<p>The foremost objective of the recent analysis is to scrutinize the non-swirling flow of magnetized viscous fluid amid two narrowly flat permeable disks employed with the influence of thermal dependent viscosity, Hall effect, and Darcy–Forchheimer flow accompanied by the exploration of temperature and mass transference. Due to the practical significance of disk flow in engineering and technological applications, the exact solution for the temperature, concentration, and velocity is derived by employing the Jacobi elliptic function of the first and second kinds. The graphical analysis for the examination of the impression of key controlling parameters on velocity, temperature, and concentration profiles is conducted. The asymmetric parabolic velocity profiles for both accelerating and decelerating flow are examined due to the impact of temperature-dependent viscosity with maximal velocity at the central region and least at the surface of disks. The fluid temperature escalated for the variation in the heat source/sink parameter, and Sorret–Dufour parameter, and dwindled for the increment in radiation parameter. The exact opposite behavior for the concentration profiles is observed. Streamlines are also demarcated to exhibit the flow behavior. The variations in skin friction, Nusselt number, and flow rate are discussed numerically in tabular form. The results offer insights relevant to practical applications such as thermal regulation in rotating machinery, biomedical devices like dialysis systems, microfluidic cooling systems, and porous media reactors. This work contributes to the advancement of exact analytical modeling in complex fluid flow systems relevant to both industrial and scientific domains.</p>