Channels with partial porous inserts provide improved thermal performance while effectively limiting pressure drop compared to fully porous configurations. The present study investigates steady, fully developed thermo-fluid transport of a Williamson couple stress fluid in a vertical parallel-plate channel partially filled with a Darcy–Forchheimer porous medium. The flow domain is divided into clear-fluid and porous regions, driven by a uniform pressure gradient. The channel walls are maintained at different constant temperatures, and interfacial conditions enforcing continuity of velocity, shear stress, temperature, and heat flux are imposed to ensure proper coupling between both regions.

The governing nonlinear momentum and energy equations are non-dimensionalized and solved using the Differential Transform Method (DTM). The resulting semi-analytical solutions exhibit rapid convergence and are validated against numerical results. The obtained velocity and temperature fields are further used to evaluate entropy generation, skin friction, and Nusselt number, with detailed parametric effects reported through tables and plots.

The study provides a unified mathematical framework for non-Newtonian thermofluid transport in composite porous channels and highlights the efficiency of DTM for solving strongly coupled nonlinear boundary-value problems of engineering relevance. The study is motivated by emerging applications in microfluidic transport, bioengineering systems, porous thermal management devices, polymer processing, and enhanced filtration technologies, where the simultaneous influence of microstructural effects, fluid elasticity, and inertial porous resistance becomes significant.