A solution of two-parameter asymptotic expansions for a two-dimensional unsteady boundary layer

A solution procedure based on two-parameter asymptotic expansions, in terms of a Blasius parameter and a dimensionless time, is presented for a two-dimensional, unsteady boundary layer over a flat surface. The Blasius parameter is used to scale the stretching of the boundary-layer length scale, and the dimensionless time represents the unsteadiness caused by the outer flow field. The matching conditions between the outer solutions and inner solutions are obtained according to the matching procedure from which the streamfunction, velocity and pressure are matched all at the same time. Closed-form solutions are obtained until the second-order expansions of the solution. Applications of the solution to example problems are given with comparisons to the results in the literature to show the validity and versatility of the current solution to accommodate a variety of outer flows. The solution is even valid for predicting the time and location when the flow separation first occurs in some applications.