Abstract
This paper investigates the magnetohydrodynamic (MHD) flow along with heat and mass transfer behavior of Casson-type hybrid nanofluid through aluminum oxide (\({\varvec{A}}{{\varvec{l}}}_{2}{{\varvec{O}}}_{3}\)) as well as titanium dioxide (\({\varvec{T}}{\varvec{i}}{{\varvec{O}}}_{2}\)) nanoparticles dispersed throughout engine oil, a thermally stable functioning fluid that is used in utmost industrialized thermal exchanger applications. The advanced model incorporates the collective influences of heat radiation, Darcy–Forchheimer permeable media resistance, magnetic field-heating, and activation energy under nonlinear chemical reaction circumstances. Employing similarity transformations, the intricate governing equations are streamlined to ordinary differential equations (ODEs) that are numerically solved by means of the Bvp4c solver. The numerical solutions attained via the Bvp4c algorithm are employed for training a Morlet Wavelet Neural Network with Particle Swarm Optimization as well as Neural Network Algorithm (MWNN–PSO–NNA), through improving prediction robustness as well as generality behavior. The results show that strengthening the magnetic field leads to a decrease in the velocity distribution whereas thermal radiation growth in temperature, via variations falling within the range of 15–25% across the flow domain. Raising activation energy nearly 30% is observed to regulate species concentration as well as promote a more controlled thermal response inside the porous structure. In comparison with the base fluid and single-nanoparticle suspensions, the hybrid nanofluid exhibits superior thermal performance. Moreover, the MWNN–PSO–NNA outcomes remain in close agreement with the numerical solutions, yielding error levels of the order 10⁻5–10⁻⁶, which confirms the reliability of the proposed framework for complex non-Newtonian hybrid nanofluid systems relevant to industrial thermal applications. The proposed neural network model demonstrates strong predictive capability, achieving an accuracy greater than 99% while reducing computational time by approximately 45% when compared with traditional numerical methods. An ANN is developed to rapidly predict flow, heat, and mass transfer. Trained on bvp4c data, it achieves comparable accuracy while reducing computational time by 45% compared to repeated numerical simulations. Additionally, the hybrid nanofluid formulation displays strong potential for industrial lubrication applications, thermal control of mechanical components, and energy-based cooling systems, where improved heat transfer productivity is a main performance requirement. The main motivation of this study is to address the growing demand for efficient thermal management in industrial lubrication systems involving non-Newtonian fluids. The goal is to apply a robust MWNN–PSO–NNA framework to accurately predict the flow, heat, and mass transfer characteristics of Casson hybrid nanofluid over a radially stretching surface under combined physical effects. This study is the first to integrate engine-oil-based Casson hybrid nanofluid modeling with Darcy–Forchheimer porous effects and an optimized MWNN–PSO–NNA framework, providing highly accurate thermal–fluid predictions relevant to advanced industrial heat transfer systems.
