Dual exact solutions of graphene-water nanofluid flow over stretching/shrinking sheet with suction/injection and heat source/sink: Critical values and regions with stability

Boundary layer flow of an electrically conducting fluid past a stretching/shrinking sheet inserted in graphene-water nanofluid was investigated. The detailed exact analytical solutions for the flow and heat have been extracted. A theoretical analysis was implemented to explore the regions of non-existence and existence of unique and dual solutions by providing the critical values. In addition, the asymptotic formulas for large first/second slips valid for a stretching/shrinking type, has been deduced. The profiles and distributions of the stream function, velocity, temperature, reduced skin friction coefficient and reduced Nusselt number were examined and discussed via the obtained dosed forms. With no restrictions, it was proved that the unique solutions were detected for the flow of all considered parameters over the stretching sheet. However, to obtain the dimensionless temperature solutions, a very sensitive condition on the heat source/sink parameter has to be considered. Whilst in the case of flow over the shrinking sheet, solutions turn out to be unique or dual for some combinations of the parameters. Moreover, two sensitive conditions have to be simultaneously achieved to get the dual solutions depending mainly on the magnetic and thermal radiation parameters. Further, even in the weakness of magnetic field, the dual solutions were obtained for both suction and injection cases. Furthermore, it was proved in detail that the first solutions of the stream function and temperature are stable and physically realizable, whereas those of the second ones are highly-unstable. This means that, while two solutions exist mathematically, only the first one is to be physically considered. It was also concluded that graphene nano-particles can be used as a heater on increasing the solid volume fraction, injection, thermal radiation and Eckert number. On the other hand, they act as a cooler with an increase of magnetic field, suction, first velocity slip, second velocity slip and heat source/sink. Additionally, the asymptotic formulas and behavior of the stream function and temperature for both sheets were introduced. (C) 2018 Elsevier B.V. All rights reserved.