An asymptotic matching approach to the approximate solution of the Blasius problem

Abstract

The Blasius equation dates back to the early twentieth century and describes the boundary layer caused by a uniform flow over a flat plate. Due to the asymptotic condition on the first derivative of the solution, the problem is not an initial value one. The non-linear nature of the equation has led many authors to exploit approximation strategies to obtain an estimate of the solution. Several papers deal with series expansions, but the solutions are partly numerical and do not converge rapidly everywhere. In the present paper, a strategy is proposed to obtain an approximate solution of the Blasius problem in the closed form, i.e., without the use of series expansions. Motivated by the asymptotic behavior of the second derivative, a parametric definition of the third derivative of the Blasius solution is introduced and successive integrations are performed. By imposing the expected asymptotic conditions, a final approximate solution is obtained. This latter has the notable advantage of possessing the same behavior at 0 and at ∞ as the analytical solution up to the third derivative. It will also be shown how this result can be extended to the Falkner–Skan problem with non-zero pressure gradient and provide an accurate estimation of the thermal boundary layer over a flat plate under isothermal and adiabatic conditions.