Flow and heat transfer of a second-grade hybrid nanofluid over a permeable stretching/shrinking sheet

The flow and heat transfer characteristics of a second-grade hybrid nanofluid over a permeable stretching/shrinking sheet are investigated. Using similarity transformations, the governing equations are reduced to a set of differential equations. These equations have been solved utilizing nonlinear shooting method. It is found that the region of the existence of the dual solutions becomes wider for higher values of the suction parameter, non-Newtonian parameter, magnetic field parameter and volume fraction of Cu nanoparticles. Due to the increase of the suction parameter, non-Newtonian viscosity parameter and volume fraction of Cu nanoparticles, the velocity of the fluid significantly increases, whereas the temperature decreases. The results elucidated with streamlines show that the flow field can be consisted of three layers depending on the velocity ratio parameter which is defined as the ratio of velocities of the sheet and the free stream. In addition, stability analysis is carried out to determine the stable and unstable solutions of the problem. It is found that the upper solutions are stable and physically acceptable.