Machine-learning-assisted analysis of MHD Williamson ternary-hybrid nanofluid over a permeable stretching disk under assisting/opposing flow

A steady, incompressible, axisymmetric mixed-convection model is developed for the stagnation-region flow and heat transfer of an electrically conducting Williamson Al₂O₃-Cu-TiO₂/sodium-alginate ternary-hybrid nanofluid over a vertical permeable radially stretching disk. The model assumes a single-phase effective medium in local thermal equilibrium, the Boussinesq approximation for buoyancy, no induced magnetic field, first-order velocity and thermal slip, Rosseland diffusion with linearized T⁴, and a small-Weissenberg Williamson approximation. The wall and exterior radial velocities are linear in radius, and the wall-to-ambient temperature difference changes as r 2 , which allows a similarity reduction to linked nonlinear ordinary differential equations. The equations are solved using MATLAB bvp4c and compared with published limiting instances. The generated profiles are then learned by distinct multi-output Levenberg-Marquardt backpropagation artificial neural networks (LMB-ANNs) for helping and opposing buoyancy. Quantitatively, the increase of velocity slip from A=0.1 to 1.0 decreases the reduced skin friction by approximately 59%, whereas the increase of suction from S=0.5 to 2.0 increases the reduced skin friction by around 61-63%. For the baseline thermal setting, the lowered Nusselt number for the ternary-hybrid suspension is roughly 48.8% higher than the single-nanofluid scenario. The ternary-hybrid decreased Nusselt number increases by approximately 60.4% when the radiation increases from Rd=0.25 to 1.0 and decreases by about 89% when the thermal slip increases from B=0.1 to 1.0. The best MSE values of the LMB-ANN models for velocity and temperature are 5.08×10-10-1.04×10-9 and 1.36×10-8-6.01×10-8, respectively. The findings reveal actual trade-offs between particle loading, suction, slide, and radiation. The ANN provides a quick interpolative surrogate within the measured parameter ranges.

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