A parameter-free series solution approach for differential equations in fluid flow

A. Alameer

doi:10.59400/adecp3151

A parameter-free series solution approach for differential equations in fluid flow

A. Alameer Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin 31991, Saudi Arabia

Article ID: 3151

Keywords: converging/diverging channel; Maclaurin series method; boundary value problem; fractal calculus; numerical solution

Abstract

Viscous incompressible flow across a converging/diverging channel generally known as Jeffery-Hamel flow is an important form of flow in the fluid dynamics sector that exists in a variety of engineering systems, rivers, and in the biological world. This paper proposes a novel Maclaurin Series Method (MSM) to investigate Hamel’s fractal flow pattern in a wedge-shaped region. The fundamental partial differential equations are changed by suitable transformation into the dimensionless non-linear ordinary differential equation. The resulting equation is solved through MSM. The Maclaurin series method obtains the solution of the two-dimensional incompressible viscous flow in the converging / diverging channels according to initial condition. The MSM provides an efficient and accurate alternative to traditional solution techniques. To validate the Maclaurin series method, error analysis of the solution is calculated and presented in tabular form, demonstrating excellent agreement with benchmark results. Furthermore, the MSM solution is plotted for various β values. The comparison between MSM approximate and exact solutions confirms the reliability and effectiveness of the method. Overall, the results indicate that the suggested approach is an effective and reliable tool for solving fluid flow problems.

Published

2025-09-30

How to Cite

Alameer, A. (2025). A parameter-free series solution approach for differential equations in fluid flow. Advances in Differential Equations and Control Processes, 32(3). https://doi.org/10.59400/adecp3151

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Issue

Vol. 32 No. 3 (2025)

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Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]He J-H. A new fractal derivation. Thermal Science. 2011; 15(suppl. 1): 145–147. doi: 10.2298/TSCI11S1145H

[2]He J-H. Fractal calculus and its geometrical explanation. Results in Physics. 2018; 10: 272–276. doi: 10.1016/j.rinp.2018.06.011

[3]Franc DGE, Khan Y, Tchangou TI. The fractal and piecewise structure of some chaotic neural networks using a generalized model. Fractals. 2022; 30(8): 2240228. doi:10.1142/S0218348X22402289

[4]Shi X-J, Yu W-D. Fractal phenomenon in micro-flow through a fiber bundle. International Journal of Nonlinear Sciences and Numerical Simulation. 2009; 10(7). doi: 10.1515/IJNSNS.2009.10.7.861

[5]Yang X-J, Baleanu D, Khan Y, et al. Local fractional variational iteration method for diffusion and wave equations on Cantor sets. Romanian Journal of Physics. 2014; 59(1–2): 36–48. Available online: https://rjp.nipne.ro/2014_59_1-2/RomJPhys.59.p36.pdf

[6]Heydari MH, Atangana A, Avazzadeh Z, et al. Numerical treatment of the strongly coupled nonlinear fractal-fractional schrödinger equations through the shifted chebyshev cardinal functions. Alexandria Engineering Journal. 2020; 59(4): 2037–2052. doi: 10.1016/j.aej.2019.12.039

[7]He C-H, Shen Y, Ji F-Y, et al. Taylor series solution for fractal bratu-type equation arising in electrospinning process. Fractals. 2020; 28(01): 2050011. doi: 10.1142/S0218348X20500115

[8]Fardi M, Khan Y. A novel finite difference–spectral method for fractal mobile/immobile transport model based on the Caputo–Fabrizio derivative. Chaos, Solitons & Fractals. 2021; 143: 110573. doi: 10.1016/j.chaos.2020.110573

[9]Doungmo GEF, Khan Y. A new auto-replication in systems of attractors with two and three merged basins of attraction via control. Communications in Nonlinear Science and Numerical Simulation. 2021; 96: 105709. doi: 10.1016/j.cnsns.2021.105709

[10]He J-H. A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes. Journal of Electroanalytical Chemistry. 2019; 854: 113565. doi: 10.1016/j.jelechem.2019.113565

[11]Atangana A, Goufo EFD. The Caputo-Fabrizio fractional derivative applied to a singular perturbation problem. International Journal of Mathematical Modelling and Numerical Optimisation. 2019; 9(3): 241. doi: 10.1504/IJMMNO.2019.100486

[12]Jeffery GBL. The two-dimensional steady motion of a viscous fluid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1915; 29(172): 455–465. doi: 10.1080/14786440408635327

[13]Hamel G. Spiral-shaped Movements of Fluids. nnual Report of the German Mathematical Association. 1916; 25: 34–60. Available online: https://eudml.org/doc/145468 (in German)

[14]Axford WI. The magnetohydrodynamic jeffrey-hamel problem for a weakly conducting fluid. The Quarterly Journal of Mechanics and Applied Mathematics. 1961; 14(3): 335–351. doi: 10.1093/qjmam/14.3.335

