfer of magnetohydrodynamic hiemenz flow through porous media. Acta Mechanica, 130(3):147-158, 1998. https://doi.org/10.1007/BF01184307
S. Zhu, Q. Wu and X. Cheng. Numerical solution of the Falkner-Skan equation based on quasi-linearization. Applied Mathematics and Computation, 215(7):2472-2485, 2009. https://doi.org/10.1016/j.amc.2009.08.047
S. Zhu, H. Zhu, Q. Wu and Y. Khan. An adaptive algorithm for the Thomas-Fermi equation. Numerical Algorithms, 59(3):359-372, 2012. https://doi.org/10.1007/s11075-011-9494-1
N.S. Asaithambi. A numerical method for the solution of the Falkner-Skan equation. Applied Mathematics and Computation, 81(2):259-264, 1997. https://doi.org/10.1016/S0096-3003(95)00325-8
R.E. Bellman and R.E. Kalaba. Quasilinearization and Nonlinear Boundary-Value Problems. American Elsevier Publishing Company, New York, 1965.
H. Blasius. Grenzschichten in Flussigkeiten mit kleiner Reibung. Zeitschrift fur angewandte Mathematik und Physik, 56:1-37, 1908.
T. Cebeci and H. Keller. Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equations. Journal of Computational Physics, 7(2):289-300, 1971. https://doi.org/10.1016/0021-9991(71)90090-8
R. Cortell. Numerical solutions of the classical Blasius at-plate problem. Applied Mathematics and Computation, 170(1):706-710, 2005. https://doi.org/10.1016/j.amc.2004.12.037
V.M. Falkner and S.W. Skan. LXXXV. solutions of the boundary-layer equations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 12(80):865-896, 1931. https://doi.org/10.1080/14786443109461870
T. Fang, W. Liang and C.F. Lee. A new solution branch for the Blasius equation - A shrinking sheet problem. Computers & Mathematics with Applications, 56(12):3088-3095, 2008. https://doi.org/10.1016/j.camwa.2008.07.027
T. Fang and J. Zhang. An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching. International Journal of Non-Linear Mechanics, 43(9):1000-1006, 2008. https://doi.org/10.1016/j.ijnonlinmec.2008.05.006
R. Fazio. The iterative transformation method for the Sakiadis problem. Computers & Fluids, 106(SupplementC):196-200, 2015. https://doi.org/10.1016/j.compuid.2014.10.007
D. Hartree. On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer. Mathematical Proceedings of the Cambridge Philosophical Society, 33(2):223-239, 1937. https://doi.org/10.1017/S0305004100019575
K. Hiemenz. Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder. Polytechnisches Journal, 326:391-393, 1911.
F. Homann. Der Einuss grosser Zahigkeit bei der Stromung um den Zylinder und um die Kugel. Zeitschrift fur Angewandte Mathematik und Mechanik, 16(3):153-164, 1936. https://doi.org/10.1002/zamm.19360160304
L. Howarth. On the solution of the laminar boundary layer equations. Proceedings of the Royal Society of London A : Mathematical, Physical and Engineering Sciences, 164(919):547-579, 1938. https://doi.org/10.1098/rspa.1938.0037
S. Liao. A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite at plate. Journal of Fluid Mechanics, 385:101-128, 1999. https://doi.org/10.1017/S0022112099004292
S. Liao. Beyond Perturbation. Introduction to the Homotopy Analysis Method. Chapman Hall/CRC, Boca Raton, 2003. https://doi.org/10.1201/9780203491164
C.-S. Liu and J.-R Chang. The Lie-group shooting method for multiple-solutions of Falkner-Skan equation under suction-injection conditions. International Journal of Non-Linear Mechanics, 43(9):844-851, 2008. https://doi.org/10.1016/j.ijnonlinmec.2008.05.005
S. Mukhopadhyay, I.C. Mondal and A.J. Chamkha. Casson fluid flow and heat transfer past a symmetric wedge. Heat Transfer-Asian Research, 42(8):665-675, 2013. https://doi.org/10.1002/htj.21065
K. Pohlhausen. Zur naherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht. Journal of Applied Mathematics and Mechanics (ZAMM), 1(4):252-268, 1921. https://doi.org/10.1002/zamm.19210010402
C.S.K. Raju, M.M. Hoque and T. Sivasankar. Radiative ow of Casson uid over a moving wedge filled with gyrotactic microorganisms. Advanced Powder Technology, 28(2):575-583, 2017. https://doi.org/10.1016/j.apt.2016.10.026
C.S.K. Raju, S.M. Ibrahim, S. Anuradha and P. Priyadharshini. Bio-convection on the nonlinear radiative flow of a Carreau fluid over a moving wedge with suction or injection. The European Physical Journal Plus, 131(11):409-409, 2016. https://doi.org/10.1140/epjp/i2016-16409-7
C.S.K. Raju and N. Sandeep. A comparative study on heat and mass transfer of the Blasius and Falkner-Skan flow of a bio-convective Casson fluid past a wedge. The European Physical Journal Plus, 131(11):405, 2016. https://doi.org/10.1140/epjp/i2016-16405-y
C.S.K. Raju and N. Sandeep. Nonlinear radiative magnetohydrodynamic Falkner-Skan flow of Casson fluid over a wedge. Alexandria Engineering Journal, 55(3):2045-2054, 2016. https://doi.org/10.1016/j.aej.2016.07.006
C.S.K. Raju and N. Sandeep. MHD slip flow of a dissipative Casson fluid over a moving geometry with heat source/sink: A numerical study. Acta Astronautica, 133:436-443, 2017. https://doi.org/10.1016/j.actaastro.2016.11.004
P.L. Sachdev, R.B. Kudenatti and N.M. Bujurke. Exact analytic solution of a boundary value problem for the Falkner-Skan equation. Studies in Applied Mathematics, 120(1):1-16, 2008. https://doi.org/10.1111/j.1467-9590.2007.00386.x
A.A. Salama. Higher-order method for solving free boundary-value problems. Numerical Heat Transfer, Part B: Fundamentals, 45(4):385-394, 2004. https://doi.org/10.1080/10407790490278002
I. Sher and A. Yakhot. New approach to solution of the Falkner-Skan equation. American Institute of Aeronautics and Astronautics, 39(1):965-967, 2001. https://doi.org/10.2514/2.1403
H. Temimi and M. Ben-Romdhane. An iterative finite difference method for solving Bratu's problem. Journal of Computational and Applied Mathematics, 292(SupplementC):76-82, 2016. https://doi.org/10.1016/j.cam.2015.06.023
K.A. Yih. The effect of uniform suction/blowing on heat transfer of magnetohydrodynamic hiemenz flow through porous media. Acta Mechanica, 130(3):147-158, 1998. https://doi.org/10.1007/BF01184307
S. Zhu, Q. Wu and X. Cheng. Numerical solution of the Falkner-Skan equation based on quasi-linearization. Applied Mathematics and Computation, 215(7):2472-2485, 2009. https://doi.org/10.1016/j.amc.2009.08.047
S. Zhu, H. Zhu, Q. Wu and Y. Khan. An adaptive algorithm for the Thomas-Fermi equation. Numerical Algorithms, 59(3):359-372, 2012. https://doi.org/10.1007/s11075-011-9494-1