An Improved Numerical Analysis of Couette-Poiseuille Flow Using Hybrid Shooting-Differential Transform Method
DOI:
https://doi.org/10.4314/njtd.v22i1.2938Keywords:
Couette-Poiseuille flow, Differential Transform Method, Modified Shooting Method, Fluid Mechanics, Nonlinear Differential EquationsAbstract
Couette-Poiseuille flow is a fundamental concept in fluid mechanics that involves the combined effects of shear and pressure-driven forces in a channel that has a wide range of engineering applications such as lubrication, microfluidics, and heat exchangers. Appropriate methodological mechanism is important in the analysis of the effects and behaviour of such systems. Conventional numerical methods frequently encounter stability and convergence issues, particularly when dealing with nonlinear or stiff-flow problems. This study presents the application of the Modified Shooting Method (MSM) to solve and analyse the differential equation resulting from the Couette-Poiseuille Flow. The DTM reduces differential equations to a set of algebraic equations, providing improved efficiency and accuracy. On the other hand, the MSM boosts stability and convergence compared to conventional shooting procedures. The hybrid shooting-differential transform method combines the two techniques and leverages the strengths of both, creating a strong solution framework for the nonlinear differential equations that drive Couette-Poiseuille flow. The equations were first transformed into algebraic equations using the appropriate transforms. Finally, the shooting numerical scheme was deployed to solve the transformed equations. The results of our study show that the hybrid method achieved a percentage error of < 10-8% and an improved computing efficiency of  >70%.
References
Abouzeid, M., and Ibrahim, M. (2024). Multi-step differential transform method for both Hall currents and mixed convection effects on MHD flow of non-Newtonian fluid with Al2O3 nanoparticles. Egyptian Journal of Chemistry, 67 (6): 225-232.
Ali, K.; A. A. Faridi; N. Khan; K. S. Nisar and S. Ahmad. (2023). On the suitability of differential transform method for solving the selfâ€similar channel flow problems. ZAMMâ€Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 103(1), e202100358.
Al-Rozbayani, A. M. and Shammar, A. H. (2023, February). Applied reduced differential transform method to the Wu-Zhang equation in (1+1)-dimensional. In American Institute of Physics Conference Series, 2414 (1): 040004. https://doi.org/10.1063/5.0114833.
Anwar, M. I.; H. Firdous; A. A. Zubaidi; N. Abbas and S. Nadeem. (2022). Computational analysis of induced magnetohydrodynamic non-Newtonian nanofluid flow over nonlinear stretching sheet. Progress in Reaction Kinetics and Mechanism, 47, 14686783211072712.
Arikoglu, A. and Ozkol, I. (2005). Inner–outer matching solution of Blasius equation by DTM. Aircraft Engineering and Aerospace Technology, 77 (4): 298–301.
Arikoglu, A. and Ozkol, I. (2006). Solution of differential-difference equations by using differential transform method. Applied Mathematics and Computation, 181 (2): 1532-1542.
Ascher, U. M.; R. M. Mattheij and R. D. Russell. (1995). Numerical solution of boundary value problems for ordinary differential equations. SIAM.
Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge University Press.
Botti, M.; D. C. Quiroz; D. A. Di Pietro and A. Harnist. (2021). A hybrid high-order method for creeping flows of non-Newtonian fluids. ESAIM: Mathematical Modelling and Numerical Analysis, 55 (5): 2045-2073.
Derikvand, M.; F. Farhatnia and D. H. Hodges. (2023). Functionally graded thick sandwich beams with porous core: Buckling analysis via differential transform method. Mechanics Based Design of Structures and Machines, 51 (7): 3650-3677. https://doi.org/10.1080/15397734.2021.1931309
El-Zahar, E. R. (2013). Approximate analytical solutions of singularly perturbed fourth order boundary value problems using differential transform method. Journal of King Saud University-Science, 25 (3): 257-265.
Filipov, S. M.; I. D. Gospodinov and I. Faragó (2019). Replacing the finite difference methods for nonlinear two-point boundary value problems by successive application of the linear shooting method. Journal of Computational and Applied Mathematics, 358: 46-60.
Gie, G. M.; C. Y. Jung and Lee, H. (2023). Semi-analytic shooting methods for Burgers’ equation. Journal of Computational and Applied Mathematics, 418, 114694.
Hołubowski, R. and Jarczewska, K. (2023). A two-parameter multiple shooting method and its application to the natural vibrations of non-prismatic multi-segment beams. Applied Mathematics and Mechanics, 44 (12): 2243-2252.
Kumar, M. (2020). Study of differential transform technique for transient hydromagnetic Jeffrey fluid flow from a stretching sheet. Nonlinear Engineering, 9 (1): 145-155.
Kumar, M.; G. J. Reddy; N. N. Kumar and O. A. Bég (2019). Application of differential transform method to unsteady free convective heat transfer of a couple stress fluid over a stretching sheet. Heat Transfer—Asian Research, 48 (2): 582-600.
Nayfeh, A. H. (2011). Introduction to perturbation techniques. John Wiley & Sons.
Nadeem, S.; B. Ishtiaq; J. Alzabut and S. M. Eldin. (2024). Implementation of differential transform method on the squeezing flow of trigonometric non-Newtonian fluid between two heated plates. International Journal of Modern Physics B, 38 (24): 2450326.
Nouri, R.; D. D. Ganji and M. Hatami. (2013). MHD nanofluid flow analysis in a semi-porous channel by a combined series solution method. Challenges in Nano and Micro Scale Science and Technology, 1 (2): 124-137.
Press, W. H.; S. A. Teukolsky; W. T. Vetterling and B. P. Flannery. (2007). Numerical recipes: The art of scientific computing. Cambridge University Press.
Sadiq Murad, M. A. and Hamasalh, F. K. (2023). Numerical study for fractional-order magnetohydrodynamic boundary layer fluid flow over stretching sheet. Punjab University Journal of Mathematics, 55 (2).
Stoer, J. and Bulirsch, R. (2002). Introduction to numerical analysis. Springer.
Wazwaz, A. M. (2009). The variational iteration method for analytic treatment for linear and nonlinear ODEs. Applied Mathematics and Computation, 212 (1): 120-134.
Wellot, Y. A. S. (2022). Application of the Reduced Differential Transform Method to Solve the Navier-Stokes Equations. Pure and Applied Mathematics Journal, 7 (1): 96-101.
White, F. M. (2006). Viscous fluid flow. McGraw-Hill.
Zhao, Q.; W. Liu; W. Yu, and F. Cai. (2023). Dynamics of a fluid-conveying pipe by a hybrid method combining differential transformation and Galerkin discretization. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 1-13.
Zhou, J. K. (1986). Differential transformation and its applications for electrical circuits. Huarjung University Press.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Nigerian Journal of Technological Development
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
In accordance with the Copyright Act of 1976, which became effective January 1, 1978, the following statement signed by each author must accompany the manuscript submitted: "I, the undersigned author, transfer all copyright ownership of the manuscript referenced above to the Nigerian Journal of Technological Development, in the event the work is published. I warrant that the article is original, does not infringe upon any copyright or other proprietary right of any third party, is not under consideration by another journal, and has not been published previously. I have reviewed and approve the submitted version of the manuscript and agree to its publication in the Nigerian Journal of Technological Development." A copyright transfer form should be downloaded from the NJTD Website ( http://njtd.com.ng/index.php/njtd). Author(s) will be consulted, whenever possible, regarding republication of material. All authors must have access to the data presented and the authors and sponsor (if applicable) must agree to share original data with the editor if requested.
