Journal of Computational Applied Mechanics

Magnetic Hydrodynamic Flow and Heat Transfer of Williamson Nanofluids in a Porous Medium Impact of Chemical Reactions and Melting Effects

Document Type : Research Paper

Authors

1 Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Mardan, Pakistan

2 School of Information Technology and Systems, University of Canberra, Canberra ACT 2617, Australia

3 College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait

4 Scuola Internazionale Superiore di Studi Avanzati Via Bonomea 265, 34136 Trieste, Italy

5 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Abstract

Radiation and chemical reaction effects on the steady magnetohydrodynamic (MHD) boundary layer flow of Williamson nanofluid through a porous medium over a horizontally linearly stretching sheet are numerically investigated, incorporating coupled influences of melting heat transfer and nanoparticle dispersion. The governing partial differential equations are reduced to a system of nonlinear ordinary differential equations using similarity transformations and solved via the fourth-order Runge-Kutta (RK-4) method to generate reference datasets. A novel supervised machine learning framework, Feed-Forward Neural Network optimized with the Backpropagated Levenberg-Marquardt Algorithm (FFNN-BLMA), is proposed, trained on 1001 data points with 70% training, 15% validation, and 15% testing splits. The FFNN-BLMA yields exceptional predictive accuracy with absolute errors ranging from 10⁻⁸ to 10⁻¹⁰ across velocity temperature  and concentration  profiles, validated through 10-fold cross-validation, error histograms, regression analysis, and curve superposition. Parametric studies reveal that increasing the melting parameter  enhances velocity and reduces temperature, while the chemical reaction parameter  diminishes concentration trends consistent with prior literature. Skin friction, Nusselt, and Sherwood numbers are computed to quantify engineering performance. The FFNN-BLMA outperforms traditional RK-4 and analytical methods in accuracy, convergence, and computational efficiency, establishing a robust, discretization-free paradigm for solving complex non-Newtonian multi-physics flow problems with potential extension to fractional-order systems.

