Analysis of magnetohydrodynamic ternary hybrid nanofluid flow in variable Darcy–Forchheimer porous medium with thermal and velocity slip effects using local non-similarity approach and machine learning

This study investigates the unsteady flow and heat transfer of a ternary hybrid nanofluid (zinc oxide, silicon dioxide, and cobalt ferrite in water or ethylene glycol) over a stretching surface. Embedded in a variable porous medium, the system is analyzed under a transverse magnetic field, velocity and thermal slip, viscous dissipation, Joule heating, and temperature-dependent thermal conductivity. Motivated by advanced thermal-management needs, this work captures spatially heterogeneous porous structures by allowing wall-normal variations in permeability and porosity. Consistent with the unsteady-stretching framework, transient scalings are assigned to the magnetic field strength, reference permeability, and the Forchheimer inertial-drag coefficient; the latter is uniquely introduced so inertial resistance shares a common temporal behavior with other momentum contributions. The governing equations are transformed using a third-level local non-similarity formulation to preserve critical streamwise dependencies. Solved numerically via MATLAB’s bvp4c routine, the scheme yields validation errors below 0.04%. Concurrently, Bayesian-regularized deep neural networks trained on 8,000 solutions are deployed as rapid surrogates for skin-friction and local heat-transfer rates, achieving a test-set correlation of R = 1.0000 with generalization confirmed by minute point-wise errors. Quantitative analysis reveals that variable permeability reduces skin-friction magnitude by approximately 31%, whereas variable porosity increases it by 45%. Increasing the magnetic parameter raises skin friction by 32% but reduces heat transfer by 17%. Notably, thermal slip dominates thermal performance, reducing the local heat-transfer rate by over 80%. Finally, despite similar velocity profiles, the higher Prandtl number of ethylene glycol (approximately 204, versus 6.2 for water) produces substantially higher heat-transfer rates. Ultimately, this validated numerical and deep-learning framework enables highly efficient predictions of heat-transfer characteristics in magnetically controlled systems.