Dual solutions of an unsteady magnetohydrodynamic stagnation-point flow of a nanofluid with heat and mass transfer in the presence of thermophoresis

The unsteady two-dimensional magnetohydrodynamic stagnation point flow of a nanofluid with thermophoresis effect is investigated numerically. The technique of similarity transformation is implemented to obtain the self-similar ordinary differential equations and then the self-similar equations are solved numerically using shooting method. This analysis explores the conditions of the existence, non-existence, uniqueness, and duality of the solutions of self-similar equations numerically. Dual solutions of velocity, temperature and concentration profiles are reported for different values of the each parameter involved for two types of nanoparticles, namely copper (Cu) and gold (Au) in the water-based fluid. It is found that the dual solutions exist for negative values of unsteady parameter A, whereas for positive values of unsteady parameter, the solution is unique. The results also indicate that the nanoparticle volume fraction reduces the skin friction coefficient, the heat transfer rate as well as mass transfer rate. Further, due to increase of thermophoresis parameter, the concentration inside the boundary layer reduces and the mass transfer rate enhances. In addition, to validate the present numerical results, comparison with published results is made and found to be in excellent agreement.