Flow of non-Newtonian fluid with convective conditions in Darcy-Forchheimer media: an unsteady case

This research investigates the transient hydromagnetic behavior and heat transfer attributes of a non-Newtonian Casson nanoliquid embedded with microorganisms, flowing past a stretched surface in a Darcy-Forchheimer medium. The effect of a magnetic field, oriented at an angle $\alpha$ with the boundary surface, Joule dissipation, and convective boundary conditions are considered to determine the flow behavior, heat transfer, nanoparticle concentration, and microorganism density. To solve the non-dimensionalized system of coupled and nonlinear partial differential equations, the bivariate spectral quasi-linearization method (BSQLM) is employed. This numerical scheme has proven to be both convergent and accurate. Outcomes are compared with the results available in the literature and found good agreement. Variations in flow, heat transfer, distribution of nanoparticles, and microorganisms are illustrated by reproducing the numerical results in graphical form, whereas Nusselt and Sherwood numbers are displayed in tables. The Casson parameter uniformly diminishes the velocity and temperature inside the boundary layer region. Angle of inclination ($\alpha$) boosts the temperature profile near the boundary and decreases the fluid velocity and nanoparticle concentration. The Prandtl number gives a rise in temperature near the wall and reveals an opposite effect away from the thermal boundary layer region. The Lewis number exerts a diminishing impact on the nanoparticle concentration field. Eckert number thickens the thermal boundary layer region. The microbe density field is a decreasing function of Peclet number. Solutal, thermal, and microorganism biot number exert, respectively, an enhancing effect on nanoparticle concentration, a diminishing influence on temperature profile, and a microbe density. This model is valuable for understanding the applications of solar energy in thermal engineering processes and has direct implications for industries such as glass and polymer manufacturing, thermal exchangers, homogenization, biomedical engineering, nuclear reactors, and metallic plate cooling.