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

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 u(0)(t). The effects of the accelerating parameter a and unsteadiness parameter beta 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 beta 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 beta. This relationship essentially gives us the conditions needed for the solutions that exhibit flow separation (where (a + beta)< 0) and those conditions that exhibit only flow reattachment (where (a + beta)> 0). 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 beta. 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 (u(0)(t) = 0), this USSP flow is found to be separated for all values of a and beta 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 beta.