Numerical and artificial neural network prediction of double-stratified magnetized nanofluid flow and heat transfer over a stretchable cylinder with Brownian motion and thermal slip.
7 affiliations
- 1Department of Mathematics and Statistics, The University of Faisalabad, University Town, Faisalabad, 38000, Pakistan.
- 2Center for Scientific Research and Entrepreneurship, Northern Border University, Arar, 73213, Saudi Arabia. [email protected].
- 3Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah, 51452, Saudi Arabia.
- 4Department of Computer Engineering, Biruni University, Istanbul, 34010, Turkey.
- 5Mathematics Research Center, Near East University, Nicosia, 99138, Cyprus.
- 6Department of Mathematics, University of Aden, Aden, Yemen. [email protected].
- 7Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, 71491, Saudi Arabia.
Abstract
The current study aims to investigate the effects of double stratification on magneto-hydrodynamic (MHD) nanofluid flow over a stretching cylinder in the presence of thermal radiation. The model incorporates Brownian motion, thermophoresis, and velocity, thermal, and concentration slip conditions to provide a realistic description of the transport phenomena. The problem is formulated in the form of PDEs are converted to ODEs by applying suitable similarity variables. The transformed system is solved numerically using bvp4c solver in MATLAB. A Levenberg–Marquardt Algorithm (LMA) based Artificial Neural Network (ANN) is developed to validate and improve the predictive capability of the numerical results. The ANN trained using the numerical dataset generated by bvp4c solution, with the dataset 70% for training, 15% for validation, and 15% for testing. The suggested hybrid numerical-ANN framework delivers an efficient and accurate predictive model that reveals excellent agreement with the numerical solutions. The ANN model demonstrates excellent predictive accuracy and computational efficiency, showing close agreement with the numerical results. The influences of various physical parameters on temperature, velocity, and concentration distributions are analyzed graphically. It is pragmatic that the velocity profile decreases with an increasing velocity slip parameter, whereas thermal stratification reduces the temperature field and thermophoresis enhances the concentration distribution. Numerical outcomes of friction factor, heat transfer rate, and Sherwood number are premeditated and discussed in detail. Comparison with previously published results in limiting cases endorsements the accuracy and validity of the current methodology. The findings highlight the effectiveness of the hybrid bvp4c–ANN framework for analyzing complex nanofluid transport problems arising in engineering and industrial applications.
Graphical abstract: This image presents four panels related to a computational model involving boundary layers and neural networks. The top-left panel illustrates three boundary layers—thermal, concentration, and momentum—developing along a cylindrical surface with a magnetic field applied in the positive radial direction. It includes equations defining temperature and concentration in terms of boundary layer variables.The top-right panel depicts a neural network architecture with three input parameters labeled M, R, and Le, feeding into a hidden layer with weights and biases, followed by an output layer producing three outputs labeled f prime of eta, theta of eta, and phi of eta.The bottom-left panel contains four scatter plots comparing predicted outputs to target values for training, validation, test, and all data sets. Each plot shows data points closely aligned with a fit line and a reference line where output equals target, with correlation coefficients equal to one.The bottom-right panel shows a flowchart outlining a six-step iterative process to solve an initial value problem. Steps include converting a boundary value problem, guessing initial conditions, solving using a specific numerical scheme, calculating residuals, checking if residuals are below tolerance, and updating the initial guess if needed, leading to a final solution.
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BibTeX
@article{majeed2026numerical,
title = {{Numerical and artificial neural network prediction of double-stratified magnetized nanofluid flow and heat transfer over a stretchable cylinder with Brownian motion and thermal slip.}},
author = {Aaqib Majeed and Muhammad Junaid Zia and Taoufik Saidani and Parvez Ali and Yakup Yildirim and Jihad Younis and Mounirah Areshi},
journal = {Discover nano},
year = {2026},
month = oct,
doi = {10.1186/s11671-026-05096-2},
url = {https://europepmc.org/article/MED/42961598},
note = {CC BY-NC-ND},
}
RIS
TY - JOUR
TI - Numerical and artificial neural network prediction of double-stratified magnetized nanofluid flow and heat transfer over a stretchable cylinder with Brownian motion and thermal slip.
AU - Aaqib Majeed
AU - Muhammad Junaid Zia
AU - Taoufik Saidani
AU - Parvez Ali
AU - Yakup Yildirim
AU - Jihad Younis
AU - Mounirah Areshi
PY - 2026
DA - 2026/10/30
JO - Discover nano
DO - 10.1186/s11671-026-05096-2
UR - https://europepmc.org/article/MED/42961598
AB - The current study aims to investigate the effects of double stratification on magneto-hydrodynamic (MHD) nanofluid flow over a stretching cylinder in the presence of thermal radiation. The model incorporates Brownian motion, thermophoresis, and velocity, thermal, and concentration slip conditions to provide a realistic description of the transport phenomena. The problem is formulated in the form of PDEs are converted to ODEs by applying suitable similarity variables. The transformed system is solved numerically using bvp4c solver in MATLAB. A Levenberg–Marquardt Algorithm (LMA) based Artificial Neural Network (ANN) is developed to validate and improve the predictive capability of the numerical results. The ANN trained using the numerical dataset generated by bvp4c solution, with the dataset 70% for training, 15% for validation, and 15% for testing. The suggested hybrid numerical-ANN framework delivers an efficient and accurate predictive model that reveals excellent agreement with the numerical solutions. The ANN model demonstrates excellent predictive accuracy and computational efficiency, showing close agreement with the numerical results. The influences of various physical parameters on temperature, velocity, and concentration distributions are analyzed graphically. It is pragmatic that the velocity profile decreases with an increasing velocity slip parameter, whereas thermal stratification reduces the temperature field and thermophoresis enhances the concentration distribution. Numerical outcomes of friction factor, heat transfer rate, and Sherwood number are premeditated and discussed in detail. Comparison with previously published results in limiting cases endorsements the accuracy and validity of the current methodology. The findings highlight the effectiveness of the hybrid bvp4c–ANN framework for analyzing complex nanofluid transport problems arising in engineering and industrial applications. Graphical abstract: This image presents four panels related to a computational model involving boundary layers and neural networks. The top-left panel illustrates three boundary layers—thermal, concentration, and momentum—developing along a cylindrical surface with a magnetic field applied in the positive radial direction. It includes equations defining temperature and concentration in terms of boundary layer variables.The top-right panel depicts a neural network architecture with three input parameters labeled M, R, and Le, feeding into a hidden layer with weights and biases, followed by an output layer producing three outputs labeled f prime of eta, theta of eta, and phi of eta.The bottom-left panel contains four scatter plots comparing predicted outputs to target values for training, validation, test, and all data sets. Each plot shows data points closely aligned with a fit line and a reference line where output equals target, with correlation coefficients equal to one.The bottom-right panel shows a flowchart outlining a six-step iterative process to solve an initial value problem. Steps include converting a boundary value problem, guessing initial conditions, solving using a specific numerical scheme, calculating residuals, checking if residuals are below tolerance, and updating the initial guess if needed, leading to a final solution.
SN - 2731-9229
ER -