This paper examines the influence of the unsteadiness parameter on the maximum wall shear stress in an unsteady stagnation-point boundary layer over a vertical flat plate with oscillatory wall blowing or suction. The analysis considers a two-dimensional, incompressible, viscous flow impinging on the plate, with time-periodic variations in both the free-stream velocity and the wall blowing/suction rate. The formulation follows the approach of Blyth and Hall. Using similarity transformations, the time-dependent Navier–Stokes equations are reduced to a single nonlinear ordinary differential equation governing the boundary layer. This equation is solved numerically using a fully implicit finite-difference scheme. Wall shear stress is obtained from the second derivative of the similarity function, and its maximum value is tracked over time to characterize the flow dynamics. Results show that increasing the blowing parameter raises the peak shear stress, whereas larger unsteadiness reduces it. For each set of flow conditions, the shear stress reaches a maximum at a specific oscillation frequency before asymptotically approaching a steady-state trend. Higher unsteadiness delays the occurrence of this peak and diminishes its magnitude. In certain parameter ranges, the shear stress exhibits erratic behavior, indicating incipient divergence............... Keywords:.............. similarity solution, stagnation-point flow, unsteadiness, oscillation, blowing.