On nonlinear coupled differential equations for corrugated backward facing step (CBFS) with circular obstacle: AI-neural networking

Khalil Ur Rehman; Wasfi Shatanawi; Weam G. Alharbi; Khalil Ur Rehman; Wasfi Shatanawi; Weam G. Alharbi

doi:10.3934/math.2025212

AIMS Mathematics

2025, Volume 10, Issue 3: 4579-4597. doi: 10.3934/math.2025212

Previous Article Next Article

Research article Special Issues

On nonlinear coupled differential equations for corrugated backward facing step (CBFS) with circular obstacle: AI-neural networking

1.
Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, 71491, Saudi Arabia

Received: 24 October 2024 Revised: 12 February 2025 Accepted: 19 February 2025 Published: 03 March 2025
MSC : 35A25, 65M06, 76D05

Nonlinear mathematical formulations provide an accurate representation of intricate phenomena, including turbulence, vortices, and chaotic flow behavior. The solution of nonlinear differential equations for narrating flow fields is a challenging task. In this regard, the present article offers a solution remedy by conjecturing finite element method (FEM) outcomes with artificial intelligence-based neural networks. More precisely, a backward-facing step (BFS) is being treated as the study domain. The two corresponding triangular ribs make BFS corrugated, and the inlet has a parabolic pattern. We derive the differential system for the flow field within a BFS rooted with a circular obstacle. The solution is obtained by using the FEM. The artificial neural networks (ANNs) model is created with an input layer containing the viscosity, density, characteristics length, and mean inflow velocity, and it has lift coefficient (LC) to be output in the last layer. We choose 67 (70%) values for training and the remaining data points are taken for validation and testing as a 14 each. ANN has 10 neurons in the hidden layer and is trained with the Levenberg-Marquardt algorithm. Mean square error and regression analysis are performed to validate the model. It is concluded that the ANN design will act as the most accurate forecasting model of hydrodynamic force on circular obstruction in BFS for an extensive range, except normal parameters where classical methodologies were unable to predict.
- nonlinear PDEs,
- neural networking,
- hydrodynamic force,
- circular obstacle,
- BFS,
- finite element method
Citation: Khalil Ur Rehman, Wasfi Shatanawi, Weam G. Alharbi. On nonlinear coupled differential equations for corrugated backward facing step (CBFS) with circular obstacle: AI-neural networking[J]. AIMS Mathematics, 2025, 10(3): 4579-4597. doi: 10.3934/math.2025212

Related Papers:

Abstract

Nonlinear mathematical formulations provide an accurate representation of intricate phenomena, including turbulence, vortices, and chaotic flow behavior. The solution of nonlinear differential equations for narrating flow fields is a challenging task. In this regard, the present article offers a solution remedy by conjecturing finite element method (FEM) outcomes with artificial intelligence-based neural networks. More precisely, a backward-facing step (BFS) is being treated as the study domain. The two corresponding triangular ribs make BFS corrugated, and the inlet has a parabolic pattern. We derive the differential system for the flow field within a BFS rooted with a circular obstacle. The solution is obtained by using the FEM. The artificial neural networks (ANNs) model is created with an input layer containing the viscosity, density, characteristics length, and mean inflow velocity, and it has lift coefficient (LC) to be output in the last layer. We choose 67 (70%) values for training and the remaining data points are taken for validation and testing as a 14 each. ANN has 10 neurons in the hidden layer and is trained with the Levenberg-Marquardt algorithm. Mean square error and regression analysis are performed to validate the model. It is concluded that the ANN design will act as the most accurate forecasting model of hydrodynamic force on circular obstruction in BFS for an extensive range, except normal parameters where classical methodologies were unable to predict.

