The construction of special explicit solutions of the boundary-layer equations. Steady flows

A new approach is given for finding explicit similarity solutions of partial differential equations and applied to the incompressible boundary-layer equations. The method involves the usual direct substitution of a similarity form in a given partial differential equation but does away with the usual requirement of reducibility to an ordinary differential equation. Instead, the partial differential equation is reduced to an overdetermined system of ordinary differential equations which can be solved in closed form. The method produces solutions having some special forms but, in return, it allows similarity reductions containing free parameters which in a number of cases are arbitrary functions of the longitudinal coordinate. Each such solution can represent any one of a variety of special solutions of specific boundary-layer problems differing in the form of main stream and boundary conditions. A number of new explicit solutions of boundary-layer problems have been obtained in this way. These do not usually represent solutions with similar velocity profiles, but the particular cases include some explicit solutions of the common self-similar form, as well as some explicit self-similar solutions of the Navier-Stokes equations. Arguments are presented showing that the similarity reductions produced cannot be obtained using other methods for finding similarity solutions of partial differential equations, and in particular the standard Lie-group method of infinitesimal transformations and its generalizations.