A new multifold series general solution of the steady, laminar boundary layers. III - Extension of the solution to the case of not smoothly joining mainstream-velocities

In the preceding reports, a new theory of multifold series expansion for steady, laminar boundary layers has been presented. The solution is given in a series of the multiparameters which involve respectively different order derivatives of the characteristic function beta (xi) = (2 xi/U.)(dU./d xi), so that the theory covers an extremely wide range of mainstream velocity distributions. The only problem remaining unsolved is the case of the mainstream U(x) which is described by a function of class C to the k-power, where k is in the zero-to-infinity range (that is, the case when the derivative d to the k + 1 power U (x)/dx to the k + 1 power indicates a discontinuity at some x-position. The present paper extends the solution to this problem by replacing the original singular characteristic function with a smooth function which is substantially equivalent to the original one for determining the boundary layer flow characteristics. The theory can be applied widely and gives good result even for a very difficult problem, such as a flow containing separation and reattachment.