References

[1]	D. Demarco, E. N. Dvorkin, Modeling of metal forming processes: Implementation of an iterative solver in the flow formulation, Comput. Structures, 79 (2001), 1933–1942. https://doi.org/10.1016/S0045-7949(01)00118-3 doi: 10.1016/S0045-7949(01)00118-3
[2]	J. Vimmr, Mathematical modelling of compressible inviscid fluid flow through a sealing gap in the screw compressor, Math. Comput. Simul., 61 (2003), 187–197. https://doi.org/10.1016/S0378-4754(02)00075-7 doi: 10.1016/S0378-4754(02)00075-7
[3]	H. Nogami, M. Chu, J. Yagi, Multi-dimensional transient mathematical simulator of blast furnace process based on multi-fluid and kinetic theories, Comput. Chem. Eng., 29 (2005), 2438–2448. https://doi.org/10.1016/j.compchemeng.2005.05.024 doi: 10.1016/j.compchemeng.2005.05.024
[4]	D. S. Sankar, K. Hemalatha, Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries–A mathematical model, Appl. Math. Model., 31 (2007), 1497–1517. https://doi.org/10.1016/j.apm.2006.04.012 doi: 10.1016/j.apm.2006.04.012
[5]	S. U. Siddiqui, N. K. Verma, S. Mishra, R. S. Gupta, Mathematical modelling of pulsatile flow of Casson's fluid in arterial stenosis, Appl. Math. Comput., 210 (2009), 1–10. https://doi.org/10.1016/j.amc.2007.05.070 doi: 10.1016/j.amc.2007.05.070
[6]	S. K. Pandey, M. K. Chaube, D. Tripathi, Peristaltic transport of multilayered power-law fluids with distinct viscosities: A mathematical model for intestinal flows, J. Theoret. Biol., 278 (2011): 11–19.https://doi.org/10.1016/j.jtbi.2011.02.027
[7]	M. Tessarotto, C. Cremaschini, Mathematical properties of the Navier–Stokes dynamical system for incompressible Newtonian fluids, Phys. A, 392 (2013), 3962–3968. https://doi.org/10.1016/j.physa.2013.04.054 doi: 10.1016/j.physa.2013.04.054
[8]	M. A. Sumbatyan, A. E. Tarasov, A mathematical model for the propulsive thrust of the thin elastic wing harmonically oscillating in a flow of non-viscous incompressible fluid, Mech. Res. Comm., 68 (2015), 83–88. https://doi.org/10.1016/j.mechrescom.2015.02.005 doi: 10.1016/j.mechrescom.2015.02.005
[9]	A. D. Obembe, M. E. Hossain, K. Mustapha, S. A. Abu-Khamsin, A modified memory-based mathematical model describing fluid flow in porous media, Comput. Math. Appl., 73 (2017), 1385–1402. https://doi.org/10.1016/j.camwa.2016.11.022 doi: 10.1016/j.camwa.2016.11.022
[10]	M. Aleem, M. I. Asjad, M. S. R. Chowdhury, A. Hussanan, Analysis of mathematical model of fractional viscous fluid through a vertical rectangular channel, Chinese J. Phys., 61 (2019), 336–350. https://doi.org/10.1016/j.cjph.2019.08.014 doi: 10.1016/j.cjph.2019.08.014
[11]	H. R. Patel, S. D. Patel, R. Darji, Mathematical study of unsteady micropolar fluid flow due to non-linear stretched sheet in the presence of magnetic field, Int. J. Thermofluids, 16 (2022), 100232. https://doi.org/10.1016/j.ijft.2022.100232 doi: 10.1016/j.ijft.2022.100232
[12]	N. S. M. Hanafi, W. A. W. Ghopa, R. Zulkifli, M. A. M. Sabri, W. F. H. W. Zamri, M. I. M. Ahmad, Mathematical formulation of Al₂O₃-Cu/water hybrid nanofluid performance in jet impingement cooling, Energy Rep., 9 (2023), 435–446. https://doi.org/10.1016/j.egyr.2023.06.035 doi: 10.1016/j.egyr.2023.06.035
[13]	D. Thenmozhi, M. E. Rao, R. Punithavalli, P. D. Selvi, Analysis on mathematical model of convection system of micropolar fluid as darcy forchheimer flow undergoes heterogeneous and homogeneous chemical reaction, Forces Mech., 12 (2023), 100214. https://doi.org/10.1016/j.finmec.2023.100214 doi: 10.1016/j.finmec.2023.100214
[14]	S. Gholampour, H. Balasundaram, P. Thiyagarajan, J. Droessler, A mathematical framework for the dynamic interaction of pulsatile blood, brain, and cerebrospinal fluid, Comput. Methods Prog. Bio., 231 (2023), 107209. https://doi.org/10.1016/j.cmpb.2022.107209 doi: 10.1016/j.cmpb.2022.107209
