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Abstract The addition of nanoparticles in the base fluid increases the thermal conductivity of the fluid but the stability of the fluid gets affected due to the presence of nanoparticles in the base fluid. The addition of motile microorganisms in the nanofluid increases the stability of the nanofluid by enhancing thermal conductivity as well as mass transport. In this perspective, the current investigation deals with the study of the behavior of microorganisms in bioconvection over the fluid variables in the non-Darcy background. A two-dimensional Prandtl-nanofluid moving along an inclined stretching flat surface subject to inlet and outlet rate of flow to the surface under the action magnetic drag force normal to the surface. The non-linear partial differential equations are transmuted to ODEs through appropriate similarity transformations. The MATLAB inbuilt bvp4c solver has been employed to generate the solution of the transformed ODEs. The stability of the finite difference scheme is demonstrated by a stability test, and the influence of various parameters on the nanofluid velocity, thermal as well as concentration, and density of motile microorganisms is depicted using graphs for each of them. Additionally, interested physical quantities, shearing stresses, rate of thermal diffusion, rate of mass diffusion, and rate of motile microorganism at the surface, have been plotted by varying the controlling parameters. Growth in the shear stress as well as the rate of thermal diffusion has been detected for the increasing values of δ and β . However, the Sherwood number decays for augmented values of δ . The findings could be utilized to increase the thermal efficiency of heat exchangers in order to maintain thermal balance management in small heat-density equipment and gadgets, as well as the thermal efficiency of microbial fuel cells, enzyme biosensors, and microfluidics devices. [ABSTRACT FROM AUTHOR]

The two-dimensional boundary layer flow of a Prandtl nanofluid was explored in the presence of an aligned magnetic field over an inclined stretching/shrinking sheet in a non-Darcy permeable medium. To transform the PDEs of the leading equations into ODEs, a coupled boundary value problem was formed and suitable similarity functions were used. To obtain numerical answers, an efficient code for the Runge–Kutta technique with a shooting tool was constructed with a MATLAB script. This procedure is widely used for the solution of such problems as it is efficient and cost-effective with a fifth-order accuracy. The significance of immersed parameters on the velocity, temperature, concentration, and bioconvection is shown through figures. Furthermore, the physical parameters of the skin friction coefficient and the Nusselt numbers are demonstrated in tables. The declining behavior of the flow velocity against the porosity parameter K p and the local inertia co-efficient F r is shown, and the both parameters of the Darcy resistance and Darcy–Forchheimer resistance are responsible for slowing the fluid speed. The increasing values of the Schmidt number S c decrease the concentration of the nano entities. [ABSTRACT FROM AUTHOR]