On the axially symmetric flow of the viscous incompressible fluid around a needle

We consider the stationary spatial axially symmetric flow of the viscous incompressible fluid around a semi-infinite needle. The flow is described by the Navier-Stokes system of equations, which is reduced to a partial differential equation for a stream function. The boundary conditions are set at infinity and at the needle. The support of the equation consists of 5 points. Its convex hull is the trapezoid $\Gamma$. To each its edge and each its vertex there corresponds a truncated equation. We prove that no truncated equation has a solution satisfying both boundary conditions. We consider also two truncated equations corresponding to two edges of the trapezoid $\Gamma$. They have matched self-similar solutions depending from one parameter. For some its value, the matched solutions satisfy both boundary conditions. However, in a layer the pressure tends to $-\infty$. Hence, the solution has no physical meaning.