[15]	Q. Raza, X. Wang, M. Z. A. Qureshi, S. M. Eldin, A. M. A. Allah, B. Ali, et al., Mathematical modeling of nanolayer on biological fluids flow through porous surfaces in the presence of CNT, Case Stud. Therm. Eng., 45 (2023), 102958. https://doi.org/10.1016/j.csite.2023.102958 doi: 10.1016/j.csite.2023.102958
[16]	N. S. Akbar, S. Akhtar, D. L. C. Ching, M. Farooq, I. Khan, Non-Newtonian fluid model analysis due to metachronal waves of cilia: A physiological mathematical model, Partial Differ. Equ. Appl. Math., 12 (2024), 101022. https://doi.org/10.1016/j.padiff.2024.101022 doi: 10.1016/j.padiff.2024.101022
[17]	E. S. Baranovskii, E. Y. Prosviryakov, S. V. Ershkov, Mathematical analysis of steady non-isothermal flows of a micropolar fluid, Nonlinear Anal. Real World Appl., 84 (2025), 104294. https://doi.org/10.1016/j.nonrwa.2024.104294 doi: 10.1016/j.nonrwa.2024.104294
[18]	P. Deepalakshmi, A. Darvesh, H. AL Garalleh, M. Sánchez-Chero, G. Shankar, E. P. Siva, Integrate mathematical modeling for heat dynamics in two-phase casson fluid flow through renal tubes with variable wall properties, Ain Shams Eng. J., 16 (2025), 103183. https://doi.org/10.1016/j.asej.2024.103183 doi: 10.1016/j.asej.2024.103183
[19]	P. Sanil, R. Choudhari, M. Gudekote, H. Vaidya, K. V. Prasad, Impact of chemical reactions and convective conditions on Peristaltic mechanism of Ree-Eyring Fluid in a porous medium–A mathematical model, Int. J. Thermofluid, 26 (2025), 101086. https://doi.org/10.1016/j.ijft.2025.101086 doi: 10.1016/j.ijft.2025.101086
[20]	A. K. Hilo, A. R. A. Talib, A. A. Iborra, M. T. H. Sultan, M. F. A. Hamid, Effect of corrugated wall combined with backward-facing step channel on fluid flow and heat transfer, Energy, 190 (2020), 116294. https://doi.org/10.1016/j.energy.2019.116294 doi: 10.1016/j.energy.2019.116294
[21]	K. Ur Rehman, N. Maan, E. M. Sherif, H. Junaedi, Y. P. Lv, On magnetized Newtonian liquid suspension in single backward facing-step (SBFS) with centrally translated obstructions, J. Mol. Liq., 337 (2021), 116265. https://doi.org/10.1016/j.molliq.2021.116265 doi: 10.1016/j.molliq.2021.116265
[22]	A. R. A. Talib, A. K. Hilo, Fluid flow and heat transfer over corrugated backward facing step channel, Case Stud. Thermal Eng., 24 (2021), 100862. https://doi.org/10.1016/j.csite.2021.100862 doi: 10.1016/j.csite.2021.100862
[23]	J. Lee, V. T. Nguyen, Effects of various backward-facing steps on flow characteristics and sediment behaviors, Phys. Fluids, 35 (2023), 065124. https://doi.org/10.1063/5.0152852 doi: 10.1063/5.0152852
[24]	M. Alhasan, H. Hamzah, A. Koprulu, B. Sahin, Couette-Poiseuille flow over a backward-facing step: Investigating hydrothermal performance and irreversibility analysis, Case Stud. Thermal Eng., 53 (2024), 103954. https://doi.org/10.1016/j.csite.2023.103954 doi: 10.1016/j.csite.2023.103954
[25]	Z. Asghar, U. Khalid, M. Nazeer, H. S. Rasheed, A. Kausar, Computational study of flow and heat transfer analysis of Ellis fluid model in complicated divergent channel, Modern Phys. Lett. B, 38 (2023), 2450119. https://doi.org/10.1142/S0217984924501197 doi: 10.1142/S0217984924501197
[26]	M. A. Javed, Z. Asghar, H. M. Atif, M. Nisar, A computational study of the calendering processes using Oldroyd 8-constant fluid with slip effects, Polym. Polym. Compos., 31 (2023). https://doi.org/10.1177/09673911231202888
[27]	M. Hussain, M. Imran, H. Waqas, T. Muhammad, S. M. Eldin, An efficient heat transfer analysis of MHD flow of hybrid nanofluid between two vertically rotating plates using Keller box scheme, Case Stud. Thermal Eng., 49 (2023), 103231. https://doi.org/10.1016/j.csite.2023.103231 doi: 10.1016/j.csite.2023.103231