Keywords

Main Subjects

[1]          J. Zeng, Y. Xuan, Analysis on interaction between solar light and suspended nanoparticles in nanofluids, Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 269, pp. 107692, 2021.
[2]          A. M. Alqahtani, M. Bilal, M. Usman, T. R. Alsenani, A. Ali, S. R. Mahmuod, Heat and mass transfer through MHD Darcy Forchheimer Casson hybrid nanofluid flow across an exponential stretching sheet, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 103, No. 6, pp. e202200213, 2023.
[3]          M. F. Endalew, S. Sarkar, Numerical exploration of forced convection hydromagnetic hyperbolic tangent nanofluid flow over a permeable wedge with melting heat transfer, Scientific Reports, Vol. 13, No. 1, pp. 3515, 2023.
[4]          F. Mabood, G. Bognár, A. Shafiq, Impact of heat generation/absorption of magnetohydrodynamics Oldroyd-B fluid impinging on an inclined stretching sheet with radiation, Scientific Reports, Vol. 10, No. 1, pp. 17688, 2020.
[5]          A. M Megahed, Williamson fluid flow due to a nonlinearly stretching sheet with viscous dissipation and thermal radiation, Journal of the Egyptian Mathematical Society, Vol. 27, No. 1, pp. 1-10, 2019.
[6]          S. Parvin, S. S. P. Mohamed Isa, N. M. Arifin, F. Md Ali, The inclined factors of magnetic field and shrinking sheet in Casson fluid flow, heat and mass transfer, Symmetry, Vol. 13, No. 3, pp. 373, 2021.
[7]          H. Maaitah, A. N. Olimat, O. Quran, H. M. Duwairi, Viscous dissipation analysis of Williamson fluid over a horizontal saturated porous plate at constant wall temperature, International Journal of Thermofluids, Vol. 19, pp. 100361, 2023.
[8]          Z. Khan, A. Ali, S. Aldahmani, H. Nordmark, B. Lausen, Customer lifetime value modelling via two stage selected trees ensembles, IEEE Access, 2025.
[9]          S. Gul, D. Muhammad Khan, S. Aldahmani, Z. Khan, Margin weighted robust discriminant score for feature selection in imbalanced gene expression classification, PLoS One, Vol. 20, No. 6, pp. e0325147, 2025.
[10]        M. R. Khan, M. A. Elkotb, R. Matoog, N. A. Alshehri, M. A. Abdelmohimen, Thermal features and heat transfer enhancement of a Casson fluid across a porous stretching/shrinking sheet: Analysis of dual solutions, Case Studies in Thermal Engineering, Vol. 28, pp. 101594, 2021.
[11]        A. Ali, Z. Khan, H. Du, S. Aldahmani, Double weighted k nearest neighbours for binary classification of high dimensional genomic data, Scientific reports, Vol. 15, No. 1, pp. 12681, 2025.
[12]        A. Ali, Z. Khan, S. Aldahmani, Optimized feature selection in high-dimensional gene expression data using weighted differential gene expression analysis, Applied Soft Computing, pp. 113329, 2025.
[13]        K. Prasad, H. Vaidya, K. Vajravelu, G. Manjunatha, M. Rahimi-Gorji, H. Basha, Heat transfer analysis of three-dimensional mixed convective flow of an oldroyd-B nanoliquid over a slippery stretching surface, in Proceeding of, Trans Tech Publ, pp. 164-182.
[14]        Z. Khan, A. Ali, D. M. Khan, S. Aldahmani, Regularized ensemble learning for prediction and risk factors assessment of students at risk in the post-COVID era, Scientific reports, Vol. 14, No. 1, pp. 16200, 2024.
[15]        Z. Khan, A. Ali, S. Aldahmani, Feature selection via robust weighted score for high dimensional binary class-imbalanced gene expression data, Heliyon, Vol. 10, No. 19, 2024.
[16]        E. O. Titiloye, J. A. Gbadeyan, A. T. Adeosun, Heat and mass transfer of MHD dissipative Casson nanofluid flow over a stretching or shrinking sheet with multiple slip boundary conditions, in Proceeding of, Trans Tech Publ, pp. 103-120.
[17]        A. Ali, Z. Khan, D. M. Khan, S. Aldahmani, An optimal random projection k nearest neighbors ensemble via extended neighborhood rule for binary classification, IEEE Access, Vol. 12, pp. 61401-61409, 2024.
[18]        S. Aldahmani, O. Kortbi, M. Mesfioui, Copula-Based Regression with Mixed Covariates, Mathematics, Vol. 12, No. 22, pp. 3525, 2024.