[15]Rosenhead L. The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 1940; 175(963): 436–467. doi: 10.1098/rspa.1940.0068

[16]Khan Y. A novel Laplace decomposition method for non-linear stretching sheet problem in the presence of MHD and slip condition. International Journal of Numerical Methods for Heat & Fluid Flow. 2013; 24(1): 73–85. doi: 10.1108/HFF-02-2012-0048

[17]He J-H, Latifizadeh H. A general numerical algorithm for nonlinear differential equations by the variational iteration method. International Journal of Numerical Methods for Heat & Fluid Flow. 2020; 30(11): 4797–4810. doi: 10.1108/HFF-01-2020-0029

[18]Khan Y. Two-dimensional boundary layer flow of chemical reaction MHD fluid over a shrinking sheet with suction and injection. Journal of Aerospace Engineering. 2014; 27(5): 04014028. doi: 10.1061/(ASCE)AS.1943-5525.0000274

[19]He J-H, Moatimid GM, Sayed A. Effect of mass and heat transfer on EHD stability of two dusty liquid layers between two inclined rigid plates. International Journal of Modern Physics B. 2024; 38(1): 2450013. doi: 10.1142/S0217979224500139

[20]He J-H. Frequency–amplitude relationship in nonlinear oscillators with irrational nonlinearities. Spectrum of Mechanical Engineering and Operational Research. 2025; 2(1): 121–129. doi: 10.31181/smeor21202535

[21]He J-H, Moatimid GM, Mohamed MAA, et al. Unsteady MHD flow in a rotating annular region with homogeneous–heterogeneous chemical reactions of Walters’ B fluids: Time-periodic boundary criteria. International Journal of Modern Physics B. 2024; 38(14): 2450169. doi: 10.1142/S0217979224501698

[22]Liu Y-P, He J-H. A fast and accurate estimation of amperometric current response in reaction kinetics. Journal of Electroanalytical Chemistry. 2025; 978: 118884. doi: 10.1016/j.jelechem.2024.118884

[23]Domairry G, Mohsenzadeh A, Famouri M. The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery–Hamel flow. Communications in Nonlinear Science and Numerical Simulation. 2009; 14(1): 85–95. doi: 10.1016/j.cnsns.2007.07.009

[24]Joneidi AA, Domairry G, Babaelahi M. Three analytical methods applied to Jeffery–Hamel flow. Communications in Nonlinear Science and Numerical Simulation. 2010; 15(11): 3423–3434. doi: 10.1016/j.cnsns.2009.12.023

[25]Esmaili Q, Ramiar A, Alizadeh E, et al. An approximation of the analytical solution of the Jeffery–Hamel flow by decomposition method. Physics Letters A. 2008; 372(19): 3434–3439. doi: 10.1016/j.physleta.2008.02.006

[26]Ganji ZZ, Ganji DD, Esmaeilpour M. Study on nonlinear Jeffery–Hamel flow by He’s semi-analytical methods and comparison with numerical results. Computers & Mathematics with Applications. 2009; 58(11–12): 2107–2116. doi: 10.1016/j.camwa.2009.03.044

[27]Makinde OD, Mhone PY. Hermite–Padé approximation approach to MHD Jeffery–Hamel flows. Applied Mathematics and Computation. 2006; 181(2): 966–972. doi: 10.1016/j.amc.2006.02.018

[28]Motsa SS, Sibanda P, Awad FG, et al. A new spectral-homotopy analysis method for the MHD Jeffery–Hamel problem. Computers & Fluids. 2010; 39(7): 1219–1225. doi: 10.1016/j.compfluid.2010.03.004

[29]Wang J, Liu Y, Yan L, et al. Fractional sub-equation neural networks (fSENNs) method for exact solutions of space–time fractional partial differential equations. Chaos. 2025; 35(4): 043110. doi: 10.1063/5.0259937

[30]Zhang H, Zhang R, Liu Q. A novel multi-modal neurosymbolic reasoning intelligent algorithm for BLMP equation. Chinese Physics Letters. 2025; 42(10): 100002. doi: 10.1088/0256-307X/42/10/100002

[31]Na TY. Computational methods in engineering boundary value problems. Mathematics in science and engineering. Elsevier; 1979. doi: 10.1016/S0076-5392(08)X6096-5

[32]Schlichting H. Boundary Layer Theory. McGraw-Hill; 1968.

[33]Marinca V, Herişanu N, Bota C, et al. An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Applied Mathematics Letters. 2009; 22(2): 245–251. doi: 10.1016/j.aml.2008.03.019

[34]Ain Q, He J-H. On two-scale dimension and its applications. Thermal Science. 2019; 23(3 Part B): 1707–1712. doi: 10.2298/TSCI190408138A

[35]He J-H, Ji F-Y. Two-scale mathematics and fractional calculus for thermodynamics. Thermal Science. 2019; 23(4): 2131–2133. doi: 10.2298/TSCI1904131H

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