This article investigates the significance of varying radius of copper nanoparticles for non-Newtonian nanofluid flow due to an extending sheet in the presence of a magnetic field and porous medium. The modern technological applications of non-Newtonian nanofluids have attracted researchers in the current era. So, the impacts of the radius of nanoparticles with micropolar fluid have been taken into consideration. Three-dimensional leading equations (PDEs) for momentum, concentration, and temperature are transformed into ODEs by applying the appropriate similarity transformation. The numerical approach bvp4c is applied to obtain the problem’s solution numerically. The influence of the nanoparticles’ radius and various physical parameters on the microrotation, velocity, and temperature profile are analyzed. The velocity profile decreases against the magnetic field (M), rotational parameter ($\Gamma$), and Forchheimer number (Fr), but the temperature distribution has increasing behavior for these parameters, and the microrotation is augmented for rising inputs of the magnetic parameter and boundary parameter ($\beta$). It is also observed that the temperature reduces against the material parameter (∇) and Forchheimer number ($Fr$). The skin friction coefficients and Nusselt number decrease against the growing strength of the Forchheimer number $\left(Fr\right)$. At the stretching surface, the skin friction factor and Nusselt number are numerically and graphically calculated.

Hybrid nanofluids’ enhanced thermophysical properties make them applicable in a plethora of mechanical and engineering applications requiring augmented heat transfer. The present study focuses on a three-dimensional Copper-Aluminium Oxide $$\left( Cu\text{- }Al_{2}O_{3}\right)$$Cu-Al2O3 -water based hybrid nanofluid flow within the boundary layer with heat transfer over a rotating exponentially stretching plate, subjected to an inclined magnetic field. The sheet rotates at an angular velocity $$\Omega$$Ω and the angle of inclination of the magnetic field is $$\gamma$$γ . Employing a set of appropriate similarity transformation reduces the governing PDEs to ODEs. The resulting ODEs are solved with the finite difference code with Shooting Technique. Primary velocity increases at large rotation but the secondary velocity reduces as the rotation increases. In addition, the magnetic field is found to oppose the flow and thereby causing a reduction in both the primary and secondary velocities. Increasing the volume fraction reduces the skin friction coefficient and enhances the heat transfer rate.

Differential transform method (DTM) as a method for approximating solutions to differential equations have many theorems that are often used without recourse to their proofs. In this paper, attempts are made to compile these proofs. This paper also proceeds to establish the convergence of the DTM for ordinary differential equations. This paper establishes that if the solution of an ordinary differential equation can be written as Taylors' expansion, then the solution can be obtained using the DTM. This is also demonstrated with some numerical examples.