[28]	C. Sowmiya, B. R. Kumar, MHD Maxwell nanofluid flow over a stretching cylinder in porous media with microorganisms and activation energy, J. Magn. Magn. Mater., 582 (2023), 171032. https://doi.org/10.1016/j.jmmm.2023.171032 doi: 10.1016/j.jmmm.2023.171032
[29]	T. Salahuddin, G. Fatima, M. Awais, M. Khan, B. Al Awan, Adaptation of nanofluids with magnetohydrodynamic williamson fluid to enhance the thermal and solutal flow analysis with viscous dissipation: A numerical study, Results Eng., 21 (2024), 101798. https://doi.org/10.1016/j.rineng.2024.101798 doi: 10.1016/j.rineng.2024.101798
[30]	A. Jafari, M. Vahab, P. Broumand, N. Khalili, An eXtended finite element method implementation in COMSOL Multiphysics: Thermo-hydro-mechanical modeling of fluid flow in discontinuous porous media, Comput. Geotech., 159 (2023), 105458. https://doi.org/10.1016/j.compgeo.2023.105458 doi: 10.1016/j.compgeo.2023.105458
[31]	Y. S. Mohamed, O. Hozien, M. M. Sorour, W. M. El-Maghlany, Heat transfer simulation of nanofluids heat transfer in a helical coil under isothermal boundary conditions using COMSOL Multiphysics, Int. J. Thermal Sci., 192 (2023), 108396. https://doi.org/10.1016/j.ijthermalsci.2023.108396 doi: 10.1016/j.ijthermalsci.2023.108396
[32]	M. Schafer, S. Turek, F. Durst, E. Krause, R. Rannacher, Benchmark computations of laminar flow around a cylinder, In: Flow simulation with high-performance computers II, 48 (1996) 547–566. https://doi.org/10.1007/978-3-322-89849-4_39
[33]	M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686–707. https://doi.org/10.1016/j.jcp.2018.10.045 doi: 10.1016/j.jcp.2018.10.045
[34]	J. Yu, L. Lu, X. Meng, G. Em Karniadakis, Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems, Comput. Methods Appl. Mech. Eng., 393 (2022), 114823. https://doi.org/10.1016/j.cma.2022.114823 doi: 10.1016/j.cma.2022.114823
[35]	W. Wu, S. Duan, Y. Sun, Y. Yu, D. Liu, D. Peng, Deep fuzzy physics-informed neural networks for forward and inverse PDE problems, Neural Netw., 181 (2025), 106750. https://doi.org/10.1016/j.neunet.2024.106750 doi: 10.1016/j.neunet.2024.106750
[36]	M. Adamu, A. B. Çolak, Y. E. Ibrahim, S. I. Haruna, M. F. Hamza Prediction of mechanical properties of rubberized concrete incorporating fly ash and nano silica by artificial neural network technique, Axioms, 12 (2023), 81. https://doi.org/10.3390/axioms12010081 doi: 10.3390/axioms12010081
[37]	M. Alhazmi, Z. Shah, M. A. Z. Raja, N. A. Albasheir, M. Jawaid, A multi-layer neural network-based evaluation of MHD radiative heat transfer in Eyring–Powell fluid model, Heliyon, 11 (2025), e41800. https://doi.org/10.1016/j.heliyon.2025.e41800 doi: 10.1016/j.heliyon.2025.e41800
[38]	A. Rauf, M. Omar, T. Mushtaq, S. Aslam, S. A. Shehzad, M. K. Siddiq, Application of artificial neural network in the numerical analysis of Reiner–Rivlin fluid flow with Newtonian heating, Multiscale Multidiscip. Model. Exp. Des., 8 (2025), 130. https://doi.org/10.1007/s41939-024-00719-6 doi: 10.1007/s41939-024-00719-6
[39]	H. Qureshi, U. Khaliq, Z. Shah, H. Abutuqayqah, M. Waqas, S. Saleem, et al., Artificial intelligence-based analysis employing Levenberg Marquardt neural networks to study chemically reactive thermally radiative tangent hyperbolic nanofluid flow considering Darcy-Forchheimer theory, J. Radiation Res. Appl. Sci., 18 (2025), 101253. https://doi.org/10.1016/j.jrras.2024.101253 doi: 10.1016/j.jrras.2024.101253
[40]	P. Kumar, F. Almeida, A. R. Ajaykumar, Q. Al-Mdallal, Artificial neural network model using Levenberg Marquardt algorithm to analyse transient flow and thermal characteristics of micropolar nanofluid in a microchannel, Partial Differ. Equ. Appl. Math., 13 (2025), 101061. https://doi.org/10.1016/j.padiff.2024.101061 doi: 10.1016/j.padiff.2024.101061

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)