  We investigate the incompressible and compressible boundary layer with heat
conduction applying the two dimensional self-similar Ansatz. We found analytic
solutions for the incompressible case which can be expressed with special
functions. The parameter dependencies are studied and discussed in details.
  For a better physical understanding we compare our results to the
Rayleigh-B\'enard (RB) convection because the boundary layer equations can be
considered as a simplification of the RB flow. Our obtained results have a
strong indication that B\'enard cells can build up in heated boundary layers as
well which is the main message of our analysis. In the last part of our study
we present the ordinary differential equation (ODE) system obtained for
compressible boundary layers.
