The present paper examines thermal and hydrodynamic behavior of the incompressible laminar flow of a non-Newtonian magnetic nanofluid in a vertical rectangular channel numerically using two-phase mixture model, Carreau model, and finite volume method. The non-uniform transverse magnetic field is created by an electric current-carrying wire located along the channel. The Schiller-Naumann model is employed to calculate the slip velocity between the solid and liquid phases. The flow pattern and nanofluid temperature is assessed by changing effective parameters such as Reynolds number, the magnetic field strength, flow rate, mean axial temperature, and channel heat transfer. It is observed that the transverse secondary flow increases by increasing the magnetic strength due to Kelvin force. The hot fluid is transferred more from the sidewall to the center of the channel and the cold fluid moves from the center of the channel towards the wall, leading to an increase in heat transfer. Also, at low Reynolds numbers, more fluctuations occur in the velocity profile due to the dominance of Kelvin force over inertial force. Manuscript profile