Radiative flow of clay nanoparticles on the lubricity of Williamson drilling fluids across a vertical surface in a Darcy-Brinkman porous medium

Drilling fluids are essential for extracting gases and oils from rocks and soil. To increase drilling fluid efficiency, clay nanoparticles are crucial. The use of clay particles in drilling fluids increases their thermal conductivity, viscosity, and boiling point, hence giving resilience to high temperatures and controlling fluid costs. Therefore, this research examines the convection radiative flow and heat transfer incorporated in the Williamson nanofluid with a heat source/sink. The effective thermophysical properties of clay nanofluid are represented quantitatively by using Maxwell-Garnett and Brinkman's formulas. The leading PDEs with physical boundary conditions that control the flow phenomena are predetermined. These PDEs are converted into ODEs using the similarity method, and dual solutions are then found by using an effective bvp4c solver. The effects of mixed convective, permeability, Williamson constraint, heat source/sink, nanoparticle volume fraction, and radiation parameters were all thoroughly studied quantitatively and theoretically. The Nusselt number and skin friction are calculated and displayed in tabular form as well as graphical form along with the velocity and temperature profiles. Multiple solutions are observed in the shrinkable sheet as well as the buoyancy assisting flow. The findings demonstrate that the Nusselt number rises noticeably when volume concentration increases. In addition, the permeability parameter expands the boundary layer thickness in the lower solution, while the contrary behavior is observed in the upper solution.