Oscillatory Flow Near a Stagnation Point

The classical Hiemenz solution describes incompressible two-dimensional stagnation point flow at a solid wall. We consider an unsteady version of this problem, examining particularly the response close to the wall when the solution at infinity is modulated in time by a periodic factor of specified amplitude and frequency. While this problem has already been tackled in the literature for general frequency in cases when the amplitude of the time-periodic factor is either large or small, we compute the flow for arbitrary values of both these parameters. For any given amplitude, we find that there exists a threshold frequency above which the flow is regular and periodic, with the same period as the modulation factor, and beneath which the solution terminates in a finite time singularity. The dividing line in parameter space between these two possibilities is identified and favorably compared with the predictions of asymptotic analyses in the small and large frequency limits.