This study investigates anomalous heat transfer in a radiative mixed convection flow of water-based drilling nanofluid (NF) immersed in a Darcy–Brinkman permeable medium. Since the underground fluids are exposed to anomalous temperatures, the extraction of gases and petroleum from rocks and soil requires the drilling fluids to have temperature-dependent viscosity and thermal conductivity. The variable thermal conductivity and viscosity of the drilling fluid provide resistance to high temperatures and pressure. Therefore, the objective of this study was to investigate anomalous heat transfer subject to frictional heating, melting heat, thermal radiation, and magnetic fields. Using the Brinkman–Maxwell–Garnett model, the effective thermophysical properties of drilling NF are considered. The computational performance of drilling NF is investigated using the Adams–Bashforth predictor and corrector approach. The findings suggest that the volume percentage and free convection parameter increases the Nusselt number considerably. The thickness of the boundary layer is increased by a higher permeability parameter, whereas the Darcy number exhibits the opposite tendency. Fluid velocity and skin friction are reduced by the magnetic number, whereas the temperature profile is raised by increases in the radiation parameter and the volume percentage of nanoparticles. The rate of heat transmission becomes significant in the scenario of variable characteristics.

This investigation systematically examined mixed convection flow of a fractional micropolar fluid over an oscillating plate, incorporating thermal radiation and memory effects through Caputo fractional derivatives. The governing equations of the proposed problem were non-dimensionalized using appropriate dimensionless variables. Exact solutions for velocity, microrotation, and temperature distributions were derived via the Laplace transform method. The obtained exact solutions were expressed in terms of Wright functions to preserve memory characteristics. Special cases, including Stokes' first problem and fractional viscous fluids, were demonstrated, which showed the model's versatility. The influences of key parameters, such as fractional order ( $\alpha$), micropolar material parameter ( $\beta$), Grashof number ( $Gr$), Prandtl number ( $Pr$), and radiation parameter ( $R$) on flow and heat transfer characteristics, were analyzed and presented in various graphs. Graphical results illustrate parametric trends emphasizing memory effects, while tabulated data quantified skin friction, wall couple stress, and Nusselt number variations. The results yielded: (1) a generalized fractional micropolar model capturing memory effects, (2) new insights into radiation's role in thermal boundary layer modulation under non-local dynamics, and (3) benchmark solutions for microfluidic device design. This work unified fractional calculus (with its inherent memory effects), micropolar theory, and oscillatory boundary conditions, establishing a foundation for advanced fluid mechanics research.