Thermal irreversibility analysis of radiative hybrid nanofluid flow in porous media using Darcy–Forchheimer model

Thermodynamic irreversibility is analyzed through a partial differential equation (PDE) framework describing Darcy–Forchheimer thermal transport of a water-based hybrid nanofluid containing copper and silver nanosolid suspensions over a convectively heated, porous, and bidirectional slippery surface. The governing PDE system incorporates nonlinear radiation, magnetic field effects, and internal heat generation/absorption. By employing similarity transformations, the PDEs are reduced to a coupled nonlinear ordinary differential equation system, which is subsequently solved numerically using the finite difference–based Keller-box scheme. Numerical simulations yield detailed predictions of velocity profiles, temperature fields, Nusselt number, drag force, irreversibility distributions, and Bejan number. Validation against benchmark results from the literature confirms the accuracy of the present formulation. The analysis reveals that thermodynamic irreversibility and the Bejan number increase with higher concentrations of copper and silver nanoparticles, while temperature is enhanced with larger Biot and Eckert numbers. Furthermore, the hybrid nanofluid demonstrates superior thermal conductivity and density compared to its base nanofluid and pure water counterparts, highlighting the role of PDE-based modeling in quantifying complex transport mechanisms.