Integration of the boundary-layer equations

The equations of the boundary layer are integrated by an expression of the form f(x) = ∫x0 e-F(x)ϕ(x)dx, where F(x) is a positive function with x = 0 as the stationary point; ϕx) is slowly varying; the integral contains an unknown parameter which is found from the condition f(x) = 1. The integral is evaluated by the method of steepest descent. The expressions obtained are usually divergent, except in few cases which include Blasius’s equation; the divergent expressions are summed by Euler’s transformation. To check the procedure it is applied to Falkner & Skan’s equation. The results obtained are very striking; few terms in the expansions are sufficient to obtain close agreement with Hartree’s laborious numerical computations. The method is also applied to the general boundary-layer equation for the case of flow past an elliptic cylinder, measured by Schubauer. The results obtained are in close agreement with Schubauer’s measurements for the velocities, almost up to separation, for the position of the separation point; and in satisfactory agreement downstream of separation.