[19]        B. Gireesha, M. Umeshaiah, B. Prasannakumara, N. Shashikumar, M. Archana, Impact of nonlinear thermal radiation on magnetohydrodynamic three dimensional boundary layer flow of Jeffrey nanofluid over a nonlinearly permeable stretching sheet, Physica A: Statistical Mechanics and its Applications, Vol. 549, pp. 124051, 2020.
[20]        M. Hamraz, A. Ali, W. K. Mashwani, S. Aldahmani, Z. Khan, Feature selection for high dimensional microarray gene expression data via weighted signal to noise ratio, PloS one, Vol. 18, No. 4, pp. e0284619, 2023.
[21]        N. Gul, W. K. Mashwani, M. Aamir, S. Aldahmani, Z. Khan, Optimal model selection for k-nearest neighbours ensemble via sub-bagging and sub-sampling with feature weighting, Alexandria Engineering Journal, Vol. 72, pp. 157-168, 2023.
[22]        S. Shehzad, T. Hayat, A. Alsaedi, MHD flow of Jeffrey nanofluid with convective boundary conditions, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 37, No. 3, pp. 873-883, 2015.
[23]        A. Ali, M. Hamraz, N. Gul, D. M. Khan, S. Aldahmani, Z. Khan, A k nearest neighbour ensemble via extended neighbourhood rule and feature subsets, Pattern Recognition, Vol. 142, pp. 109641, 2023.
[24]        R. Abdullah Mohamed, A. Mahmoud Aly, S. Elsayed Ahmed, M. Sayed Soliman, MHD Jeffrey nanofluids flow over a stretching sheet through a porous medium in presence of nonlinear thermal radiation and heat generation/absorption, Challenges in Nano and Micro Scale Science and Technology, Vol. 8, No. 1, pp. 9-22, 2020.
[25]        N. Younas, A. Ali, H. Hina, M. Hamraz, Z. Khan, S. Aldahmani, Optimal causal decision trees ensemble for improved prediction and causal inference, IEEE Access, Vol. 10, pp. 13000-13011, 2022.
[26]        M. Hamraz, Z. Khan, D. M. Khan, N. Gul, A. Ali, S. Aldahmani, Gene selection in binary classification problems within functional genomics experiments via robust Fisher score, IEEE Access, Vol. 10, pp. 51682-51692, 2022.
[27]        D.-C. Lu, M. Ramzan, M. Bilal, J. D. Chung, U. Farooq, Upshot of chemical species and nonlinear thermal radiation on Oldroyd-B nanofluid flow past a bi-directional stretched surface with heat generation/absorption in a porous media, Communications in Theoretical Physics, Vol. 70, No. 1, pp. 071, 2018.
[28]        N. Gul, M. Aamir, S. Aldahmani, Z. Khan, A weighted k-Nearest neighbours ensemble with added accuracy and diversity, IEEE Access, Vol. 10, pp. 125920-125929, 2022.
[29]        T. Hayat, M. B. Ashraf, S. Shehzad, N. N. Bayomi, Mixed convection flow of viscoelastic nanofluid over a stretching cylinder, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 37, No. 3, pp. 849-859, 2015.
[30]        Z. Khan, M. Naeem, U. Khalil, D. M. Khan, S. Aldahmani, M. Hamraz, Feature selection for binary classification within functional genomics experiments via interquartile range and clustering, IEEE Access, Vol. 7, pp. 78159-78169, 2019.
[31]        Y. B. Kho, A. Hussanan, M. K. A. Mohamed, M. Z. Salleh, Heat and mass transfer analysis on flow of Williamson nanofluid with thermal and velocity slips: Buongiorno model, Propulsion and Power Research, Vol. 8, No. 3, pp. 243-252, 2019.
[32]        S. Aldahmani, H. Dai, Q.-Z. Zhang, Hybrid Graphical Least Square estimation and its application in portfolio selection, Statistics and Its Interface, Vol. 12, No. 4, pp. 631-645, 2019.
[33]        K. Swain, S. M. Ibrahim, G. Dharmaiah, S. Noeiaghdam, Numerical study of nanoparticles aggregation on radiative 3D flow of maxwell fluid over a permeable stretching surface with thermal radiation and heat source/sink, Results in Engineering, Vol. 19, pp. 101208, 2023.
[34]        K. Ren, G. Wu, G. Thomas, Wave excited motion of a body floating on water confined between two semi-infinite ice sheets, Physics of Fluids, Vol. 28, No. 12, 2016.
[35]        J. H. Lienhard, Heat transfer in flat-plate boundary layers: A correlation for laminar, transitional, and turbulent flow, Journal of Heat Transfer, Vol. 142, No. 6, pp. 061805, 2020.
[36]        R. Sharma, N. Acharya, K. Das, On the impact of variable thickness and melting transfer of heat on magnetohydrodynamics nanofluid flow past a slendering stretching sheet, Indian J. Geo-Mar. Sci, Vol. 49, pp. 641-648, 2020.
