Axisymmetric boundary layer on a needle

We consider the stationary spatial axisymmetric flow of the viscous compressible heat conductive fluid along a semi-infinite needle. It is described by a system of three partial differential equations with boundary conditions in infinity and on the needle. Its truncated system, describing the flow in the boundary layer, was selected by methods of Power Geometry. After introducing self-similar coordinates, the truncated system is reduced to a system of two ordinary differential equations. It has an invariant manifold, where it can be reduced to a second order differential equation. Analysis of its solutions made by methods of Power Geometry and numerically, shows the existence of solutions satisfying all boundary conditions and having a power or logarithmic singularity near the needle.