On two solutions of the boundary-layer equations

Summary

The boundary layer equations for the steady two-dimensional incompressible laminar flow past a porous boundary may be reduced to an ordinary third-order non-linear differential equation when the stream velocity is proportional to some power of the distance along the surface. The most general equation for these so-called similar solutions is derived for the first time and two particular cases of it considered in detail, no general discussion being given in this paper. One case leads to an explicit analytical expression for the velocity distribution while the other leads to a simple first-order equation. Both solutions are unique among others in that they relate in closed form conditions at infinity to those at the boundary and curious properties of the equation are demonstrated. In physical terms, it appears that in the first case zero skin friction occurs for the minimum suction velocity allowed by conditions at infinity while in the second zero skin friction is incompatible with proper behaviour at infinity. Numerical results are given.