In this study, we aim to deal with the flow behavior betwixt a pair of rotating circular plates filled with Carreau fluid under the suspension of nanoparticles and motile gyrotactic microorganisms in the presence of generalized magnetic Reynolds number. The activation energy is also contemplated with the nanoparticle concentration equation. The appropriate similarity transformations are used to formulate the proposed mathematical modeling in the three dimensions. The outcomes of the torque on both plates, i.e., the fix and the moving plate, are also contemplated. A well-known differential transform method (DTM) with a combination of Padé approximation will be implemented to get solutions to the coupled nonlinear ordinary differential equations (ODEs). The impact of different nondimensional physical aspects on velocity profile, temperature, concentration, and motile gyrotactic microorganism functions is discussed. The shear-thinning fluid viscosity decreases with shear strain due to its high velocity compared to the Newtonian and shear-thickening case. The impact of Carreau fluid velocity for shear-thinning , Newtonian case , and shear-thickening cases on axial velocity distribution has been discussed in tabular form and graphical figures. For the validation of the current methodology, a comparison is made between DTM-Padé and the numerical shooting scheme.

This article deals with the unsteady flow in rotating circular plates located at a finite distance filled with Reiner-Rivlin nanofluid. The Reiner-Rivlin nanofluid is electrically conducting and incompressible. Furthermore, the nanofluid also accommodates motile gyrotactic microorganisms under the effect of activation energy and thermal radiation. The mathematical formulation is performed by employing the transformation variables. The finalized formulated equations are solved using a semi-numerical technique entitled Differential Transformation Method (DTM). Padé approximation is also used with DTM to present the solution of nonlinear coupled ordinary differential equations. Padé approximation helps to improve the accuracy and convergence of the obtained results. The impact of several physical parameters is discussed and gives analysis on velocity (axial and tangential), magnetic, temperature, concentration field, and motile gyrotactic microorganism functions. The impact of torque on the lower and upper plates are deliberated and presented through the tabular method. Furthermore, numerical values of Nusselt number, motile density number, and Sherwood number are given through tabular forms. It is worth mentioning here that the DTM-Padé is found to be a stable and accurate method. From a practical point of view, these flows can model cases arising in geophysics, oceanography, and in many industrial applications like turbomachinery.