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Abbreviations
- \(u, v\) :
-
Velocity components along x- and y-directions (m/s)
- \(T\) :
-
Fluid temperature (K)
- \({T}_{w}\) :
-
Wall temperature (K)
- \({T}_{\infty }\) :
-
Ambient temperature (K)
- \(C\) :
-
Nanoparticle concentration (kg/m3)
- \({C}_{w}\) :
-
Wall concentration (kg/m3)
- \({C}_{\infty }\) :
-
Ambient concentration (kg/m3)
- \(\rho\) :
-
Density of fluid or nanofluid (kg/m3)
- \(\mu\) :
-
Dynamic viscosity (Pa·s)
- \(\nu\) :
-
Kinematic viscosity (m2/s)
- \(k\) :
-
Thermal conductivity (W/m·K)
- \({C}_{p}\) :
-
Specific heat capacity at constant pressure (J/kg·K)
- \(\alpha\) :
-
Thermal diffusivity (m2/s)
- \({D}_{B}\) :
-
Brownian diffusion coefficient (m2/s)
- \({D}_{T}\) :
-
Thermophoresis diffusion coefficient (m2/s)
- \({B}_{0}\) :
-
Magnetic Field Strength T(Tesla)
- \(R\) :
-
Universal gas constant (J/mol. K)
- \(\sigma\) :
-
Electrical conductivity (S/m.(\({\Omega }^{-1}\cdot {m}^{-1}\)))
- \(\phi\) :
-
Nano-particles volume fractions
- \({q}_{r}\) :
-
Radiative heat flux (W/m2)
- \({\sigma }^{*}\) :
-
Stefan–Boltzmann constant
- \({k}^{*}\) :
-
Mean absorption coefficient
- \({h}_{f}\) :
-
Convective heat transfer coefficient (W m-2 K−1)
- \(\Psi\) :
-
Dimensionless stream function
- \(\theta (\eta )\) :
-
Dimensionless temperature
- \(\phi (\eta )\) :
-
Dimensionless concentration
- \(\eta\) :
-
Similarity variable
- \(\beta\) :
-
Casson fluid parameter
- \(n\) :
-
Nonlinearity index (power-law index)
- \(M\) :
-
Magnetic field parameter
- \(K\) :
-
Permeability parameter
- \(Fr\) :
-
Forchheimer (inertial drag) number
- \(\lambda\) :
-
Velocity slip parameter
- \(Ec\) :
-
Eckert number (viscous dissipation parameter)
- \(Pr\) :
-
Prandtl number
- \({R}_{d}\) :
-
Thermal radiation parameter
- \({Q}^{*}\) :
-
Heat source
- \(Nb\) :
-
Heat generation due to nanoparticle concentration
- \({M}_{1}\) :
-
Magnetic heating parameter
- \(Sc\) :
-
Schmidt number
- \({K}_{r}\) :
-
Chemical reaction rate constant
- \({E}_{1}\) :
-
Activation energy parameter
- \(\gamma\) :
-
Temperature-dependent activation modifier
- \(Re\) :
-
Reynolds number
- \(\delta\) :
-
Heat source/sink parameter
- \({M}_{2}\) :
-
Magnetic energy coefficient
- \({k}_{hnf}\) :
-
Convective heat transfer coefficient (W/m2 K)
- \({\mu }_{hnf}\) :
-
Dynamic viscosity of Hybrid nanofluid (PaS)
- \({\sigma }_{hnf}\) :
-
Electrical conductivity of Hybrid nanofluid \(\Omega^{-1}{m}^{-1}\)
- \({\rho }_{hnf}\) :
-
Density of Hybrid nanofluid (Kg m-3)
- \({\psi }_{m}\) :
-
Morlet wavelet function
- \(W\) :
-
Neural network weight vector
- \(E(T)\) :
-
Fitness function
- \({P}_{LB},{P}_{GB}\) :
-
PSO personal and global best positions
- \(w\) :
-
Inertia weight (in PSO)
- \({c}_{1}{,c}_{2}\) :
-
Acceleration coefficients (PSO)
- \({u}_{1},{u}_{2}\) :
-
Random numbers uniformly distributed in [0, 1]
- \({T}_{MWNN}\) :
-
MWNN-predicted temperature
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Acknowledgements
The authors extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant No. R.G.P2/339/46.
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Mohammad Ayman-Mursaleen: Conceptualization, Investigation, Software Syed Tauseef Saeed: Conceptualization, Investigation, Software, Project Administration, Supervision, Writing – original draft. Saja Mohammad Almohammadi: Software, Validation, Writing – original draft. Khalid Arif: Investigation, Methodology, Software Muhammad Imran: Conceptualization, Investigation, Software, Project Administration.
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Ayman-Mursaleen, M., Saeed, S.T., Almohammadi, S.M. et al. A deep neural network model for heat transfer in darcy–forchheimer hybrid nanofluid flow with activation energy. Sci Rep (2026). https://doi.org/10.1038/s41598-026-39536-x
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DOI: https://doi.org/10.1038/s41598-026-39536-x
