A RELIABLE ALGORITHM FOR SOLVING BLASIUS BOUNDARY VALUE PROBLEM
Abstract views: 26 / PDF downloads: 61
Keywords:
Nonlinear differential equations, Blassius equation, Approximate solutions, Adomian decomposition method, laminar flowAbstract
The Blasius equation is a well-known third-order nonlinear
ordinary differential equation that can be found in some fluid dynamics boundary layer problems.
In this paper, we convert the nonlinear differential equation to an integral equation, this integral equation has
a shifted kernel. Our goal is to propose an efficient modification of the
standard Adomian decomposition method, combined with the Laplace
transform, for solving the Blasius equation. The main impediment to
solving the Blasius equation is the absence of the second derivative at
zero. Once this derivative has been correctly evaluated, an analytical
solution to the Blasius problem can be easily found; as a result, we use
our approximate solution to estimate the value of y′′(0), also known asthe
Blasius constant. Understanding the Blasius constant is essential
for calculating shear stress at a plate. Furthermore, once this value
is determined, we have the initial value problem, which can be solved
numerically
References
H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys. 56: 1–37 (1908).
J. He, Approximate analytical solution of Blasius’ equation. Communi. in Nonlinear Sci. and Numerical Simul. 3(4):
–263 (1998).
J.-H. He, A simple perturbation approach to Blasius equation, Appl. Math. Comput. 140: 217 (2003). https://doi.
org/10.1016/S0096-3003(02)00189-3
A. Khanfer, L. Bougofa, S. Bougoufa, Analytic Approximate Solution of the Extended Blasius Equation with
Temperature‐Dependent Viscosity, J of Nonlinear Math. Phys. 30: 287–302 (2023). https://doi.org/10.1007/
s44198-022-00084-3
S. Mziou, L. Bougoffa, R.C. Rach. Generalized Blasius equation, uniqueness and analytical approximation solution,
TJMM. 10(2): 95–111 (2018).
G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Computers Math.
Appl. 21(5): 101–127 (1991).
R. Khandelwal, Y. Khandelwal, Solution of Blasius Equation Concerning with Mohand Transform, Int. J. Appl.
Comput. Math. 6: 128 (2020). https://doi.org/10.1007/s40819-020-00871-w
L. Bougoffa, Wazwaz, A.-M, New approximate solutions of the Blasius equation, Inter. J of Numerical Methods for
Heat and Fluid Flow. 25(7): 1590–1599 (2015). https://doi.org/10.1108/HFF-08-2014-0263
H. Aminikhah, Analytical Approximation to the Solution of Nonlinear Blasius’ Viscous Flow Equation by LTNHPM,
International Scholarly Research Network ISRN Mathematical Analysis. Volume 2012, Article ID 957473, 10 pages.
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, 1994.
M.M. Rahman, S. Khan, M.A. Akbar, Numerical and analytical solutions of new Blasius equation for turbulent flow.
Heliyon. 9: e14319 (2023).
A. Asaithambi, Numerical Solution of the Blasius Equation with Crocco-Wang Transformation, J of Applied Fluid
Mech. 9(5): 2595–2603 (2016).
F.M. Allan, M.I. Syam, On the analytic solutions of the nonhomogeneous Blasius problem, J. Comput. Appl. Math. 182:
(2005). https://doi.org/10.1016/j.cam.2004.12.017
J. Parlange, R. Braddock, G. Sander, Analytical approximations to the solution of the Blasius equation, Acta Mech. 38:
(1981). https://doi.org/10.1007/BF01351467
J. Zheng, X. Han, Z. Wang, C. Li, J. Zhang, A globally convergent and closed analytical solution of the Blasius equation
with beneficial applications, AIP Adv. 7: 065311 (2017).
D. Durante, E. Rossi, A. Colagrossi, G. Graziani, Numerical simulations of the transition from laminar to chaotic
behaviour of the planar vortex flow past a circular cylinder, Commun. Nonlinear Sci. Numer. Simul. 48: 18–38 (2017).
https://doi.org/10.1016/j.cnsns.2016.12.013
A. Jafarimoghaddam, M. Soler, A. Simorgh, The optimal decomposition method (ODM) for nonlinear problems,
J. Comput. Sci. 62: 101690 (2022). https://doi.org/10.1016/j.jocs.2022.101690
K. Al-Khaled, M.S. Ajeel, I. Abu-Irwaq, H.K. Al-Khalid, An Efficient Approximate Method for Solving Bratu’s Boundary
Value Problem. Inter. J of Elect. and Comp. Eng. 14(5): 5738–5743 (2024).
M. Rahmanzadeh, T. Asadi, M. Atashafrooz, A Numerical Algorithm based on the RCW method to solve a set of First-
Order Ordinary Differential Equations, J of the Serbian Soc. for Comput. Mechan. 14(1): 63–74 (2020). https://doi.org/10.24874/jsscm.2020.14.01.06
S. Abbasbandy, A numerical solution of Blasius equation by Adomian’s decomposition method and comparison
with homotopy perturbation method, Chaos, Solitons & Fractals. 31(1): 257–260. (2007). https://doi.org/10.1016/j.
chaos.2005.10.071
A. Ebaid, N. Al-Armani, A New Approach for a Class of the Blasius Problem via a Transformation and Adomian’s
Method, Abstract and Appl. Analy. Volume 2013, Article ID 753049, 8 pages http://doi.org/10.1155/2013/753049
M. Rahmanzadeh, T. Asadi, M. Atashafrooz, The Development and Application of the RCW Method for the Solution of
the Blasius Problem, J of Appl. and Comput. Mechan. 6(1): 105–111 (2020).
F.M. White, Fluid Mechanics, McGraw-Hill (1986).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Results in Nonlinear Analysis
This work is licensed under a Creative Commons Attribution 4.0 International License.