The boundary layers of an unsteady separated stagnation-point flow of a viscous incompressible fluid over a moving plate

Abstract

In this paper we have investigated the boundary layer analysis of an unsteady separated stagnation-point (USSP) flow of an incompressible viscous fluid over a flat plate, moving in its own plane with a given speed . The effects of the accelerating parameter a and unsteadiness parameter β on the flow characteristics are explored numerically. Our analysis, based on the similarity solution of the boundary layer equations, indicates that the governing ordinary differential equation, which is non-linear in nature, has either a unique solution, dual solutions or multiple solutions under a negative unsteadiness parameter β with a given value of a. Whatever the number of solutions may be, these solutions are of two types: one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). A novel result which emerges from our analysis is the relationship between a and β. This relationship essentially gives us the conditions needed for the solutions that exhibit flow separation (where ) and those conditions that exhibit only flow reattachment (where ). Another noteworthy result which arises from the present analysis is the existing number of non-zero stagnation-points inside the flow for the given values of a and β. It is found that this number is exactly two when the velocity gradient at the wall is positive; otherwise this number will only be one. For a stationary plate , this USSP flow is found to be separated for all values of a and β in both cases of AFS and RFS. Finally, we have also established that in the case of AFS flow over a stationary plate, no stagnation-point exists inside the flow, even though the flow becomes separated for all values of a and β.