[37]        B. Prasannakumara, B. Gireesha, P. Manjunatha, Melting phenomenon in MHD stagnation point flow of dusty fluid over a stretching sheet in the presence of thermal radiation and non-uniform heat source/sink, International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 16, No. 5, pp. 265-274, 2015.
[38]        S. Hasan, Numerical Investigation of Convective Heat and Mass Transfer Flow along a Vertical Porous Plate with Soret and Dufour Effects in the Presence of Induced Magnetic Field,  Thesis, Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh., 2017.
[39]        A. K. Dash, S. Mishra, MHD stagnation point flow of micropolar fluid past on a vertical plate in the presence of porous medium, European Journal of Electrical Engineering, Vol. 19, No. 1-2, pp. 43, 2017.
[40]        D. Sahu, R. K. Deka, Influences of thermal stratification and chemical reaction on MHD free convective flow along an accelerated vertical plate with variable temperature and exponential mass diffusion in a porous medium, Heat Transfer, Vol. 53, No. 7, pp. 3643-3666, 2024.
[41]        Z. Lv, L. Qiao, J. Li, H. Song, Deep-learning-enabled security issues in the internet of things, IEEE Internet of Things Journal, Vol. 8, No. 12, pp. 9531-9538, 2020.
[42]        D. Pal, G. Mandal, Mixed convection–radiation on stagnation-point flow of nanofluids over a stretching/shrinking sheet in a porous medium with heat generation and viscous dissipation, Journal of Petroleum Science and Engineering, Vol. 126, pp. 16-25, 2015.
[43]        G. S. Seth, R. Tripathi, M. Rashidi, Hydromagnetic natural convection flow in a non-Darcy medium with Soret and Dufour effects past an inclined stretching sheet, Journal of Porous Media, Vol. 20, No. 10, 2017.
[44]        N. A. Khan, M. Sulaiman, F. S. Alshammari, Heat transfer analysis of an inclined longitudinal porous fin of trapezoidal, rectangular and dovetail profiles using cascade neural networks, Structural and Multidisciplinary Optimization, Vol. 65, No. 9, pp. 251, 2022.
[45]        M. Sulaiman, Z. Hussain, F. S. Alshammari, G. Laouin, Computational Analysis of Natural Convection Heat Transfer in Nanofluids Under a Uniform Magnetic Field Using Levenberg–Marquardt Backpropagation Neural Networks, Journal of Computational Applied Mechanics, Vol. 57, No. 1, pp. 41-62, 2026.
[46]        N. Sharif, M. Alam, H. U. Molla, Dynamics of nonlinear pendulum equations: Modified homotopy perturbation method, Journal of Low Frequency Noise, Vibration and Active Control, Vol. 44, No. 3, pp. 1460-1473, 2025.
[47]        H. Y. Lam, G. Ceruti, D. Kressner, Randomize Low-Rank Runge–Kutta Methods, SIAM Journal on Matrix Analysis and Applications, Vol. 46, No. 2, pp. 1587-1615, 2025.
[48]        A. Khan, M. Sulaiman, H. Alhakami, A. Alhindi, Analysis of oscillatory behavior of heart by using a novel neuroevolutionary approach, IEEE Access, Vol. 8, pp. 86674-86695, 2020.
[49]        T. Demir, Unified Theory and Intelligent Simulation Platform for Coupled Multi-Physics, Multi-Scale, and Fractional-Order Systems: Foundations, Algorithms, and Digital Twin Applications.
[50]        M. Sulaiman, M. Masihullah, Z. Hussain, S. Ahmad, W. K. Mashwani, M. A. Jan, R. A. Khanum, Implementation of improved grasshopper optimization algorithm to solve economic load dispatch problems, Hacettepe Journal of Mathematics and Statistics, Vol. 48, No. 5, pp. 1570-1589, 2019.
[51]        R. Kumar, R. Mehta, A. Bhatnagar, I. Ozsahin, B. Uzun, B. Almohsen, H. Ahmad, Numerical study of Williamson fluid flow over a stretching sheet with Newtonian heating embedded in a porous medium in presence of thermal radiation and heat source/sink, Journal of Nonlinear Mathematical Physics, Vol. 32, No. 1, pp. 4, 2025.
[52]        M. Krishnamurthy, B. Prasannakumara, B. Gireesha, R. S. R. Gorla, Effect of chemical reaction on MHD boundary layer flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, an International Journal, Vol. 19, No. 1, pp. 53-61, 2016.
Journal of Computational Applied Mechanics
Volume 57, Issue 2
April 2026
Pages 173-194
  • Receive Date: 29 October 2025
  • Revise Date: 17 November 2025
  • Accept Date: 29 November 2025