The proposed study demonstrates the flow phenomenon and thermo-variation of a magnetized stretching sheet induced-radiative nanofluid flow. By incorporating the response surface methodology, the heat transfer rate of the thermally convective flow of nanofluid is optimized. The graphene nanomaterial is used in the water-based nanofluid. A dynamic magnetic field, thermal radiation, and the Cattaneo–Christov heat flux model have used to represent the thermal behavior of the nanofluid. The simulation utilizes experimentally estimated values for the nanomaterial’s thermal conductivity and viscosity. To further reveal the thermal enhancement of the flow, the impact of nanoparticle diameter and the solid-liquid interfacial layer is proposed at the molecular level. The response surface methodology and the sensitivity analysis has used to examine the effects of the nanoparticle volume fraction, Biot number, and magnetic parameter on the rate of heat transfer statistically. A set of equations is formed from the governing partial differential equations by implementing suitable similarity transformations. The bvp4c approach is used to solve the problem numerically. The effect of various parameters has displayed through tables, graphs, and surface plots on heat transfer, mass transfer, and the local Nusselt number. It is discovered that as the Biot number increases, so does the concentration and temperature profile. An excellent accord between the present and previously existing solutions is establishing the validity of the achieved results.

This study focused on the analysis of two-dimensional incompressible magnetohydrodynamic Carreau nanofluid flow across a stretching cylinder containing microorganisms with the impacts of chemical reactions and multiple slip boundary conditions. Moreover, the main objective is concerned with the enhancement of thermal transportation with the effect of heat source and bioconvection. By assigning pertinent similarity transitions to the governing partial differential equations, a series of equations (ODES) is generated. An optimum computational solver, namely the bvp5c software package, is utilized for numerical estimations. The impact of distinct parameters on thermal expansion, thermophoresis, and the Nusselt number has been emphasized, employing tables, diagrams, and surface maps for both shear thinning (n < 1) and shear thickening (n > 1) instances. Motile concentration profiles decrease with $L_b$ and the motile microorganism density slip parameter. It is observed that with increasing values of $Pr$, both the boundary layer thickness and temperature declined in both cases. The Weissenberg number demonstrates a different nature depending on the type of fluid; skin friction, the velocity profile and Nusselt number drop when n < 1 and increase when n > 1. The two- and three-dimensional graphs show the simultaneous effect of involving parameters with physical quantities. The accuracy of the existing observations is evidenced by the impressive resemblance between the contemporary and preceding remedies.