Intelligent computing and sensitivity analysis of unsteady darcy-forchheimer nanofluid flow over an inclined permeable disk subject to thompson-troian slip conditions

The present study numerically investigates the unsteady three-dimensional electromagnetohydrodynamic (EMHD) flow of a dual-fractional nanofluid (DFNF) comprising cobalt ferrite (CoFe 2 O 4 ) and water flowing over an inclined porous rotating disk (PRD) using a combination of numerical simulation and soft computing techniques. Both the velocity slip and the thermal convective boundary conditions are complex non-linear. Time-dependent similarity transformations are applied to the governing partial differential equations to convert them to a system of coupled ordinary differential equations, and then they are solved by the Parametric Continuation Method (PCM). The grid independence test demonstrates that full numerical stability and asymptotic convergence are achieved at an optimal grid mesh of N = 80 steps, with an extremely high-precision maximum residual error floor of 1.42 × 10 − 7 . A localized finite-difference sensitivity analysis is formulated to assess the mathematical robustness of the model by plotting the engineering output responses as a function of a parameter variation of ± 20%. The sensitivity matrices show that the thermal radiation parameter has the highest positive sensitivity index (+ 0.10315) with regard to the heat transfer term, and the electric field factor has the highest negative sensitivity index (-0.05405) with regard to skin friction, indicating that the electric field factor can reduce the boundary layer skin friction. In order to complement the deterministic results, an artificial neural network (ANN), optimized by the Levenberg-Marquardt back-propagation algorithm (LMBPA), is introduced over a generated dataset of 1001 points throughout the extended domain (0 ≤ η ≤ 10). The soft computing validation results in an outstanding predictive performance, with a mean squared error (MSE) value of 1.883 × 10 − 9 and an absolute linear regression metric ( R = 1) value, demonstrating that the network reproduces the complex boundary layer mechanics, velocity crossover zones, and multi-physical coupling profiles without systematic drift.